Pd1

´ gico y de Estudios Superiores de Monterrey Instituto Tecnolo Campus Toluca

Preface

Product design: techniques for robustness, reliability and optimization

Class Notes Dr. Jos´ e Carlos Miranda V.

It is widely recognized that to develop successful products, systems or services it is extremely important to follow a structured product development process. Although each company follows a process tailored to its specific needs, in general the start of a product development process is the mission statement for the product. It identifies the target markets for the product, provides a basic functional description of the product, and specifies the key business goals of the effort. The end of the development effort occurs when the product is launched and becomes available for purchase in the market place. The different activities that take place during the product development process can be grouped into five phases: Concept development, system-level design, detail design, testing and refinement, and production ramp-up. During the detailed design and the testing and refinement phases, product optimization, robustness and reliability becomes critical. As many powerful techniques have appeared to make a product more optimal, robust and reliable, it is necessary to know how they work and how can they be applied to design products that exceed customer expectations and minimize costs. The present notes have been prepared for the courses of Design Methodologies and Product Design that I teach. Although these notes are far from complete and therefore may contain many mistakes and inaccuracies, they evolve term after term and with the help and suggestions of my students are continuously improved. Once these notes are mature, it is my desire to publish them to reach a wider audience and receive further comments. If you have any feedback, suggestions or have detected any mistakes, or simply would like to assist me or contribute in this effort, please do not hesitate to contact me. I will be very happy to hear from you.

v. Spring 2006

c Copyright 2006 Dr. Jos´ e CarlosMiranda. Todos los derechos reservados.

Jos´e Carlos Miranda Research Center for Automotive Mechatronics [email protected] c Copyright 2006 Dr. Jos´ e CarlosMiranda. Todos los derechos reservados.

CHAPTER

1

The Engineering Design Process

Part I The product design process

1.1

Definition of design

The word design has had different meanings over the last decades. While sometimes a designer is considered to be the person drafting at the drawing board or in the computer, the word design really conveys a more engineering and analytical sense. Design is much more than just drafting. Suh (1990) defines design as the creation of synthesized solutions in the form of products, processes or systems, that satisfy perceived needs through the mapping between functional requirements and design parameters. In the scope of the previous definition, functional requirements (FRs) respond to the question of what a product must do or accomplish. On the other hand, design parameters (DPs) respond to the question of how the functional requirements will be achieved. What relates the domain of functional requirements to the domain of design parameters is design (see figure 1.1). It should be noted that although design parameters should fulfill the functional requirements, the mapping between them is not unique. For a set of functional requirements may be several design parameters that fulfill those functional requirements. Another, less technical, definition of design is the one promulgated by ABET (Accreditation Board for Engineering and Technology): c Copyright 2006 Dr. Jos´ e CarlosMiranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e CarlosMiranda. Todos los derechos reservados.

3

1.2 The design process WHAT?

1.2 The design process

4

HOW? Mechanical Engineering

List of Functional Requirements

design

List of Design Parameters

Electronic Engineering

Purchasing Product Design

Manufacture Engineering

Marketing

Figure 1.1: Design is the process of mapping functional requirements to design parameters.

“Engineering design is the process of devising a system, component, or process to meet desired needs. It is a decision making process (often iterative) in which the basic sciences, mathematics and engineering sciences are applied to convert resources optimally to meet a stated objective. Among the fundamental elements of design process are the establishment of objectives and criteria, synthesis, analysis, construction, testing and evaluation. . . It is essential to include a variety of realistic constraints such as economic factors, safety, reliability, aesthetics, ethics and social impacts.”

Industrial Design

Figure 1.2: Engineering design core disciplines. 1.2.1. Design process Probably the most simple model of the design process models is the one shown in figure 1.3, where only four general stages are outlined. Another relatively simple model is presented by Ullman (1992) who suggest to view the design as problem solving. When solving a given problem, five basic actions are taken: 1. Establishment of need or realize there is a problem to be solved.

Although several definitions of design may be found, the last one highlights one of the main difficulties associated with design: its truly multidisciplinary nature. Design involves several, if not all, different departments in a given company (see figure 1.2). Design engineers should always be aware of this condition, involving in the design process the expertise of people of different disciplines.

1.2

The design process

2. Understanding of the problem. 3. Generation of potential solutions for it. 4. Evaluation of the solutions by comparing the potential solutions and deciding on the best one. 5. Documentation of the work. While it is possible to see design as problem solving, it is important to realize that most analysis problems have one correct solution whereas most design problems have many satisfactory solutions.

There are many different maps or models of the design process. Some of these models describe steps and their sequence as they occur in the design process. Some other models try to define or prescribe a better or more appropriate pattern of activities. Cross (1994) describe some of these models.

A more detailed model, which involves all steps of the design process, is presented in figure 1.4. As shown, this model divides design process in 5 phases: Concept development, System-Level design, Detail design, Testing and refinement and Production. Each phase has one or more steps. It is important to realize that this model is general and may be necessary to follow different paths in one or more phases depending on the project at hand.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

5

1.2 The design process

1.2 The design process Phase 1: Concept Development

Recognition of need

Exploration

Conceptualization Phase 2: System-Level Design

Generation

6

Feasibility assessment

Phase 3: Detail Design

Preliminary design cost analysis / redesign

Evaluation Development testing

Phase 4: Testing and Refinement

Communication

Detailed design Qualification testing

Production planning and Tooling design

Phase 5: Production Acceptance testing

Production

Figure 1.3: A simple model of the design process with 4 stages. Figure 1.4: Detailed model of the design process. Independently of the model, it is generally agreed that the design process should always start with the recognition of a need. After the need has been recognized it is necessary to consider alternatives for its solution, which is done during the concept development phase. Here the statement of the problem is taken and broad solutions to it are generated. This phase presents the greatest chance for improvements and hence is specially imperative to be objective, open to new ideas and recognize when changes are needed. Once the best ideas have been selected, preliminary design may start to further evaluate those ideas. In this phase testing may be of great help to differentiate good ideas from regular ones. After a design has been finally selected, detailed design begins to incorporate every feature that the design may need to incorporate. Hence, a very large number of small but essential points should be decided. After the detailed design has been re-evaluated and tested, production planning may be started and final products tested for final acceptance. In what follows the different steps in the design process are discussed more in depth. 1.2.2. Identifying customer needs

The need to design a new product may come from different sources: consumers, organizations or governments. The need may also sometimes

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

be substituted for an idea of a product with possibilities of becoming commercially successful. Eide et al. (1988) state that in industry, it is essential that products sell for the company to survive. Inasmuch as most companies exist to make a profit, profit can be considered to be the basic need. Hence, a bias toward profit and economic advantage should not be viewed as a selfish position because products are purchased by people who feel that they are buying to satisfy a need which they perceived as real. The consumers are ultimately the judges of whether there is truly a need. Identifying the needs of the costumer is one of the most important steps in the design process and is, at the same time, one of the most difficult since is not unusual to find that the customer does not know exactly what features the product must have. Once the needs have been specified together with the costumer, this information is used to guide the design team in establishing design parameters, generating concepts and selecting the best one of them. According to Ulrich & Eppinger (2000) the process of identifying customer needs includes five steps: 1. Gather raw data from customers. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

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1.2 The design process Metric

Value

The product must be . . . easy to install

Average time for installation < x seg.

durable

Must withstand 10x cycles

easy to open

Opens with a force of max. x newtons.

able to resist impacts

Withstand drops from x meters.

able to work in cold weather

Operation possible at -x◦ C.

1.2 The design process

8

1.2.4. Concept generation According to French (1985) in this phase the statement of the design problem is taken and broad solutions are generated in the form of schemes. It is the phase that makes the greatest demands on the designer, and where there is more scope for striking improvements. It is the phase where engineering science, practical knowledge, production methods and commercial aspects need to be brought together. It is also the stage where the most important decisions are taken.

2. Interpret the raw data in terms of customers needs.

In the scope of design, a concept is an abstraction, an idea that can be represented in notes and/or sketches and that will eventually become a product. It is generally recognized that, for a given product, several ideas (sometimes hundreds of them) should be generated. From this pool of ideas, a couple of them will merit serious consideration for further evaluation and development.

3. Organize the needs into a hierarchy of primary and secondary needs.

The concept generation stage can be divided into 4 steps:

Table 1.1: Examples of metrics and their value.

4. Establish the relative importance of the needs. 5. Reflect on the results and the process.

1. Clarification of the problem. 2. Gathering of information.

1.2.3. Establishing the design requirements

As was briefly discussed above, when the design engineer is first approached with a product need, it is very unlikely that the customer will express clearly what is needed. In most occasions it is only know what is wanted in a very general way without idea of the particularities involved. Hence, the starting point for a design engineer is to turn an ill-defined problem with vague requirements into a set of requirements that are clearly defined. This set of product requirements may change as the project advances, so it is convenient to clarify them at all stages of the design process. For the product requirements to be helpful, they must be translated to technical specifications that are precise, easily understood and can be measure by means of one or more design variables. Ulrich & Eppinger state that “A specification consists of a metric and a value.” Table 1.1 shows some examples of metrics and their values. Several tools can be used to establish product specifications. Although simple to apply, the objectives tree and decision tree methods offer a clear and useful starting format for such a statement of requirements and their relative importance. As will be discussed later, other more sophisticated and more useful method is Quality Function Deployment (QFD). c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

3. Use and adaptation of design team’ s knowledge. 4. Organization of team’s thinking. Although concept generation is an inherently creative process, it is possible to use some techniques to improve it like functional decomposition and generation of concepts from functions. Although sources for conceptual ideas come primarily from the designer’s own expertise, it can be enhanced through the use of books, experts, lead engineers, patent search, brainstorming and current designs. 1.2.5. Concept selection The purpose of concept selection is assessing the feasibility of concepts to ensure that they are achievable technically and economically. The feasibility of the concept is based on the design engineer’s knowledge. As in the generation of concepts, the design engineer can rely in tools –like the decision-matrix method– to compare and evaluate concepts. The importance of the concept selection phase cannot be understated. It is known that decisions made during the design process have the greatest effect on the cost of a product for the least investment. In figure 1.5, the cost of design and its influence in manufacturing cost for an automotive project c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

1.2 The design process Overhead 100

Labor Material

Percent

80

Design

1.2 The design process Design changes

9

10 Company A

60

Company B

40 20 0 Final Manufacturing Cost

Influence on Final Manufacturing Cost

Figure 1.5: Design influence on manufacturing cost (After Ullman, 1992).

is shown. From the figure it can be stated that the decisions made during the design process have the greatest effect on the cost of a product for the least investment. Typically, around 70% to 80% of the manufacturing cost is committed by the end of the conceptual phase of the design process. Hence the importance of concept evaluation. Also, the generation and evaluation of concepts have a great effect on the time it takes to produce a new product. Figure 1.6 shows the number of design changes made by two automobile companies with different design strategies. Company A made many changes during the early stages of the design process as a result of the iterative process of generation and evaluation of concepts. Company B made just a few changes in the initial stage, but was still making changes later in the process, even when the product was released for production. The advantage gained by company A is clear since changes made late in the process are far more expensive than changes made in early stages. The evaluation of concepts to find its viability may occur not only during concept development, but throughout the design process. This will lead to the so called Design process paradox (Ullman, 1992). The design process paradox states that during the design process, the knowledge about the design increases as the project runs in time and the design team gains understanding of the problem at hand. Hence, the knowledge of the design team is at its top when c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Begin design

Release for production

Time

Figure 1.6: Engineering changes in automobile development (After Ullman, 1992).

the design process is at its end. Although this seems natural, it is important to realize that, by the end of the process, most decisions have already been made. This increased knowledge at the end of the project tempt most design teams to feel the need of re-doing the project now that they fully understand it. Unfortunately, economics almost always drive the design process, and second chances rarely exist. Figure 1.7 shows the dilemma above. At the beginning of the process, the design team has the most freedom since no decisions have been made. As time goes by, knowledge increases as a result of the design time efforts, but freedom is lost since decisions have been made and changes are increasingly expensive to perform. 1.2.6. Concept testing Concept testing is closely related to concept selection. It is used to gather opinions and information from potential customers about one or more of the selected concepts that may be pursued. It can also be used gather information about how to improve an specific product and to estimate the sales potential of the product. Ulrich & Eppinger (2000) suggest to divide the concept testing into 6 steps: c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

11

1.2 The design process 100

Knowledge about the design problem

1.2 The design process

12

integration between different engineering disciplines involved on the design effort.

80

The preliminary design helps to obtain more precise design requirements involving analysis, benchmarking, literature search, experience, good judgment and, if necessary, testing. The refinement of the project also helps to have a better estimate of the project cost and required time for completion.

Percent

60 40 Design freedom

20 0 Time into design process

Figure 1.7: The design process paradox (After Ullman, 1992). 1. Definition of the purpose of the concept test. 2. Choosing of a survey population. 3. Choosing of a survey format. 4. Communication of the concept. 5. Measurement of customer response. 6. Interpretation of results. Both concept selection and concept testing are used to narrow the possible concepts under consideration. Concept selection relies in the work and judgment of the development team. Concept testing is based in data gathered directly from potential customers.

1.2.8. Detailed design After the preliminary design stage has been carried out, it is necessary to go into the details of the design in order to better understand the concepts. Detail design is mostly concerned with the design of the subsystems and components that make up the entire design. Because of the latter, this stage is sometimes divided into two independent parts, System-level design and the detail design itself. In the system-level design the product arquitecture is defined and decomposition of the product into subsystems and components takes place. These components may be integrated circuits, resistors, shafts, bearings, beams, plates, handles, seats, etc., depending on the nature of the product under development. Here, geometric layouts of the product and functional specifications for each subsystem are stated. The detail design phase includes the complete specification of each independent part such as geometry, materials and tolerances and identifies all those parts that will be purchased from suppliers. In this stage, the control documentation of the product is generated, including technical drawings, part production plans and assembly sequences. 1.2.9. Production planning

1.2.7. Preliminary design

The preliminary design stage or embodiment design stage, fills the gap between design concept and detailed design. According to French, in this phase the schemes are worked up in greater detail and, if there is more than one, a final choice between them is made. There is (or should be) a great deal of feedback from this phase to the conceptual design phase. Is during this stage of the design process that the overall system configuration is defined. Extensive engineering documentation in the form of schemes, diagrams, layouts, drawings, notes or other types of documents is generated to provide control over the project and to ensure better communication and c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

This stage initiates with the identification of the machines, tooling and processes required to manufacture the designed product. Technical data such as dimensions, tolerances, materials and surface finishes among others are evaluated to determine the appropriate assembly sequence for the manufacturing operations. According to Ertas and Jones (2000), typical tasks included in the production planning include: 1. Interpretation of design drawings and specifications. 2. Selection of material stock. 3. Selection of production processes. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

13

1.2 The design process

1.3 Quality Function Deployment

14

4. Selection of machines to be used in production. With QFD Effort

5. Determination of the sequence of operations. 6. Selection of jigs, fixtures, tooling and reference datum.

Traditional approach

7. Establishment of tool cutting parameters, such as speed, depth and feed rate. 8. Selection of inspection gauges and instruments. 9. Calculation of processing time. 10. Generation of process documentation and numerically controlled machine data. Once the production planning has been made and all the decisions regarding production have been taken, a production ramp-up is made using the intended production system. The purpose of the production ramp-up is to evaluate the correctness of the production plan, the tooling and the assembly sequences to follow as well as to identify possible flaws before going to a full-scale production. 1.2.10. Documentation

Engineers feel most of times burdened with the idea of documenting their designs. The preparation of documents describing the design process and the reasons behind decisions taken is oftenly seen as as an activity that does not directly contribute to the design. Other times documentation is seen as an unattractive task that does not involve any challenge at all. Nevertheless, documentation is as important as any other in the task in the design process. Product documentation is important not only in terms of instructions to user, maintainers or others, but is imperative for purposes like legal protection or future product redesign. Hence, keeping track of the ideas developed and decisions made in a design notebook is essential. It is advisable to keep, for patent or legal purposes, a notebook with dated pages that is sequentially numbered and signed. In this notebook, all information related to the design such as sketches, notes, calculations and reasons behind decisions should be included. The notebook does not have to be neat, but certain order has to be kept. When design information like plots, photocopies, drawings or results of analyses are too large or bulky to keep in the notebook, a note stating what the document is, a brief summary of its contents and where it is filed should be written. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Time Design

Details

Process

Production

Figure 1.8: Traditional vs. QFD design approaches (After Ouyang et al. When the design effort has concluded, standard drawings or computer data files of components showing all the information necessary for the production of the product have to be generated. This drawings usually include written documentation regarding manufacturing, assembly, quality control, inspection, installation, maintenance and, retirement.

1.3

Quality Function Deployment

It is not uncommon that designers find themselves working a problem only to find out later that they were solving the wrong one. An efficient designer must try by all possible means to define the correct problem at the beginning or discover the problem at earliest possible moment. The Quality Function Deployment technique provides a methodological way to do it. Quality Function Deployment (QFD) originated in Japan as a help to translate customer requirements into technical requirements throughout the development and production of a product. It originated in Japan in the 1970’s as the Kobe supertanker company wanted to develop the logistics for building complex cargo ships. Professors were asked to create a technique that would ensure that each step of the construction process would be linked to fulfilling a specific customer requirement. Using this technique, Toyota was able to reduce the costs of bringing a new car c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

15

1.3 Quality Function Deployment

1.3 Quality Function Deployment

16

Design Requirements

Details

Process

HOWS

3

WHATS

2

3. Determining relative importance of the requirements.

Process Requirements

Product Requirements

HOWS

2. Determining customer requirements.

WHATS

HOWS

Parts Requirements

Parts Requirements

Design

WHATS

1

1. Identifying the customer(s).

Design Requirements

WHATS

Customer Requirements

HOWS

The QFD technique uses six steps to do this translation:

Production Requirements

4

4. Competition benchmarking. 5. Translating customer requirements into measurable engineering requirements. 6. Setting engineering targets for the design.

Production

Figure 1.9: The four phases of QFD. From customer requirements to client satisfaction. The hows on each House of Quality becomes the whats in the next.

Each step will be reviewed in more detail, but before going any further is convenient to highlight that: • No matter how well a design team thinks it understand a problem, it should employ the QFD method.

model to the market by 60 percent and to decrease the time required for its development by one third. As shown in figure 1.8, QFD requires more effort on the design stage, but as most design flaws are catched early in the design process, later stages are less prone to fail or require adjustments or redesigns.

• Customer requirements must be translated into clear engineering targets involving measurable quantities.

According to Ouyang et al., Qualify Function Deployment has four distinct phases: design, details, process and production. As shown in figure 1.9, in the Design phase, the customer helps to define the requirements for the product or service. In the Details phase, design parameters (hows) carried over the design phase become the functional requirements (whats) of individual part details. In the Process phase, the processes required to produce the product are developed. Once more, the design parameters of the details phase become the functional requirements of the process phase. Finally, in the Production phase, the design parameters of the process phase become functional requirements for production.

• It is important to first worry about what needs to be designed and, once the problem is fully understood, to worry about how it will be designed.

• The QFD technique may be applied to the whole design as well as to subsystems or subproblems.

1.3.1. Identification of costumers

Sometimes is not only not clear what the customer wants, but also who the customer is. Furthermore, is very common to find that there is more than one

customer to satisfy.

As discussed above, QFD can be applied all the way through the design process from concept to production using the same principles on each phase. It is generally agreed that the QFD technique is most valuable at the early design stages where customer requirements have to be translated to engineering targets.

Independently of how many customers may be, it is essential to realize that the customer, and not the engineer, is the one driving the product development process. Many times the engineer has a mental picture of how the product should be like and how it should perform, picture that may be very different from what the customer really wants. On the other hand, may products have been poorly received by the customers simply because the engineer failed to identify accurately the customers’ desires.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

17

1.3 Quality Function Deployment

1.3.2. Determination of The determination of customer requirements should costumers requirements be made through customers surveys or evaluation of similar existing products. Customer requirements should be made in the customer’s own words such as “fast”, “easy”, “durable”, “light”, “strong”, etc. As much as possible, customer requirements should be stated in positive terms. In order to facilitate understanding, requirements may be grouped in types like performance requirements, appearance requirements, safety requirements, and so on. If the customer has specific preference for one given type, determining the relative importance of different requirements may be easier to do.

1.3.3. Determination of Not all requirements will be regarded as equally relative importance of the important to customers. For example, “easy to requirements use” may be more important for the customer than “easy to maintain”, and “easy to maintain” may be regarded as more more important than “good looking”. On the other hand, some requirements like “safe to use”, may be regarded as absolute requirements rather than relative preferences. In order to design effectively, the design team should know which attributes of their product design are the ones that most heavily affect the perception of the product. Hence, it is necessary to establish the relative importance of those attributes to the customers themselves.

1.3.4. Competition benchmarking

Sometimes customers often make judgment about product attributes in terms of comparisons with other products. One screwdriver, for example, may have better grip than others or another screwdriver may seem more durable. Given that customers are not generally experts, they may compare different attributes by observation of what some products achieve.

1.4 Some important design considerations

18

1.3.5. Conversion of Once a set of customer requirements have been customer needs into selected due to its importance, it is necessary to engineering requirements develop a set of engineering requirements that are measurable. Some of these engineering requirements, or design specifications, may be cleared defined from the beginning. One example is the weight that a chair must withstand. Others, may be more difficult to characterize as will be measurable by different means. In the case of a chair that is to be “easily assembled” by the customer, “easily” may be measured in terms of the number of tools needed for the assembly, the number of parts to be assembled, the number of steps needed for the assembly or the time needed for the assembly. In this step, every effort should be made in order to find all possible ways in which a customer requirement may be measured. 1.3.6. Setting engineering The last step in the process is setting the engitargets neering targets. For each engineering measure determined in the previous step, a target value will be set. This target values will be used to evaluate the ability of the product to satisfy customer requirements. Two actions will be needed, to examine how the competition meets the engineering requirements, and to establish the value to be obtained with the new product. Best targets are established using specific values. Less precise, but still usable, are those targets set within some range. Another type, extreme values, are targets set to a minimum or maximum value. Although extreme type targets are measurable, they are not the best since they give no clear information of when the performance of a new product is acceptable. Here, evaluation of the competition can give at least some range for the target value.

1.4

Some important design considerations

If the product is to be well positioned in a competitive market, the design team must ensure that its product will satisfy customer requirements better than competitor products. Therefore, the performance of the competition of those product attributes that are weighted high in relative importance should be analyzed.

When designing products, several considerations must be taken into account. For the inexpert designer, this considerations may or may not be obvious sources for requirements, parameters and targets. In what follows, three design considerations, whose importance may depend on the project at hand, are briefly discussed.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

19

1.4 Some important design considerations

1.5 Good design practices

Most of the times, when designing a new product, the design team does not pay much attention in how the product will be distributed. Decisions regarding packaging, transportation and shelf stocking are taken after the product has been designed. Nevertheless, design features that could be avoided may increase the distribution cost due to the need of special packaging, transportation or shelfs. Design teams must do everything at their hands to avoid this situations that unnecessarily increase the cost of the product.

process from early stages.

1.4.1. Product distribution

Taking into account the distribution of the product is specially important when redesigning a product. Generally speaking, companies looking for an existing product of better features are unwilling to make extensive modifications to the existing distribution infrastructure. In this cases, product distribution will be a major source of requirements. 1.4.2. Design for after life

It is normally assumed for most engineering products, that after it has completed its useful life, the product will be removed from its original installation, retired and dispose of. Nevertheless, in many occasions the product is put to some second use that is different from its original purpose. Consider for example, an empty 20 lts. (4 Gal.) bucket that is used as a step. The problem arises as this second use was not included in the initial design specifications and is therefore not accounted for in the design process. The result may be failure and personal injury leading to product liability litigation. The fact that a certain product was used in a way never intended by the original design may not be of importance on the court. Courts seem to focus on whether the failure was foreseeable and not whether there was negligence or ignorance. The best the design team can do is to try to foresee both use and misuse an make provision in the design for credible failures.

In order to translate functional requirements into design parameters, the study of ergonomics has produced a body of anthropometric (human measure) data that can be used in designing anything that involves interaction between a human and a product. As anyone will agree, humans bodies come in a variety of shapes and sizes, which makes somewhat difficult to design a product to fit absolutely everybody. Nevertheless, human measure can be well represented as normal distributions. This last feature makes it possible to define parameters to fit, let say, 90% percent of the population. In many occasions, to be able to design for such a high percentage of the population it is required to include adjustable features to the product. One typical example is the way in which seat and steering wheel positions can be altered in many cars to adapt the height and size of the driver. Other three ways in which humans may interact with products is as a source of power (for example when opening a door), as a sensor (for example reading a dashboard) or as a controller (for example the operating a CD player). In the first case, information about the average force that a human can provide (or is expected to provide) is vital toward a successful product. In the second case, if the human is expected to be able to read information is important that the person has only one way to interpret the data. In the third way, a product must be designed so there is no ambiguities in the form in which the product operates. For the product to be easy to interact with, there must be only one obviously correct thing to do for every action that is required.

1.5 1.4.3. Human factors in design

Almost every product that is designed will interact with humans whether during manufacture, operation, maintenance, repair or disposal. Operation is probably the most important since it will involve the largest span of interaction.

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Good design practices

Considering operation, a good product will be the one that becomes an extension of the user’s motor and cognitive functions. To achieve this, human– machine interaction features should be included as parameters in the design

1.5.1. Good design versus The goal for the introduction of models for the bad design design process is to provide a guideline to help the engineer/designer to achieve a better product through the use of good design practices. As experience would tell, in most occasions it is not difficult to tell either as engineer or consumer, a good design from a bad design. Table 1.2 show some general characteristics of good design versus bad design.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

21

1.5 Good design practices

Good Design

Bad Design

1. Works all the time

1. Stops working after a short time

2. Meets all technical requirements

2. Meets only some technical requirements

3. Meets cost requirements

3. Costs more than it should

4. Requires little or no maintenance

4. Requires frequent maintenance

5. Is safe

5. Poses a hazard to user

6. Creates no ethical dilemma

6. Fulfills a need that is questionable

Table 1.2: Characteristics of good design versus bad design. After Horenstein (1999). 1.5.2. Good design Horenstein (1999) highlights the traits of good engineer versus bad design design engineer and bad design engineers. Acengineer cording to Horenstein, a good engineer: • Listens to new ideas with an open mind. • Considers a variety of solution methodologies before choosing a design approach. • Does not consider a project complete at the first sign of success, but insists on testing and retesting. • Is never content to arrive at a set of design parameters by trial and error. • Use phrases such as “I need to understand why” and “Let’s consider all the possibilities”. A Bad Engineer:

1.5 Good design practices

22

• Equates pure trial and error with engineering design. Green (1992) summarizes skills that seem to mark the expert designer in domains of routine design. Supplying context. The requirements seldom provide enough information to create a design. This occurs in part because the client himself does not know precisely what he/she wants. However, another problem is that the stated requirements imply several other, unstated, requirements. The expert can “read between the lines” and supply context that reduces the search space. Decision ordering. Strategic knowledge is a major part of the designers’ expertise. The expert designer is able to make decisions in the correct order to avoid spending much time in backtracking and revising. Decision ordering is important because it rank constraints. The expert’s decision ordering set constraint values in some optimal sequence. Heuristic classification. Although the overall design problem may be ill-structured, it usually contains some well-structured components. Some decisions fell into the heuristic classification paradigm (here, heuristic means problem-solving techniques that utilize self-education techniques, as the evaluation of feedback, to improve performance). The designer begins by listing requirements, both stated and unstated, and maps them to design parameters which enables him/her to choose a set of design classes. Parameter abstraction. Much of routine design requires to simultaneously manage a large collection of variable values. This can be a very complex cognitive task since it requires the expert to maintain a large amount of information in working memory. Experts are able to reduce the complexity of the problem by abstracting only the most important parameters, treating related parameters as single entities whenever possible.

• Thinks he/she has all the answers; seldom listens to the ideas of others. • Has tunnel vision; pursues with intensity the first approach that comes to mind.

References

• Ships the product out the door without thorough testing.

1. Cross, N. (1994) Engineering Design Methods, John Wiley & Sons. 2. Eide, A., Jenison, R., Mashaw, L. & Northup, L. (1998) Introduction to Engineering Design. McGraw-Hill. 3. Ertas A. & Jones, J. (1996) The Engineering Design Process, second ed.,

• Use phrases such as “good enough” and “I don’t understand why it won’t works; so-and-so I it this way.” c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

23

1.5 Good design practices

John Wiley & Sons. 4. Horenstein, M. (1999) Design Concepts for Engineers, Prentice-Hall. 5. Otto, K. & Wood, K. (2001) Product Design - Techniques in Reverse Engineering and New Product Development, Prentice-Hall. 6. Ouyang, S., Fai, J., Wang, Q. & Johnson, K. Quality Function Deployment. University of Calgary Report. 7. Pugh, S. (1990) Total Design, Addison Wesley. 8. Suh, N. (1990) The Principles of Design. Oxford University Press. 9. Ullman, D. (1992) The Mechanical Design Process, McGraw-Hill. 10. Ulrich, K. & Eppinger, S. (2000) Product Design and Development. Irwin McGraw-Hill.

CHAPTER

2

Identifying customer needs

If a new or redesign product is to be successful, it should fulfill the needs of the customer. Unfortunately, the process of finding which are the real needs to be fulfilled is not a straightforward one. According to Ulrich & Eppinger (2000), the goals of a method for comprehensively identifying a set of customer needs should be: 1. Ensure that the product is focused on customer needs. 2. Identify latent or hidden needs as well as explicit needs. 3. Provide a fact base for justifying the product specification. 4. Create an archival record of the needs activity of the development process. 5. Ensure that no critical customer need is missed or forgotten. 6. Develop a common understanding of customer needs among members of the development team. The main purpose of identifying customer needs is to create a direct information link between customers and developers. The involvement of members of the design team (specially engineers and industrial designers) results essential as they must have a clear view of how the product will be used by the end c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

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25

2.1 Customer satisfaction

2.1 Customer satisfaction

In this chapter, the next 5 steps to effectively identify customer needs will be discussed:

Customer Satisfaction

user. This direct experience will help the design team not only to discover the true needs of the customer, but also to create better concepts and to evaluate them in a more accurate form.

26 Delighted e

urv

c an orm

eC

rf

d cte pe Ex

Delighted Performance Curve

Pe

Fully Implemented

1. Gather raw data from customers. 2. Interpret the raw data. 3. Organize the needs into a hierarchy.

Function

Absent

Basic Performance Curve

4. Establish the relative importance of the needs. 5. The review of the process and its results. Disgusted

2.1

Customer satisfaction

In order to satisfy customers, a given product must fulfill customer expectations about it. Even when finding which features are wanted by the customer is a difficult task since customers usually not mention them directly, customer satisfaction translates to the implementation in a given product as much desired features as possible. In order to better understand this relationship, the Kano diagram may be of help. 2.1.1. The Kano diagram

The Kano model shown in figure 2.1, shows the relationship between customer needs and satisfaction in an easy to appreciate diagram ranking the customer satisfaction from disgusted to delighted.

The lower curve in Kano’s diagram is called the basic performance curve or expected requirements curve. It represent the essentially basic functions or features that customers normally expect of a product or service. They are usually unvoiced and invisible since successful companies rarely make catastrophic mistakes. However, they become visible when they are unfulfilled.

Figure 2.1: Kano diagram of customer satisfaction. After Otto & Wood (2001).

They satisfy customers when fulfilled. But they do not leave customers dissatisfied when left unfulfilled. And they are invisible to customers since they are not even known. The center line of the Kano diagram is called the one-to-one quality or linear quality line. It represents the minimum expectation of any new product development undertaking. It is related also to performance type issues such as “faster is better.” These represent what most customers talk about. Thus, they are visible to the company and its competitors. The expected requirements and exciting requirements provide the best opportunity for competitive advantage. Hence, ways to make hem visible and then deliver on them are needed.

The upper curve in Kano’s diagram is called the delighted performance curve or exciting requirements curve. They are a sort of “out of the ordinary” functions or features of a product or service that cause “wow” reactions in customers.

Kano’s diagram is often interpreted simply as a relationship model of expected quality vs. excited quality. What is really important, however, is that the target of customer satisfaction can not only invisible but also moving. Customer expectations increase over time. This calls for a more complex analysis and deeper market understanding.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

27

2.1 Customer satisfaction

2.2 Gather data from customers

28

According to Otto & Wood (2001) customer needs may be profitably considered in general categories based on how easy the customer can express them and how rapidly they change. They can be classified in three categories: first, direct and latent needs which consider observability, second, constant and variable needs which consider technological changes and finally, general and niche needs which consider variance in the consumers.

2.2

Direct needs These are the needs that, when asked about the product customer have no trouble declaring as something they are concerned about. These are easily uncovered using standard methods as the one that will be described hereafter.

Interviews One or more members of the design team interview a number of customers, one at a time. Interviews are generally carried out in the environment of the costumer where the product is used. They typically last for one to two hours.

Latent needs These are the needs that typically are not directly expressed by the customer without probing. Customer typically do not think in modes that allow themselves to express these needs directly. Latent needs are better characterized as customer needs, not of the product, but of the system within which the product operates. Other products, services or actions currently satisfy the needs directly. Yet, these needs might be fulfilled with a developing product, and doing so can provide competitive advantage.

Questionnaires A list of important concerns, questions and criteria is prepared by the design team and sent to selected customers. Although this type of survey is quite useful at later stages of the design process, at this stage they do not provide enough information about the use environment of the product. It is also important to notice that not all needs may be revealed using this method.

2.1.2. Types of customer needs

Constant needs These needs are intrinsic to the task of the product and always will be. When a product is used, this need will always be there. Such needs are effective to examine with customer needs analysis, since the cost can be spread over time. Variable needs These needs are not necessarily constant; if a foreseeable technological change can happen, these needs go away. These needs are more difficult to understand through discussions with the customer, since the customer may not understand them yet. General needs These needs apply to every person in the customer population. It is necessary for a product to fulfill these needs if it is to compete in the existing market. Niche needs These needs apply only to a smaller market segment within the entire buying population.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Gather data from customers

In order to obtain information from customers, several methods are available: interviews, questionnaires, focus groups, observing the product in use and finally, be the customer oneself. In what follows, a brief description of each one together with pros and cons is given.

Focus groups A group of 8 to 12 customers participate in a discussion session facilitated by a moderator. Focus groups are typically conducted in a special room equipped with a two-way mirror allowing several members of the development team to observe the group. It is desired for the moderator to be a professional market researcher, but a member of the development team can also perform as moderator. Observing the product in use When watching a customer using an existing product or perform a task for which a new product is intended, details about customer needs can be reveled. Observation may be passive, leaving the customer to use the product without any direct interference or can be carried out along with one of the design team members allowing the development of firsthand experience about the use of the product. Be the customer In many situations, members of the design team may perform as users of existing competitor products or, in later stages of the design process, of prototypes. Although this method is very cost effective and relatively easy to perform as no persons outside the design team are involved, it posses two main problems. First, members of the design team may not have the required skills or experience to accurately evaluate the product, and second, they may feel biased towards certain characteristics of the product. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

29

2.2 Gather data from customers Lead Users

Users

Retailer or Sales Outlet

Service Centers

Occasional User Frequent User Heavy−duty User

Figure 2.2: Customer selection matrix. After Ulrich & Eppinger (2000).

2.2 Gather data from customers

30

2.2.2. Conducting Interviews

Ulrich and Eppinger provide some general hints for effective customer interaction. First, they suggest to sketch an interview guide that help to obtain an honest expression of needs. This can not be stressed enough, the goal of the interview is to obtain customer needs, not to convince the customer of what he or she really wants. Some helpful questions and prompts to use are: • When and why do you use this type of product? • Walk us through a typical session using the product. • What do you like about the existing products? • What do you dislike about the existing products?

From the above methods, research carried out by Griffin and Hauser (1993) reports that conducting interviews is the most cost and effort effective method. According to their report, one 2-hour focus group reveals about the same number of needs as two 1-hour interviews. They also report that interviewing nine customers for one hour each will obtain over 90% of the customer needs that would be uncovered when interviewing 60 customers. These figures where obtained when a single function product was being considered, and may change when considering multi-function products. According to Ulrich & Eppinger, as a practical guideline for most products, conducting fewer than 10 interviews is probably inadequate and 50 interviews are probably too many.

2.2.1. Selecting customers

Selecting customers is not always a straightforward activity as many different persons may be considered a “customer”. Consider, for example, all those products that are purchased by one person and used by another. In all cases, it is important to gather information from the end user, and then gather information from other type of customers and stake-holders. A customer selection matrix like the one shown in figure 2.2, is useful for planning exploration of both market and customer variety. It is recommended that market segments be listed on the left side of the matrix while the different types of customers are listed across the top. The number of intended customer contacts is entered in each cell to indicate the depth of coverage.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

• What issues do you consider when purchasing the product? • What improvements would you make to the product? Second, they suggest the following general hints for effective interaction with customers: • Go with the flow. If the customer is providing useful information, do not worry about conforming to the interview guide. The goal is to gather information data on customer needs, not to complete the interview guide in the allotted time. • Use visual stimuli and props. Bring a collection of existing and competitors’ products, or even products that are tangentially related to the product under development. At the end of a session, the interviewers might even show some preliminary product concepts to get customers’ early reactions to various approaches. • Suppress preconceived hypotheses about the product technology. Frequently customers will make assumptions about the product concept they expect would meet their needs. In these situations, the interviewers should avoid biasing the discussion with assumptions about how the product will eventually be designed or produced. When customers mention specific technologies or product features, the interviewer should probe for the underlying need the customer believes the suggested solution would satisfy. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

31

2.2 Gather data from customers • Have the customer demonstrate the product and/or typical tasks related to the product. If the interview is conducted in the use environment, a demonstration is usually convenient and invariably reveals new information.

2.2 Gather data from customers Customer Data: Project/Product Name Customer: Address: Willing to do follow up? Type of user: Question

Customer Statement

32 Interviewer(s): Date: Currently uses: Interpreted Need

Importance

• Be alert for surprises and the expression of latent needs. If a customer mentions something surprising, pursue the lead with follow-up questions. Frequently, an unexpected line of questioning will reveal latent needs important dimensions of the customers’ needs that are neither fulfilled nor commonly articulated and understood. • Watch for nonverbal information. The design process is usually aimed at developing better physical products. Unfortunately, words are not always the best way to communicate needs related to the physical word. This is particularly true of needs involving the human dimensions of the product, such as comfort, image or style. The development team must be constantly aware of the nonverbal messages provided by customers. What are their facial expressions? How do they hold competitors’ products? 2.2.3. How to document interactions

There are four main methods for documenting interactions with customers:

Figure 2.3: Customer data template. After Otto & Wood (2001).

Notes Handwriting notes are the most common method of documenting an interview. If a person is designated as notetaker, other person can concentrate in effectively questioning the customer. The notetaker should try to capture the answers of the customer in a verbatim form. If the notes from the interview are transcribed inmediately after it, a very close account of the interview can be obtained.

the advantage of being inexpensive and easy to do.

Audio recording Audio recording is probably the easiest way of documenting and interview. Unfortunately, many customers feel intimidated by it. Another disadvantage is that transcribing the recording into text is very time consuming. Video recording Video recording is the usual way of documenting focus group sessions. It is also very useful for documenting observations of the customer in the use environment and the performance of existing products.

In the first column, the question prompted is recorded. In the second column, a verbatim description of the answer and comments given by the customer is recorded. In the third column, the customer needs implied by the raw data are written. Special attention must be given to clues that may identify potential latent needs like humorous remarks, frustrations or non-verbal information. In the last column, linguistic expressions of importance that the customer may have used are recorded. The importance may be expressed in terms of words like must, good, should, nice or poor.

Still photography Even when dynamic information cannot be captured by it, still photography can be used to capture high quality images. It also has

According to Otto & Wood, a must is used when a customer absolutely must have this feature, generally when it is a determining criterion in purchasing

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

One useful aid in the collection of data from a customer interview is a customer data template. A customer data template, like the one shown in figure 2.3, helps to record questions, answers and comments. The template can be filled during the interview or inmediately afterwards.

33

2.3 Interpret raw data

the product. Must ratings should be used very sparingly; only a few must’s per customer interview is a good rule. A very important customer need should have a good importance rating. Needs that are presumed should have at least a should rating. If the customer feels the product should satisfy this requirement, it is important enough for the design team to consider it. The nice category is for customer needs that would be nice if the product satisfied them but are not critical.

2.3

2.4 Organization of needs Guideline

Customer Statement

Need Statement - Right

Need Statement - Wrong

“What” not “how”

“Why don’t you put protective shields around the battery contacts?”

The screwdriver battery is protected from accidental shorting.

The screwdriver battery contacts are covered by a plastic sliding door.

Specificity

“I drop my screwdriver all the time.”

The screwdriver operates normally after repeated dropping.

The screwdriver is rugged.

Positive not negative

“It doesn’t matter if it’s raining; I still need to work outside on Saturdays.”

The screwdriver operates normally in the rain.

The screwdriver is not disabled by the rain.

An attribute of the product

“I’d like to charge my battery from my cigarette lighter.”

The screwdriver battery can be charged from an automobile cigarette lighter.

An automobile cigarette lighter adapter can charge

Avoid “must” and “should”

“I hate it when I don’t know how much juice is left in the batteries of my cordless tools.”

The screwdriver provides an indication of the energy level of the battery.

The screwdriver should provide an indication of the energy level of the battery.

Interpret raw data

At this point, customer needs are expressed in terms of verbatim written statements. Every customer comment or observation as expressed in the customer data template may be translated into any number of customer needs. It has been found that multiple analysts may translate the same interview notes into different needs, so it is convenient for more than one team member to be involved in this task. Ulrich & Eppinger provide five guidelines for writing need statements. They recognize the first two as fundamental and critical to effective translation, and the remaining three as guidelines to ensure consistency of phrasing and style across all team members. Table 2.1 shows examples to illustrate each guideline.

34

Table 2.1: Examples illustrating the guidelines for writing need statements for a cordless screwdriver (After Ulrich & Eppinger, 2000). • Express the need as an attribute of the product. Wording needs as statements about the product ensure consistency and facilitates subsequent translation into product specifications. • Avoid the words must and should. The words must and should imply a level of importance for the need.

• Express the need in terms of what the product has to do, not in terms of how it might do it. Customers often express their preferences by describing a solution concept or an implementation approach; however, the need statement should be expressed in terms independent of a particular technological solution.

After all the customer comments have been translated into need statements, the design team ends up with a group of maybe tens or even hundreds of need statements. At this point, some may be similar, other may not be technological feasible, and others may express conflicting needs. In the following section, methods for organizing and classifying these needs are presented.

• Express the need as specifically as the raw data. Needs can be expressed at many different levels of detail. To avoid loss of information, express the need at the same level of detail as the raw data.

2.4

• Use positive, not negative, phrasing. Subsequent translation of a need into a product specification is easier if the need is expressed as a positive statement. This may not apply in those occasions when the statement is expressed more naturally in negative terms. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Organization of needs

2.4.1. Classification of needs

In order to work effectively with all the customer needs, it is necessary to classify them in groups of equal or similar statements. Each group may be subsequently sorted out in a list according to the relative

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35

2.4 Organization of needs

2.5 Design brief

importance of each need in the group. Each group list typically consists of a set of primary needs, each one of which will be characterized by a set of secondary needs and if needed, tertiary needs.

Importance

This process of sorting and classification is normally performed by the design team. Nevertheless, it also exists the possibility of leaving this task to a group of selected customers. According to Otto and Wood, this approach prevents the customer data from being biased by the development team. The classification of needs can be done without many difficulties following the next steps:

36 Ranking 1

Ranking 2

Must

9

1.0

Good

7

0.7

Should

5

0.5

Nice

3

0.3

not mentioned

0

0

Table 2.2: Two different ranking systems for the importance of needs. interpreted importance rank of the ith customer need can be obtained from

1. Write each need on a small card.

Ri =

2. Group similar needs eliminating redundant statements. 3. Choose a descriptive name for each group. 4. Review the process and consider alternative ways of grouping the statements. When working with different customer segments, cards with different color labels can be used to distinguish between them. The sorting and classification process can also be done separately for each customer segment observing differences in both the needs themselves and their organization. The latter approach is best suited when the segments are very different in their needs and when there is the question if just one product may suit the needs of all segments. 2.4.2. Determination of relative importance of needs

As of now, the classification of needs does not provide any information regarding the relative importance that the customer place on different needs. Each customer need has an importance expressed by the own customer during the interview. It is expected that different customers will feel different regarding the importance of features according to their own use of the product.

number of times mentioned number of subjects

(2.1)

It is important to have in mind that the above method may raise inconclusive results as it mainly measures the obviousness of the need as opposed to its importance. Therefore, needs that may be obvious but not important may be ranked high as opposed to important needs that may not be obvious. A more correct approach, is to include in the ranking the importance statements given by the customer during the interview. In order to do so, it is necessary to convert the subjective importance ratings into numerical equivalents. A typical transformation is shown in table 2.2. Once the mapping has been carried out, the importance assigned to each customer need can be calculated as: average rating × number of times mentioned (2.2) Ri = number of subjects Although a better method of ranking customer needs, the previous method has also its own flaws as it still may hide important needs that were reveled by only few customers but were not seen by the rest.

2.5

Design brief

An elementary approach to establish the relative importance of needs is to first construct a set of normalized weightings by comparing the number of subjects who mention a need versus the total number of subjects. Hence, the

After grouping and ranking customer needs, a better idea of the design problem is at hand. To keep a clear idea of the direction of the design process, the design

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37

2.6 Clarifying customer needs

team may issue what is called a design brief or mission statement. A design statement includes a brief description of the product, key business goals, target markets, assumptions and constraints and stakeholders: • Description of the product A brief description typically includes the key customer benefits of the product avoiding implying a specific product concept. • Key business goals. These goals generally include goals for time, cost, quality and market share. Other goals may be added as deem appropriate. • Target markets. Identifies the primary as well as secondary markets that should be considered during the design process. • Assumptions and constraints. In some projects is necessary to make assumptions in order to keep a project of manageable scope and size. In other occasions, time, cost or even feature constraints are known from the beginning of the product. • Stakeholders. It is always convenient to list all the stakeholders in order to handle subtle issues that may appear during the development process. Stakeholders are all the groups of people who are affected by the success or failure of the product. The list usually begins with the end user and the customer who makes the buying decision about the product. Stakeholders also include the customers residing within the firm such as the sales force, the service organization and the production departments.

2.6

Clarifying customer needs

One step further in the determination of customer needs is to try to clarify all the customer need that were grouped, classified and prioritized. In fact, it is very helpful to have the clearest possible idea of the customer needs at all stages of the design process. These customer needs, that will guide the design process, should be expressed in a form which is easily understood and which can be agreed by both, client and designer. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

2.6 Clarifying customer needs

38

2.6.1. The objectives tree The objectives tree method offers a clear and usemethod ful format for such a clarification of customer need statements in form of objectives. It also shows in a diagrammatic form the ways in which different objectives are related to each other and the hierarchical pattern in which they are organized. As with many methods in the design process, the objectives tree is not as important as the procedure for arriving at it. One way to start making vague statements more specific is to try to simple specify what it means. Consider the following example provided by Cross (1994) where an objective for a machine tool must be ‘safe’. This objective might be expanded to mean: 1. Low risk of injury to operator. 2. Low risk of operator mistakes. 3. Low risk of damage to work-piece or tool 4. Automatic cut-out on overload. These different statements can be generated simply at random as the design team discusses about the objective. The types of questions that are useful in expanding and clarifying objectives are simple ones like ‘why do we want to achieve this objective?’ and ‘what is the problem really about?’. Some authors also include questions like ‘How can we achieve it?’ starting to give some insight about how the objectives may be accomplished. This gives way to statements like ‘automatic cut-out on overload’ which are not objectives by themselves but means of achieving certain objectives. Nevertheless, it is difficult to avoid making concessions reducing the scope of the possible solutions that may be generated in later stages of the design process. For this reason, in the approach followed here, everything related to the ‘how to’ accomplish objectives will be left to the concept generation stage. As the list of objectives is expanded, it becomes clear that some are at higher levels of importance than others. This relative importance may be represented in a hierarchical diagram of relationships as shown in figure 2.4. In some cases, the relative position of each statement in the diagram may be a source of disagreement between the different members of the design team. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

39

2.6 Clarifying customer needs

2.6 Clarifying customer needs

40

Machine must be safe

Provide opening

How Low risk of injury to operator

Low risk of operator mistakes

Low risk of damage to workpiece or tool

Enable in/out

Pivot door Open door

Why

Push/pull door

Automatic cut−out on overload

Close door

Figure 2.4: Hierarchical diagram of relationships. After Cross (1994).

Keep weather out Povide seal

However, exact precision of relative levels is not important, and most people can agree when only a few levels are being considered. At this point, it is important to notice that the level of importance of the statement should not be confused with the level of importance of the customer need. Here, importance is related to the statements written to try to clarify one objective, which correspond to one customer need.

When open

Provide protection

Correct amount

Safe force

In many cases, different people will draw different objectives trees for the same problem or the same set of objective statements. The tree diagram simply represents one perception of the problem structure. It is only a temporary representation, which will probably change as the design process proceeds. One more elaborated example of an objective tree is shown in figure 2.5 where the objectives tree for the design of a car door is shown.

Safe direction

Against injury

When closing

Resist impact Resist damage

Provide safety

Safe interior

When closed

The procedure of building an objectives tree can be summarized using the following steps:

Strong latch

Provides latch Latches securely

Against theft

Secure handle Inaccessible lock

Figure 2.5: Objectives tree for a car door. After Pugh (1991).

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

41

2.6 Clarifying customer needs

2.6 Clarifying customer needs

42

1. Prepare a list of design objectives. Black Box

2. Order the list into sets of higher-level and lower-level objectives. 3. The expanded list of objectives and sub-objectives is grouped roughly into hierarchical levels.

Inputs

Function

Outputs

Figure 2.6: A ‘black box’ system model. After Cross (1994).

4. Draw a diagrammatic tree of objectives showing hierarchical relationships which suggest means of achieving objectives.

Transparent Box

From the objectives tree method, it is clear that design problems can have different levels of generality or detail. Hence, the level at which the problem is defined is crucial and it is always appropriate to question the level at which the design problem is posed. On the other hand, focusing too narrowly on a certain level may hide a more radical or innovative solution. 2.6.2. The functional decomposition method

In any way, it is useful to have means of considering the problem level at which a design team is to work. It is also very useful if this can be done considering the essential functions that a solution will be required to satisfy. This approach leaves the design team free to develop alternative solution proposals that satisfy the functional requirements. The function decomposition method offers such means of considering essential functions and the level at which the problem is to be addressed. The essential functions are those that the device, product or system to be design must satisfy, independently what physical components might be used to fulfill them.

Subfunction

Subfunction

Inputs

Subfunction

Subfunction

Function

Outputs

Figure 2.7: A ‘transparent box’ model. After Cross (1994). are the outputs for?, what is the next stage of conversion?, etc. can be made to the customer. Usually the conversion of the set of inputs into the set of outputs is a complex set of tasks occurring inside the black box. This complex set of tasks must be broken down into sub-tasks or sub-functions which linked together by their inputs and outputs satisfy the overall function of the product or device being designed. As this necessary sub-functions are establish, the black box is redraw as a ‘transparent box ’ (see figure 2.7).

The starting point of this method is to clarify what is the main purpose of the design. As it has been up to now, it is important what has to be achieved by the new design and not how is going to be achieved. The most simple way of representing this main purpose is to draw a ‘black box ’ which converts certain inputs into desired outputs (see figure 2.6). This black box contains all the functions which are necessary for converting inputs into outputs.

According to Pahl and Beitz (2001), anyone setting up a function structure ought to bear the following points in mind:

At this point, it is preferable to try to make this overall function as broad as possible, avoiding to start with a narrow function that limits the range of possible solutions. In order to establish in an accurately way the required inputs and outputs as well as the ‘system boundary’ which defines the function of the product or device, questions like where do the inputs come from?, what

1. First derive a rough function structure with a few sub-functions from what functional relationships you can identify in the requirements list, and then break this rough structure down, step by step, by the solution of complex sub-functions. This is much simpler than starting out with

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43

2.6 Clarifying customer needs more complicated structures. In certain circumstances, it may be helpful to substitute a first solution idea for the rough structure and then, by analysis of that first idea, to derive other important sub-functions. It is also possible to begin with subfunctions whose inputs and outputs cross the assumed system boundary. From these it is possible to determine the inputs and outputs for the neighboring functions, in other words, work from the system boundary inwards.

2.6 Clarifying customer needs

44

• Storing energy – for instance, storing kinetic energy. Conversion of material: • Changing matter – for instance, liquefying a gas. • Varying material dimensions – for instance, rolling sheet metal. • Connecting matter with energy – for instance, moving parts.

2. If no clear relationship between the sub-functions can be identified, the search for a first solution principle may, under certain circumstances, be based on the mere enumeration of important sub-functions without logical or physical relationships, but if possible, arranged according to the extent to which they have been realized.

• Connecting matter with signal – for instance, cutting off steam.

3. Logical relationships may lead to function structures through which the logical elements of various working principles (mechanical, electrical, etc.) can be anticipated.

• Storing material - for instance, keeping grain in a silo.

• Connecting materials of different type – for instance, mixing or separating materials. • Channelling material - for instance, mining coal.

Conversion of signals:

4. Function structures are not complete unless the existing or expected flow of energy, material and signals can be specified. Nevertheless, it is useful to begin by focusing attention on the main flow because, as a rule, it determines the design and is more easily derived from the requirements. The auxiliary flows then help in the further elaboration of the design, in coping with faults, and in dealing with problems of power transmission, control, etc. The complete function structure, comprising all flows and their relationships, can be obtained by iteration, that is, by looking first for the structure of the main flow, completing that structure by taking the auxiliary flows into account, and then establishing the overall structure.

• Changing signals – for instance, changing a mechanical into an electrical signal, or a continuous into an intermittent signal.

5. In setting up function structures it is helpful to know that, in the conversion of energy, material and signals, several sub-functions recur in most structures and should therefore be introduced first. Essentially, the generally valid functions are described next. Conversion of energy:

• Channelling signals – for instance, transferring data.

• Changing energy – for instance, electrical into mechanical energy. • Varying energy components – for instance, amplifying torque. • Connecting energy with a signal – for instance, switching on electrical energy. • Channeling energy – for instance, transferring power. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

• Varying signal magnitudes – for instance, increasing a signal’s amplitude. • Connecting signals with energy – for instance, amplifying measurements. • Connecting signals with matter – for instance, marking materials. • Connecting signals with signals – for instance, comparing target values with actual values. • Storing signals – for instance, in data banks. 6. In the case of mechanical devices, table 2.3 can be a good starting point to identify functions. 7. For the application of micro-electronics, it is useful to consider signal flows as shown in figure 2.5. This results in a function structure that suggests clearly the modular use of elements to detect (sensors), to activate (actuators), to operate (controllers), to indicate (displays) and, in particular, to process signals using microprocessors. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

45

2.6 Clarifying customer needs

Operate

User

Indicate

2.6 Clarifying customer needs

46

Absorb/remove

Dissipate

Release

Actuate

Drive

Rectify

Amplify

Hold or fasten

Rotate

Assemble/disassemble Increase/decrease Secure Process (control)

Detect

Technical system

Activate

Figure 2.8: Basic signal flow functions for modular use in micro-electronics. After Pahl and Beitz (2001). 8. From a rough structure, or from a function structure obtained by the analysis of known systems, it is possible to derive further variants and hence to optimize the solution, by: • braking down or combining individual sub-functions; • changing the arrangement of individual sub-functions; • changing the type of switching used (series switching, parallel switching or bridge switching); and • shifting in the system boundary. Because varying the function structure introduces distinct solutions, the setting up of function structures constitutes a first step in the search for solutions. 9. Function structures should be kept as simple as possible, so as to lead to simple and economical solutions. To this end, it is also advisable to aim at the combination of functions for the purpose of obtaining integrated function carriers. There are, however, some problems in which discrete functions must be assigned to discrete function carriers, for instance, when the requirements demand separation, or when there is a need for extreme loading and quality.

Change

Interrupt

Shield

Channel or guide

Join/separate

Start/stop

Clear or avoid

Lift

Steer

Collect

Limit

Store

Conduct

Locate

Supply

Control

Move

Support

Convert

Orient

Transform

Couple/interrupt

Position

Translate

Direct

Protect

Verify

Table 2.3: Typical mechanical design functions. After Ullman (2003). The procedure to follow to establish the required functions and the system boundary of a new design can be stated using the following steps: 1. Express the overall function for the design in terms of the conversion of inputs and outputs. 2. Break down the overall function into a set of essential subfunctions. 3. Draw a block diagram showing the interaction between subfunctions. 4. Draw the system boundary. The system boundary defines the functional limits for the product or device to be designed. In order to effectively apply the functional decomposition method, the following guidelines should be followed: 1. Document what not how. 2. Use standard notation when possible. 3. Consider logical flows.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

47

2.6 Clarifying customer needs Cold water

2.6 Clarifying customer needs

48

Hot tea

(measured quantity) Tea begin BREWED Tea leaves (measured quantity)

Tea leaves (waste)

Figure 2.9: Black box model of the tea brewing process. After Cross (1994). 4. Match inputs and outputs in the functional decomposition. (a)

One simple example that can be used to illustrate the process of functional decomposition is that of a tea maker (Cross, 1994). The fundamental process to be achieved by such a machine is to convert cold water and tea leaves into hot tea as illustrated in figure 2.9.

Water is heated

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Tea is infusing

Tea and water are separated

Tea leaves

(b)

Water

Leaves

Tea Water is heated

Tea leaves are immersed

Energy Tea leaves

(c)

1. Cross, N. (1994) Engineering Design Methods, John Wiley & Sons. 2. Otto, K. & Wood, K. (2001) Product Design - Techniques in Reverse Engineering and New Product Development, Prentice-Hall. 3. Pahl, G. and Beitz W. (2001) Engineering Design - A systematic Approach. Second Ed. Springer. 4. Ullman, D. (2003) The Mechanical Design Process. Third Ed. McGraw-Hill. 5. Ulrich, K. & Eppinger, S. (2000) Product Design and Development. Second Ed. Irwin McGraw-Hill.

Water and tea united

Energy

Some transparent box models of the tea maker are shown in figure 2.10. These models represent three alternative processes by which the overall function can be achieved. After considering them, the designer settled on the first process where various necessary auxiliary functions became apparent, specially regarding the control of the heating and brewing processes.

References

Tea

Water

5. Break the function down as finely as possible.

Water Water is heated

Tea leaves are wetted

Leaves

Concentrate and water are united

Tea

Energy

Figure 2.10: Three alternatives to the transparent box model for the tea brewing process. After Cross (1994).

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

3.1 Benchmarking

50

they understand their product by mere self-inspection, they are closing doors to a wide array of alternative possibilities.

CHAPTER

3

Benchmarking and Product Specifications

Benchmarking the competition as an activity in the product development process overlaps many of the other activities as it generates data that is important to understand a product and forecast its future development. This activity cannot be understated, product developers must learn from competitors. Companies must avoid the Not-Invented-Here (NIH) syndrome that presents when engineers at a company choose not to use technology developed outside it as it is considered to not be of any good. This may cause a product to fail, as it leaves the design teams and companies behind as new technology emerges at the marketplace. Design teams must understand the importance of newly introduced technology by competitors and be ready to respond. Benchmarking allows to meet this goal. It is also an important step in establishing engineering specifications.

3.1

Benchmarking

A famous example of understanding the competition is that of Xerox Corporation. When in 1979 Xerox marketshare in the copy machines segment was rapidly decreasing, its engineers pondered the following question: “How in the world could the Japanese manufacture in Japan, ship it over to the United States, land it, sell it to a distributor who sells it to a dealer who marks up the cost to the final customer, and the price the customer pays is about what it would cost us to build the machine in the first place?” (Jacobson and Hillkirk, 1986). Even when at the time Xerox was not able to analyze and understand competitor’s product, production and distribution, they have now competitive benchmarking activities. These activities allows them to focus on how to be successful, rather than how competitors can be better than them. In order to understand the competition, design teams must tear down and analyze competitive products. This activity must be done periodically, not only supporting new design efforts but also developing a continuous understanding of trends and directions in technology development. Many large companies have entire departments devoted only to benchmarking activities. These departments provide insight not only on new technological developments, but also in the position of the company’s products in the marketplace in terms of quality, value and performance. Benchmarking activities are vital at all stages of the product development as they: • provide a way to understand what needs other products are satisfying • provide means to establish product specifications ensuring that products goals superpass existing competition • help in the concept generation stage providing best-in-class concepts • help to incorporate in the detailed design new and improved design features of the best-in-class products

There are two main purposes for studying existing competitive products: first, creates an awareness of what products are already available, and second, reveal opportunities to improve what already exists. Design teams must be aware not only on what other products offer, but also how other competitors provide similar products. As Otto & Wood (2001) clearly state, when engineers think

According to Otto & Wood (2001), product benchmarking can be carried out following the next steps:

c Copyright 2006 Dr. Jos´ e CarlosMiranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

• help to find the best-in-class components and suppliers

51

3.1 Benchmarking 1. Form a list of design issues 2. Form a list of competitive or related products 3. Conduct an information search 4. Tear down multiple products in class 5. Benchmark by function 6. Establish best-in-class competitors by function 7. Plot industry trends

3.1.1. Form a list of design A list of issues must be developed for comparaissues tive benchmarking. Further, this list should be continually revised and updated. With a focus for benchmarking efforts, an efficient exploration path may be pursued. The result is a reduction in wasted time and resources. Considering the design issues and product function 3.1.2. Form a list of competitive or related in product development, the next step is to examproducts ine retailer stores and sales outlets for products that demonstrate these issues. For a product, it is necessary to list all competitors and their different product models. In addition, all related products in their portfolio should be listed. If the competitors have a family of products under a common platform (they use identical components for some aspects of each product but different components for niche demands), detailed information about this should be included as it indicate the competitor’s preferred market segments and compromises made for other market segments. This step should only be an identification of the competitors in the form of company names and product names. With a complete set of different products, vendors and suppliers to examine, the list should be screened by highlighting the particular competitors that appear most crucial for the design team to fully understand. This step serves as basis for the next step, conducting an information search. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

3.1 Benchmarking

52

3.1.3. Conduct an The information search is a step of great importance. In information search order to benchmark a piece of hardware, the design team must gather as much information about the product as possible. Any printed article that mentions the product, its features, its materials, the company, manufacturing locations or problems, customers, market reception or share, or any other information will be useful. Because of the proliferation of computerized databases and the World Wide Web, a good library is essential. There is a generous amount of information available about all business operations. Before starting any design activity, a team must understand the market demand for product features and what the competition is doing to meet it. A design team should gather information on • the products and related products • the functions they perform • the targeted market segments All keywords associated with these three categories should be formed and used in informational searches. Sources of information can be quite varied. Most businesspersons are perfectly happy to discuss the market and noncompetitive business units. Although most businesspersons will not provide strategic information about their own companies, many people are happy to tell all about their competitors. Suppliers will usually discuss their customers as they can, if it appears that the requester might provide an additional sale. The key is always to be open, honest and ethical when questioning for information. Once people understand that a design team is designing a new product or redesigning an existing one, they naturally want to get involved with new orders and will help the team as far as they legally can. Pursuit of information beyond that point is unethical and not necessary. Most people are happy to share information, and so simple honesty and a friendly attitude can get team members along way. Sources of information can be divided in two main groups: public sources that are freely accessible, and market research databases that are accessible through a fee. Public sources of product information include: c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

53

3.1 Benchmarking

Libraries University libraries are filled with technical engineering modeling references. Many libraries that does not have a large book count, have access to other larger libraries where information may be found and retrieved through inter-library loans. Thomas Register of Companies This set of documents is a “yellow pages” for manufacturing-related business. The Thomas Register list vendor by product name (http://www.thomasregister.com). Consumer Reports Magazines These magazines survey and test a number of common consumer products. Useful data are available for customer needs, qualitative benchmarking, engineering specifications, and warranty andmaintenance information. If a given product is not covered in the magazines, other products can provide analogies as a starting point. (http://www.consumerreports.com/, http://www.profeco.gob.mx/new/html/revista.htm). Trade Magazines Consumer trade magazines such as Car and driver, Byte, Consumer Electronics, JD Powers and Associates, and others provide comparative studies of products within a field. Such studies are very useful to understand how a given product compares with the competition and to understand important customer and technical criteria. Patents After examining trade journals and uncovering which competitors have new innovations, gathering the patents on these new innovations explains much. Patent searches based on company names are difficult since companies typically “bury” their patents by filing them under the individual names of designers. Uncovering the individual patents is usually through refined topical searches, and hence, as much information as possible should be at hand when doing the research. Patent information may be obtained from the Classification and Search Support System (CASSIS) of from Web sites such as http://www.patents.ibm.com/. Market Share Reporter Published every year by International Thomson Publishers, this book summarizes the previous market research of Gale Research, Inc. It is composed of market research reports from the periodicals literature. It includes corporate market shares, institutional shares and brand market shares. National Bureau of Standards This U.S. government branch provides, among other things, national labor rates for all major countries. This inforc Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

3.2 Setting product specification

54

mation proves very useful for determining competitors manufacturing costs. Census of Manufactures Taken every 5 years by the U.S. Department of Commerce, this census includes statistics on employment, payroll, inventories, capital expenditures, and selected manufacturing costs. Also, the supplemental Current Industrial Reports lists production and shipment data on industries and some products. Moody’s Industry Review Taken every 6 months, this survey provides key financial information, operating data, and ratios on about 3,500 companies. Companies as an industry group may be compared with one another group and against industry average. 3.1.4. Some comments about benchmarking

Even when benchmarking can help to understand the market, forecast trends and identify key innovations and technology, one complaint about it is that always provide lagging information. Hence, it is argued that market leaders can find little or no information at all through this practice. Nevertheless, it should be realized that very few market leaders constantly produce leading technology in a market. Markets are always evolving and the opportunity for a competitor to produce new exciting technology is always latent. One way market leaders can benefit from benchmarking is from focusing it on components rather than in products. Components benchmarking may allow them to introduce new technology in components that are not directly developed by them. One problem is commonly associated with benchmarking is the “chasing the competition” syndrome. This problem presents when benchmarking is only used to see what the competition is doing rather than to help the development of new competitive products.

3.2

Setting product specification

After benchmarking, one next step is to use the information gathered up to this point to set targets for a new product development effort. Specifications for a new product are quantitative, measurable criteria that the product should be designed to satisfy. In order to be useful, each specification should consist of a c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

55

3.2 Setting product specification

metric and a value. This value can be a specific number or a range. Examples are: 50 Hz, 30-40 N, > 10 dB, etc. In general terms, specifications fall into two categories, functional requirements and constraints. As discussed before, functional requirements or engineering design specifications are statements of the specific performance of a design, what the device should do. On the other hand, constraints are external factors that limit the selection of the characteristics of the system or subsystem. Constraints are not directly related to the function of the system, but apply across the set of functions for the system. In many situations, constraints can drive the design process of a product and should be established only after critical evaluation. Setting specifications is generally not a straightforward task, and specifications are usually checked several times during the design process. Several concepts may be derived from a customer requirement giving rise to different engineering specifications. Take for example a lid that can be either screwed or pushed to close a container. Both solutions will give way to different engineering specifications since in the first case to screw is related to torque and in the second one to push is related to force. In this case, early concept-independent criteria such as “opening ease” may be refined later into performance specifications for the selected concept. In those specifications that are not expected to change during the design process, margins in target values of ±30% at the beginning of the design process are commonly expected. In any case, it is primordial for each specification should be measurable, and testing and verification of it should be possible at any stage. If for any reason, a specification is not testable and quantifiable, it is not a specification. Ulrich and Eppinger (2000) suggest to consider a few guidelines when constructing the list of specifications: • Specifications should be complete. Ideally each customer need would correspond to a single specification, and the value of that specification would correlate perfectly with satisfaction of that need. In practice, several specifications may be necessary to completely reflect a single customer need. • Specifications should be dependent, not independent, variables. As do customer needs, specifications also indicate what the product must do, not how the specifications will be achieved. Designers use many types c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

3.2 Setting product specification

56

of variables in product development; some are dependent, such the mass of a product and other are independent, such as the material used to manufacture the product. In other words, designers cannot control mass directly because it arises from other independent decisions the designer will make, such as dimensions and material choices. Metrics specify the overall performance of a product and should therefore be the dependent variables in the design problem. By using dependent variables for the specifications, designers are left with the freedom to achieve the specifications using the best approach possible. • Specifications should be practical. It does not serve the team to devise a specification for a given product that can only be measured by a scientific laboratory at a cost of several thousand dollars. Ideally, specifications will be directly observable or analyzable properties of the product that can be easily evaluated by the team. • Some needs cannot easily be translated into quantifiable specifications. Needs like “the product instills pride” may be critical to success, but are difficult to quantify. In this cases the team simply repeats the need statement as a specification and notes that the metric is subjective and would be evaluated by a panel of customers. • The specifications should include the popular criteria for comparison in the marketplace. Many customers in various markets buy products based on independently published evaluations (see examples of sources in the previous section). If the team knows that its product will be evaluated by the trade media and knows what the evaluation criteria will be, then it should include specifications corresponding to these criteria. 3.2.1. Specification Lists

With the above guidelines, a specification list like the ones shown in tables 3.1 and 3.2 can be generated. In order to help with the search for relevant design specifications, an approach known as Specification List Generation can be of some help. Specification List Generation uses the decomposition method to obtain a list of general specifications from latent needs such as safety, regulations and environmental factors. Each specification can be labeled as a required demand or a desirable wish to communicate its level of importance. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

57

3.2 Setting product specification

3.2 Setting product specification

58

To identify specifications, the table 3.3 devised by Franke (1995) provides a good starting point. In order to apply Franke approach follow the next steps: 1. Compile specifications and constraints and label them accordingly. Start with specifications and follow with constraints. 2. Determine if each of the functional requirements and constraints is a demand or wish. 3. Determine if each of the functional requirements and constraints are logically consistent. Check for obvious conflicts. Check that specifications are technically and economically feasible. 4. Quantify wherever possible. 5. Determine detailed approaches for ultimately testing and verifying the specifications during the product development process. 6. Circulate specifications for comment and/or amendment inside and outside the development team. 7. Evaluate comments and amendments. 3.2.2. Quality function deployment

Up to this point, several pieces of information are available to the design team. Without proper guidance, the team may feel that is “lost in a see of information”. One technique that is commonly used to help in the design process is Quality Function Deployment (QFD). One of the main advantages of the QFD method is that it is organized to develop the major pieces of information necessary to understand a design problem:

M.

N.

Metric

1

1,3

2

2,6

Attenuation from dropout to handlebar at 10 Hz Spring preload Maximum value from the Monster Minimum descent time on test track Damping coefficient adjustment range Maximum travel (26 in. wheel) Rake offset Lateral stiffness at the tip Total mass Lateral stiffness at brake pivots Headset sizes Steertube length Wheel sizes Maximum tire width Time to assemble to frame Fender compatibility Instills pride Unit manufacturing cost Time in spray chamber without water entry Cycles in mud chamber without contamination Time to disassemble/assemble Special tools required for maintenance UV test duration to degrade rubber parts Monster cycles to failure Japan Industrial Standards test Bending strength (frontal loading)

3

1,3

4

1,3

5

4

6

5

7

5

8

6

9

7

10

8

11

9

12

9

13

9

14

9

15

10

16

11

17

12

18

13

19

14

20

15

21

16,17

22

17,18

23

19

24

19

25

20

26

20

Imp

Units

3

dB

3

N

5

g

5

s

3

N-s/m

3

mm

3

mm

3

kN/m

4

kg

2

kN/m

5

in

5

mm

5

List

5

in

1

s

1

list

5

Subj.

5

US

5

s

5

cycles

3

s

3

list

5

hours

5

cycles

5

binary

5

kN

Table 3.1: List of metrics for a mountain bike suspension. The relative importance of each metric and the units for the metric are shown. “M.” and “N.” are abbreviations for the number of specification and the need it comes from. “Subj.” is an abbreviation indicating that a metric is subjective. (Adapted after Ulrich & Eppinger, 2000).

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

59

3.2 Setting product specification

3.2 Setting product specification

60

1. The specifications of the product. Date

Demand/

Design specification

Wish

Test/ Verification

Functional Requirements

3. What is important from the point of view of the customer.

1/25

D

Provide thrust for maximum height (velocity > (20 m/s)

of momentum analysis

1/25

D

Maintain stable vertical flight path (less

Flight tests with prototype

than 0.25 m deviation from vertical path)

Bernoulli and conservation

Design of experiments

Constraints 1/25

D

Rocket length ≤ 15 cm

Verify with engr. drawings during concept generation, detail design, etc.

1/26

D

2. How the competition meets the goals.

No detachable part less than

Verify with dimensional

5 cm in diameter

check of engr. drawings

Table 3.2: Specification sheet template, example of a toy rocket product (partial). Adapted after Otto & Wood (2001).

4. Engineering specifications to work toward. There are two points that are worth considering before applying QFD to a design problem. First, no matter how well it is taught that a design problem is understood, the design team should employ the QFD method for all original design or redesign projects. Second, the QFD technique can be applied to an entire product and its sub-systems. To apply the QFD methodology, the following steps should be followed: 1. Identify the customers. 2. Determine the requirements of the customers.

Specification category

Description

Geometry

Dimensions, space requirements, . . .

Kinematics

Type and direction of motion, velocity, . . .

Forces

Direction and magnitude, frequency,load imposed by, energy type, efficiency, capacity, conversion, temperature

Material

3. Determine the relative importance of the requirements. 4. Perform a benchmarking activity to determine how competition satisfy the customers. 5. Generate engineering specifications.

Properties of final products, flow of materials, design for manufacturing

Signals

Input and output, display

Safety

Protection issues

Ergonomics

Comfort issues, human interface issues

Production

Factory limitations, tolerances, wastage

Quality Control

Possibilities for testing

Assembly

Set by DFMA or special regulations or needs

Transport

Packaging needs

Operation

Environmental issues such as noise

Maintenance

Servicing intervals, repair

Costs

Manufacturing costs, material costs

Schedules

Time constraints

6. Set engineering targets. 7. Relate the requirements of the customers to engineering specifications. 8. Identify relationships between engineering requirements. Applying the above steps builds what is known as the house of quality. This house provides in a single picture all the pieces of information gathered by the design team and their relationships. As shown in figure 3.1, the house has many rooms, each containing valuable information.

Table 3.3: Categories for searching and decomposing specifications (After Franke, 1995).

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c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

61

3.2 Setting product specification

3.2 Setting product specification

62

The first step for documenting information in the house of quality is to determine the customer requirements and its relative importance. This information can be registered in the first room of the house: customer requirements. This room relates to what the customers want. The next step is to write down the information regarding the benchmarking activities carried out in the second room of the house: Customer targets and ratings. This room relates to now vs. what or how the customer are currently being satisfied.

Correlation Matrix How vs How

In this step, each competing product must be compared with the requirements of customers, rating each existing design on a scale of 1 to 5:

Relationship Matrix What vs How

1. 2. 3. 4. 5. Customer Targets and Ratings Now vs. What

Customer Requirements WHAT

Importance Rating

Engineering Design Specifications HOW

The The The The The

product product product product product

does not meet the requirement at all. meets the requirement slightly. meets the requirement somewhat. meets the requirement mostly. fulfills the requirement completely.

The benchmarking step is very important as it shows opportunities for both product improvement and gain in market share. If all the competition rank low on one requirement, that is clearly an opportunity, specially if the customer ranked that specific requirement as essential. After the engineering specifications have been generated, each one can be written in the third room of the house: engineering design specifications. This room relates how customer requirements will be measured to ensure satisfaction.

Targets How Much

Figure 3.1: Template for the House of Quality.

Hand in hand with the previous room is the targets room, which specify how much should be achieved. In this room all the target values related to each one of the engineering design specifications are stated. In many cases, extreme values for the delighted and disgusted states of customer satisfaction are also included for each specification. After the previous steps have been carried out, only two more steps are missing, to relate the requirements of the customers to engineering specifications and to identify relationships between engineering requirements. To relate the requirements of the customers to engineering specifications, the room at the center of the house, the relationship matrix, is used. In this matrix,

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c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

63

3.2 Setting product specification Indicator Meaning

Strength



Strong relationship

9



Some relationship

5



Small relationship

3

Indicates no relationship

0

Blank

Table 3.4: Symbols used to indicate the level of relationship between customer requirements and engineering design specifications. Indicator

Meaning

Strength

⊕

Strong positive correlation

9

+

Some positive correlation

3

-

Some negative correlation

-1



Strong negative correlation

-3

Table 3.5: Symbols used to indicate the level of correlation between engineering design specifications.

3.2 Setting product specification

down the design process, but it does not. Time spent developing information is returned in time saved later in the process. Finally, it should be kept also in mind that QFD is a tool to build consensus. It is a tool to ensure that a variety of specifications from different areas converge to a successful product.

References 1. Jacobson, G. & Hillkirk, J. (1986) Xerox: American Samurai. 2. Franke, H. J. (1975) Methodische Schritte beim Klaren konstruktiver Aufgabenstellungen. Konstruktion. 27, 395-402. 3. Otto, K. & Wood, K. (2001) Product Design - Techniques in Reverse Engineering and New Product Development, Prentice-Hall. 4. Ullman, D. (2001) The Mechanical Design Process. Third Ed. McGrawHill. 5. Ulrich, K. & Eppinger, S. (2000) Product Design and Development. Second Ed. Irwin McGraw-Hill.

each cell represents how an engineering specification relates to a customer requirement. Although many parameters can measure more than one customer requirement, the strength of the relationship can vary. The strength of the relationship is represented through the specific symbols shown in table 3.4. To finish with the procedure, the roof of the quality house, the correlation matrix is filled. Here, the relationship between different engineering specifications is shown. The idea of the roof is to show that as one works to meet one specification, you may be having a positive or negative effect on others. For this purpose, the symbols shown in table 3.5 may be used. As the above steps are completed, the house of quality fills up. Figures 3.2 and 3.3 show two different examples of houses of quality for two different products. 3.2.3. Comments on QFD and the house of quality

One hint for effectively using the House of Quality is that the matrix should not grow too large. If the house is larger than 50 rows and columns, then the design team should operate at different levels in the product. Another is to devote QFD as much time as needed. It may appear that QFD slows c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

64

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

3

Brew larger amount

2

3

2

4

4

3

Contain steam

2

4

4

2

5

1

1

3

Technical Difficulty

3

4

Measurements Units

C

sec cup C

Object Target values

98 8.0

3

2

3

2

sec sec sec ft3

qt

44

0

30 0.2

?

98 << 4.75 88

5

20 0.4 2

1

20 0.2

5

20 0.4 1.5

3

3

Objective

West Bend

Measures

Mr. Coffee

?

8

WB Coffee Maker

98

na

0

Old Fashion Way

99

5

na hot

5

240 na

Powered Tea

na

na

na

25

5

60

Technical

Absolute

83

81

63 45

45

Importance

Relative

1

2

3

4

4

4.75 ? ?

3

2

100 ?

Mountain Bike

5

Recumbent

5

BikeE CT

3

Amount of change in damping

4

Amount of change in spring rate

3

# of tools to adjust

3

# number of tools to adjust

4

Rider height range

2

Rider weight range

3

Largest size of brew

3

Total Volume

3

Time to clean product

3

2

Time needed to add tea

2

2

Volume of water in tank

2

3

Riders that notice pogoing

Easy to store

2

Max accel on 5.0 cm standard pothole

3

1

Max accel on 2.5 cm standard pothole

Easy to clean

5

Max acceleration on standard street

3

5

66

Energy transmitted on standard road

3

5

1

3.2 Setting product specification

Environment Adjustability Performance

Easy to add ice Easy to add tea

West Bend Iced Tea Maker

4

5

Powdered Tea

9

Old Fashion Way

Stronger tea

West Bend Coffee Maker

Mr. Coffee Iced Tea Maker

Hottest temperature outside container

Temperature of exiting hot tea

Time water is in contact with tea

3.2 Setting product specification

Temperature of water in sleeping basket

65

Smooth ride on streets

1

3

3

Eliminate shock from bumps

1

2

4

No pogoing

5

2

1

Easy to adjust for different weights

5

3

2

Easy to adjust for different heights

5

3

3

Easy to adjust ride hardness

1

1

3

No noticeable temperature effect

5

4

4

No noticeable dirt effect

5

4

4

No noticeable water effect

5

5

5

Units

%

gs

BikeE CT

gs

gs

%

lbs in

#

#

%

%

95

0.4 1.6 3.0 0.0 100 6.0 0.0 0.0 0.0 0.0

Mountain Bike

35

0.1 0.4 0.5 20 30

4.0 2.0 1.0 0.0 0.0

na 100

Recumbent

50

0.1 0.7 0.9 40 40

6.0 1.0 1.0 0.0 0.0

na

na

25

Target (delighted)

30

0.1 0.4 0.5 100 100 6.0 0.0 1.0 0.0 0.0

27

36

27 18

Target (disgusted)

50

0.2 0.7 1.0 50 50

6

5

6

?

3.0 1.0 1.0 0.0 0.0

8

Figure 3.2: House of quality for iced tea maker (partial). After Otto & Wood (2001). c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Figure 3.3: House of quality for suspension system (partial). Adapted from Ullman (2003). c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

4.1 Brainstorming

68 Basic Methods

CHAPTER

Concept generation methods

4

The Morphological Chart

Logical methods

Concept Generation

Brainstorming 6−3−5 Method

TRIZ (TIPS) Axiomatic design

Figure 4.1: Some concept Generation methods. creates a tendency for designers to take their first idea and start to refine it toward a product. This is considered a very weak practice.

Up to now, all energies have been focused to understand the design problem and to develop its specifications and requirements. The goal now is to generate concepts that will lead to a quality product. A concept may be defined as an idea that is sufficiently developed to evaluate the physical principles that govern its behavior (Ullman, 2003). Hence, concepts must be refined enough to evaluate their form and the technologies needed to realize them. Concepts can be represented in rough sketches, flow diagrams, set of calculations or textual notes. In any case, each concept must contain enough details so the functionality of the idea can be ensured. Sometimes, design begins with a concept to be developed into a product. This is considered to be a weak philosophy and generally does not lead to quality design. In order to minimize changes later in the process, it is normally expected that the concept generation scheme should take 20-25 percent of the design process. It is very important for the design team to generate as many concepts as possible, following the old advice: Generate one idea, and it will probably be a poor one. Generate twenty ideas, and you may have a good one.

In order to avoid the previous problems, various methods aimed to generate concepts are presented here. Some may be more complicated to follow or have their particular value. In any case, it is a decision of the design team which to follow. Figure 4.1 shows the methods that will be discussed in this chapter. It is important to recall once more the importance of information gathering. Considered the first method for concept generation, this activity should be really the starting point of any concept generation method. This activity will include the search for documented ideas on solving problem functions that will increase the scope of possible solution generated by the design team. In the last chapter several sources of information were reviewed in the scope of the benchmarking activity. This sources are also valid here, and figure 4.2 shows a summary of them.

4.1

Brainstorming

It is a natural tendency to generate concepts as the design process progress, as it is naturally to associate ideas with things that we already known. This

Brainstorming is a group-oriented technique aimed to generate as many concepts as possible. The procedure is quite simple and has the advantage that a committed team can create a large number of ideas from different points of view. The guidelines for brainstorming are as follows:

c Copyright 2006 Dr. Jos´ e CarlosMiranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

69

4.1 Brainstorming Benchmarking

4.2 The 6-3-5 method

70

5. Avoid confining the group to experts in the area, that may limit the introduction of new ideas. Nature

Analogies

Product Function Product Architecture Patents Journals

Information Sources

Published media

Product Information Textbooks Consumer Products Periodicals Goverment Reports Professionals in Field

People

Customers Experts

Figure 4.2: Information sources for concept generation. Many of this information can be found through the World Wide Web. 1. Form a group with 5 to 15 people.

6. Avoid hierarchically structured groups. Bosses, supervisors and managers should not be included in many of the sessions. Some hints may be used to stimulate new thinking and the generation of new ideas: • Make analogies, think what other devices solve a related problem, even if they are applied to an unrelated area of application. • Wish and wonder. Think wild. Sometimes silly, impossible ideas, give way to useful ones. • Use related and unrelated stimuli. First, use photos of objects or devices that are related to the problem at hand. Next, use photos of objects unrelated to the problem. This activity usually gives way to new ideas. • Set quantitative goals. Set a reasonable number of concepts and do not leave the session until you have achieve them. For a group session, individual targets of 10 to 20 concepts is reasonable.

2. Designate a person to work as facilitator to prevent judgments and encourage participation by all. Although some authors state that the facilitator should also be a contributor. Other suggest that the facilitator should only guide and record the session avoiding further participation. The latter allows the designation of the most participate person of the team as facilitator to encourage other team members to participate actively.

4.2

3. Brainstorm for 30-25 minutes. The first 10 minutes are generally devoted to introduce the problem at hand. The next 20-25 minutes sees the most generation of ideas, and during the last 10 minutes a sharp decline in ideas may happen.

The guidelines for the 6-3-5 method are also very simple:

4. Do not allow the evaluation of ideas, just the generation of them. This is very important. Ignore any comments about the usefulness, validity or value of any idea. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

The 6-3-5 method

One of the two main disadvantages with brainstorming is that first, all ideas are conveyed by words. Second, the generation of ideas can be dominated by one or two team members. The 6-4-5 method forces equal participation by all.

1. Arrange teams around a table. Although 6 members are optimal, a number between 3 and 8 should suffice. 2. Establish a specific function of the product to work with. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

71

4.3 TRIZ 3. Ask each member to draw in a sheet of paper two lines in order to create three columns. After that, ask each member to write, 3 ideas, one on each column, about how the function could be fulfilled. Ideas can be communicated by words, sketches or both. 4. After 5 minutes of working in the concepts, pass the sheets of papers to the right. 5. Give the participants another 5 minutes to add other three ideas to the list. 6. After completing a cycle stop to discuss the results and find the best possibilities.

It is important to mention that there should be no verbal communication in this technique until the end. This rules forces interpretation of the previous ideas only from what it is on the paper.

4.3 TRIZ

Level

72 Degree of inventiveness

Percent of solutions

Source of knowledge

Approximate number of solutions to consider

1

Apparent solution

32%

Personal Knowledge

10

2

Minor improvement

45%

Knowledge within company

100

3

Major improvement

18%

Knowledge within industry

1000

4

New concept

4%

Knowledge outside industry

100,000

5

Discovery

1%

All that is knowable

1,000,000

Table 4.1: Levels of Inventiveness. 4. identifying new principles 5. identifying new product functions and solving them with known or new principles.

4.3

TRIZ

The Teoriya Resheniya Izobreatatelskikh Zadatch (TRIZ) or Theory of Inventive Problem Solving (TIPS), was developed by Genrikh S. Altshuller in the former U.S.S.R. at the end of the 1940’s. The TRIZ theory is based on the idea that many of the problems that engineers face contain elements that have already been solved, often in a completely different industry for a totally unrelated situation that uses an entirely different technology to solve the problem. Based on this idea, Altshuller collaborated with an informal collection of academic and industrial colleagues to study patents and search for the patterns that exist.

The first two categories were designated as “routine design”, meaning that they do not exhibit significant innovations beyond the current technology. The last three categories represent designs that included inventive solutions. He also noted that as the importance of the innovation increased, the source of the solution required broader knowledge and more solutions to consider before an ideal one could be found. Table 4.1 summarizes this idea.

After spending 1500+ person-years studying at first around 400,000 patents (today the database extend up to 2.5 million patents), Altshuller discovered that they could be classified into five categories: 1. basic parametric advancement 2. change or rearrangement in a configuration 3. identifying conflicts and solving them with known physical properties c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

73

4.3 TRIZ

PROBLEM TO SOLVE

SOLUTION

Find contradictions

Apply Inventive Principles

Contradiction Matrix

1 2 3 4 n Inventive Principles

TRIZ Figure 4.3: TRIZ methodology. Based on his studies, Altshuller observed some trends in historical inventions: • Evolution of engineering systems develops according to the same patterns, independently of the engineering discipline or product domain. These patterns can be used to predict trends and direct search for new concepts. • Conflicts and contradictions are the key drivers for product invention. • The systematic application of physical effects aids invention, since a particular product team does not know all physical knowledge. In this regard, Altshuller noticed that almost all invention problems involved in one way or another the solution to a contradition. By contradition it is understood a situation in which the improvement of one feature means detracting another. The quality of the invention was in most occasions related to the quality of the solution to the contradiction.

4.3 TRIZ

74

1

Weight of movable object

21

2

Weight of stationary object

22

Power Waste of energy

3

Lenght of movable object

23

Loss of substance

4

Lenght of stationary object

24

Loss of information

5

Area of movable object

25

Waste of time

6

Area of fixed object

26

Quantity of substance

7

Volume of movable object

27

Reliability

8

volume of stationary object

28

Measurement accuracy

9

Speed

29

Manufacturing precision

10

Force

30

Harmful action at object

11

Stress or pressure

31

Harmful effect caused by object

12

Shape

32

Ease of manufacture

13

Stability of the object’s composition

33

Ease of operation

14

Strength

34

Ease of repair

15

Durability of a moving object

35

Adaptation

16

Durability of a stationary object

36

Device complexity

17

Temperature

37

Measurement or test complexity

18

Illumination intensity

38

Degree of automation

19

Use of energy by moving object

39

Productivity

20

Use of energy by stationary object

Table 4.2: TRIZ 39 design parameters. Then, use the 40 inventive principles of TRIZ to generate ideas to overcome this problem. The 40 inventive principles were found by Altshuller to be the underlying principles behind all patents. This procedure is depicted in figure 4.3. Applying TRIZ principles allows the innovation without having to wait for inspiration. Practitioners of the TRIZ theory have a very high rate of developing new, patentable ideas.

Based on this premise, Altshuller devised TRIZ. The goal of using TRIZ is to find those contradictions that makes the design problem hard to solve. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

29 34

38 34

40 28

1

8 15 35 28

17 7

40 29

10 40 14 15

4

6. Area of stationary object

18 30

14 18 7. Volume of movable object

2 26

8. Volume of stationary object 9. Speed

8

1 28

9 36

1

19

4

8 40 26

40 15 27 15. Durability of a movable object

3

5

1

5

7 37

1 15 15 14

18. Illumination intensity 19. Use of energy by moving object 20. Use of energy by stationary object

4 34 35

7 29 34 1 18 15

9

2 36 13 28

6 35 35 24

5 34 11

2

3. Length of movable object

28 2

2

10 27

19 6

8

35 19

29 34 14

4. Lenght of a stationary object

28 26 5. Area of movable object

3

15 6

40

7. Volume of movable object

9

4

36

7

2 34 28

35

35 40

35 4

35 33

15 10 2

33 1

4 10

15 22 35

34 18 37 40 10 14

18 4

2 11 39

28 10 34 28 33 15 10 35 2

13

19 39 35 40 28 18 21 16 40 9 14

8 13 10 18 10 3

14

7 17 15 26 14

10

2

19

5

1 40 35

16

27

28 25 35 35 23

6

2 28 35 10 35 39 14 22 36 30

35 19 32

19 32

2

32

16

26

10

19

12 28

15 19

35 13

8

25

18

3 21 19

1 35

32 30 32 3

6

3

35 16 26 23 14 12 2

19 13

35 21 2

17 24

36 37

27

29

27 4 29 18

3

19

13

10 14 30 16 10 35 2

18

6 7

18 39

4

15 36 39 2

35 16 35 3

35 34

32 18

6

35 38

38 2

3

40 27

19 2

10 37

22 14 13 15 2

10 40

19 32 32

34 14

35 1

13 19 27 4

19 17 1

32 3

16 19 35 14 15 8

29 38

35

10 37 14 29

5

36

25

3 37

14

35 29

14 34 36 22

2

3

5

10 17

32 35 14 2

2

14

35 27 15 32

4

6

36 10 36

18 36

37 36 10 14 36 4

14

6

29 18 27 31 39 6

30 10 35 19 19 35 35 40

40

10 35 2

29 30

10 19

19 35 28 38

36 37 18 37

27 15

6

19 35 14 20 10 13 13 26

30 14 14 26

10 35 35 23 32

19

15

22 2

18 40 4

10 39

12. Shape

30 40

10 26 35

35 28

35 29 3

29 10

10

35 28

31 40

28 10 27

27 3

19 35 2

19 28 6

19 10

28 27 10

20 10 3

10

39

35 35 18

35 38

4

19 18

16. Durability of a stationary object

18. Illumination intensity

26. Quantity of substance 6

39

30

10

17. Temperature

23. Loss of substance

22. Waste of energy

17. Temperature

21. Power

29 30

32 18 30 26 2

4

14 24

26

14

19 10 15 17 10 35 30 26 26 4

34 10 7

21

15. Durability of a movable object

29 35

29

13 18 13 16 34 10

35 39

27 3

35 26 18 26 24 15 2

10 18 2

18 19 3

15

29 1

28 10 28 24 26 30 29

30

14 27

9 13 27 39 3

4

35

9

17

10 15 10 20 19 6

34 39 10 13 35

28 30 10 13 8

9 25

6

17 32 17 7

11. Stress or pressure

3 18

16

36 40

28 20 3

18 38

10 16 31

19 18

32 30 19 15

2

22 40 39

36 40

21 16 3

17 25 35 38 29 31

35 19 2

19

6

32 19

32

35

19

19. Use of energy by moving object

5

19 28 35

19 24 2

9

35 6

3

20. Use of energy by stationary object

35

18

15

14 19

17

1 32 35 32 1

15

14 21 17 21 36 19 16 13 1 1

6

35

28 18 10 40

27 16 10

10 30 19 3

27 25

19 13

35 34 35 6 38

2

3 26

20 28 18 31

24 35

15 15 32 19 32

35 10

2 19 32 32

10 13 26 19

35 7

12 8

35 10 19 2

14. Strength

39 3

40 18 4

25

10. Force

10 30 13 17

35 34 3 35 35 38 34 39 35

14

35

35 39 23 10

36 2

18 4

3 14 26 13 3

4

8

10 24 10 35

18 22 28 15 13 30 35 1

26 14 35 5

13. Stability of the object’s composition

38 39 18

13

2 19

35

10 35 39

14 6 35

3 14 18 40 35 40 35

3 35 19

19 30

35 22 1

9 8

9. Speed

40

2 35 15 35 10 34 15

9 40 10 15

2

35 3

16

17 15

1 18

7

3 34

8. Volume of stationary object

40 34 21

21

24

38 18

2

15 7

1 39

14

4

19 3

40 14 6. Area of stationary object

8

35 3

28 1

10 15 32 1

5 35

18 19 15 19 18 19 5

32 22 32

19 30 38 1 15 28 10

6 18 35 15 28 33

11

27 28 19 35 19

2

24. Loss of information

18 31 34 19 3 31

20. Energy expense of fixed object

13 36 6

34 31

16. Duration of fixed object’s operation

25 12

38 32

28 27 5

25. Waste of time

15. Duration of moving object’s operation

29 19 1

4

What is deteriorated? 2. Weight of a stationary object

4 13 39 2 38

6

14. Strength

13. Stability of the object’s composition

12. Shape

6 35

6 35 36 35

2

6

35

18 21 10 35 35 10

36 37 12 37 18 37 15 12

3 17

9

8

7

15 19 38 40 18 34

32

19 9

11. Stress or pressure

13 28

19 1

2 8 31

15

2 36 28 29

4 15 35

8 35 28 26 40 29 28

9

12 18

10. Force

9. Speed

4

1

1 14 13 14 39 37 35

2 19 6 27

1. Weight of a movable object

1 40

5 34

1 28

19 16 6 38 32

28 10

29 30 19 30 10 15

14 16 36 28 36 37 10

36 22 22 35 15 19 15 19

17. Temperature

8

2 14

2 18 24 35

1 10 15 10 15

9

34 31

16. Durability of a stationary object

35

8

34

18 40 31 35

What should be improved?

10 29 15 34

37

15

4 10

1 35

35 36 36 37

34

21 35 26 39 13 15 1

35

1

1 18 10 15

29 30

8

29 40 26

14. Strength

4

8

13 14

10 36 13 29 35 10 35

2 39

8

8

38 34 36 37 36 37 29

8 10 15 10 29 34 13 14

13. Stability of the object’s composition

4 35

2 14

37 40 10 18 36 12. Shape

1

7

1 18 13 17 19 28 10 19 10

37 18 11. Stress or pressure

4 17 10

29

35 10 19 14 35

13 38 10. Force

13

4 17

2 28

19 35 10 18 29 14

7 17

7

19 14

8 10 10 36 10 14

8 10 13 29 13 10 26 39

2

9 39 1

29 40

14

17 26

8

15 38 18 37 37 40 35 40 19 39

7 14

4

2

2 5 35

2

4

2 17 29

13 15 17

29 34

2

35 30

29 35

5. Area of movable object

8. Volume of fixed object

29

10

4. Lenght of a stationary object

7. Volume of movable object

29 17

2. Weight of a stationary object 3. Length of movable object

6. Area of fixed object

8

76

19. Energy expense of movable object

15

4.3 TRIZ

18. Illumination

1. Weight of a movable object

5. Area of movable object

4. Lenght of a stationary object

3. Length of fixed object

What should be improved?

2. Weight of a fixed object

What is deteriorated?

4.3 TRIZ

1. Weight of a movable object

75

35 28 3

35 17

21 18 30 39 1

6

6

19 1

1

19

26 17

19 12 22 35 24

37 18 15 24 18 5

35 38 34 23 19 18 16 18

19 2

28 27

3

35 32

18 31

31

Figure 4.4: TRIZ contradiction matrix.Continued.

Figure 4.5: TRIZ contradiction matrix. Continued.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

35

3

28

35 17 22 37 1

4

39 9

15 17 15 1

29 37 17 24

15 29 32 28

5. Area of movable object

29

6. Area of stationary object

32 35 26 28 2 40 4

32 3

7. Volume of movable object

14 1

25 26 25 28 22 21 17 2

28

3

2 32 1

28 1 29 27 2

22 1

28 3

19 27 35 28 2

16

35 22 2 37

40

30 12

2

24 35 13 32 28 34 2

33

1

13 12 28 27 26

15 37 1 3

35

15. Durability of a movable object

11 2

3

16. Durability of a stationary object

34 27 10 26

1

17. Temperature

19 35 32 19 24

30

22

29 10 16 34 2 3

3

35 10 1

27

13

1

2

3 10 24

35 2

11 15 3

2 37

2 24

16

10 2

32 39 28 26 19

13 27 3

19 21 3

1

35

2 35 28 26 19 35 1

11 27 32

6

27

6

20. Use of energy by stationary object

10 36

10 2

23

22 37 18

30

19 29 6

14 19

18 2

15 15 17

6

16 38

17 3

35

15 28 35 31 19 16 35

16 6

20 10

13

26 2

10

2 29 35 38 32 2

25

16

19 35

1

6

14 29 10 28 35 2

29 3

39 6

39 10 13 14 15

3 36 29 35 2

10 31 39 31 30 36 18 31 28 38 18 40 37 10 3

10 24 10 35 1

30 26 30 16

2

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

5

10 20 10 20 15 2

30 24 26 4

37 35 26 5

14 5

35 6

29

27 26 29 14

8

5

10 35 2

16 17 4

22 26 32

15 14 2 29

5

35 16

18 15 20

40 4

10 37 36 37 4

34 10 32 18

36 5

4

10 35 3

34 17 22 5

35 29 35 14 10 36 35 14 15 2

29

3

10 15 9

15 29 17 10 32 35 3

10 40 8

28 14 4

28 11 14 16 40 4

16

28 32 28 35 10 28 13 18 27 9

34 28 3

3 16 32 3

32 3

2 32 28 33 2

29 37 10

10 2 35

14 24 24 6

21 35 8

14 3

17 40

28 10 24 35 1

29 32 28 25 10 10 28 28 19

29 32 18 36 2

35

32

27 39 13 24 39 4

33 28 39 35 37 35 19 27 35 28 39 18 37 17 2

15 39 1

39 16 22

18 39 40

28 29 1

27 1

1

18 22 1

29 15 17 13 1

15 16 36 13 13 17 27

26 12

25 2

1

6

13 1

17

13 15 1

25 13 12

2

27 1

27 2

1

36. Device complexity

28 3

30 18 35 28 35 28 2

40

35 4

16 40 13 29 35 1

32

1

29 16 29 2

16

29 7

19 26

14 1

1

1

27 26 6

2

38. Degree of automation

28 26 28 26 14 13 23

17 14

35 13

18 35 35 10 28 17

13

16

39. Productivity

35 26 28 27 18 4

13 16 17 26 26 24

13 2

28 38 26 7

34 31 17 7

11 13

28 11 13

29 28 30 1

13 2

2

4

2

1

18 3

35 4

35

22

28 15 17 19 39 39 30

35 13 35 15 32 18 1 1

34 10 10 2

14

36 28 35 36 27 13 11 22

28 10 2 35 37

8

29 13 2

16 26 31 16 35 40 19 37 32 1

6

35 40 27 39

32 15 34 32 35

12

20

28

39 29 1

30 14 10 26 10 35 2

1

35

37 13 27 1

16 34 10 26 16 19 1

6

18 17 30 16 4

3

35 10 15 17 35 16 15 37 35 30 14

37. Measurement or test complexity

30 18 35 24

33 35 1

10

36 34 26 1

13 16

1 35

34 9

29 6

40

40 27 18

18 18 13 28 13 2

35 35 30 15 16 15 35

34 36 35 39 26 24

24 37 15 3

8

16 4

23 1

35 11

19 15 35 1

28 13 28 1

3

13

22 1

35 13 35 12 35 19 1

40

18 15 13 16 25 25 2

6

26 1

17 2

17 18 16 1

15 8

26 30 2

22 1

22 23 34 39 21 22 13 35 22 2

13 16 15 39 35 15 39 31 34

35 11 35 11 10 25 31 35. Adaptation

27 2

28 32 35

32

34 36

19 22 35 22 17 15

22 17 1

28 6

3 35 32 30 30 18

22 21 2

34. Ease of repair

14

30 40

5

30. Harmful action at object

33. Ease of operation

13. Stability of the object’s composition

10. Force

14 2

23 40 22 32 10 39 24 26 26

40 15 31

38

13

28 15 10 37 10 10 35 3 10 36 14

16 25

Figure 4.6: TRIZ contradiction matrix. Continued.

12. Shape 2

16 35 36 38

17 30 30 18 23 10 18 1

9. Speed

6. Area of fixed object

5. Area of movable object

35 6

11. Stress or pressure

4. Lenght of a stationary object

3. Length of fixed object

2. Weight of a fixed object

What is deteriorated?

13 7

18 7

32

12 28 35

6

7

32 24

27 26 2

32 32 15 2

15 26 17 7

26 28 25 26 5

10 35 17

25 14 1

17 28 13 16 27 28 4

10 14

19 28 18 9 35 6

6 38

35 19 16 11

29 35

15 40

2

6

23 35 40 3

15

29 35 39 35

13 16 19

19. Use of energy by moving object

19 22 1

39. Productivity

38. Degree of automation

22 26 39 23 35

7

11 28 10 3

32. Ease of manufacture

8

19 6

22 10 29 14 35 32

36 35 35

28 13 32 2

34 10

32

2

32 35 28 35 28 26 32 28 26 28 26 28 32 13

1

35 35 22 1

15 35 26 2

25

28. Measurement accuracy

17 26

28 39

30 6

13 38 38

3

36 35 24 10 14 35 37

19 38 17 32 35 6

27. Reliability

3 28 35 37

10

18 31 18 35 35 18

16 29 15 13 15 1

35 10 4

27

26. Quantity of substance

31. Harmful effect caused by the object

28

32 15 19 35 19 19 35 28 26 15 17 15 1

32 1

37. Measurement or test complexity

36. Device complexity 19 1

40 33 22 33 22 35 26 27 26 27 4

25. Waste of time

29. Manufacturing precision 2 35

15 6

35

3 34 10 18

35

15 35 30 2

12 27 29 10 1

16 40 33 28 16 22 4

6 40 24

35

32 40 27 11 15 3 2 2

10 32 28 2

27 22 15 21 39 27 1

10 2

18 20 10 18 10 19

2

24. Loss of information

35 37

4 34 27 16

32 32 15 2 13 1 1 15

27 18 35 15 35 11 3

17 1

2 17

15 17 26 35 36 37

17 28 26

22 2

17 7 16 24 34

25 11

1 35 11

26

29 26 35 34 10 6 2

28 15 1

2

37 1

31

23. Loss of substance

10 15

30 18

15 10 10 28

27 18 16 35 1

35 23

26

35 24 35 40 35 19 32 35 2

27 3

1

22. Waste of energy

36 14 30 10 26

26 18 28 23 34 2

1 18 2

15 29 26 1

1

40

13

13 15 16

2

36 19 26 1

38 31 17 27 35 37

30 14

4

18

3

35. Adaptation

34. Ease of repair

15 13 10

1

16

18. Illumination intensity

16

13 3

26 26

15 17 15 13 15 30 14 1

8

21. Power

17 26 14 4

7

18 30 27 39 11 3

14. Strength

33. Ease of operation

35 1

4

35 13 3

10 40 28 32 32 30 22 1

13. Stability of the object’s composition

15 35

1

25 3

19 35 1

What should be improved?

1 28

16 26 24 26 24 24 16 28 29

29 1

24 32 25 35 23 35 21 8

19 35 25 12. Shape

26 39 17 15 35

36

16 27 35 40 1

3 35 35 10 28 29 1 10 13 6

2 26

1

4 10

40 16 16 4

35 10 34 39 30 18 35 25

10 25 28

35

13 21 23 24 37 36 40 18 36 24 18 1 11. Stress or pressure

32 28 11 29

18 36 39 35 40

11 35 28 32 10 28 1

10. Force

27 19 15 1

18 39 26 24 13 16 10 1

2

27 28 1

1

36 34 26 32 18 19 24 37

1 28 14 15 1

13 1

2

35

13 2

26 30 28 29 26 35 35 3

27 32 22 33 17 2

40 11 28 16

27 29 5

6

15 17 2

10

9 26 28 2 32 3

9. Speed

18

2

24 28 11 15 8

29 15 29

17

4. Lenght of a stationary object

8. Volume of stationary object

32. Ease of manufacture

31. Harmful effect caused by object

2 19 35 22 28 1

10 14 28 32 10 28 1 29 40

36 2

8. Volume of fixed object

10 28 18 26 10 1 8

3. Length of movable object

35 26 26 18 18 27 31 39 1

78

7. Volume of movable object

27 2. Weight of a stationary object

30. Harmful action at object

3 11 1 28 27 28 35 22 21 22 35 27 28 35 3

4.3 TRIZ

1. Weight of a movable object

1. Weight of a movable object

28. Measurement accuracy

27. Reliability

What should be improved?

29. Manufacturing precision

4.3 TRIZ

What is deteriorated?

77

Figure 4.7: TRIZ contradiction matrix. Continued.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

34 40 22 39

22. Waste of energy

26 10 19 35 16

2

28

17 25 19

10 38

26

14 16

19 38 1 7

23. Loss of substance

31 40 3

6 16

38

13 6

10

3

18 38

38

10 6

37

32 7

18

21. Power 22. Waste of energy

25

15 18 6

24 5

31

35 10 10 24

10 19 19 10

24 26 24 28

12 31 18 38 2

3

29 3

20 10 28 20 35 29 1

19 35 38 1

35 20 10 5

28 18 28 18 10 16 21 18 26 17 19 18 26. Quantity of substance

14 35 3

35 3

35 3

34 10 10 40 31 27. Reliability

11 28 2

28. Measurement accuracy

28 6 32

3

29. Manufacturing precision

3

10 26 6

32

24 27

35 35

18 35 22 15 17 1 37 1

13 19 6

1

27 19

26 31 35

3

3

6

28 24 32

32

19 26 3

32 32 2

22 33 1

33 28 40 33 35 2

19 1

32 13 6

15 35 15 22 21 39 22 35 19 24 2 22 2

33 31 16 22 2

32. Ease of manufacture

1

27 1

6

32 32 2

10 32 4 3

28 8

34. Ease of repair

1

11 11 29 1

2

9

36. Device complexity

1

25 25

35 19 22 2 18

27 1

28 32 32

13 32 35 31

32 26 32 30

2

28 18

35 2

19 40 2

35 21 35 10 1

10 21 1

22 3

29

24

22 2

27 1

19 35 15 34 32 24 35 28 35 23 33

39 1

18 16 34 4

19 28 32 4

10 4

1

10 13

2

24 27 22 10 34

15 1

15 10 15 1

2

35

13

28 16

32 2

32 19 34 27

10 25 10 25

22 19 35

19 1

18 15 15 10

35 28 3

29

1

35

3

35 26 1

13 10 4

2

17 24 17 27 2

28

28 15

27 3

19 29 25 34 3

13

38. Degree of automation

25 13 6

6

13 27 2

29 13

26 2

8

29 28

19

19

30 34 13 2 16

32

13

2 32 13

35 3

6

1

27

35 20 28 10 28 10 13 1

10 18 2

10

38 19

35

27

29 18

29 35 35 23 5

23

1

15 34 32 28 2

4

28 32 28 18 34

18 3

3

2

27. Reliability

33 30 35 33

29. Manufacturing precision 30. Harmful action at object 31. Harmful effect caused by the object

2

39. Productivity

38. Degree of automation

37. Measurement or test complexity

36. Device complexity

35. Adaptation

24 34 27 2

28 24 10 13 18

10 23

35 33 35

13 23

4 28 32 1

18 39 34 4

10 34 10

3 35 29 1

35 29 2

35 28 6

27 17 1

40 40 26

32 10 35 30 32 15 3

35 1

13 3

27 8

35 13 29 3

27 35

8

27 10 29 18 24 1

28

27 29 38

1

23

22 26 39 10 25 18 17 34 13 11 2

10 34 32 28 10 34 28 32

11 32

26 28 4

26 2

26 28 10 18

1

10 36 34 26

18

18 23 32 39

17

1 24 35 2

2

2

24 2

3

33 4

2

32 2

25

2

34 35 23 28 39 2

1

16

35 11

1

8

32 31

31

26 26 24 22 19 19 1

10 34 32

1

29 40

37. Measurement or test complexity

27 40 26 24

38. Degree of automation

11 27 28 26 28 26 2

28 8

1

32 28 10 34 18 23

35 1

35 1

22 35

2

10 32 1

22 19 2 29 28 33 2

1 35 1

32 1

12

13 15 34 1 1

16 7

27 26 27 1

21 5

1

13 9 26 24 28 2

5

13

24 7

28 1

10 28 34 15 1

12 3

28

34 35 1

16

13 11

7

16

15 29 1

27 34 35 28

4

37 28

35

13 29 15 28 37

13

1 1

16 12 17

28 35 1

35 1

12 26 1

12 1

34 3

22 35 35 22 35 28 1

10 38 34 28 18 10 13 24 18 39 2

11 1

1 32 1

6

37 28

28 34 21 35 18 24 5

15 24 34 27

35 10

37

12 17

15 15 10

35 27 4

32

13 10

15 10 15 1

37 28

26 1

18 39

28 8

714

11 29 1

27 1

13 27 26 6 1

21 2

12 26 15 34 32 25

11 10 26 15

35 13 35 5 10

5

12

25 10 35 10

16 13

13 35 2

13 24

13 16 11 19 15

11 10 10 2

24 1

5

3 22 31

19 1 3 2 1 24 2

10 34

22 31 29 40 29 40 34

17

35

34. Ease of repair

1

25 35 10 35 11 22 19 23 19 33

28 39 2

12 18 40 2

32 25 10

34 26

40 39 26 1

32 13 35 27 35 26 24 28 2

35 23

27 24 28 33 26 28 40 23 26 10 18

13 1

3

11 13 35 13 35 27 40 11 13 1

40

33 6

29 18 28 24 28

28 24 3

17 27 25 13 1

39. Productivity

29 35

11

33. Ease of operation

36. Device complexity

28 10

5

32. Ease of manufacture

35. Adaptation

2

23 35 3

15

11 32 27 35 35 2

11 23 1

34

7

35 15 10 35 10 35 18 35 10 28 35

29 31 40 39 35 27 10 25 10 25 29

32 3

17

22

10 30 24 34 24 26 35 18 35 22 35 28

28. Measurement accuracy

34. Ease of repair

33. Ease of operation

32. Ease of manufacture

31. Harmful effect caused by the object

30. Harmful action at object

29. Manufacturing precision

28. Measurement accuracy

23 25. Waste of time

19

28 35

30 34 16 15 23

2

32 35 38

22

19 17 20 19 19 35 28 2

10 34 34

27 22

35 30

29 28 35 10 20 10 35 21 26 17 35 10 1 18 16 38 28 10 19 1

28

23 28 35 10 35 33 24 28 35 13 18 5

1

2

10

22 10 10 21 32

27 10

18 35 33 18 28 3

16 10 15 19 10 24 27 22 32 9 28 2

35 2

8 2

29 13 3

28 29

35

34

39 35 31 28 24 31 30 40 34 29 33

15

20 19 10 35 35 10

24 35 38 19 35 19 1

35 35 16 26

2

32 1

35 32 2

24

28 12 35

35 26 10 26 35 35 2

18

10 28

29 31

18

2

34

34

1

13 1

9

27

10 24

21 22 21 35

3 6

10 15 1

32 6

15 28 25 39 6

24 34 2

35 34 2

24

40

26 32 10 16 31 28

11 10 32

28 40 28

13

16 27 2

24

4

4

31 2

24. Loss of information

18 16

29 39

19 22 2

26 31 2

10 29 16 34 35 10 33 22 10 1

26. Quantity of substance

24 28 35 38

10 24 35

12 24

13

35 3 2

3

4

37. Measurement or test complexity

39. Productivity

27 1

16 26 27 13 17 1

28 27 2

18 6

19 22 21 22 33 22 22 10 35 18 35 33

27 22 37 31 2

24 39 32 6

18

32 40 29 3

35. Adaptation

24 10 2

35 16 27 26 28 24 28 26 1

33. Ease of operation

18 16

35 11 32 21 17 36 23 21 11 10 11 10 35 10 28 10 30 21 28

31. Harmful effect caused by the object

3

35 38

25

40 30. Harmful action at object

35 18 24 26

18 32 10 39 28 32 7

16 18 31

40 10

28 6

27 3

34 29 3

39

35 34 27 3 25 6

17

10 6

19 24 32 15 32 2

23. Loss of substance

28 32 35 25. Waste of time

27. Reliability

What should be improved? 34

35 18 28 27 28 27 35 27

19

80

19

35 27 19 10 10 18 7 2

4.3 TRIZ

What is deteriorated?

26. Quantity of substance

25. Waste of time

24. Loss of information

23. Loss of substance

22. Waste of energy

21. Power

20. Energy expense of fixed object

10 35 28 27 10 19 35 20 4

19 37

18 18 38 39 31 13

10

6

32 15

35 28 28 27 27 16 21 36 1

24. Loss of information

19. Energy expense of movable object

18. Illumination

17. Temperature

16. Duration of fixed object’s operation

14. Strenght

What should be improved? 21. Power

15. Duration of moving object’s operation

4.3 TRIZ

What is deteriorated?

79

35 12 17 35 18 5

19 10 25 28 37 28 24 27 2

Figure 4.8: TRIZ contradiction matrix. Continued.

Figure 4.9: TRIZ contradiction matrix. Continued.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

12

35 26

25 12

35 26

81

4.3 TRIZ

4.3.1. 39 design parameters

From the patents studied, Altshuller extracted 39 design parameters that cause conflict. These 39 parameters are listed in table . To effectively use these parameters, it is necessary to find those two that are cause of conflict in a given design. Consider for example that a non-moving mechanical component needs to be lighter but remain as strong. From the 39 design parameters, find the principle that needs to be changed, in this case, “Weight of stationary object” (principle #2). Then find the parameter that is negatively affected, in this case, “Strength” (principle #14). Then, using the contradictions matrix shown in figure 4.9, find those inventive principles that are candidates to solve the conflict. From the contradictions matrix, the inventive principles that solve the contradiction between “Weight of a stationary object” and “Strenght” are 2, 10, 27 and 28.

4.3 TRIZ

82

• To scare birds from buildings and airports, reproduce the sound of a scare bird using a tape recorder. • Hovercraft. 3. Principle of local quality • Change the structure of the object or environment from homogeneous to non-homogeneous. • Have different parts of the object carry out different functions. • Place each part of the object under conditions most favorable for its operation. Examples:

4.3.2. Forty inventive Once the matrix has been used to find those inventive principles principles candidates to solve the engineering contradiction, they can be applied to generate solutions for the problem at hand. These inventive principles can also be used independently of the contradiction matrix as a source of ideas to solve conflicts. The forty TRIZ design principles to solve engineering conflicts are: 1. Principle of segmentation • Divide an object into independent parts that are easy to disassemble. • Increase the degree of segmentation as much as possible. Examples: • Sectional furniture, modular computer components, folding wooden ruler, food processor. • Garden hoses can be joined together to form any length needed. Drill shafts. 2. Principle of removal • Remove the disturbing part or property of the object. • Remove the necessary part or property of the object. Examples: c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

• Fuselage skin of commercial airplanes. • Stapler. A pencil and an eraser in one unit. 4. Principle of asymmetry • Make an object asymmetrical. • Increase the object asymmetry. Examples: • Eccentric weight on motor creates vibration. 5. Principle of joining • Merge homogeneous objects or those intended for contiguous (adjacent) operations. • Combine in time homogeneous or contiguous operations. Examples: • TV/VCR, Cassette tape heads. • The working element of a rotary excavator has special steam nozzles to defrost and soften frozen ground in a single step. 6. Principle of universality • Let one object perform several different functions. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

83

4.3 TRIZ • Remove redundant objects. Examples: • Sofa which converts from a sofa in the daytime to a bed at night. Fingernail clipper. 7. The nesting principle • Place one object inside another, which in turn is placed in a third, etc. • Let an object pass through a cavity into another. Examples:

4.3 TRIZ

84

• Set up the object such that they can perform their action immediately when required. Examples: • Cutter blades ready to be snapped off when old. • Correction tape. 11. Principle of introducing protection in advance • Compensate for the low reliability of an object by introducing protections against accidents before the action is performed. Examples:

• Telescoping antenna, stacking chairs.

• Fuses, electric breakers. Shaft couplers.

• Mechanical pencil with lead stored inside.

• Shoplifting protection by means of magnetized plates in products.

8. Principle of counterweight • Compensate for the weight of an object by joining it with another object that has a lifting force. • Compensate for the weight of an object by interaction with an environment providing aerodynamic or hydrodynamic forces. Examples: • Boat with hydrofoils, hot air balloon. • Rear wings in racing cars to increase the pressure from the car to the ground. 9. Principle of preliminary counteraction • Perform a counter-action to the desired action before the desired action is performed. Examples: • Reinforced concrete column or floor. Reinforced shaft. 10. Principle of preliminary action • Perform the required action before it is needed. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

12. Principle of equipotentiality • Change the conditions such that the object does not need to be raised or lowered. Examples: • Pit for change oil, Loading dock, airport gate. 13. Principle of opposite solution • Implement the opposite action of what is specified. • Make a moving part fixed and the fixed part mobile. • Turn the object upside down. Examples: • Abrasively cleaning parts by vibrating the parts instead of the abrasive. • Lathe, Mill. 14. Principle of spheroidality • Switch from linear to curvilinear paths, from flat to spherical surfaces, etc. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

85

4.3 TRIZ • Make use of rollers, ball bearings, spirals. • Switch from direct to rotation motion. • Use centrifugal force. Examples: • Computer mouse. • Screw lift.

15. Principle of dynamism • Make the object or environment able to change to become optimal at any stage of work. • Make the object consist of parts that can move relative to each other. • If the object is fixed, make it movable. Examples: • A flashlight with flexible neck. • Bicycle drivetrain and derailer. 16. Principle of partial or excessive action • If it is difficult to obtain 100% of a desired effect, achieve somewhat more or less to greatly simplify the problem.

4.3 TRIZ

86

• A computer mouse where a 2D screen is transformed into a horizontal mouse pad. • A composite wing where loads are in only one direction per layer. 18. Use of mechanical vibrations • Make the object vibrate. • Increase the frequency of vibration. • Use resonance, piezovibrations, ultrasonic, or electromagnetic vibrations. Examples: • Vibrating casting molds. • Quartz clocks. 19. Principle of periodic action • Use periodic or pulsed actions, change periodicity. • Use pauses between impulses to change the effect. Examples: • Hammer drill. • Emergency flashing lights. 20. Principle of uninterrupted useful effect

Examples: • Raincoats, snowboards. 17. Principle of moving into a new dimension • Increase the degrees of freedom of the object. • Use a multi-layered assembly instead of a single layer. • Incline the object or turn it on its side. • Use the other side of an area. Examples: c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

• Keep all parts of the object constantly operating at full power. • Remove idle and intermediate motions. Examples: • Steam turbine, mechanical watch. 21. Principle of rushing through • Carry out a process or individual stages of a process at high speeds. Examples: c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

87

4.3 TRIZ • Cutting thin wall plastic tubes at very high speeds so cutting action occurs before deformation.

22. Principle of turning harm into good

4.3 TRIZ

88

• Nail resistant tires. 26. The copying principle

• Use harmful factor to obtain a positive effect.

• Instead of unavailable, complicated or fragile objects, use a simplified cheap copy.

• Remove a harmful factor by combining it with other harmful factors.

• Replace an object by its optical copy, make use of scale effects.

• Strengthen a harmful factor to the extent where it ceases to be harmful.

• If visible copies are used, switch to infrared or ultraviolet copies.

Examples: • Medical defibrillator. Use of high frequency current to heat the outer surface of metals for heat treatment. 23. The feedback principle • Introduce feedback. • If feedback already exists, reverse it. Examples: • Air conditioning systems. • Noise canceling devices. 24. The go between principle • Use an intermediary object to transfer or transmit the action. • Merge the object temporarily with another object that can be easily taken away. Examples: • Gear trains. 25. The self service principle • The object should service and repair itself. • Use waste products from the object to produce the desired actions. Examples: c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Examples: • Rapid prototyping. Crash test dummies. • Measure shadows instead of actual objects. 27. Cheap short life instead of expensive longevity • Replace an expensive object that has long life with many cheap objects having shorter life. Examples: • Inkjet printer heads embedded in ink cartridges. Cardboard box. 28. Replacement of a mechanical pattern • Replace a mechanical pattern by an optical, acoustical or odor pattern. • Use electrical, magnetic or electromagnetic fields to interact with the object. • Switch from fixed to movable fields changing over time. • Go from unstructured to structured fields. Examples: • CD player. • Microwave oven. Crane with electromagnetic plate. 29. Use of pneumatic or hydraulic solutions • Replace solid parts or an object by gas or liquid. Examples: c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

89

4.3 TRIZ • Power steering. Bubble envelopes.

30. Using flexible membranes and fine membranes • Replace customary constructions with flexible membranes and thin film. • Isolate an object from outside environment with thin film or fine membranes. Examples: • Dome tent. High Altitude Balloon. 31. Using porous materials • Make the object porous or use porous elements. • If the object is already porous, fill the pores in advance with some useful substance. Examples: • Running shoe soles. Air filters. 32. The principle of using color • Change the color or translucency of an object or its surroundings. • Use colored additives to observe certain objects or processes. • If such additives are already used, employ luminescence traces. Examples: • Transparent bandage. Roadway signs. 33. The principle of homogeneity • Interacting objects should be made of the same material, or material with identical properties. Examples: • Shaft and bushing. 34. The principle of discarding and regenerating parts c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

4.3 TRIZ

90

• Once a part has fulfilled its purpose and is no longer necessary, it should automatically be discarded or disappear. • Parts that become useful after a while should be automatically generated. Examples: • Multistage rockets. Bullet castings. 35. Changing the aggregate state of an object • Change the aggregate state of an object, concentration or density, the degree of flexibility or its temperature. Examples: • Heat packs. Light sticks. 36. The use of phase changes • Use phenomena occurring in phase changes like change of volume and liberation or absorption of heat. Examples: • Fire extinguisher. 37. Application of thermal expansion • Use expansion or contraction of materials by heat. • use materials with different thermal expansion coefficients. Examples: • Thermometers. Bimetallic plates. 38. Using strong oxidation agents • Replace air with enriched air or replace enriched air with oxygen. • Treat the air or oxygen with ionizing radiation. • Use ionized oxygen or ozone. Examples: c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

91

4.4 The morphological chart • Metal forming ovens. Torch cutting.

Convert Electrical Energy to Translational Energy

39. Using an inert atmosphere • Replace the normal environment with an inert one. • Carry out the process in a vacuum. Examples: • Aluminum cans for beverages. Arc welding. 40. Using composite materials • Replace a homogeneous material with a composite one. Examples: • Steel belted tires. wings.

4.4

Tennis racquets.

4.4 The morphological chart

High performance aircraft

The morphological chart

The aim of the morphological chart is to generate a complete range of alternative design solutions for a product widening the search for potential new solutions. It is based on the use of identified functions to foster ideas and has two parts. First, to generate as many concepts as possible. Second, to combine the individual concepts into overall concepts that meet all functional requirements. The procedure to create and use a morphological chart is quite simple and can be summarized as follows: 1. List the features or functions that are essential to the product. Each function will be a row of the chart. The list of functions should contain all the features that are essential to the product, at an appropriate level of generalization. If a QFD procedure has been already performed, the list of customer requirements can be used as the list of functions. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

92

Rotary motor with transmission

Linear Motor

Accumulate Energy

Spring

Moving mass

Apply Translational Energy to Nail

Simgle impact

Multiple impacts

Solenoid

Rail gun

Push nail

Figure 4.10: Morphological table for a hand-held nailer. Adapted from Ulrich & Eppinger (2000). 2. For each feature or function, list the means by which it may be achieved. These lists will be the columns of the chart. Lists might include new ideas as well as known solutions. 3. After the chart has been filled out, identify feasible combinations of subsolutions. Each combination will be a possible solution to identify. Figure 4.10 shows an example of a morphological chart for a hand nailer presented by Ullrich & Eppinger (2000). The rows in the table correspond to the functions identified by the design team: convertion of electrical energy to translational energy, acummulation of translational energy and application of translational energy to the nail. The entries in each column correspond to possible solutions for the function at hand. It is important to notice that in order for the chart to be most useful, the items in the list of functions should all be at the same level of generality, and they should be as independent of each other as possible. The list should not be too long, however, and no more than 10 functions should be considered. Some authors advice to use no more than 4 functions at a time. If some functions are to be disregarded for this matter, the development team must clearly understand the risks and tradeoffs of not taking them into consideration. It is also advisable to arrange solution principles so that the columns create logical grouping, for example, of mechanical type, of electrical type, etc. Also, sketches should be used whenever possible to convey as much information as possible. Finally, consideration should be given only to solutions that meet the estimated engineering specifications. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

93

4.4 The morphological chart

Once the chart is filled with solutions to all the specific functions listed, the next step is to consider combinations from the range of all possible solutions. Usually a large number of combinations is possible, although restrictions apply as not all combinations of solutions are possible, for example, combinations that have intrinsic incompatibilities should be discarded. It is essential to analyze very carefully each option before rejecting it. The design team must have in mind that, initially many combinations may not seem to provide a practical solution to the problem at hand, specially to the inexperienced designer. In the example shown in figure 4.10, 24 combinations can be found from the concepts generated ( 4×2×3). Figures 4.11, 4.12 and 4.13 show the sketch of four possible solutions arising from the combination of concepts. The first solution, shown in figure 4.11, is due to the combination the concepts “solenoid”, “spring” and “Multiple impacts”. The second solution, shown in figure 4.12, results from the combination of “rotary motor with transmission”, “spring” and “multiple impacts”. The third, fourth and fifth solutions, shown in figure 4.13, arise from the combinations of concepts “rotary motor with transmission”, “spring” and “single impact”.

4.4 The morphological chart Convert Electrical Energy to Translational Energy

94

Rotary motor with transmission

Linear Motor

Accumulate Energy

Spring

Moving mass

Apply Translational Energy to Nail

Simgle impact

Multiple impacts

Solenoid

Rail gun

Push nail

111111 000000 000000 111111 000000 111111 111111 000000 000000 111111 000000 111111 000000 111111 111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 111111 000000 000000 111111 111111 000000 000000 111111

Figure 4.11: Concept 1. Solenoid compressing a spring which is then released repeatedly in order to drive the nail with multiple impacts. Adapted from Ulrich & Eppinger (2000). Convert Electrical Energy to Translational Energy

Rotary motor with transmission

Linear Motor

Accumulate Energy

Spring

Moving mass

Apply Translational Energy to Nail

Simgle impact

Multiple impacts

11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111

Solenoid

Rail gun

Push nail

1 0 1 0 0 1

Figure 4.12: Concept 2 showing a possible combination of a motor with a transmission, a spring and multiple impacts. The motor repeatedly winds and releases the spring, storing and delivering energy over several hits. Adapted from Ulrich & Eppinger (2000). c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

95

4.4 The morphological chart

4.4 The morphological chart

96

Feature

Convert Electrical Energy to Translational Energy

Rotary motor with transmission

Accumulate Energy

Spring

Linear Motor

Rail gun

Solenoid

Moving mass

Support

Wheels

Track

Air cushion

Slides

Propulsion

Driven wheels

Air thrust

Moving cable

Linear induction

Power

Electric

Petrol

Diesel

Bottled gas

Steam

Gears and shafts

Belts

Chains

Hydraulic

Flexible cable

Steering

Turning wheels

Air thrust

Rails

Stopping

Brakes

Reverse thrust

Hydraulic ram

Rack and pinion

Screw

Chain or rope hoist

Seated at rear

Standing

Walking

Transmission

Lifting

Apply Translational Energy to Nail

Simgle impact

Multiple impacts

Push nail Operator

0 1 0 1 0 1 0 1

MOTOR

1 0 0 1 0 1 0 1

1 0 0 1 0 1 0 1 0 1

1111 0000

TRIGGER

11111111 00000000 00 11 11 00 11 00 00 11 00 11 00 00 11 11

Means

Seated at front

Pedipulators

Ratchet

Remote control

Figure 4.14: Morphological chart for a forklift truck, with one possible combination of sub-solutions picked out by the dashed line (After Cross, 1994). Two examples of morphological charts are presented by Cross (1994). The first one is concerned with finding alternative versions of the conventional forklift truck used for lifting and carrying loads. In the second one, alternatives for the design of a welding positioner are explorer.

CAM

Regarding the finding of alternative versions of a lifting truck, the essential features of the truck are: 1. Means of support which allows movement.

Figure 4.13: Concept 3, 4 and 5 showing possible combinations of a motor with a transmission, a spring and a single impact. The motor winds a spring, accumulating potential energy which is then delivered to the nail in a single hit. Adapted from Ulrich & Eppinger (2000).

2. Means of moving the vehicle. 3. Means of steering the vehicle. 4. Means of stopping the vehicle. 5. Means of lifting loads. 6. Location of the operator. The morphological chart generated for the above functions is shown in figure 4.14 where one possible solution is highlighted. It is interesting to note that

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

97

4.4 The morphological chart

4.4 The morphological chart

98

there are 90 000 possible combinations in the chart, although, some of them obviously, are not possible or include incompatible concepts. In the second example, alternatives for the design of a welding positioner, a device used to support and hold a workpiece and locating it in a suitable position are explored. Figure 4.15 shows the morphological chart for this case where several concepts are described by means of sketchs and text. One possible combination of concepts is indicated by the zig-zag line through the chart.

References 1. Altshuller, G. S. (1984) Creativity as an exact science. Gordon and Breach Science Publishers, New York, U.S.A. 2. Cross, N. (1994) Engineering Design Methods, John Wiley & Sons. 3. Otto, K. & Wood, K. (2001) Product Design - Techniques in Reverse Engineering and New Product Development, Prentice-Hall. 4. Pahl, G. and Beitz W. (2001) Engineering Design - A systematic Approach. Second Ed. Springer. 5. Ullman, D. (2001) The Mechanical Design Process. Third Ed. McGrawHill. 6. Ulrich, K. & Eppinger, S. (2000) Product Design and Development. Second Ed. Irwin McGraw-Hill.

Action Principles / Families of Function carriers Partial Functions ENABLE connection with workpiece

ENABLE rotational movement

ENABLE tilting movement

ENABLE height adjustment

1

2

DRIVE (by hand)

CONTROL of movement

4

5

6

Force locking (friction)

Form interlocking

mechanical screw or bolt

wedge

pneumatic

hydraulic

magnetic

Rotational guidance sliding journal bearing

rolling bearing sphere

cylinder

11111 00000 00000 11111

fulcrum pin position

0000 1111 111 0000 000 1111 000 111 1010 10 screw thread

straight line guidance bearing, sliding or rolling

LOCK state

3

hold directly by hand, weight of workpiece

form interlocking

hang from above

lever mechanism

11 00 00 11

force locking (friction) screw screw with wedge, washer brake block

hole − pin

ratchet mechanism

within guidance

Direct

gear wheel pair

rack and pinion

helical gears (crossed)

through drive mechanism

through locking (ratchet)

optical

electronic

with mechanical advantage device

mechanical

Show position line scale

pointer scale

band, lever worm and rope, eccentric worm−wheel chain cam

mechanical stop

Figure 4.15: Morphological chart for a welding positioner, with one possible combination of sub-solutions picked out by the zig-zag line (After Cross, 1994).

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

100 Concepts

Design Brief Generation

CHAPTER

Generation and Screening

Screening and Scoring

Screening and Scoring

Development and Testing

Final Concept Testing

5

Concept Selection

In the previous chapter some methods to generate potential solutions for a design problem where reviewed. Normally, a design team should generate tens or even hundred of ideas. Clearly not all ideas will lead to a successful product. However, at this point in time, with few information at hand, it is not possible to say which concept is best. In concept selection the goal is to expend the least amount of time and resources on deciding which concepts have the best chances to become a successful product. Concept selection requires the evaluation, of concepts with respect to some criteria comparing their relative strengths and weaknesses in order to select one or more concepts for further evaluation and testing. Here, evaluation should be understood as the process of comparison and decision making. The concept selection phase usually requires at least three steps: 1. Estimate the technical feasibility of the concepts. All those concepts that are regarded as not feasible or ill-conceived are quickly discarded. 2. Concept screening. Concepts are compared roughly in relative terms against a common reference concept. Those that do not offer any advantage or fail to fulfill the requirements of the customer are discarded. 3. Concept scoring. A more detailed comparison is carried out including more information about the concepts for finer resolution. c Copyright 2006 Dr. Jos´ e CarlosMiranda. Todos los derechos reservados.

Feasibility Judgment

Figure 5.1: Concept selection is an iterative process that goes through different phases and is closely related to concept generation and concept testing.

The last to steps are usually done in an iterative fashion, where concepts are discarded and some other are combined to generate new concepts. After sufficient iterations have been carried out, the team goes to the next stage: concept testing. Figure 5.1 depicts the above procedure. The iterative behavior of the design cycle is perhaps better represented by the design-build-test cycle shown in figure 5.2. In the diagram, two different design cycles are represented by the inner and outer loops. The first loop represent the design cycle when new or complex technologies are being use. In this case, building a physical model and testing it is the only approach possible. The outer loop represent a more common approach where no physical devices are build until the very end of the process. Here, the time and expense of building physical models is eliminated by developing analytical models and simulating the concept before anything is build. All the iterations occurs without building any prototypes as all ideas are represented by means of analytical models and graphical representations, usually with the help of computers. Regardless of which design path is chosen, several benefits arises when a structured approach is followed to select concepts. Probably the most most important is that because concepts are compared against customer needs, the selected concept is likely to be focused on the customer. Other benefits may include a reduced time to product introduction and effective decision making. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

101

5.1 Estimating Technical Feasibility DESIGN

Iterate

Iterate

Build prototypes with each closer to the final product

Analytical models and graphical drawings to refine concept and product

Test physical protoypes

TEST Build final product

Figure 5.2: Design evaluation cycles. After Ullman (2003).

5.1

102

represent a problem to estimate as different people will give values that can vary by orders of magnitude.

Simulatable technology

Design prototypes

BUILD

5.1 Estimating Technical Feasibility

Estimating Technical Feasibility

When concepts are generated, members of the design team may experience feelings about the idea that can be grouped in three main reactions: 1. It will never work 2. It may work depending on something else 3. It is an idea worth considering The above judgments regarding technical feasibility are based on the experience of the design team and the individual engineers and their ability to estimate correctly. In general, it is safe to say that the more experience, the more chances the decision will be reliable at this point. Fortunately, estimating is a skill that any person can learn and cultivate to a very good degree. According to Otto and Wood (2001), the estimating skill of an engineer is dependent mainly on familiarity with dimensional units and with the different values along the dimensions. This familiarity takes place in two different levels of abstraction. First, perceived units like length or mass usually represent no problem as everyone can associate their dimensions with day-to-day experiences. On the other hand, derived units like energy or power, usually c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

What allows an engineer to become more familiar with derived dimensions is to associate them with known values. For example, one might realize that 2000W is 3hp, the common power for a lawnmower; or that 0.1MPa is 1atm, the atmospheric pressure at sea level. It is said that a skilled engineer will have at least three readily understood reference levels for every dimensional unit such as power, energy, pressure, force, acceleration, etc. Table 5.1 shows some approximated values for different units to be used as reference. The “gut feeling” reactions to the generated concepts are worth exploring since they have different implications and may induce the individual or design team to discard potentially good ideas or adapt potentially dangerous ones. It will never work. Before discarding concepts that appears to be infeasible or unworkable, consider it briefly from different points of view before reject it. As a guideline, before rejecting the concept answer the following questions: • Why it is not technologically feasible? • Does it meet the requirements of the customer? • Is the concept different from the rest? • Is the concept an original idea? To answer the first two cases, where more attention is deserved, the methods described later in this chapter will be of help. In the case of the last two, it is worth analyzing if the reaction is a product of resistance to change or the “not invented here” syndrome. It may work depending on something else. This reaction is product of a doubt in the design team due to internal or external requirement that may be judged to be either non-existent or not ready for consideration. Typical question to be made in order to get insight of this reaction are: • Is the technology needed available? • Is the technology ready for production? • Is all information needed readily available? • Is is dependent on other parts of the product? c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados. plant: 1GW

Electrical power

town: 1MW

to a small

Electrical power

2000W

engine:

mower

Small lawn

100-1000W

appliance:

household

Typical

bulb: 100W

Bright light

10W

Small flashlight:

LED: 40mW

1cm/s: 33μW

a wall at

Ant crawling up

(W)

Power

A small apple

m/s2 . Belly flopping

stopping in sand, Head-on car collision

Pressure to create a diamond: 5GPa Deep ocean trench:

Two small people, weight: 1.5kN

0.5×109kg

91,400 tons @ GJ kinetic energy

30 knots: 9.9

Aircraft carrier:

107×106kg

Ocean liner:

300,000kg

loaded:

A 747 fully

5000kg

Elephant:

diving board jump,

Statue of

vacuum: 3×108m/s

light trapped in a black hole: 2×1013 m/s2

0.2 GN thrust

20 ×109 MPa

Speed of light in

Centrifugal acceleration of

Center of the sun:

Saturn V or Space Shuttle:

Dallas, TX to outer space: 17km/s

gun: 800km/s2

3.84×109m

Earth to moon:

1000km

Pittsburgh PA:

Boston MA to

Denver, CO or

Voyager 1 traveling in

Projectile fired from a rail

106 MPa

town: 5km

sound: 1km/s

Center of the Earth: 0.40×

Width of a small

3 times the speed of

Liberty: 93m

thrust.

from a rifle: 60km/s2

10km/s2 . Bullet fired

occupant deceleration:

Marble dropped from 1m

100m/s .

2

causing broken bones:

Soccer field

Jetliners: 250m/s

Height of the

length: 100m

Highway speed: 30m/s

Boeing 747: 1MN

0.11GPa

Humans black out: 40 hard in water from a 10m

pressure: 3.5MPa

1.3 MPa

Piston engine firing

USS Nimitz

Battery: 5MJ

Automotive

energy

1 MJ kinetic

Car @ 130km/h:

80kJ

at sea level: 9.8m/s2

compression pressure:

100N

1300kg

battery: 1kJ

2m

1.5m/s

car: 7m/s2 . Earth gravity

Piston engine

appliance buttons: 7N

Bag of potatoes, weight:

70kg

extractable

D-sized battery:

Person’s height:

1m/s. Walking speed:

3m/s2 . Hard braking

1atm = 0.10MPa

Finger force for

low to high: 0.1 mm/s Falling body after 1/10s:

10−3 MPa Fast car acceleration:

thickness: 2mm

10m underwater: 0.10 MPa

Speed of tide rising from

minute hand: 1mm/s

Small apple, weight: 1N

2

a regular clock minute

Centripetal acceleration of Tip speed of a wrist watch

person:

Mid-sized car:

from a AA

30μm

0.1μm/s Book cover

thickness:

snow accumulation rate

hand: 0.3μm/s2 over 2 winter months:

hour hand: 20μm/s. 10’

a regular clock hour

Human hair

(m)

Length

hand: 1 mm/s

10−3 MPa. Mars

(m/s)

(m/s2 ) Centripetal acceleration of Tip speed of a wrist watch

Velocity

Acceleration

atmosphere: 0.8×

Blood pressure: 16 ×

MPa

0.04N

10

Piece of paper, weight:

atom: 0.08μN

proton in hydrogen

between electron and

Moon surface: 0.13 ×

Electrostatic attraction −9

(MPa)

Pressure

(N)

Force

Average

Energy effectively

energy

114J kinetic

140km/hr fast ball:

kinetic energy

falling 1m: 1J

1kg

large snack:

lifted 1m in gravity: 1J.

Small meal or a

Penny: 3g

Grape: 10g

A small apple

energy

2 mJ kinetic

Bee in flight:

×10−6 kg

of paper: 40

0.56μJ kinetic energy

1” × 1” piece

(kg)

Mass

Moving 5g snail:

(J)

Energy

Table 5.1: Approximate reference values on different dimensions (adapted). After Otto & Wood (2001).

109

10

6

103

100

1

10−3

10

−6

103 5.1 Estimating Technical Feasibility 5.2 Concept screening

5.2

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

104

In the first two cases, the technology needed should be carefully examined to see if it is available and if it is already mature or can be by the time the product reach production. In the third case, serious consideration should be given to decide if it is worth waiting for more information to become available. Finally, in the last case, it should be pondered if other parts of the product could be modified to accommodate the intended design without causing a delay in the design process.

It is an idea worth considering. This is generally the hardest case to evaluate, since knowledge and experience are an important part for the evaluation of its feasibility. The methods described later in the chapter will help to develop a deeper knowledge about the concept in order to evaluate it.

Concept screening

Concept screening is a technique based on a method developed by Stuart Pugh and is also known as Pugh concept selection method (Pugh, 1990). The main idea of the technique is to narrow the number of concepts quickly comparing the concepts between themselves based on common criteria and to improve the concepts whenever possible. Concept screening is based on the following steps: 1. Choose the criteria for comparison.

2. Choose which concepts will be evaluated.

3. Decide on a reference concept to be used as a datum.

4. Prepare the selection chart.

5. Rate the concepts.

6. Rank the concepts.

7. Combine and improve the concepts.

8. Select one or more concepts.

Step 1: Choose the criteria for comparison. To start, it is necessary to know the basis on which the different concepts will be compared with each

105

5.2 Concept screening

other. During QFD, an effort was made to develop a set of customer requirements. This requirements are generally well suited to be used as a criteria for comparison. In some cases, when the concepts are well refined, engineering targets may be used instead. Step 2: Choose which concepts will be evaluated. After concept generation several options where available. These options where narrowed down discarding those concepts that were not technically feasible. From the options left, choose the group to be evaluated. If more than 12 concepts are to be considered, the design team can vote to select the 12 concepts that will be compared.

5.2 Concept screening

106 Concepts

Selection Criteria

Datum

Concept B

Concept C

Concept D

Concept E

Concept F

Criterion 1 Criterion 2 Criterion 3 Criterion 4 Criterion 5 Criterion 6 Criterion 7

Step 3: Decide on a reference concept to be used as a datum. To select a reference concept or datum, the design team can follow several approaches. If the company already has a current product, it may serve well as a well understood concept. Other option is to use a competitive product that the team wish to superpass. Pugh (1990) recommends using as a datum the concept that the team vote best.

Sum +’s Sum 0´s Sum −’s Net Score Rank Continue?

Step 4: Prepare the selection chart. Once the criteria for comparison, the concepts that will be evaluated and the datum all have been chosen, the next step is to prepare the selection chart. For that purpose the template shown in figure 5.3 can be of help. Step 5: Rate the concepts. To rate the concepts, compare them against the datum using a very simple scale. It is recommended to use a + if the concept is better than the datum for the current criterion, a − if the concept is worse than the datum and a 0 or an S (“same”) if the concept is judged to be about the same as the datum or there is some ambivalence. If the decision matrix is carried out in a spreadsheet use +1, 0 and −1 for scoring. It is advisable to to rate every concept on one criterion before moving to the next. Step 6: Rank the concepts. After all the concepts have been rated for each one of the criterion, four scores are generated: the number of +’s, the number of −’s, the number of 0’s and the net score. The net score is obtained subtracting the number of −’s from the number of +’s. To rank the concepts, simple use the one with the best net score as 1, the next one as 2 and so forth. Step 8: Combine and improve the concepts. After the concepts have been rated and ranked, the design team should verify the validity of the results. Some recommendations for the interpretation of results are pointed out by c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Figure 5.3: Template for the Pugh selection chart. Ullman (2003): • If a concept or group of similar concepts has a good overall total score or a high + total score, it is important to notice what strengths they exhibit, that is, which criteria they meet better than the datum. Likewise, groupings of − scores will show which requirements are especially hard to meet. • If most concepts get the same score on a certain criterion, examine that criterion closely. It may be necessary to develop more knowledge in the area of the criterion in order to generate better concepts. In many occasions, concepts can be combined to improve them. Here, to help visualize if concepts can be combined, Ullrich and Eppinger suggest to answer the following questions: • Is there a generally good concept which is degraded by one bad feature? c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

107

5.2 Concept screening

5.2 Concept screening

• Can a minor modification improve the overall concept and yet preserve a distinction from the other concepts?

108 Removable Unit

Removable Chamber

Removable Blade

Washable

Scraper

Cost

0

+

_

+

0

Store Grinder

0

+

0

0 _

+ _

+ _

0

Put in Beans

_

_

0

0 _

0

+

0

• Are the two concepts which can be combined to preserve the “better than” qualities while annulling the “worse than” qualities? If any improved concepts arose from combination, these are added to the selection chart and ranked along the original concepts. Step 9: Select one or more concepts. Once the above steps have been carried out, and the design team is satisfied with their understanding of each concept, its strengths and weaknesses, it is time to decide which concepts should be selected for further refinement and analysis. The design team should also clarify if issues need to be investigated further before a final decision can be made. In addition, decisions should be made if the screening matrix has provided enough resolution and if another round of concept screening should be performed. If concept screening has not provided enough resolution, concept scoring should be applied next. la An example of a Pugh chart for the redesign of a coffee grinder is shown in figure 5.4. In this example, presented by Otto & Wood (2001), the goal was to evaluate different concepts all restricted to the use of a chopper. Several ideas were developed to improve the grinder, focusing on cleaning functions. The criteria for the redesign evaluation gathered directly from customer needs and engineering specifications are as follows: • Cost: unit manufacturing cost (development and delivery costs were not considered). Measured in $.

Selection Criteria

0 _

Take Out Coffee

0

Power Setup

0

Cleanable

0

0 _

Development Risk

0

+

0 _

Sum +’s

0

3

1

3

0

Sum 0´s

7

2

2

1

6

Sum −’s

0

2

4

3

1

Net Score

0

1

−3

0

−1

Rank

2

1

4

2

3

0

Figure 5.4: Pugh chart for coffee mill redesign concepts regarding cleanability. Adapted from Otto & Wood (2001).

• Store grinder: facility to put away in a cabinet out of sight. Measured in cm3 .

• Cleanable: Time or steps needed from the point where the coffee has been taken out until the point of being spotless. Measured in number of steps or seconds.

• Put in beans: Time elapsed between the beans are in a bag until the chopper switch can be activated. Measured in seconds.

• Development risks: Difficulty getting a working alpha prototype. Measured in number of potential faults or difficulties.

• Take out coffee: Time elapsed between removing all grounds until all the coffee is poured into a coffee maker. Measured in seconds. • Power setup: Time elapsed between the grinder is plugged in until the switch can be activated. Measured in seconds.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

From the chart some conclusions could be drawn. First, the power setup criteria does not distinguish between concepts as all of them were about the same. Therefore, although it was an important criterion for the product, it did not impact cleanability, and was dropped from further discussion. Next, the removable blade concept was clearly ahead of the rest and was a natural c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

109

5.3 Concept scoring

5.3 Concept scoring

110

candidate for further development.

5.3

Concepts

Datum

Concept scoring Selection Criteria

Concept scoring is a technique very similar to concept screening and it is used when increased resolution will better differentiate among concepts. In this method, the teams weight the relative importance of the selection criteria and focuses on more refined comparisons with respect to each criterion. The steps to use the method are as follows:

Weight

Rating

Weighted Score

B Rating

C Weighted Score

Rating

D Weighted Score

Rating

Weighted Score

Ease of use Readability of settings Ease of handling Dose metering accuracy Durability Ease of manufacture Portability Total Score Rank

1. Choose the criteria for comparison. 2. Choose which concepts will be evaluated. 3. Decide on whether only one concept will be used as a datum or, if different concepts will be used as reference for different criteria. 4. Prepare the selection chart and decide the weight for each criterion. 5. Rate the concepts. 6. Rank the concepts. 7. Combine and improve the concepts. 8. Select one or more concepts. As most of the steps are identical to the concept screening ones, only those different will be discussed next. Step 3: Decide on whether only one concept will be used as a datum or if different concepts will be used as reference for different criteria. Although a single reference concept can be used for the comparative ratings of all criteria as in the screening method, this is not always appropriate. Unless by pure coincidence the reference concept is of average performance relative to all criteria, the use of the same reference concept for the evaluation of each criterion may lead to what is known as the “scale compression effect”. Consider, for example, that the reference concept to be used as datum is better than the rest in 1 criterion. If this is the case, all the concepts could be evaluated only as “same as” or “worse than”, effectively compressing the evaluation scale to 2/3. This effect applies independently of the scale used, as c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Continue?

Figure 5.5: Template for a concept scoring matrix.

it will be seen later. For this reason, many times different concepts are used as reference for different criteria. Step 4: Prepare the selection chart and decide the weight for each criterion. The selection charts for the scoring method is very similar to Pugh charts with two exceptions. First, for each criterion, it includes its weight. Second, the chart includes two columns per concept: rating and weighted score. A template for an scoring chart is shown in figure 5.5. The weight for each criterion is usually defined as the percentage of importance that the criterion has relative to the other criteria. Each percentage is defined such that the sum of all different percentages is 100%. An example illustrating the use of weights is shown in figure 5.6 where three different cars are compared in base to four different criteria: fuel consumption, cost of spare parts, simplicity of servicing and comfort. Each criterion has its own weight defined by some chosen rules: fuel consumption weight is 50%, the cost of spare parts has a weight of 20%, easy to maintain 10% and finally, comfort 10%. It is easy to see in this example that the sum of all weights is 100%. In many ocassions, the selection of the right weighting factors can be a cumbersome task, specially if many different criteria have to be taken into account. One alternative is to use a objectives tree that includes weighting for each criterion. To show how objectives trees are constructed, consider the objective c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

111

5.3 Concept scoring Car A

Selection Criteria

Weight

Parameter

Low fuel consumption

50%

Miles per galon

Low cost of spare parts

20%

Easy to maintain High comfort

Value

Car B

Rating Score

Value

1

Rating Score

Value

1.0

40

4

2.0

36

3

1.5

Cost of 5 typical parts

£18

7

1.4

£22

5

1.0

£28

2

0.4

10%

Simplicity of servicing

Very simple

0.5

Com− plicated

2

0.2

Average

3

0.3

20%

Comfort rating

0.4

Very good

5

1.0

Good

4

0.8

Rank

Poor

2

1.0

Rating Score

2

Total Score

112

Car C

33

5

5.3 Concept scoring

3.3

4.2

3.0

2

1

3

11

12

13

0.25 0.25

0.60 0.60

0.15 0.15

Figure 5.6: Scoring matrix for three alternative motorcars. Adapted from Cross (1994).

tree in figure 5.7. Each criterion in the objectives tree is represented by a circle or box with three numbers on it. At the top of each box, a number represents the level of the criterion. For example, the set of criteria is level 1, representing a weight of 100% or 1. If there are three main criteria, then the first would be represented by the number 11, the second by the number 12, the third by the number 13, an so on. If the second criterion, number 12 has two criterions that must be considered, then the first one will be identified by the number 121 and the second one by the number 122. If the criterion identified by the number 121 has to be divided into two different criteria, then the first would be 1211 and the second one 1212. The objectives three can have as many levels as necessary. The second number, at the lower left side of the box, indicates the weight of the factor to whom it belongs. The third number, at the lower right end, is result of the multiplication of the weight of the criterion times the weight factor of its parent box. This product gives the contribution of the criterion to the total 100%.

1.0

1.0 = 0.25

121

122

131

132

0.75 0.45

0.25 0.15

0.30 0.05

0.70 0.11

1211

1212

0.50 0.22

0.50 0.22

+ 0.22

+

0.22

+ 0.15

+

0.05

+

0.11

Figure 5.7: An objetive tree with weighting factors.

for 0.25, mechanical behavior for 0.60 and cost of manufacturing for 0.15.

This procedure is better explained through an example. The above objectives tree was constructed to aid in the selection of weighting factors for the selection of a mechanical component. The main factors for the selection of the component were specified as how safe the component was (criterion 11), its mechanical behaviour (criterion 12) and and its cost of manufacturing (criterion 13). Since these three factors must add 100%, or 1 for short, then 1.0 have to be divided between these factors. It was decided that safety accounted

Mechanical behavior was divided in two criteria, first, strenght (121) which accounted for 0.75 of the original 0.60 specified for mechanical behavior, and second, freedom from resonance (122) which accounted for 0.25 of the original 0.60. It is important to stress at this point that the sum of weights of factors which have the same parent and are at the same level must be 1.0 or 100% (here 0.75 + 0.25 = 1.0). It was considered to divided strenght further into two more specific criteria, both with the same weight, stiffness (1211) and maximum allowable stress in the component (1212). Carrying out the products, the final weight for stiffness is 0.22, which is the same value for the maximum stress criterion.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

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5.3 Concept scoring CONCEPTS

TEST RIG BOTTOM−RIGHT HINGE SCORING MATRIX Selection Criteria

5.3 Concept scoring

Weight

B

Reference Rating

Weighted Score

Rating

C Weighted Score

D Weighted Score

Rating

Rating

Weighted Score

Selection Criteria

Weight

114 Do Nothing

Renuzit Air Freshner

Rating Score

Rating Score

Baking Soda

Cedar Chips

Rating

Score

Rating

Vented Walls

Score

Rating

Activated Carbon

Score

Rating

Score

Safety

0.25

Performance (olfactory distance − ft)

50%

0

0

70

35

70

35

80

40

20

10

90

45

Stiffness

0.22

Cost

25%

100

25

52

13

76

19

84

21

100

25

20

5

Max. allowable stress

0.22

Freedom from resonance

0.15

Ease of replacement

13%

100

12.5

70

9

90

11

40

5

100

13

20

3

12%

100

12.5

0

0

50

6

67

8

100

13

67

8

Cost of tooling

0.05

Cost of materials

0.11

Frequency of Replacement

Total Score Rank

Total Score

50

57

71

74

61

61

Rank

6

5

2

1

3

4

Continue?

Figure 5.8: An objetive tree with weighting factors. Finally, the cost of manufacturing was divided into two more specific costs: cost of tooling (131) and cost of materials (132). Cost of tooling had a weight of 0.30 whereas the cost of materials had a weight of 0.70. Hence, cost of tooling final weight is 0.05 and the final weight for cost of materials is 0.11. As final observation about objectives trees, it is important to notice that the sum of weights of each level is always 1 as shown in figure 5.7 for the final level. In figure 5.8 the final template for the scoring matrix for this problem is presented. Step 5: Rate the concepts. Similar to the procedure followed in the screening method, here each concept is rated using a simple comparative scale. As more detail is needed, a more detailed rating scale is generally used. A common option is to use a 5 levels scale:

Relative performance

Rating

Much worse than reference

1

Worse than reference

2

Same as reference

3

Better than reference

4

Much better than reference

5

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Figure 5.9: Concept scoring matrix for the selection of odor control alternatives. Adapted from Otto & Wood (2001).

Sometimes a 10 or more levels scale is used, but its use is discouraged as it requires more time and effort. For example, figure 5.10 presents a scoring matrix for an outpatient syringe where four different concepts are been evaluated. Note that in this example a 5 levels scale is being used and that different concepts serve as datum for different criteria. This example is different from the one presented in figure 5.6 where a 10 levels scale is used. Step 6: Rank the concepts. Once the ratings have been specified for each criterion, the weighted score is obtained multiplying the rating for the weight of the criteria. The total score for each concepts is simply the sum of all weighted scores. Finally, each concept is given a rank corresponding to its total score. Another example showing a more elaborated scoring matrix is presented by Otto & Wood (2001). The matrix, shown in figure 5.9, helped the design team to evaluate between different alternatives to control the odors in a cat litter box. The careful reader should notice that the rating scale used in this example goes from 0 to 100, which at first sight may look unnecessary. Nevertheless, in this case a 0-100 scale was chosen as the team had high-quality information about the relative performance of each concept regarding each one of the criteria. This high-quality information is usually gathered directly from testing and experimentation, and it is not influenced by the opinion of the design team members. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

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5.4

5.4 Concept Testing

Concept Testing

5.4 Concept Testing

116

• Which of several alternatives should be pursued? • How the concept may be improved to better meet customer needs?

In the concept selection process, it is very likely that some form of customer’s response will be needed in order to further discuss the possibilities of the proposed concepts. In order to communicate the idea of the concept and to measure the response of the customer, in some cases simple verbal descriptions or drawings will suffice. In other cases, there is no other choice but to create physical prototypes of the product. This testing will give a better idea on the feasibility of the concepts and the sales potential of the product. Concept testing is carried out to facilitate decision-making during final concept selection stages, generally after some detailed design has been done. Concept testing is not necessary when: • time required to test the concept is large relatively to the product life cycle. • cost of testing is large relative to the cost of actually launching the product. Ulrich & Eppinger (2000) presents a 6 steps methodology for testing product concepts: 1. Definition of the purpose of the concept test. 2. Choosing of a survey population. 3. Choosing of a survey format.

• Approximately how many units are likely to be sold? • Should development be continued? 5.4.2. Choosing a survey It is necessary to define the number of possible population customers to survey and in what market segments they will be. This selection is carried out in a similar fashion as the selection of customers in the “Identifying customer needs phase”. It is important, however, to have in mind that concept testing is a much more expensive activity. The most important question to answer is how large the survey population should be. Some useful guidelines are outlined next. Factors favoring a smaller sample size: • Test occurs early in the concept development process. • Test is primarly intended to gather qualitative data. • Surveying potential customers is relatively costly in time and money. • Required investment to develop and launch the product is relatively small. • A relatively large fraction of the target market is expected to value the product.

4. Communication of the concept. 5. Measurement of customer response. 6. Interpretation of results. 5.4.1. Concept testing purpose

In this initial step, the design team should clarify what questions they want to answer with the test. It is essential to define what the test or experiment is for. Some typical questions are: c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Factors favoring a larger sample size: • Test occurs later in the concept development process. • Test is primarily intended to assess demand quantitatively. • Surveying customers is relatively fast and inexpensive. • Required investment to develop and launch the product is relatively high. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

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5.4 Concept Testing

• A relatively small fraction of the target market is expected to value the product. Concept tests can be done in the early stages of the development process to solicit feedback on the basic concept. 5.4.3. Choosing a survey The following formats are commonly used in conformat cept testing:

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118

5.4.5. Measure customer response

Most concept test surveys first communicate the product concept and then measure customer response. Although is good practice to include questions to measure customer response to product concepts, in many occasions concept test generally attempt to measure purchase intent. A useful scale to measure purchase intent may be: • Definitely would buy

• face-to-face interaction

• Probably would buy

• telephone

• Might or might not buy

• postal mail

• Probably would not buy

• electronic mail

• Definitely would not buy

• internet It is important to realize that each of these formats presents risks of sample bias. The way in which the concept will be surveyed, is closely related to the way in which the concept will be communicated. Communication of the concept can be carried out by the following means:

5.4.6. Interpreting results Usually interpretation is straightforward if the design team is just comparing two or more concepts. It is important, though, to be sure that customers understood the key differences among concepts.

5.4.4. Communicating the concept

• verbal descriptions • sketch • photos and renderings • storyboard • video • simulation

References 1. Cross, N. (1994) Engineering Design Methods, John Wiley & Sons. 2. Otto, K. & Wood, K. (2001) Product Design - Techniques in Reverse Engineering and New Product Development, Prentice-Hall. 3. Pugh, S. (1990) Total Design, Addison Wesley. 4. Ullman, D. (2003) The Mechanical Design Process, Third Edition. McGrawHill. 5. Ulrich, K. & Eppinger, S. (2000) Product Design and Development. Irwin McGraw-Hill.

• interactive multimedia • physical appearance models • working prototypes c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

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5.4 Concept Testing

CHAPTER Embodiment design

Concepts

Selection Criteria

Weight

A

B

C

D

Master Cylinder

Lever Stop

Swash Ring

Dial Screw

Rating

Weighted Score

5%

3

0.15

Readability of settings

15%

3

Ease of handling

10%

2

Dose metering accuracy

25%

Durability

Rating

Weighted Score

Rating

Weighted Score

Rating

Weighted Score

3

0.15

4

0.2

4

0.2

0.45

4

0.6

4

0.6

3

0.45

0.2

3

0.3

5

0.5

5

0.5

3

0.75

3

0.75

2

0.5

3

0.75

15%

2

0.3

5

0.75

4

0.6

3

0.45

Ease of manufacture

20%

3

0.6

3

0.6

2

0.4

2

0.4

Portability

10%

3

0.3

3

0.3

3

0.3

3

0.3

Ease of use

6

Total Score

2.75

3.45

3.10

Rank

4

1

2

3.05 3

Continue?

No

Develop

No

No

Figure 5.10: Concept scoring matrix for an outpatient syringe. The reference points for each criterion are signified by bold rating values. Adapted from Ulrich & Eppinger (2000).

As a design task, concept embodiment is perhaps the one that is most identified with engineers as in this phase of the design process the choice of components, interfaces, materials, dimensions, shapes, tolerances, surface finishes, union methods, manufacturing and assembly processes, etc., are carried out. In order to make wise choices, engineers should be able to understand throughly the design, its functionality, objectives and constraints. Is in this stage where engineers apply their skills in mathematics and basic science. Regardless of size, complexity or cost, products must be effectively modeled, tested and, whenever possible, refined. Methods for concept embodiment must aid in this process.

6.1

Product Architecture

Product architecture is the the definition of the layout of systems, sub-systems and components according to their functional purposes. This definition of the layout of the product must also deal with what interfaces are necessary between components, sub-systems and systems. Product architecture allows the design to be divided so building blocks can be assigned to individuals, teams or suppliers in order to permit parallel detailed design, testing and refinement. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

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6.1 Product Architecture Product Architecture for the Product and Common Derivatives

Product Architecture Product

Type Integral

Modular

6.1 Product Architecture

Derivatives

Type

Fisted vegetable peeler

Fixed unsharing

Shares no common components

Wooden pencil

#1, #3 lead pencils

Modular platform

Differing lead hardness

Kitchen knife

Kitchen knife set

Modular platform

Parametric handle size

Swiss Army Knife

Complex knives

Modular platform

Expanding width

PC computer

More RAM, devices

Adjustable for purchase

Standard interfaces

Black and Decker cordless

VersaPak line of power

drill

tools

Tinkertoys

Theme sets

Modular platform Adjustable for use

Pros

Characteristics

Handheld vegetable peeler

122 Cons

Improves device reconfigurability

May make devices look too similar

Increases the device variety and speed of

Makes imitation of device easier by competitors

introduction for new devices Modular architecture

Improves maintainability and serviceability

Reduces device performance

of device Decouples development tasks (and

Common battery

manufacturing to some extent)

Modular design may be more expensive than integral design

Common motor Standard interfaces Component variety

Integral architecture

Figure 6.1: Product architecture examples. After Otto & Wood (2001).

Harder for competitors to copy design

Hinders change of design in production

Tighter coupling of team with less interface

Reduces the variety of devices that can be

problems

produced

Increases system performance Possible reduction in system cost

Product architecture is also related what is called portfolio architecture. Portfolio Architecture relates to a group or family of products, where design strategy revolves around how to share components or subsystems across products in the portfolio. Figure 6.1 shows some examples of product and portfolio architecture.

Figure 6.2: Comparison of modular and integral architectures. After Otto & Wood (2001).

6.1.1. Types of product architecture

Slot-modular architecture. Each of the interfaces between modules in a slot-modular architecture is of a different type from the others, so that the various modules in the product cannot be interchanged. An example of this type of architecture are products that are to be assembled by the customer and are constructed in such a way that any given module can fit in only one place.

In general terms, product architecture can be divided in two main types: integral architecture and modular architecture. Each type has its own advantages and disadvantages as shown in figure 6.2. A product has an integral architecture when no attempt is made to divide functions into components or systems resulting in on a very small number of physical elements carrying out all functions of the product. Integral architecture is common for high-volume products where cost is not reduced through sharing components but through easiness of assembly. This results in products with fewer components but much function sharing. Modular product architecture is the result of dividing product functions into a similar number of blocks or modules that perform a limited set of functions. Ideally, a one-to-one correspondence between modules and functions is achieved. In practice, modularity is not strict, and generally speaking, products are neither fully modular or fully integral. Rather, a given product will present more or less modularity than another comparative product.

types: slot, bus and sectional. These three types, shown in figure 6.3, are explained next.

Bus-modular architecture. In a bus-modular architecture, there is a common bus to which the other modules connect via the same type of interface. An example of bus-modular architecture are the floppy drive, DVD, CDRW and battery that connects to a bay in a laptop using the same interface. Sectional-modular architecture. In a sectional-modular architecture, all interfaces are of the same type, but there is no single element to which all the other modules attach. The assembly is built up by connecting the modules each other via identical interfaces. An example may be modular office furniture that can be arrange in different ways depending on the modules used.

According to Ulrich (1995), modular architecture can be classified in three

Slot-modular architectures are the most common type of modular architecture because for most products, different modules require different interfaces to

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6.1 Product Architecture

6.1 Product Architecture

124

Product variety

Slot−Modular Architecture

Bus−Modular Architecture

Sectional−Modular Architecture

Figure 6.3: Three types of modular architectures. After Ulrich & Eppinger (2000) accommodate unique interactions with the rest of the system. 6.1.2. Implications of the architecture

Even when the architecture of a product is initially defined, at least informally, since the concept generation stage, formal decisions are made during the embodiment design phase. Product architecture is one of the development decisions that plays a major impact in the ability to deliver a variety of products with standard components that allows better product performance, manufacturability and maintenance.

Product variety refers to the amount of different products that any given company can manufacture over a period of time. Product variety generally respond to market needs, as consumers want distinctive products. Product architecture can help to achieve a large product variety for a minimum overhead in its cost. An example is Swatch watches, where hundreds of different combinations can be achieved choosing different components during assembly.

Component standardization Component standardization is the use of the same components or modules in multiple products. This standardization allows the manufacturer to minimize cost and increase quality through the production of larger volumes and the refined design of such common components. An example are cars within same or sister companies that share many parts and subsystems.

Product Performance Product change Modules are the building blocks of the product and the architecture defines how this blocks relate to the function of the product and how the blocks interact with each other. If each module is responsible for certain isolated functions, it would be possible to replace or change any given module without affecting the rest of the product. The contrary is true for an integral architecture, where changing one part of the product may have an influence in the functions carried out by the rest of it. Some of the motives for product change are: • Upgrades • Add-ons • Adaptation • Wear • Consumption • Flexibility in use and reuse. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Product performance is related to how well the final product meets customer requirements in terms of intended functions. Some examples of typical performance measures are speed, acceleration, efficiency, life, accuracy and noise. Here, architecture can facilitate the optimization of performance characteristics by means of integration and function sharing. Function sharing refers to the implementation of multiple functions through a single module or component. Function sharing can help to optimize a design, but trade offs in the advantages of modular architecture have to be considered by the design team.

Manufacturability As discussed above, product architecture can influence the manufacturing cost through product variety and component standardization. In addition, many decisions regarding the architecture of a product influence the easiness of manufacturing as many complicated modules can be produced in larger volumes to reduce cost or many functions can be implemented in a single module to reduce either parts or manufacturing operations. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

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6.1 Product Architecture

Due to the importance of product architecture in subsequent steps of the product design process, it should be throughly discussed by the team design and established in a cross-functional fashion. At the end of this step, an approximate geometric layout of the product, description of the major modules and documentation about the interaction between modules should be obtained. Ulrich & Eppinger (2000) suggest to follow a four steps approach that will be illustrated using a DeskJet printer as an example. The four recommended steps are:

6.1 Product Architecture

126

6.1.3. Establishing the architecture

1. 2. 3. 4.

Create a schematic of the product. Cluster the elements of the schematic. Create a rough geometric layout. Identify the fundamental and incidental interactions.

Step 1: Create a schematic of the product. A schematic is a diagram showing the constituent elements of a product as understood by the design team. It is important that the schematic reflects the best understanding of the team, although great detail is not necessary at this step. For the purpose of establishing product architecture, some authors recommend to aim for fewer than 30 elements in the schematic. It also should be realized that there is not a unique schematic for any given product, so the team should generate several alternatives to select from. In figure 6.4 The schematic for a DeskJet printer is shown. Many elements in the schematic represent physical components such as the print cartridge, while other elements represent functional elements such as store output. Functional elements are those that have not yet been reduced to physical concepts or components, requiring further discussion by the design team in order to achieve a final decision about how they will be implemented. On the other hand, components that have been reduced to physical components are generally those that are central to the basic product concept that the team has generated and selected. Step 2: Cluster the elements of the schematic. In this step, the objective is to assign the different elements in the schematic into specific modules. As in the previous step, the assignment of elements into modules is not unique, and the design team will be faced with different viable alternatives that can range from the few to the hundreds. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Enclose Printer Print Cartridge Provide Structural Support Position Cartridge in X−Axis

Store Output

Accept User Inputs

Display Status

Position Paper in Y−Axis Control Printer

Store Blank Paper

Flow of forces or energy Flow of material

"Pick" Paper

Supply DC Power

Communicate with Host

Command Printer

Flow of signals or data Connect to Host

Figure 6.4: Schematic of the DeskJet printer. Note the presence of both functional elements (e.g., “Store Output”) and physical elements (e.g., “Print Cartridge”). For clarity, not all connections among elements are shown. After Ulrich & Eppinger (2000)

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6.1 Product Architecture

6.1 Product Architecture

128

One method to manage such complexity is for each element to be assigned to its own module and then to cluster elements when advantageous. Some factors worth considering when clustering elements are: • Geometric integration and precision. In some cases, it is convenient to cluster elements that control certain functions that are related between themselves. Elements requiring precise location or close geometric integration can often be best designed if they are part of the same module. In the case of the DeskJet, this principle would suggest clustering the elements associated with positioning the cartridge and the paper.

Enclosure Enclose Printer Print Cartridge Provide Structural Support

User Interface Board Position Cartridge in X−Axis

Chassis

Store Output

Accept User Inputs

Position Paper in Y−Axis Control Printer

Store Blank Paper Paper Tray

Flow of forces or energy Flow of material

Display Status

"Pick" Paper

Power Cord and "Brick" Supply DC Power

Print Mechanism

Communicate with Host

Flow of signals or data Connect to Host

Command Printer Host Driver Software

Logic Board

Figure 6.5: Clustering the elements into modules. Nine modules make up this proposed architecture for the DeskJet printer. After Ulrich & Eppinger (2000)

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

• Function sharing. When a single device can implement several different functions, it is best to cluster the related components together. For the DeskJet it was believed that that the status display and the user controls could be incorporated into the same component. • Capability of vendors. If a specific vendor is know for its capacity in developing and manufacturing certain components, it is best if those components are cluster together. This will help the vendor to integrate more efficiently the said components. • Similarity of design or production technology. When two or more components are designed or manufactured using the same or similar technology is best to cluster them in order to save costs. An typical example of the application of this principle is the clustering of several electronic devices into a single circuit board. • Localization and change. If the design team anticipates that a component will suffer several changes over time, it is best to isolate that component into its own module. In the case of the DeskJet, the engineers decided that the printer would suffer cosmetic shape modifications and decided to isolate the enclosure into its own module. • Accommodate variety. If some components of the product will be changed to satisfy different market or operative conditions, it is best to isolate these components in a module that can be easily replaced. For the Deskjet, the engineers decided to isolate the components associated with the DC power supply as the printer was going to be sold in different parts of the world with different power standards. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

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6.1 Product Architecture

6.1 Product Architecture

• Enabling standardization. When a component or components can be used in different products, it is best to isolated them in separate module or modules. This allows the higher production of the elements in the module. An example of this standardization is the printer cartridge in the DeskJet printer.

Step 3: Create a rough geometric layout. The next step once the general components have been arranged in modules, is to generate a general geometric layout to analyze if the proposed distribution is physically possible. This rough sketch can be made out of foam or cardboard, or even as a rough 3-D computer model, as it is not necessary to include great detail. Nevertheless, it should be sufficient to decide whether component and interface distributions are possible. An example of a geometric layout for the DeskJet printer is shown in figure 6.6. In this example, the design team realized that there was a trade off between the height of the machine and how much paper could be stored in the paper tray. In many cases, the design team may decide that the geometric layout or the clustering chosen are not feasible. In this cases, components may be assigned to different modules. Step 4: Identify the fundamental and incidental interactions. It is common practice to divide the design so each module can be assigned to an specific person or team. Since the different modules interact in one way or another, different persons or teams have to constantly coordinate their activities and exchange information. To facilitate this interaction, interactions between modules have to be identified. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

User interface board

Logic board print cartridge

Paper tray

Print mechanism

Chassis

Enclosure

Logic board

• Portability of the interfaces. Some interactions are more easily transmitted over large distances than others. For example, it is easier to transmit electric or light signals over a distance than mechanical forces and motions. It is also true for the transmission of fluid connections. As a result, it is easier to separate elements with electronic and fluid interactions. In the case of the DeskJet, the flexibility of electrical interactions allowed the design team to cluster control and communication functions into the same chunk. On the other hand, the design team was constrained by the geometric and mechanical interactions of the paper handling mechanism.

130

Print cartridge Roller/guide Paper Paper tray Chassis

Figure 6.6: Geometric layout of the printer. After Ulrich & Eppinger (2000)

According to Ulrich & Eppinger (2000) there are two types of interactions between modules. First, fundamental interactions are those corresponding to the lines on the schematic that connect the chunks to one another (see Figure 6.4). This interaction is planned and is fundamental to the operation of the system. Second, incidental interactions, are those that arise because of particular geometric or physical implementation of modules. In the Deskjet example, the vibration from the actuator in the paper tray could interfere with the precise location of the print cartridge in the x−axis. Even when the principal interaction between modules were described in the schematic, incidental ones should be documented apart. When the system includes a reasonable small number of incidental interactions (less than 10), c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

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6.1 Product Architecture

6.2

User Interface Board

Enclosure

Styling

Paper Tray

Vibration

Print Mechanism

Thermal Distortion RF Shielding

Chassis

Thermal Distortion

Logic Board

Host Driver Software

RF Interference

Power Cord and "Brick"

Figure 6.7: Incidental interaction graph. After Ulrich & Eppinger (2000) an incidental interaction graph is convenient. Figure 6.7 shows an example of an interaction graph regarding the DeskJet example. This graph shows that vibration and thermal distortion are two incidental interactions that may affect the performance of the print mechanism. The design team should be careful to address these issues. To define the interactions between modules, flows in material, energy and signals must be investigated and refined at each module boundary. These flows usually define the interactions and the boundaries define the interfaces. According to Cutherell (1996), four types of interactions are typically investigated: 1. Material interactions: solid, liquids, or gases that flow from one module to the next. 2. Energy interactions: energies that must be transmitted or shielded between modules. 3. Information interactions: signals (tactile, acoustic, electrical, visual, etc.) that must be processed from one module to the next, and 4. Spatial interactions: geometrical dimensions, degrees-of-freedom, tolerances, and constraints that must be maintained between modules.

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6.2 Geometry and layout refinement

132

Geometry and layout refinement

In the quest of creating a robust product, two main activities take place once rough concepts have been generated and selected: 1. refining the geometry and architecture of the product, 2. systems modeling toward detail design. Take for example the electric wok presented by Otto & Wood (2001) shown in figure 6.8. In this case the design team was faced with the task of improving an existing product. As shown, the original concept of the wok evolved to a new one that included more advanced controls and configurations accommodating improved product cleaning and storage. As a result of the embodiment design phase, components, parts, assemblies and interfaces were clearly defined, from both geometrical, and functional points of view. Another example of the result of embodiment design is shown in figure 6.9, where an exploded view of the PrestoTM hot air popcorn popper is shown. In any case, the embodiment process include the following tangible documentation: • • • • • • • • •

detailed drawings, exploded views, assembly diagrams, tool designing, manufacturing process plans, tolerance design, packaging, maintenance and warranty information and user’s manual.

In order to generate the above documentation, some guidelines described next may be followed. The main objective of these guidelines is to transform a concept sketch into refined geometry and material choices focusing on the functional performance of the product, including all relevant engineering specifications. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

133

6.2 Geometry and layout refinement

6.2 Geometry and layout refinement

134

Figure 6.8: Embodiment example of a new electric wok concept. (a) Original wok concept. (b) Original product realization. (c) Evolved wok concept. (d) Realization of new product concept. After Otto & Wood (2001) In the embodiment design phase, specific layouts and parameters are generated in order to logically chose a given concept from a number of solution alternatives that have been developed. Ideally, the result of this phase is a single developed concept, in its definitive form, for the product or each subsystem defined including: • • • •

geometric layout material composition quality and manufacturability issues economics

In practice, the design team may be faced with the situation that further refinement of the selected concepts is needed before commitment for a single solution occurs.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Figure 6.9: Embodiment example of the PrestoTM hot air popcorn popper. After Otto & Wood (2001)

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135

6.2 Geometry and layout refinement

6.2 Geometry and layout refinement

136

Prepare parts lists (BOM), assembly drawings, part drawings, product process plans Develop possible layouts for supporting functions

4. Definitive Layout

Check for robustness and errors Inventory required supporting functionality

Engineer the dimensions of components

In order to deal with these complex characteristics of embodiment design, Pahl & Beitz (1996) suggest a general process to iteratively refine the geometry and layout of a product from an abstract form to a well-defined one. Figure 6.10 illustrates this general process.

3. Preliminary Layout

Evaluate against customer, technical, robust, safety, and business case criteria

Without doubt, the main challenge of embodiment design is that when parameters in the different subsystems or modules change, they usually affect other subsystems or modules, they propagate. This behavior is the result of having parameters that are highly coupled between product subsystems/modules. This scenario means that embodiment design activities of the different subsystems/modules must be carried out simultaneously and iteratively. As one change is made, its effects in the other subsystems/modules are studied and, if acceptable, the change is approved and its effects mitigated. The process stop when the performance of the product becomes acceptable.

Complete form design with detailed drawings

In this situation, parallel development of the concepts should be carried out. It is important to try to select one final concept as soon as possible in order to direct more resources for the development of the final product.

• • • •

maximum/minimum dimensions of the product clearance between relative subsystems/modules installation paths general arrangement of components relative to one another.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Select an effective layout

Fill in the spatial form with scale drawings

2. Concept Layout

Select and effective layout for module placement

Develop preliminary layouts possibilities for the module placement

Identify the main functional modules

Define the spatial form with preliminary scale drawings

After critical specifications have been selected, the next step is to draw a scale sketches of the product. These sketches should have enough detail to incorporate all critical aspects of the alternatives but care should be taken to avoid over-constraining the models. These drawings includes the following items:

Using the engineering specifications, identify crucial requirements

• size and geometric specifications • material specifications • arrangement specifications

1. Concept

The process begins defining customer needs and the engineering specifications that fulfill them. Critical specifications/requirements that will drive the embodiment process are identified. Some examples of critical specifications are:

Figure 6.10: A general process for concept embodiment. After Pahl & Beitz (1996)

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6.2 Geometry and layout refinement

Once the sketches have been completed, it should be verify that each product subsystem, module, part or assembly fulfills its intended function completely. Also, the different sketches should be checked for possible geometry simplifications and function sharing.

6.2 Geometry and layout refinement Embodiment

Checklist issue

Function

Are the customer needs satisfied, as measured by the target values?

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Is the stipulated product architecture and function(s) fulfilled? What auxiliary or supporting functions are needed? Working principles Do the chosen form solutions (architecture and components per function) produce the desired effects and advantages?

Based on the results, alternative sketches should be generated if needed. Next, ranges of geometric, materials, and other variables should be established and listed for each subsystem/module. Also, decisions regarding the possible use of standard components in each subsystem/module should be made. This process stage is finished by choosing between alternative layouts using the product specifications. The scale drawings are updated with the choices made.

What disturbing noise factors may be expected? What byproducts may be expected? Layout, geometry

Do the chosen layout, component shapes, materials, and dimensions provide

and materials

minimal performance variance to noise (robustness), adequate durability (strength), efficient material usage (strength-to-mass ratio), suitable life (fatigue), permissible deformation (stiffness), adequate force flows (interfaces and strength concentrations),

As the next stage, additional functions that may be needed to carry out support and auxiliary requirements should be identified. Then, rough layouts for these additional functions should be developed ensuring the compatibility of all subassembly interfaces. This task usually requires the use of standards, mathematical models, design guidelines and experimentation in order to determine all appropriate parameters. At the same time, the product should be evaluated against customer, technical, robust, safety an economic criteria and the layout should be checked to estimate potential faults.

adequate stability, impact resistance, freedom from resonance, unimpeded expansion and heat transfer, and acceptable corrosion and wear with the stipulated service life and loads? Energy and

Do the chosen layout and components provide

kinematics

efficient transfer of energy (efficiency), adequate transient and steady state behavior (dynamics and control across energy domains), and appropriate motion, velocity and acceleration profiles?

Safety

Have all of the factors affecting the safety of the user, components, function, operation, and the environment been taken into account?

The embodiment process concludes with the testing of physical prototypes and the design of appropriate tooling.

Ergonomics

Have the human-machine relationships been fully considered? Have unnecessary human stress or injurious factors been predicted and avoided? Has attention been paid to aesthetics and economic analysis of the production process, capability, and suppliers?

6.2.1. Embodiment checklist

A second method to supplement the general embodiment process, is the application of the embodiment checklist developed by Pahl & Beitz shown in table 6.1. This table provides a systematic approach to apply proven design principles during the embodiment phase. The objective of the list is to ensure robustness, clarity, simplicity and safety in a product.

Quality control Assembly

Can all internal and external assembly operations be performed simply, repeatedly, an in the correct order (without ambiguity)? Can components be combined (minimize part count) without affecting modular architectures and functional independence of the product?

Transport

Have the internal and external transport conditions and risks been identified and solved? Have the required packaging and dunnage been designed?

Operation

The checklist involves categories of possible engineering specifications, where each category has a set of basic questions that should be exhaustively applied to the product and the different subsystems/modules as they are being detailed. It should be noted that mathematical models or physical prototypes may be needed to effectively answer each of the questions.

Have standard product tolerances been chosen (not too tight)? Have the necessary quality checks been chosen (type, measurements, and time)?

Have all of the factors influencing the product’s operation, such as noise, vibration, and handling been considered?

Life cycle

Can the product, its components, its packaging be reused or recycled? Have the materials been chosen and clumped to aid recycling? Is the product easily disassembled?

Maintenance

Can maintenance, inspection, repair, and overhaul be easily performed and checked? What features have been added to the product to aid in maintenance?

Costs

Have the stipulated cost limits been observed?

Schedules

Can the delivery dates be met, including tooling?

Will additional operational or subsidiary costs arise? What design modifications might reduce cycle time and improve delivery?

Table 6.1: Checklist for embodying a product concept. Adapted from Pahl & Beitz (1996). c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

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6.3

6.3 Trends for the design process

Trends for the design process

6.3 Trends for the design process Market

Specifications

140

Concept Design

Detail Design

Manufacture

Sell

Iterations

In the last decades technology has brought profound changes in the way engineers design. Computational tools together with methods to increase the communication with all parts involved in the life cycle of a product have shortened significantly the amount of time needed to put a initial concept or idea into the market as a final product. From all the techniques that are or have been applied, concurrent engineering and computer aided tools have been most significant. 6.3.1. Concurrent Traditionally, design, manufacturing and marketing acengineering tivities have taken place sequentially rather than concurrently or simultaneously. The designer team would spend large amounts of time analyzing components and preparing detail drawings for a new product. After the design was considered satisfactory, the design team forwarded the information to the manufacturing team who would, once more, spend large amounts of time figuring out how to manufacture the product according to design specifications. After facilities and manufacturing processes were ready for production, the marketing team began to prepare a marketing strategy based on the product features. Although may seem logical at first, this linear scheme proved to be inadequate. In many occasions the design team ended up with a product that was difficult to manufacture and even difficult to sell. Great efforts were wasted (and sometimes still are) doing re-designs to improve manufacturing and to add features to improve the marketing of the product. To avoid the previous problems it is best to include members of the manufacturing and marketing divisions into the design team from the conceptual stages of the design process. The inclusion of the members will help to achieve a better decisions to avoid design features that are difficult to manufacture or no desirable from the marketing point of view. This approach, that may also include members from other areas like distribution and disposal, is called Concurrent Engineering.

Figure 6.11: Depiction of concurrent engineering in the design process. After Pugh (1991).

Life cycle means that all aspect of the product, such as design, development, production, distribution, use, disposal, and recycling, are considered simultaneously. The basic goals of concurrent engineering are to reduce changes in a product’s design and engineering to reduce the time and costs involved in taking the product from its design concept to its production and its introduction into the marketplace. Figure 6.11 shows a simple design process models that makes emphasis in the interaction between phases due to the use of concurrent engineering principles. 6.3.2. Design for As discussed above, while designing a product, manufacture and assembly several disciplines must be taken into account. One of those disciplines that is specially bounded to the design process is manufacturing. Many times new products have been designed only to find out that the technology needed for its manufacturing was not readily available. Hence, each component of the product must be designed not only to fulfill engineering requirements but also to be easily and cheaply manufactured. This emphasis is called design for manufacture, and it groups selection of materials, manufacturing methods, planning, assembly, testing and quality assurance. The design team must be capable of evaluating the impact that design changes have in manufacturing processes.

Kalpakjian & Schmidt (2001) define Concurrent Engineering as a systematic approach integrating the design and manufacture of products, with a view towards optimizing all elements involved in the life cycle of the product.

After the individual parts have been manufactured, they usually have to be assembled to make the final product. The importance of the assemblies cannot be understated, in many products assembly takes the largest time of the manufacturing process. Much can be done during the design phase to make the assembly as simple and fast as possible. Figure 6.12 shows some good and bad design practices regarding assemblies.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

141

6.3 Trends for the design process

111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 Bad

111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 Good

1111 00 00 000 111 000 111 00 11 00 11 000 111 000 111 0011 11 00 000 111 000 111 00 11 00 11 000 111 000 111 0011 11 00 000 111 000 111 0011 11 00 000 111 000 111 Bad

1111 00 00 00 11 00 0011 11 00 11 00 11 00 0011 11 00 11 0011 11 00 Good

6.3 Trends for the design process Material extraction

142

Material Processing

Manufacturing

Use

Recycle

Remanufacture

Reuse

Waste managment

Bad

Good

Figure 6.12: Design considerations for assembly. Figure 6.13: Stages of a product life cycle. After Otto & Wood (2001).

6.3.3. Design for the environment

During the centuries the impact that humans have in the environment has grown steadily as both, populations and its needs, increase. In the last decades, the awareness about the consequences of extracting resources and dumping waste without control has modified the way engineers design. If populations is to keep enjoying the advantages of technological advances and higher standards of living for the centuries to come, products must have little or no impact in the environment. Design for environment (DFE) is a product design approach for reducing the impact of products on the environment. Most of the times, the impact of products into the environment is thought of in terms of their disposal. Nevertheless, products have an impact during all of its life cycle from the extraction of the materials it is made from up to their disposal (see figure 6.13). Products can have adverse impact on the environment during their manufacture through the use of polluting processes, the use of high amounts of raw materials, or the need of high quantities of energy. They can also have different levels of impact on their disposal due to large half-lives or the need of large amounts of energy for their destruction. As shown in figure 6.13 there are many opportunities for recycling, remanufacturing and reuse to reduce environmental impact. Unfortunately, products that are designed without this vision in mind are difficult to remanufacture, reuse or even recycle. Designers must use all their knowledge and creativity to create products that are environmentally friendly products throughout their manufacture, packaging, transportation, use and disposal. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

6.3.4. CAD, CAM, CAE Nowadays, computers are an integral part of the and CIM conceptual, refinement, evaluation and production phases of the design process either as engineering or management tools. The use of computers has greatly simplified the representation, study and construction of analytical models through Computer Aided Design (CAD), Computer Aided Engineering (CAE) and Computer Aided Manufacturing (CAM). These tools use computer software to assist in the creation and revision of engineering drawings and models (CAD), manufacturing (CAM), and analysis (CAE) of new products. The use of CAD/CAM/CAE tools avoids the need of making costly illustrations, models and prototypes, shortening the time needed to bring a new product from concept to production. Although these tools may be applied in different parts of the design process, they are better suited for certain parts of the process (see figure 6.14). Regarding Computer Integrated Manufacturing (CIM), Egan and Greene (1989) state that the appearance of CIM is based on the recognition that steps in the development of a manufactured product are interrelated and can be accomplished effectively and efficiently by using computers. CIM provides a mean to integrate all the steps in the manufacturing process taking into account processes, specifications, instructions and data that need to be controlled and organized.

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143

6.3 Trends for the design process

6.4

Definition of product need; marketing information

Conceptual design and evaluation; feasibility study

Design analysis; codes/standards review; physical and analytical models

Prototype production; testing and evaluation

Computer Aided Design (CAD) Computer Aided Engineering (CAE)

Production drawings; instruction manuals

Material specification; process and equipment selection; safety review

Pilot production

Production

6.4 Failure Mode and Effect Analysis

Computer Aided Manufacturing (CAM) Computer Aided Process Planning (CAPP)

Computer Integrated Manufacturing (CIM)

Inspection and quality assurance

Packaging; marketing and sales literature

Failure Mode and Effect Analysis

The notion of a reliable product comes from two different parts. First, there is the minimization of performance variation across different environments and user conditions. Second, is the assurance that the product will work as intended, without falling short of a given set of customer expectations. The first part is achieved through customer quality. The second part is achieved through the more fundamental engineering quality. With the latter, it is ensured that the product has adequate strength, reliability and failure prevention. Traditionally, reliability has been achieved through extensive testing at the end of the design process. A better idea is to design from the early design stages incorporating the concepts of quality and reliability. Historically, engineers have not been very good at designing with reliability and quality. In most occasions, engineers use a safety factor as a way of making up for all the possible failure modes that were not considered in the design. As the engineer had less idea of what could go wrong with the product, the larger the safety factor that the engineer would use. Unfortunately, as stated in the Mechanical Engineering magazine: A large safety factor does not necessarily translate into a reliable product. Instead, it often leads to an overdesigned product with reliability problems. Failure Mode and Effect Analysis (FMEA) is an analytical methodology used as means for analyzing potential reliability problems early in the product design process, where it is easier and cheaper to take corrective actions. FMEA is used to identify potential failure modes, determine their effect on the use of the product, and identify counter-actions to correct them. FMEA focuses on the entire product and not just in the different components and interfaces, although failure modes may be related to specific components or interfaces. 6.4.1. Types of FMEA

Product

Figure 6.14: The use of computer aided tools in the different steps of the design process (after Kalpakjian & Schmid, 2001). c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

144

There are several types of FMEAs, each one with its own focus and objectives. Independently of the task at hand, FMEA should always be used whenever failures would mean potential harm or injury to the user. The types of FMEA are: • System FMEA: focuses on global system functions c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

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6.4 Failure Mode and Effect Analysis

6.4 Failure Mode and Effect Analysis

• Design FMEA: focuses on components and subsystems

• Capture engineering/organization knowledge

• Process FMEA: focuses on manufacturing and assembly processes

• Emphasizes problem prevention

• Service FMEA: focuses on service functions

• Documents risk and actions taken to reduce risk

• Software FMEA: focuses on software functions

• Provide focus for improved testing and development

6.4.2. FMEA Usage

According to the Society of Automotive Engineers (2002), FMEA supports the product development process in reducing the risk of failure by: • aiding in the objective evaluation of design requirements and design alternatives • aiding in the initial design for manufacturing and assembly requirements

146

• Minimizes late changes and associated cost • Catalyst for teamwork and idea exchange between functions In order to effectively apply FMEA, the greatest challenge that the design team faces is to anticipate what might go wrong with a product. While anticipating every possible failure mode is almost always impossible, the development team should generate a detailed list of potential failures. Some questions that may help in this task are (Stamatis, 1995):

• increasing the probability that potential failure modes and their effects on system operation have been considered in the design/development process

1. What does the product do and what are its intended uses?

• providing additional information to aid in the planning of through and efficient design improvements and development testing

3. What raw materials and components are used to build the product?

• providing an open issue format for recommending and tracking risk reducing action

4. How, and under what conditions does the product interface with other products?

• providing future references to aid in analyzing field concerns, evaluating design changes, and developing advanced designs

5. What by-products are created by the product or by the use of the product?

Properly used, FMEA provides the engineer with several benefits that include (Crow, 2002):

2. How does the product perform its function?

6. How is the product used, maintained, repaired, and disposed of at the end of its useful life? 7. What are the manufacturing steps in the production of the product?

• Improve product/process reliability and quality

8. What energy sources are involved and how?

• Increase customer satisfaction

9. Who will use or be in the vicinity of the product, and what are the capabilities and limitations of these individuals?

• Early identification and elimination of potential product/process failure modes • Prioritize product/process deficiencies c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

The above questions should be aimed to gather information in order to address six basic questions: c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

147

6.4 Failure Mode and Effect Analysis

6.4 Failure Mode and Effect Analysis

148

1. What could fail or go wrong with each component of the product? 2. How or why can the part fail to meet its engineering specifications? 3. What circumstances could cause the failure? 4. To what extent might it fail? 5. What are the potential hazards produced by the failure? 6. What steps should be implemented to prevent the failure?

List of example failure modes Corrosion

Ingress

Delamination

Fracture

Vibrations

Erosion

Material Yield

Whirl

Thermal shock

Electrical short

Sagging

Thermal relaxation

Open Circuit

Cracking

Bonding failure

Buckling

Stall

Starved for lubrication

Resonance

Creep

Staining

Step 1: List each subassembly and component number, along with the basic functions or function chains of the component. The component numbers may be referenced from a product’s bill of materials. Likewise, the component functions should be consistent with the functional models and architecture developed for a product. Any functions listed for a component should concisely represent the design intent. Environmental and operational parameters, such as temperature, humidity, and pressure ranges, should be listed to clarify this intent.

Fatigue

Thermal expansion

Inefficient

Stripping

Unstable

Egress

Step 2: Identify and list the potential failures for each product component. Simple prototype models and brainstorming techniques can aid in identifying potential failure modes. Likewise, sketches, storyboards, free-body diagrams, force-flow diagrams, and process-flow diagrams can help in understanding the physics of a failure mode. Tables 6.1 and 6.2 should be used to check for typical problems with components and product systems. For any listed failure mode, the idea is that the failure could occur, but not that will necessarily occur for the product under consideration.

Wear

Loose fittings

Surge

Binding

Unbalanced

Overshooting

Enbrittlement

Ringing

Loosening

Loose

Scoring

Leaking

Radiation damage

6.4.3. Step by step Design The use of FMEA is straightforward consisting FMEA Analysis of a series of steps. Following the procedure suggested by Otto & Wood (2000), the 10 steps procedure is explained in what follows.

Step 3: List possible potential causes or mechanisms of the failure modes. Example causes include tolerance stack-up, assembly errors, poor maintenance, impact loading, overstressing, and so forth. These causes will provide insights into modeling of the failure mode. They will also indicate appropriate preventive measures that might be adopted.

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Deflections or deformations Oxidation

Fretting

Seizure

UV deterioration

Thermal fatigue

Burning

Acoustic noise

Sticking

Misalignment

Scratching and hardness Intermittent system operation

Table 6.2: Abbreviated list of example failure modes. After Otto & Wood (2000).

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6.4 Failure Mode and Effect Analysis

Step 4: List the potential effects of the failure, including impact on the environment, property, or hazards to human users. Example effects include noise, poor appearance, flying debris, unpleasant odor, erratic operation and so forth. Step 5: Rate the likelihood of occurrence (O) of the failure. The ratings should be on a scale of 1-10 as given by:

1 2/3

No effect Low (relatively few failures)

4/5/6 Moderate (occasional failures) 7/8

High (repeated failures)

9/10

Very high (failure is almost inevitable)

Step 6: Estimate the potential severity (S) of the failure and its effect. Again, a 1-10 scale should be used. The following meanings are associated with this scale:

1

No effect

2

Very minor (only noticed by discriminating customer)

3

Minor (affects very little of the system; notice by average customer)

4/5/6 Moderate (most customers are annoyed) 7/8 9/10

High (causes a loss of a primary function; customers are dissatisfied) Very high and hazardous (product becomes inoperative; customers are angered; the failure may result in unsafe operation and possible injury)

6.4 Failure Mode and Effect Analysis 1

Almost certain

2

High

3

Moderate

4/5/6 Moderate – most customers are annoyed 7/8

Low

9/10

Very remote to absolute uncertainty

Step 8: Calculate the Risk Priority Number (RPN). An RPN prioritizes the relative importance of each failure mode and effect on a scale of 1-1000. It can be calculated with the following relation: RPN = (S) × (O) × (D) A “1000” rating implies a certain failure that is hazardous and harmful and will occur, whereas a “1” rating is a failure that is highly unlikely and unimportant. Rating above “100” will occur, whereas ratings below “30” become reasonable for typical applications. It is important to notice that the RPN scale is nonlinear in risk. Step 9: Develop recommended actions for the failure modes, assign responsibilities to appropriate parties and team members, and set a schedule for implementing the actions. Corrective actions should be first developed for the highest ranked failure modes based on the RPN. Example actions include revised component or subassembly design, revised test plan or material specification, design of experiments and prototypes, etc. These actions should be specific. Step 10: Implement the corrective actions, update the S-O-D ratings, and recalculate the RPN for the updated design. The process and results of the FMEA should be documented, perhaps with the help of a template like the one shown in figure 6.15.

Step 7: List current or expected design controls/test for detecting (D) the failure before the product is released for production. A 1-10 scale is used to assess detection: c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

150

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

151

6.4 Failure Mode and Effect Analysis

6.4 Failure Mode and Effect Analysis

152

D RPN O S

Improved situation

Applied steps Suggested remedial measures O S

Current situation

Proposed test steps Failure Cause

Failure Consequence Failure Type

1. Cross, N. (1994) Engineering Design Methods, John Wiley & Sons. 2. Crow, K. (2002) Failure Modes and Effects Analysis (FMEA), DRM Associates, www.npd-solutions.com. 3. Cutherell, D. (1996) Product Architecture. Chap 16. in “The PDMA handbook of new product development”, edited by M. Rosenau, Jr. et al. New York: Wiley. 4. Kalpakjian, S. & Schmid, S. (2001) Manufacturing Engineering and Technology, fourth ed., Prentice-Hall. 5. Otto, K. & Wood, K. (2001) Product Design - Techniques in Reverse Engineering and New Product Development, Prentice-Hall. 6. Pahl, G. and Beitz W. (2001) Engineering Design - A systematic Approach. Second Ed. Springer. 7. Pugh, S. (1990) Total Design, Addison Wesley. 8. SAE (2002) Potential Failure Mode and Effects Analysis in Design (Design FMEA) and Potential Failure Mode and Effects Analysis in Manufacturing and Assembly Processes (Process FMEA) and Effects Analysis for Machinery (Machinery FMEA). SAE Standard J1739. 9. Stamatis, D. H. (1995) Failure Mode and Effect Analysis - FMEA from Theory to Execution. ASQ Quality Press. 10. Ullman, D. (2003) The Mechanical Design Process, Third Edition. McGrawHill. 11. Ulrich, K. (1995) “The role of product architecture in the manufacturing firm”. Research Policy, 24, 419-440. 12. Ulrich, K. & Eppinger, S. (2000) Product Design and Development. Irwin McGraw-Hill.

Failure Location

Name/Department/Supplier/Telephone

FAILURE MODE AND EFFECT ANALYSIS Design FMEA Process FMEA

D RPN

By (Name/Department/Telephone)

Component name

References

Figure 6.15: FMEA Template.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

CHAPTER

Part II Techniques for robustness, reliability and optimization

7

Design of Experiments using the Taguchi Method

7.1

Basics of Design of Experiments

When a person is faced with the task of solving a problem which answer is not know, many different alternatives or ideas for its solution may appear based on those factors that are regarded as important in the problem at hand. If no better ideas arise, trial and error procedures are generally employed until a satisfactory solution is found. Although trial and error may seem as a simple approach, as it is evident to any person that has tried to solve a problem in this way, the procedure has many drawbacks. First, trial and error generally lack of any structure and solution attempts are carried out changing the problem variable randomly. Second, it takes a long time to find a satisfactory solution, if one is ever found! Third, it is difficult to assess if the solution found is the optimal one. Fourth, and most importantly, trial and error procedures are prohibitively expensive to carry out as several failing attempts are usually needed to find a solution. From the above, it is obvious that trial and error procedures are not well suited for scientific purposes. Hence, a structured, reliable and efficient methodology to carry out tries or experiments is needed. The discipline of Design of Experiments, or DoE for short, is responsible for answering this question. c Copyright 2006 Dr. Jos´ e CarlosMiranda. Todos los derechos reservados.

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7.1 Basics of Design of Experiments

In order to carry a designed experiment it is recommended to follow a simple procedure: 1. Define the problem.

7.1 Basics of Design of Experiments

156

Experiment Factors 1 2

3

1

1

1

1

2

2

1

1

2. Design the experiment.

3

1

2

1

3. Setup the experiment.

4

2

2

1

4. Gather data.

5

1

1

2

5. Analyze data.

6

2

1

2

7

1

2

2

8

2

2

2

These steps will be analyzed next. 7.1.1. Define the problem In this step the problem to be solved is clearly understood. The variables, or factors, that affect its behavior are determined and the shape and value of an acceptable solution is determined. Although this step may be seen as obvious, it is the most critical part of a designed experiment. In many ocassions it is difficult not only to determine which factors are relevant and which not, but also what is the required output, or response, for the experiment. Whenever possible, attempts should be made to solve the problem analytically. Even when too many assumptions and/or simplifications were needed to solve it analytically, the formulas used can of great help providing, first, an insight into the nature of the problem and second, an idea of around where the solution should be found. This would help the scientist to determine factors and numerical values to be used in order to reduce the number of experiments to run. 7.1.2. Design the experiment

Once the problem has been fully understood, the next step it to design the experiment. Here, design the experiment involves the determination of what factors and values should be varied during the experiments and what response or responsed will be measured. It also involves the selection of how the experiment will be carried out: it will be done numerically using methods like Finite Elements or Boundary Elements, it will be done experimentally, using a full size test rig, a combination of both approaches, etc. Next, it is necessary to define how many runs will be carried out and how the inputs (factors) will be varied. One simple and effective albeit long approach, is to use a factorial design. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Table 7.1: 23 factorial design table.

In a factorial design the response to all possible combinations of factor levels is measured. The number of all possible combinations of k factors, each with n different levels to be tested, is nk . Consider for example a problem that has three different factors and it is desired to test the response of the system when each factor is varied between two different values. Since there are three different factors to take into account, k = 3. Furthermore, since each factor will take only two different values, n = 2. Hence, the number of experiments to run is 23 = 8. Table 7.1 shows all the required runs representing the low value of any factor by the number “1” and its higher value by the number “2”. The choice between two, three or more levels for each one of the factors is based on the type of response that is required from the experiment. If two levels are chosen, then making it a 2k factorial design, the response of the system will be forced to be linear. If three levels are chosen (a 3k experiment), then the response will be forced to be quadratic, and so on. It is responsability of the person designing the experiment to choose the right number of levels needed in order to mimic the nature of the response of the problem. If the response of the problem is thought to be of cubic nature, then four levels should be considered for each factor. Figure 7.1 shows graphically the different in response from a 2k and a 3k experiments. One of the main advantages of carrying out factorial designs, is that all possible c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

157

7.1 Basics of Design of Experiments quadratic response estimated with 3 factor levels (1−2−3)

Response variable

2

7.1 Basics of Design of Experiments

158

experiments which will be discussed in detail in the next section. The interested reader is encourged to consult the references by Montgomery (1997) or Berger and Maurer (2002) to review other methods.

3 b linear response estimated with 2 factor levels (a−b)

1 a

Factor Level

Figure 7.1: Difference in the response variable between a two-level factor and a three-level factor. interactions between factors —the effect of one factor depending on the value of some of the other factors— are taken into account. Their main disadvantage is the number of experiments that have to be done, consider for example, six factors with 3 levels each: 36 = 729 experiments to run! As the number of experiments needed for a factorial design increases rapidly as the number of factors increases, an alternative is needed in order to make complex experiments managable. Full nk factorial designs allow the estimation of all main effects —effects that depend on only one factor— as well as 2way interactions, 3-way interactions and so on until n-way interactions. Here interaction means interdependece between two or more main effects. One way to reduce the number of experiments needed is to give up the ability to estimate interactions, especially higher order, by confounding them with each other. Confounding two or more factors means that the effects of these factors on a response variable cannot be distinguished from one another. Factorial designs where not all runs are performed are called fractional factorial designs.

7.1.3. Setup the experiment

In this step the experiment is prepared either by means of existing equipment, a test rig or a numerical simulation. The experiment should be isolated as much as possible from all possible external influences. Also, care must be given to ensure that any factor can be modified without interfering with the others and that the response variable can be measured effectively. It is of great importance that the response variable is always something that can be quantified by just one number, as it will make the analysis more much easier. If by the nature of the problem, the response variable cannot be represented by just one number, it is strongly recommended to find an alternative way to quantify it. Consider for example that the neck of aluminum cans wrinkle during its manufacturing process as shown in figure 7.2. As it is an undesirable feature, a engineering team is assigned to determine its probable causes and to modify the process to reduce it. The team decides to design an experiment using the amount of wrinkling as the response variable. The question is now how to quantify the amount of wrinkling in a single number? Clearly, the more the final shape of the neck deviates from a perfect circle, the more the wrinkling. In this case the team decided to use the difference in the arc length between the final shape and a circle of the average radius along the neck. If there are no wrinkles, the difference would be zero. On the other hand, the more wrinkling, the greater the difference. As it shown in the example, there are times when the person or team designing

Wrinkled can

Confounding higher order interactions significantly reduces the number of experiments to carry out. Nevertheless, the question of which experiments to run has to be answered. This is not an easy question, as the selection must be specified in such a way that desired effects are not affected by interactions and undesired effects are confounded with each other. Many alternatives exists to answer this question. One alternative that has become increasingly popular for its simplicity and effectivness is the use of Taguchi’s Method for designed

Figure 7.2: Wrinkled neck in an aluminum can.

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c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Desired shape

159

7.1 Basics of Design of Experiments

Experimental

Factors A

B

C

D

1

1

1

1

1

y11

y12

y13

y1

2

1

2

2

2

y21

y22

y23

y2

3

1

3

3

3

y31

y32

y33

y3

4

2

1

2

3

y41

y42

y43

y4

5

2

2

3

1

y51

y52

y53

y5

6

2

3

1

2

y61

y62

y63

y6

7

3

1

3

2

y71

y72

y73

y7

8

3

2

1

3

y81

y82

y83

y8

9

3

3

2

1

y91

y92

y93

y9

Response

1

Response

Table 7.2: Layout of gathered data for an experiment with 4 factors (A, B and C) and 3 repetitions based on the L9 orthogonal array. the experiment must figure out how to measure and/or quantify the response variable. There is no magic formula and one form will have always advantages and disadvantages over others. It is important when desiging experiments to clearly understand the problem and the tradeoffs of decisions taken. 7.1.4. Gather data

Once the setup for the experiment is ready, the experimental runs are carried out, each one with the factor levels specified in the experimental design matrix. The output in every response variable of interest is registered on each case. If the experimental design requires repetitions, they are carried out until all experiments have been run and registered. After all experimentation is finished, all the information gathered can be conveniently arranged as shown in table 7.2. The final step is to analyze the data gathered from the experimental runs. The analysis typically answers the

following questions: • How the factor levels affect the response? • Is there any relevant interaction between factors? • How can the response be predicted from factor’s values? c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

160

3

Mean

Run

7.1.5. Analyze data

7.2 Orthogonal arrays and linear graphs 2

Figure 7.3: Linear graph for L4 array. • Which factor’s values will give the best response? In the following sections the methodology used to answer these and other questions will be discussed in detail.

7.2

Orthogonal arrays and linear graphs

Taguchi showed that if experimental runs are chosen appropriately, there is no need to run full factorial experiments. The basis for Taguchi’s method are the orthogonal arrays, which show what factors levels must be selected each time to do the fewest possible runs. The main idea is to concentrate only on those few runs that are vital for the analysis. Table 7.3 shows the most common experimental design orthogonal arrays. These arrays can be found at the end of the chapter. Orthogonal arrays are named under the convention Ln where the L comes from Latin square and the n represents the number of rows in the array. As usual, each row represent an experiment to carry out and each column represent the factors (or interactions) of interest. The simplest orthogonal array, the L4 array, is shown in table 7.4. This orthogonal is suitable to run a 3 two-level factors experiment. As it can be seen, it requires only 4 runs, compared to a full 23 factorial design that would require 8 runs. Only half the work! As experiments become more large and complex, the use of orthogonal arrays becomes more convenient. Consider for example a design to study the effects of 15 two-level factors. An orthogonal array suited for this study, the L16 requires 16 experiments whereas a full factorial design would require 32,768 experiments! A design to study the effects of 13 three-level factors using orthogonal arrays would require 27 runs using the L27 array; a full factorial design would require 1,594,323 experiments. An orthogonal array is a fractional factorial experimental matrix that is orthogonal and balanced. Here, the balance property has two meanings. First it c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

161

7.2 Orthogonal arrays and linear graphs Array

7.3 Investigating effects

Description

3

3 two-level factors

7

L4 (2 ) L8 (2 )

162 Experiment Factors 1 2

3

7 two-level factors

1

1

1

1

11

11 two-level factors

2

1

2

2

15

L16 (2 )

15 two-level factors

3

2

1

2

L31 (231 )

31 two-level factors

4

2

2

1

L9 (34 )

4 three-level factors

L18 (21 × 37 )

1 two-level and 7 three-level factors

L12 (2 )

Table 7.4: L4 (23 ) Orthogonal Array.

13 three-level factors

(or left blank as it will be discussed later) to the interaction AB.

L36 (211 × 312 )

11 two-level and 12 three-level factors

L36 (23 × 313 )

3 two-level and 13 three-level factors

L32 (21 × 49 )

1 two-level and 9 four-level factors

L16

5 four level factors

Linear graphs can also be of help choosing the right array for a given experimental setup. As shown in table 7.3, the number of standar orthogonal arrays is limited and as such, may seem very limited as not all experimental designs may fit exactly into one of the Ln arrays available. Although it is possible to create new arrays from the standard ones, in most ocassions there is no need to do so.

13

L27 (3 )

5

(4 )

L25 (56 )

6 five level factors

L50 (2 × 5 ) 1

11

1 two-level and 11 five-level factors

Table 7.3: Common orthogonal arrays

means that every column is balanced, that is, each level of the factor appears the same number of times. For example, in the L4 array any column will have 2 “1” levels and 2 “2” levels. The second meaning is that any two columns in the array are also balanced, having the same number of combinations of levels. The reader is encouraged to check both balance properties with any of the orthogonal arrays presented in section 7.6. Other important property of orthogonal arrays is that any two columns of an orthogonal array form a two-factor complete factorial design. Each orthogonal array has one or more linear graphs associated with it. The objective of the linear graph is to show in a friendly way the interaction between columns. Figure 7.1 shows the linear graph for the L4 array, where dots represent factors and lines connecting them represent interactions. In this sense, the linear graph for the L4 array shows that columns 1 and 2 should be assigned to factors, let’s say A and B, and that column 3 should be assigned c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Consider for example a experimental design with 4 two-level factors. In this case, the L4 is limited to 3 factors and the L8 is designed for seven two-level factors. In this case, the L8 array can be used. Looking at the linear graphs shown in figure 7.11, main factors can be assigned to columns 1, 2, 4 and 7. Once factors are assigned to these columns, the 8 experiments are carried out normally taking into account for each experimental run the levels of each of the 4 factors. It is important to notice that if a factor is assigned to a column reserved for interactions, its main effect cannot be estimated independently from the other factors. Nevertheless, If there is the certainty that there are no possible interactions between factors, then they can be assigned to any column independently if they are reserved for interactions or not.

7.3

Investigating effects

It has been said that the strategy behind design of experiments is to gather the most valuable data with the least amount of experimental runs. In the Taguchi methodology, the analysis of information gathered from the experiments can be carried out in four steps: c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

7.3.1. Estimation of effects

Taguchi recommends to start analyzing the experiment results using means and plots to keep simplicity. The first step of the analysis should be to calculate the average of the response variable for a given factor level. Consider for example the experiment shown in table 7.2. For this case, the mean response, or main effect, when factor A is at level 1 can be found as: A1 = (y1 + y 2 + y3 )/3

(7.1)

as the factor A takes the low level at experimental runs 1, 2 and 3. Similarly, the main effect when factor A is at level 2 can be found through the responses y 4 , y5 and y 6 as this factors takes the intermediate level at experimental runs 4, 5 and 6: A2 = (y4 + y 5 + y6 )/3

(7.2)

The main effect of any factor at any level can be found likewise. For example, the mean response when factor C is at level 3 is calculated as C 3 = (y3 + y5 + y 7 )/3

(7.3)

as the factor C takes the level 3 at runs 3, 5 and 7.

164

A1

A2

A3

No significant effect of factor A

Average response

Estimation of effects where the effect that factors have in the average response are studied. Factor influence where the amount of influence that each factor has on the response is quantified. Analysis of interactions where the amount of interaction between factors is investigated. Prediction where a mathematical model of the experiment is developed and the optimal values of factors for a given response are obtained.

7.3 Investigating effects Average response

7.3 Investigating effects

Average response

163

A1

A2

A3

Linear effect of factor A

A1

A2

A3

Non−linear effect of factor A

Figure 7.4: Effect of factor on response variable.

over the response. The second possibility is that the values for the average response forms a line with slope different from zero. In this case, factor A has a linear effect on the response. Finally, the third possibility is for the values to resemble a quadratic function, or if the factor has more than 3 levels, a non-linear function. In this case the effect of factor A on the response is non-linear. The different types of responses call for different criteria to select what factor level may be better. When the factor has no significant effect, the optimum level may be selected in terms of cost or convenience. When the effect is linear, the factor can be used to shift response towards a target. Finally, if the effect is non-linear, the value chosen may be around the flat part of the curve, so small changes in the value of the factor do not affect the overall response. The above criteria assumes that interactions between factors are insignificant, so changing the value of one factor has no effect on the others. When interactions are presumed to be present, it makes no sense to change the value of factors independently. In such a case, it is necessary to study how interactions affect the average response.

Once the mean response of the different factors at the different levels has been obtained, it is possible to determine the significance of each factor. To determine the significance of a factor it is only necessary to plot the main effects of each factor versus the values of its levels. Figure 7.4 shows the three possible types of effect of factor A on the average response. The first possibility is that the value for the average response remains almost constant along the different levels of factor A. In this case, factor A has no appreciable effect

7.3.2. Factor influence From the above discussion, it is clear that different responses represent different strategies towards obtaining a given target for the response variable. The question of how factors affect the response has been answered, but the question of how strong the effect is has been not. For example, it may be irrelevant that a factor has a

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

165

7.3 Investigating effects

non-linear effect over the response, if let say, that said factor affects the response in just 1%. Hence, it is important to see the percent influence of each factor. To show the procedure to obtain the percent influence of each factor consider the example results shown in table 7.5. Experiment Factors A

B

C

Response

1

1

1

1

6

2

1

2

2

8

3

2

1

2

12

4

2

2

1

10

7.3 Investigating effects

166

where Yi is the response for the i-th experiment. The fourth step is to obtain the factor sums of squares for each one of the factors using the formula NL  Fl2 − CF Sf = NF l l=1

where Sf is the sum of squares for the factor f , N L are the number of levels for the factor, Fl is the total effect of factor F for the level l and NF l is the number of experiments ran with factor F at level l. For the example at hand, SA = =

Table 7.5: L4 sample experiment. The first step is to calculate the total effects of factors. The total effect of a factor is obtained by adding the results containing the effects of the factor at each level. For the example being studied, A1 = 6 + 8 = 14 B1 = 6 + 12 = 18 C1 = 6 + 10 = 16

A2 = 12 + 10 = 22 B2 = 8 + 10 = 18 C2 = 8 + 12 = 20

(7.4) =

142 222 + − 324 2 2

(7.8)

B12 B2 + 2 − CF NB1 NB2 182 182 + − 324 2 2

(7.9)

= 162 + 162 − 324 = 0 SC = =

C12 C2 + 2 − CF NC1 NC2 162 202 + − 324 2 2

(7.10)

= 128 + 200 − 324 = 4

The third step is to compute the total sum of squares using the formula N 

A21 A2 + 2 − CF NA1 NA2

= 98 + 242 − 324 = 16 SB =

The second step is to compute the correction factor CF used for calculations of all sums of squares. It remains constant for all constants and is computed as T2 36 × 36 CF = = = 324 (7.5) N 4 where T is the sum of all results (6 + 8 + 12 + 10) and N is the total number of experiments.

ST =

(7.7)

Yi2 − CF

i=1

= (62 + 82 + 122 + 102 ) − CF = 344 − 324 = 20 c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(7.6)

Finally, the percent influence of the factor Pf , can be obtained as the ratio between the factor sums of squares and the total sum of squares Pf =

Sf × 100 ST

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(7.11)

PA

SA = ST =

16 × 100 20

(7.12)

= 80%

PB = =

=

(7.13)

(7.14)

= 20% As expected PA + PB + PC = 100%. The above results show that factor A has the most impact on the response and factor B has no impact whatsoever on the response. 7.3.3. Analysis of interactions

By design orthogonal arrays are unable to study all potential interactions between the different factors. Specially, third-way interactions are completely neglected. It is the responsability of the person designing the experiments to identify any interactions that may be of interest. This selection of interactions may be a consequence of prior experience or pure engineering judgment. It is best when interactions are identified before the experiment, nevertheless the knowledge about the problem itself is limited at this point in time and there is no certainty about what interactions, if any, are of interest. For this reason, Taguchi suggest to neglect all interactions and run the experiment c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

A2

A3

B3 B2 B1 A1

A2

A3

Synergistic interaction

B1 B2 B3 A1

A2

A3

Antisynergistic interaction

Figure 7.5: Possible types of interaction. normally. After the values of the response are available, a simple procedure can be followed to detect if there are any interactions of interest.

SC ST 4 × 100 20

B1

No interaction

= 0%

PC =

B2

A1

SB ST 0 × 100 20

B3

168 Average response

where for the example being studied are

7.3 Investigating effects Average response

7.3 Investigating effects

Average response

167

To detect the interactions, lets say between factors A and B, the average response of factor B at the different levels are plotted against the values of factor A. Depending on the experiment, one of three different cases may appear. These different cases are shown in figure 7.5. The first case is when there is no interaction between the two factors. In this case the lines plotted do not intersect and are parallel to each other. In the second case, a synergistic interaction appears. In this case there is some interaction as the lines are not parallel although they never intersect. Here the interaction at least gives some idea of its tendency. In the third case, an antisynergistic interaction appears, and there is no clear idea of how the interaction behaves. In this case, further study of the interaction is necessary. To explain the above procedure, consider that an experiment based on the L8 array shown in table 7.2, has been carried out. Furthermore, consider that the experiment has been done with 4 factors and that all interactions have been neglected. Once the mean response of all experimental runs y1 to y9 are available, the mean responses of two factors can be plotted against each other to check if there is any interaction between the two. Consider for example, that the interactions between factor A and factor C are to be explored. To plot the average response of factor C with respect to factor c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

169

7.3 Investigating effects

7.3 Investigating effects

170

Experiment A

B

AB

Response

1

1

1

y1

2

1

2

2

y2

3

2

1

2

y3

4

2

2

1

y4

C1

C2

C3

A1

y1

y2

y3

1

A2

y6

y4

y5

A3

y8

y9

y7

Table 7.6: Interactions AC.

Table 7.8: L4 orthogonal array with 2 factors and 1 interaction. D1

D2

D3

C1

y1

y6

y8

C2

y9

y2

y4

C3

y5

y7

y3

Table 7.7: Interactions CD. A, it is necessary to check the response of the different levels of factor C to the different levels of factor A. From table 7.2, the average response of factor C1 when factor A is at level 1 is y1 . Similarly, the average response of factor C1 when factor A is at level 2 is y6 . Finally, the average response of factor C1 when factor A is at level 3 is y8 . The procedure is repeated for the average response of factors C2 and C3 . The results of the interaction can be arranged into a matrix as shown in table 7.6. Then the average responses are plotted against each other to obtain a plot similar to the ones shown in figure 7.5. The same steps are followed to check for any possible two-way interaction. The reader is encouraged to review the interaction matrix between factors C and D. As mentioned before, in some cases it is necessary to study a given interaction in more detail. To explain how interactions are studied following Taguchi’s approach, consider now a very simple experiment consisting of two factors, namely A and B, each one with two levels each. Furthermore, consider that the interaction between the two factors, AB is of interest. For this case, the L4 orthogonal array seems adequate as from the linear graph shown in figure 7.3 column 3 may be used to estimate interactions. Hence, the experiment can be carried out using table 7.8.

As the interaction does not play a part in the setup and run of the experiment, some people recommend to leave all interaction columns blank to avoid any possible confusions. To estimate the effects of the interaction AB, the same procedure used to estimate main effects can be used. For example, the main effect when AB is at level 1 can be obtained as AB 1 = (y1 + y 4 )/2

(7.15)

Similarly, the average response when AB is at level 2 can be found as AB 2 = (y2 + y 3 )/2

(7.16)

The main effects of AB can be plotted against its levels in a similar manner as the main effects of factors are plotted. If the plot is a horizontal line, then the effect of the interaction between A and B is not significant. On the other hand, if the plot resembles a linear or non-linear effect, then the optimum values of A and B have to be determined based on the combined effect of both. 7.3.4. Prediction In most occasions the objective of a design of experiments is to estimate optimum values for the factors given a desired level of the response variable. For that matter, it is necessary to obtain a matematical model of the experiment. Typically, this is done through a multiple linear regression by least squares.

For this problem, the experiment would be carried out normally, varying the levels of factors A and B and registering the values for the response variable.

Consider that you have an experiment with four factors, namely A, B, C and D, that is ran n times. A matematical model of the experiment could be described by: (7.17) y = β0 + β1 A + β2 B + β3 C + β4 D + ε

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

171

7.3 Investigating effects

where βi are coefficients that must be found from the experimental data and ε is the error in the model.

7.3 Investigating effects The previous equations can be simplified and extended into nβˆ0

Since the previous equation should approximate all the observations done, the above model can be applied to any run i of the experiment, yi = β0 + β1 Ai + β2 Bi + β3 Ci + β4 Di + εi

(7.18)

where Ai is the value of the factor A at experiment i and the other terms are interpreted similarly. It is important to note that the coefficients are constants and do not change since only one equation should be used to model the experiment. In general, for k factors, the matematical model can be described by the equation yi = β0 + β1 Xi1 + β2 Xi2 + β3 Xi3 + · · · + βn Xik + εi k  βj Xij = β0 +

(7.19)

j=1

where Xi is the i-th factor. The least squares method consists of choosing the β’s in such a way that the sum of squares of the errors εi is minimized. The least squares functions L is described as L = =

n 

ε2i

(7.20)

i=1  n 

k 

i=1

j=1

Yi − β0 −

 βj Xij

The least squares function must be minimized with respect to β0 , β1 , . . . , βk . Therefore, the least squares estimations βˆ0 , βˆ1 , . . . , βˆk must satisfy,    n k   ∂L  = −2 βˆj Xij = 0 yi − βˆ0 − (7.21) ∂β0 βˆ0 ,βˆ1 ,...,βˆk j=1 j=1    n k   ∂L  ˆ ˆ = −2 βj Xij Xij = 0 y i − β0 − ∂βj βˆ0 ,βˆ1 ,...,βˆk j=1 j=1

j = 1, 2, ..., k

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

172

βˆ0

n 

+

Xi1 +

βˆ1 βˆ1

i=1

βˆ0

.. . n 

n  i=1 n 

Xi1 2 Xi1

+

βˆ2

+ βˆ2

i=1

Xik + βˆ1

i=1

n 

.. .

Xik Xi1 + βˆ2

i=1

n 

Xi2

+ ··· +

βˆk

n 

Xik

=

n 

i=1 n 

i=1 n 

i=1 n 

i=1

i=1

i=1

Xi1 Xi2 + · · · + βˆk

n 

.. . Xik Xi2 + · · · +

Xi1 Xik =

βˆk

i=1

.. . n 

2 Xik

i=1

=

yi

Xi1 yi

.. . n 

Xik yi

i=1

The above equations can be expressed in matrix form as ⎤⎡ ⎤ ⎡ n ⎤ ⎡ n n n     Xi1 Xi2 · · · Xik ⎥ ⎢βˆ0 ⎥ ⎢ yi ⎥ ⎢ n ⎥ ⎢ ⎥ ⎢ i=1 ⎥ ⎢ i=1 i=1 i=1 ⎥⎢ ⎥ ⎢ n ⎥ ⎢ n n n n     ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ 2 ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ X X X X · · · X X X y ˆ i1 i1 i2 i1 ik ⎥ ⎢β1 ⎥ i1 i ⎥ i1 ⎢ ⎢ ⎥ ⎢ ⎥ = ⎢ i=1 ⎥ (7.22) ⎢ i=1 i=1 i=1 i=1 ⎥⎢ . ⎥ ⎢ ⎥ ⎢ . . . . . . . ⎥⎢ . ⎥ ⎢ ⎥ ⎢ .. .. .. .. .. .. ⎥⎢ ⎥ ⎢ ⎥ ⎢ n n n n n ⎥ ⎢ ⎥ ⎢ ⎥ ⎢    .. ⎣ ⎦ ⎣ ⎦ ⎦ ⎣ 2 βˆk Xik Xik Xi1 Xik Xi2 . Xik Xik yi i=1

i=1

i=1

i=1

i=1

If the matrix multiplications are carried out, the previous equations are obtained. The careful reader will notice that the coefficient matrix is symmetric and that the elements on the main diagonal are the sums of squares of the factors and that the elements outside the diagonal are the sums of the crossproducts of factors. The above matrix system can be solved to obtain the values of the βˆk coefficients. Once these values are available, the regression model is given by yˆi = β0 +

k 

βj Xij

i = 1, 2, . . . , n

(7.23)

j=1

The difference between the observation yi and the adjusted value yˆi is called residual and is usually represented by the letter e ei = yi − yˆi c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(7.24)

173

7.4 Examples Factor A: Injection pressure (MPa)

Low

High

1.75

2.25

◦

B : Mold temperature ( C)

85

105

C : Cycle time (seconds)

25

35

Table 7.9: Factors and levels for the problem of strenght of bumpers.

The residuals are a very effective way to check the quality of the regression since the larger the residual, the poorer the quality of the model.

7.4 Examples

174

problem is as follows. For the injection pressure, factor A, A1 = (2.5 + 3.2)/2 = 2.85 A2 = (2.7 + 2.9)/2 = 2.80

(7.25)

For the mold temperature, factor B, B 1 = (2.5 + 2.7)/2 = 2.60 B 2 = (3.2 + 2.9)/2 = 3.05

(7.26)

For the time cycle, factor C, C 1 = (2.5 + 2.9)/2 = 2.7 C 2 = (3.2 + 2.7)/2 = 2.95

7.4

(7.27)

Examples

7.4.1. Strenght of bumpers

A manufacturer of automotive bumpers is trying a new material to increase the energy absorbed by the bumper during a crash. Although the manufacturing process for the new material behaves different from others known by the company, it is believed that from the different factors affecting the process, the three that are controllable and critical are: injection pressure (MPa), mold temperature (◦ C) and cycle time (seconds). As this is a first approach into the problem, it has been decided to neglect interactions. The response variable has been selected as the specific energy absorption of the bumper in kJ/kg. It has been decided to carry a 2 level experiment with the above factors. For that purpose, the L4 array seems adequate. In the current process the manufacturing process is set to a pressure of 2 MPa, a mold temperature of 95◦ C and a cycle time of 30 seconds. The selected high and low levels of each factor are described in table 7.9. With the above information the matrix for the experiment is shown in table 7.10. Note that the low and high values for the experiment have been substituted and the results of the experiment are included.

The plots of the main effects for each factor are shown in figure 7.6. From the plots is easy to see that the injection pressure has practically no effect over the specific energy absorbed by the bumper. On the other hand, the mold temperature has the larger effect followed by the cycle time. Both factors increase the energy absorbed as the their values increase. The next step is to obtain the percent influence of each one of the factors. Following the procedure shown in 7.3.2, the percent influence can be computed from the next steps: A1 = 2.5 + 3.2 = 5.7 B1 = 2.5 + 2.7 = 5.2 C1 = 2.5 + 2.9 = 5.4 Experimental run

A2 = 2.7 + 2.9 = 5.6 B2 = 3.2 + 2.9 = 6.1 C2 = 3.2 + 2.7 = 5.9

Injection Mold pressure temperature

(7.28)

Cycle time

Specific energy absorbed

1

1.75

85

25

2.5

2

1.75

105

35

3.2

3

2.25

85

35

2.7

4

2.25

105

25

2.9

Once the results of the experiments are available, it is possible to start with the analysis of the gathered data. The estimation of the main effects for this

Table 7.10: Experimental matrix for the problem of strenght of bumpers.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

7.4 Examples CF =

11.3 × 11.3 T2 = = 31.9225 N 4

ST = (2.52 + 3.22 + 2.72 + 2.92) − 31.9225 = 0.2675

SA

5.72 5.62 = + − 31.9225 2 2

7.4 Examples

176

(7.29) (7.30)

(7.31)

3.4 Specific energy absorbed (kJ/kg)

175

3

2.6

2.2

= 0.0025 1.75

5.22 6.12 + − 31.9225 2 2

(7.32) 3.4

= 0.2025

SC =

5.42 5.92 + − 31.9225 2 2

(7.33)

= 0.0625 PA =

0.0025 × 100 0.2675

Specific energy absorbed (kJ/kg)

SB =

2.25 Injection Pressure (MPa)

3

2.6

2.2 85

(7.34)

105 Mold temperature (C)

= 0.94% PB

(7.35)

= 75.70% PC

0.0625 = × 100 0.2675

(7.36)

= 23.36%

Specific energy absorbed (kJ/kg)

3.4

0.2025 = × 100 0.2675

3

2.6

2.2 25

35 Cycle time (sec)

From the percent influence, the temperature mold clearly is the factor of major influence in the process. As mentioned, it is always important to check for interactions.

Figure 7.6: Plot of the main effects of the three factors for the problem of strength of bumpers.

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c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

177

7.4 Examples

7.4 Examples

178

80 60 50 40 30 20 10 0

Mold temperature 1 Mold temperature 2

3.4 Specific energy absorbed (kJ/kg)

Influence (%)

70

A

B

3

2.6

2.2

C

1.75

2.25 Injection Pressure (MPa)

Factor

The final step of the analysis is to obtain a prediction model for the problem using a least squares regression. As there are only four factors, the mathematical model for the experiment will have the form yˆ = βˆ0 + βˆ1 A + βˆ2 B + βˆ3 C

(7.37)

The least squares regression can be performed using either a purpose-specific software as MINITAB or a spreadsheet program such as Excel, OpenOffice or Gnumeric. The reader is encourage to read and learn to use any of this software to do regressions.

Specific energy absorbed (kJ/kg)

Figure 7.8 shows the interactions plots between injection pressure and mold temperature (interaction AB), mold temperature and cycle time (interaction BC), and injection pressure and cycle time (interaction AC). From the plots it can be observed that, judging from the almost parallel lines, there is practically no interaction between the mold temperature and the cycle time. On the other hand, there is interaction between the injection pressure and the mold temperature and there is a strong interaction between the injection pressure and the cycle time. Therefore, the problem should be studied further including interactions AB and AC.

Cycle time 1 Cycle time 2

3.4

3

2.6

2.2 85

105 Mold temperature (C)

Cycle time 1 Cycle time 2

3.4 Specific energy absorbed (kJ/kg)

Figure 7.7: Percent influence of injection pressure (factor A), mold temperature (factor B) and cycle time (factor C).

3

2.6

2.2 1.75

2.25 Injection Pressure (MPa)

Figure 7.8: Plot of the interactions AB, BC and AC for the problem of strength of bumpers. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

179 From the obtained: model for

7.4 Examples regression analysis, the following values for the β coefficients are βˆ0 = 0.1375, βˆ1 = −0.1, βˆ2 = 0.0225 and βˆ3 = 0.025. Hence, the the experiment is given by

7.4 Examples

180

Experiment A

B

AB

C

AC

AD

D

1

1

1

1

1

1

1

1

2

1

1

1

2

2

2

2

(7.38)

3

1

2

2

1

1

2

2

remembering that A is the injection pressure, B is the mold temperature and C is the cycle time.

4

1

2

2

2

2

1

1

5

2

1

2

1

2

1

2

The above model fits perfectly the data of the problem, that is, the residual of between the estimated and real responses is zero. This is a consequence of having just one observation and considering only a linear behavior, which the linear least squares regression is capable of estimate without error.

6

2

1

2

2

1

2

1

7

2

2

1

1

2

2

1

8

2

2

1

2

1

1

2

yˆ = 0.1375 − 0.1 A + 0.0225 B + 0.025 C

The above analysis shows that the best response is obtained when the value for the injection pressure is kept at its lowest possible value and the mold temperature and cycle time are kept at their maximum possible values. 7.4.2. Disk brake noise

Table 7.11: Experimental matrix for the example of disk brake ingredients including interactions. Factor A is filler type, factor B is fiber type, factor C is lube and factor D is the abrasive used. Data taken from Dunlap, Riehle and Longhouse (1999).

The data for this problem was taken from Dunlap, Riehle and Longhouse (1999). Disk brake noise continues to be a problem in the automotive industry. After some use, the brakes are prone to produce different types of sounds during breaking. One of the low frequency noises occurring during decelaration is groan. To better understand the interaction of raw materiales on sustained groan, Delphi Chassis Systems engineers carried out designed experiments. One of the objectives was to find the percent influence of the different factors on the response.

placed in column 7 so the interaction filler×abrasive is set in column 6. The experimental matrix is shown in table 7.11.

The experiment was carried out taking into account four factors: filler, fiber, lube and abrasive. The response variable was chosen as the average number of stops with caliper acceleration level greater than 1g. Although the original design included the full factorial experiment, consider here that only the interactions filler×fiber, filler×lube and filler×abrasive are of interest.

As explained before, the first step to quantify the percent influence is to calculate the main effect of all factors and interactions. For example, the main effects of factor A and interaction A × B can be found from the experimental matrix shown in table 7.11 as

To carry out the experiment is necessary to set every factor to the corresponding level according to the experimental matrix. The results in the response variable are shown in table 7.12. Once the information is available, then the influence of the different factors and interactions on the response can be quantified.

A1 = y 1 + y2 + y 3 + y4 = 8.5 A2 = y5 + y 6 + y 7 + y 8 = 31

Since the experiment has four factors and three interactions, and each factor has two levels, the L8 array seems appropriate. From the corresponding linear graphs, shown in figure 7.11, the option would be to set the four factors in columns 1, 2, 4 and 7. Since the filler is part of the three interactions of interest, it should be set in column 1. If the type of fiber is set in column 2, then the interaction filler×fiber, should be placed at column 3. Similarly, if lube is placed at column 4, then the interaction between lube and filler should be set in column 5. Finally, the last factor, the type of abrasive, must be

The main effects of all other factors and interactions can be calculated similarly and are: B1 = 2, B2 = 37.5, C1 = 20, C2 = 19.5, D1 = 28, D2 = 11.5, AC1 = 12.5, AC2 = 27, AD1 = 18.5, AD2 = 21.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

AB1 = y1 + y 2 + y7 + y 8 = 31 AB2 = y 3 + y4 + y 5 + y6 = 8.5

181

7.4 Examples

run

A

B

C

D

Filler Fiber Lube Abrasive

Response

1

1

1

1

1

0.5

2

1

1

2

2

0.5

3

1

2

1

2

0.25

4

1

2

2

1

7.25

5

2

1

1

2

0

6

2

1

2

1

1

7

2

2

1

1

19.25

8

2

2

2

2

10.75

Influence (%)

Experimental

B

C

182

D

AB

AC

AD

Factor

As before, the correction factor and the total sum of squares are obtained from T 39.5 × 39.5 = = 195.03 N 8

50 45 40 35 30 25 20 15 10 5 0 A

Table 7.12: Experimental matrix for the example of disk brake ingredients excluding interactions.

2

CF =

7.5 Comments about Taguchi’s approach

(7.39)

ST = (0.52 + 0.52 + 0.252 + 7.252 + 02 + 12 + 19.252 + 10.752) −195.03 (7.40) = 345.22

Figure 7.9: Percent influence of factors and interactions for the disk brake noise problem. Factor A is the filler, factor B is the fiber, factor C is the lube and factor D is the abrasive. influence of all factors and interactions can be computed. The final results are shown graphically in figure 7.9. From the results, the fiber type has the greatest influence on the response, followed by the filler and the interaction between fiber and filler. Also, the lube has no effect whatsoever as the filler×abrasive interaction.

7.5 The sums of squares for every factor and interaction are computed accordingly, for example,

Comments about Taguchi’s approach

The sums of squares for all the other factors and interactions are: SB = 157.53, SC = 0.03, SD = 34.03, SAC = 26.28, SAD = 0.78. From these values, the

The Taguchi approach to design of experiments, albeit very convenient, is not perfect. The simplicity of its approach comes for a price. In previous discussions it has brought to the attention of the reader that orthogonal arrays neglect third-way interactions and confound two-way interactions with main effects. This confounding can lead to inaccuracies as the significance of main effects depends on the interactions. Hence, conclusions drawn from orthogonal array requires that interactions effects are insignificant. In this regard, help can be obtained from the use of linear graphs. Linear graphs provide an easy way to identify how factors, and interactions between them, should be assigned to the different columns in the orthogonal array. If the assignment of factors and interactions to columns in the array is chosen carefully, then two-way, and even sometime three-way interactions, can be studied.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

SA

8.52 3.12 = + − 195.03 4 4

(7.41)

= 63.28 SAB =

312 8.52 + − 195.03 2 2

(7.42)

= 63.28

183

7.5 Comments about Taguchi’s approach

Taguchi suggests to drop the study of interactions, especially if their behavior is not known in advance, at the first stages of the experiment. The rationale behind this suggestion being that interactions are in most occasions difficult to quantify and the effort is better spent if more factors are included rather than interactions. If interactions are found to be significant, then more detailed experiments can be carried out. Taguchi also emphasizes the use of a confirmation run to verify that the optimal values chosen for the experiment are indeed optimal. Another advantage of the Taguchi approach is that the number of runs to performe for a given experiment is minimal. Althought this is the case in most occasions, pure fractional matrices are sometimes smaller with the advantage that main effects can be kept from confounding. Unfortunately, many engineers find difficult to design their own optimal experimental matrices or do not have the time to do it. With this in mind, it is much better a non-optimal experimental design done, that an optimal one that is not carried out!

7.6 Common orthogonal arrays

7.6

184

Common orthogonal arrays

Experiment Factors 1 2

3

1

1

1

1

2

1

2

2

3

2

1

2

4

2

2

1

Table 7.13: L4 (23 ) Orthogonal Array.

As with many engineering methods, judgment should be the first tool. Never trust any method without understanding how it works and what disadvantages it provides. Remember there is no such thing as a free lunch!

References 1. Berger, P.D. and Maurer, R.E. (2002) Experimental Design with applications in management, engineering and the Sciences. Duxbury Thomson-Learning.

3 1

2

Figure 7.10: Linear graph for L4 array.

2. Dunlap, K.B., Riehle, M.A. and Longhouse, R.E. (1999) An investigative overview of automotive disc brake noise. Brake Technology and ABS/TCS Systems (SP-1413). SAE International Congress and Exposition. Detroit, USA. 3. Montgomery, D.C. (2000) Design and Analysis of Experiments. 5th Edition. Wiley Text Books, New York. 4. Phadke, M.S. (1989) Quality engineering using robust design. Prentice-Hall. 5. Roy, R.K. (2001) Design of experiments using the Taguchi approach. 16 steps to product and process improvement. John Wiley & Sons.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

185

7.6 Common orthogonal arrays

7.6 Common orthogonal arrays

186

Experiment Factors 1

2 3

4

5 6

7

1

1

1

1

1

1

1

1

1

2 3

4

2

1

1

1

2

2

2

2

1

1

1

1

1

3

1

2

2

1

1

2

2

2

1

2

2

2

4

1

2

2

2

2

1

1

3

1

3

3

3

5

2

1

2

1

2

1

2

4

2

1

2

3

6

2

1

2

2

1

2

1

5

2

2

3

1

7

2

2

1

1

2

2

1

6

2

3

1

2

8

2

2

1

2

1

1

2

7

3

1

3

2

Table 7.14: L8 (27 ) Orthogonal Array.

8

3

2

1

3

9

3

3

2

1

Experiment Factors

Table 7.15: L9 (34 ) Orthogonal Array.

1 3 3

5

7

1

5

2 4

6 2

6 (1)

4

7 (2)

3,4 1

2

Figure 7.12: Linear graph for L9 array.

Figure 7.11: Linear graph for L8 array.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

187

7.6 Common orthogonal arrays

7.6 Common orthogonal arrays

188

Experiment Factors 1

2

3 4

5

6 7

8

9 10

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

Experiment Factors 10 11

11 12 13

14 15

1

2 3

4

5 6

7

8 9

1

1

1

1

1

1

1

1

1

1

1

1

2

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

1

1

1

1

1

2

2

2

2

2

2

3

1

1

1

2

2

2

2

1

1

1

1

2

2

2

2

1

1

1

2

2

2

2

2

2

2

2

1

1

1

1

3

1

1

2

2

2

1

1

1

2

2

2

4

4

1

2

1

2

2

1

2

2

1

1

2

5

1

2

2

1

1

2

2

1

1

2

2

1

1

2

2

5

1

2

2

1

2

2

1

2

1

2

1

6

1

2

2

1

1

2

2

2

2

1

1

2

2

1

1

6

1

2

2

2

1

2

2

1

2

1

1

7

1

2

2

2

2

1

1

1

1

2

2

2

2

1

1

7

2

1

2

2

1

1

2

2

1

2

1

8

1

2

2

2

2

1

1

2

2

1

1

1

1

2

2

8

2

1

2

1

2

2

2

1

1

1

2

9

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

2

1

2

1

2

1

2

2

1

2

1

2

1

2

1

9

2

1

1

2

2

2

1

2

2

1

1

10

10

2

2

2

1

1

1

1

2

2

1

2

11

2

1

2

2

1

2

1

1

2

1

2

2

1

2

1

11

2

2

1

2

1

2

1

1

1

2

2

12

2

1

2

2

1

2

1

2

1

2

1

1

2

1

2

12

2

2

1

1

2

1

2

1

2

2

1

13

2

2

1

1

2

2

1

1

2

2

1

1

2

2

1

14

2

2

1

1

2

2

1

2

1

1

2

2

1

1

2

15

2

2

1

2

1

1

2

1

2

2

1

2

1

1

2

16

2

2

1

2

1

1

2

2

1

1

2

1

2

2

1

Table 7.16: L12 (211 ) Orthogonal Array. The interaction between any two columns is partially confounded with the rest. Do not use if interactions must be estimated.

Table 7.17: L16 (215 ) Orthogonal Array.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

189

7.6 Common orthogonal arrays

14

8

14

9

9

15

13

1

13

12

11

10

11 6

5

7

5

3

10 8

12

4

1

7 3

(2)

11

13

15

4

8

7

4

11

9 6

5

8 (4)

5

7

15

12

2

8

6

13

14

10

12

10

9

11

1

2 3

4

5

1

1

1

1

1

1

2

1

2

2

2

2

3

1

3

3

3

3

4

1

4

4

4

4

5

2

1

2

3

4

6

2

2

1

4

3

7

2

3

4

1

2

8

2

4

3

2

1

9

3

1

3

4

2

10

3

2

4

3

1

11

3

3

1

2

4

12

3

4

2

1

3

13

4

1

4

2

3

14

4

2

3

1

4

15

4

3

2

4

1

16

4

4

1

3

2

Table 7.18: L16 (45 ) Orthogonal Array.

8

4

9 3

14

13

(3)

1

15

5

14 4

10

1

3

2

12

9

2

4

6

2

(!)

6

190

Experiment Factors

3 2

7

15

1

7.6 Common orthogonal arrays

10

11

12

13

6 1

2

3

7

15

5

14

(5)

(6)

Figure 7.13: Linear graph for L16 array.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

1

3,4,5

2

Figure 7.14: Linear graph for L16 array.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

191

7.6 Common orthogonal arrays

7.6 Common orthogonal arrays

192

Experiment Factors 1

2 3

4

5 6

7

8

1

1

1

1

1

1

1

1

1

2

1

1

2

2

2

2

2

2

3

1

1

3

3

3

3

3

3

4

1

2

1

1

2

2

3

3

5

1

2

2

2

3

3

1

1

6

1

2

3

3

1

1

2

2

7

1

3

1

2

1

3

2

3

8

1

3

2

3

2

1

3

1

9

1

3

3

1

3

2

1

2

10

2

1

1

3

3

2

2

1

11

2

1

2

1

1

3

3

2

12

2

1

3

2

2

1

1

3

13

2

2

1

2

3

1

3

2

14

2

2

2

3

1

2

1

3

15

2

2

3

1

2

3

2

1

16

2

3

1

3

2

3

1

2

17

2

3

2

1

3

1

2

3

18

2

3

3

2

1

2

3

1

2

Figure 7.16: Linear graph for L25 array.

1

2

6,7

8,11 (1)

2

3,4 9 10 12 13

3,4

1

Table 7.19: L18 (21 ×37 ) Orthogonal Array. Interaction between columns 1 and 2 is orthogonal to all columns and can be estimated without sacrificing any column. Interaction between any other pair of columns is confounded partially with the remaining columns.

3,4,5,6

6,7

5

9,10

8

1 12,13

5 (2)

Figure 7.17: Linear graph for L27 array. 1

2

3

4

5

6

7

8

Figure 7.15: Linear graph for L18 array.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

11

193

7.6 Common orthogonal arrays

7.6 Common orthogonal arrays

194

Experiment Factors Experiment Factors 1 2

3

4 5

6

1

1

1

1

1

1

1

2

1

2

2

2

2

2

3

1

3

3

3

3

3

4

1

4

4

4

4

4

5

1

5

5

5

5

5

6

2

1

2

3

4

5

7

2

2

3

4

5

1

8

2

3

4

5

1

2

9

2

4

5

1

2

3

10

2

5

1

2

3

4

11

3

1

3

5

2

4

12

3

2

4

1

3

5

13

3

3

5

2

4

1

14

3

4

1

3

5

2

15

3

5

2

4

1

3

16

4

1

4

2

5

3

17

4

2

5

3

1

4

18

4

3

1

4

2

5

19

4

4

2

5

3

1

20

4

5

3

1

4

2

21

5

1

5

4

3

2

22

5

2

1

5

4

3

23

5

3

2

1

5

4

24

5

4

3

2

1

5

25

5

5

4

3

2

1

6

Table 7.20: L25 (5 ) Orthogonal Array. To estimate the interaction between columns 1 and 2, all other columns must be empty.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

1

2 3

4

5 6

7

8 9

10 11

12 13

1

1

1

1

1

1

1

1

1

1

1

1

1

1

2

1

1

1

1

2

2

2

2

2

2

2

2

2

3

1

1

1

1

3

3

3

3

3

3

3

3

3

4

1

2

2

2

1

1

1

2

2

2

3

3

3

5

1

2

2

2

2

2

2

3

3

3

1

1

1

6

1

2

2

2

3

3

3

1

1

1

2

2

2

7

1

3

3

3

1

1

1

3

3

3

2

2

2

8

1

3

3

3

2

2

2

1

1

1

3

3

3

9

1

3

3

3

3

3

3

2

2

2

1

1

1

10

2

1

2

3

1

2

3

1

2

3

1

2

3

11

2

1

2

3

2

3

1

2

3

1

2

3

1

12

2

1

2

3

3

1

2

3

1

2

3

1

2

13

2

2

3

1

1

2

3

2

3

1

3

1

2

14

2

2

3

1

2

3

1

3

1

2

1

2

3

15

2

2

3

1

3

1

2

1

2

3

2

3

1

16

2

3

1

2

1

2

3

3

1

2

2

3

1

17

2

3

1

2

2

3

1

1

2

3

3

1

2

18

2

3

1

2

3

1

2

2

3

1

1

2

3

19

3

1

3

2

1

3

2

1

3

2

1

3

2

20

3

1

3

2

2

1

3

2

1

3

2

1

3

21

3

1

3

2

3

2

1

3

2

1

3

2

1

22

3

2

1

3

1

3

2

2

1

3

3

2

1

23

3

2

1

3

2

1

3

3

2

1

1

3

2

24

3

2

1

3

3

2

1

1

3

2

2

1

3

25

3

3

2

1

1

3

2

3

2

1

2

1

3

26

3

3

2

1

2

1

3

1

3

2

3

2

1

27

3

3

2

1

3

2

1

2

1

3

1

3

2

Table 7.21: L27 (313 ) Orthogonal Array. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados. 32

31

30

29

28

27

26

25

24

23

22

21

20

19

18

17

16

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

2

2

1

1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1

1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2

1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1

30

1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2

31

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

1

1

2

2

2

2

1

1

1

1

2

2

2

2

1

1

1

1

2

2

2

2

1

1

1

1

2

2

2

2

1

2

2

1

1

1

1

2

2

2

2

1

1

1

1

2

2

1

1

2

2

2

2

1

1

1

1

2

2

2

2

1

2

2

1

1

1

1

2

2

1

1

2

2

2

2

1

1

2

2

1

1

1

1

2

2

1

1

2

2

2

2

1

1

1

1

2

2

2

2

1

1

2

2

1

1

1

1

2

2

2

2

1

1

1

1

2

2

1

1

2

2

2

2

1

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

14 15 16

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

1

2

1

2

1

2

1

2

2

1

2

1

2

1

2

1

1

2

1

2

1

2

1

2

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

2

1

2

1

2

1

2

1

17 18 19

1

2

1

2

2

1

2

1

1

2

1

2

2

1

2

1

1

2

1

2

2

1

2

1

1

2

1

2

2

1

2

1

2

1

2

1

1

2

1

2

2

]

2

1

1

2

1

2

1

2

1

2

2

1

2

1

1

2

1

2

2

1

2

1

2

1

2

1

1

2

1

2

1

2

1

2

2

1

2

1

2

1

2

1

1

2

1

2

1

2

1

2

2

1

2

1

20 21 22

2

1

2

2

1

2

1

2

1

2

1

1

2

1

2

2

1

2

1

1

2

1

2

1

2

1

2

2

1

2

1

2

2

1

1

2

2

1

1

2

2

1

1

2

2

1

1

2

2

1

1

2

2

1

1

2

2

1

1

2

2

1

1

1

2

2

1

1

2

2

1

1

2

2

1

1

2

1

2

2

1

1

2

2

1

1

2

2

1

1

2

2

1

23 24 25

2

1

1

2

2

1

1

2

1

2

2

1

1

2

2

1

2

1

1

2

2

1

1

2

1

2

2

1

1

2

2

1

2

2

1

1

2

2

1

2

1

1

2

2

1

1

2

2

1

1

2

2

1

1

2

1

2

2

1

1

2

2

1

1

1

2

1

2

2

1

2

1

1

2

1

2

2

1

2

1

1

2

1

2

2

1

2

1

1

2

1

2

2

1

26 27 28

31 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2

29 30 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1

1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1

8 9 10

1 1 1 1 1 1 1

2 1 1 2 2 2 2 2 2 2 2

3 1 1 3 3 3 3 3 3 3 3

4 1 1 4 4 4 4 4 4 4 4

5 1 2 1 1 2 2 3 3 4 4

6 1 2 2 2 1 1 4 4 3 3

7 1 2 3 3 4 4 1 1 2 2

8 1 2 4 4 3 3 2 2 1 1

9 1 3 1 2 3 4 1 2 3 4

10 1 3 2 1 4 3 2 1 4 3

11 1 3 3 4 1 2 3 4 1 2

12 1 3 4 3 2 1 4 3 2 1

13 1 4 1 2 4 3 3 4 2 1

14 1 4 2 1 3 4 4 3 i 2

15 1 4 3 4 2 1 1 2 4 3

16 1 4 4 3 1 2 2 1 3 4

17 2 1 1 4 1 4 2 3 2 3

18 2 1 2 3 2 3 1 4 1 4

19 2 1 3 2 3 2 4 1 4 1

20 2 1 4 1 4 1 3 2 3 2

21 2 2 1 4 2 3 4 1 3 2

22 2 2 2 3 1 4 3 2 4 1

23 2 2 3 2 4 1 2 3 1 4

24 2 2 4 1 3 2 1 4 2 3

2 3 i 3 3 1 2 4 4 2

2 3 2 4 4 2 I 3 3 1

2 3 3 1 1 3 4 2 2 4

2 3 4 2 2 4 3 1 1 3

2 4 1 3 4 2 4 2 1 3

2 4 2 4 3 1 3 1 2 4

2 4 3 1 2 4 2 4 3 1

32 2 4 4 2 1 3 1 3 4 2

Table 7.22: L32 (231 ) Orthogonal Array.

2

2

1

1

2

2

1

1

1

1

2

2

1

1

2

2

1

1

2

2

1

1

2

2

2

2

1

1

2

2

1

1

6 7

2

1

1

2

2

1

1

2

2

2

2

1

1

2

2

1

1

1

1

2

2

1

1

2

2

2

2

1

1

2

2

1

1

12 13

5

1

1

1

1

2

2

1

1

2

2

1

1

2

2

1

1

2

2

2

2

1

1

2

2

1

1

2

2

1

1

2

2

1

1

3 4

1

2

2

1

1

2

2

1

1

2

2

1

1

2

2

1

1

2

2

1

1

2

2

1

1

2

2

1

1

2

2

1

1

10 11

2

1

1

2

2

2

1

1

1

1

1

1

1

1

2

2

2

2

1

1

1

1

2

2

2

2

2

2

2

2

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

1

1

1

1

4

29

2 3

8 9

1

28

1

5 6

27

1

7

26

1

25

1

Experiment Factors

195 7.6 Common orthogonal arrays 7.6 Common orthogonal arrays 196

Experiment Factors

Table 7.23: L32 (21 × 49 ) Orthogonal Array.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

197

7.6 Common orthogonal arrays

7.6 Common orthogonal arrays

198

Experiment Factors

Experiment Factors 10 11

12 13 14

15 16 17

18 19 20

21 22 23

1 2

3

4 5

6

7 8

9

1

1

1

1

1

1

1

1

1

1

10 11 12 1

1

1

13 14 15 1

1

1

16 1

2

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

1

2 3

4

5 6

7

8 9

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

3

1

1

1

1

3

3

3

3

3

3

3

3

3

3

3

3

2

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

4

1

2

2

1

1

1

1

1

2

2

2

2

3

3

3

3

3

1

1

1

1

1

1

1

1

1

1

1

3

3

3

3

3

3

3

3

3

3

3

3

5

1

2

2

2

1

2

2

2

2

3

3

3

3

1

1

1

4

1

1

1

1

1

2

2

2

2

2

2

1

1

1

1

2

2

2

2

3

3

3

3

6

1

2

2

1

3

3

3

3

1

1

1

1

2

2

2

2

5

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

3

3

3

3

1

1

1

1

6

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

1

1

1

1

2

2

2

2

7

2

1

2

1

1

1

2

3

1

2

3

3

1

2

2

3

7

1

1

2

2

2

1

1

1

2

2

2

1

1

2

3

1

2

3

3

1

2

2

3

8

2

1

2

1

2

2

3

1

2

3

1

1

2

3

3

1

2

1

2

1

3

3

1

2

3

1

2

2

3

1

1

2

8

1

1

2

2

2

1

1

1

2

2

2

2

2

3

1

2

3

1

1

2

3

3

1

9

9

1

1

2

2

2

1

1

1

2

2

2

3

3

1

2

3

1

2

2

3

1

1

2

10

2

2

1

1

1

1

3

2

1

3

2

3

2

1

3

2

10

1

2

1

2

2

1

2

2

1

1

2

1

1

3

2

1

3

2

3

2

1

3

2

11

2

2

1

1

2

2

1

3

2

1

3

1

3

2

1

3

11

1

2

1

2

2

1

2

2

1

1

2

2

2

1

3

2

1

3

1

3

2

1

3

12

2

2

1

1

3

3

2

1

3

2

1

2

1

3

2

1

12

1

2

1

2

2

1

2

2

1

1

2

3

3

2

1

3

2

1

2

1

3

2

1

13

1

1

1

2

1

2

3

1

3

2

1

3

3

2

1

2

13

1

2

2

1

2

2

1

2

1

2

1

1

2

3

1

3

2

1

3

3

2

1

2

14

1

2

2

1

2

2

1

2

1

2

1

2

3

1

2

1

3

2

1

1

3

2

3

14

1

1

1

2

2

3

1

2

1

3

2

1

1

3

2

3

1

15

1

1

1

2

3

1

2

3

2

1

3

2

2

1

3

1

1

2

2

2

1

2

3

2

1

1

3

2

3

3

2

1

15

1

2

2

1

2

2

1

2

1

2

1

3

1

2

3

2

1

3

2

2

1

3

16

1

2

2

2

1

2

2

1

2

1

1

1

2

3

2

1

1

3

2

3

3

2

1

16

17

1

2

2

2

1

2

2

1

2

1

1

2

3

1

3

2

2

1

3

1

1

3

2

17

1

2

2

2

2

3

1

3

2

2

1

3

1

1

3

2

18

1

2

2

2

1

2

2

1

2

1

1

3

1

2

1

3

3

2

1

2

2

1

3

18

1

2

2

2

3

1

2

1

3

3

2

1

2

2

1

3

19

2

1

2

2

1

1

2

2

1

2

1

1

2

1

3

3

3

1

2

2

1

2

3

19

2

1

2

2

1

2

1

3

3

3

1

2

2

1

2

3

20

2

1

2

2

1

1

2

2

1

2

1

2

3

2

1

1

1

2

3

3

2

3

1

21

2

1

2

2

1

1

2

2

1

2

1

3

1

3

2

2

2

3

1

1

3

1

2

20

2

1

2

2

2

3

2

1

1

1

2

3

3

2

3

1

22

2

1

2

1

2

2

2

1

1

1

2

1

2

2

3

3

1

2

1

1

3

3

2

21

2

1

2

2

3

1

3

2

2

2

3

1

1

3

1

2

23

2

1

2

1

2

2

2

1

1

1

2

2

3

3

1

1

2

3

2

2

1

1

3

22

2

2

1

2

1

2

2

3

3

1

2

1

1

3

3

2

24

2

1

2

1

2

2

2

1

1

1

2

3

1

1

2

2

3

1

3

3

2

2

1

23

2

2

1

2

2

3

3

1

1

2

3

2

2

1

1

3

25

2

1

1

2

2

2

1

2

2

1

1

1

3

2

1

2

3

3

1

3

1

2

2

24

2

2

1

2

3

1

1

2

2

3

1

3

3

2

2

1

26

2

1

1

2

2

2

1

2

2

1

1

2

1

3

2

3

1

1

2

1

2

3

3

25

1

1

1

3

1

3

2

1

2

3

3

1

3

1

2

2

27

2

1

1

2

2

2

1

2

2

1

1

3

2

1

3

1

2

2

3

2

3

1

1

26

1

1

1

3

2

1

3

2

3

1

1

2

1

2

3

3

28

2

2

2

1

1

1

1

2

2

1

2

1

3

2

2

2

1

1

3

2

3

1

3

29

2

2

2

1

1

1

1

2

2

1

2

2

1

3

3

3

2

2

1

3

1

2

1

27

1

1

1

3

3

2

1

3

1

2

2

3

2

3

1

1

1

2

2

3

1

3

2

2

2

1

1

3

2

3

1

3

30

2

2

2

1

1

1

1

2

2

1

2

3

2

1

1

1

3

3

2

1

2

3

2

28

31

2

2

1

2

1

2

1

1

1

2

2

1

3

3

3

2

3

2

2

1

2

1

1

29

1

2

2

3

2

1

3

3

3

2

2

1

3

1

2

1

32

2

2

1

2

1

2

1

1

1

2

2

2

1

1

1

3

1

3

3

2

3

2

2

30

1

2

2

3

3

2

1

1

1

3

3

2

1

2

3

2

33

2

2

1

2

1

2

1

1

1

2

2

3

2

2

2

1

2

1

1

3

1

3

3

31

2

1

2

3

1

3

3

3

2

3

2

2

1

2

1

1

34

2

2

1

1

2

1

2

1

2

2

1

1

3

1

2

3

2

3

1

2

2

3

1

32

2

1

2

3

2

1

1

1

3

1

3

3

2

3

2

2

35

2

2

1

1

2

1

2

1

2

2

1

2

1

2

3

1

3

1

2

3

3

1

2

36

2

2

1

1

2

1

2

1

2

2

1

3

2

3

1

2

1

2

3

1

1

2

3

33

2

1

2

3

3

2

2

2

1

2

1

1

3

1

3

3

34

2

2

1

3

1

3

1

2

3

2

3

1

2

2

2

3

35

2

2

1

3

2

1

2

3

1

3

1

2

3

3

1

2

36

2

2

1

3

3

2

3

1

2

1

2

3

1

1

2

3

Table 7.24: L36 (2 × 3 ) Orthogonal Array. 11

12

Table 7.25: L36 (23 × 313 ) Orthogonal Array. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

199

7.6 Common orthogonal arrays Experiment Factors 1

2 3

4

5 6

7

8 9

1

1

1

1

1

1

1

1

1

1

10 11 1

1

12 1

2

1

1

2

2

2

2

2

2

2

2

2

2

3

1

1

3

3

3

3

3

3

3

3

3

3

4

1

1

4

4

4

4

4

4

4

4

4

4

5

1

1

5

5

5

5

5

5

5

5

5

5

6

1

2

1

2

3

4

5

1

2

3

4

5

7

1

2

2

3

4

5

1

2

3

4

5

1

8

I

2

3

4

5

1

2

3

4

5

1

2

9

1

2

4

5

1

2

3

4

5

1

2

3

10

1

2

5

1

2

3

4

5

1

2

3

4

11

1

3

1

3

5

2

4

4

1

3

5

2

12

1

3

2

4

1

3

5

5

2

4

1

3

13

1

3

3

5

2

4

1

1

3

5

2

4

14

1

3

4

1

3

5

2

2

4

1

3

5

15

1

3

5

2

4

1

3

3

5

2

4

1

16

1

4

1

4

2

5

3

5

3

1

4

2

17

1

4

2

5

3

1

4

1

4

2

5

3

18

1

4

3

1

4

2

5

2

5

3

1

4

19

1

4

4

2

5

3

1

3

1

4

2

5

20

1

4

5

3

1

4

2

4

2

5

3

1

21

1

5

1

5

4

3

2

4

3

2

1

5

22

1

5

2

1

5

4

3

5

4

3

2

1

23

1

5

3

2

1

5

4

1

5

4

3

2

24

1

5

4

3

2

1

5

2

1

5

4

3

25

1

5

5

4

3

2

1

3

2

1

5

4

26

2

1

1

1

4

5

4

3

2

5

2

3

27

2

1

2

2

5

1

5

4

3

1

3

4

28

2

1

3

3

1

2

1

5

4

2

4

5

29

2

1

4

4

2

3

2

1

5

3

5

1

30

2

1

5

5

3

4

3

2

1

4

1

2

31

2

2

1

2

1

3

3

2

4

5

5

4

32

2

2

2

3

2

4

4

3

5

1

1

5

33

2

2

3

4

3

5

5

4

1

2

2

1

34

2

2

4

5

4

1

1

5

2

3

3

2

35

2

2

5

1

5

2

2

1

3

4

4

3

36

2

3

1

3

3

1

2

5

5

4

2

4

37

2

3

2

4

4

2

3

1

1

5

3

5

38

2

3

3

5

5

3

4

2

2

1

4

1

39

2

3

4

1

1

4

5

3

3

2

5

2

40

2

3

5

2

2

5

1

4

4

3

1

3

41

2

4

1

4

5

4

1

2

5

2

3

3

42

2

4

2

5

1

5

2

3

1

3

4

4

43

2

4

3

1

2

1

3

4

2

4

5

5

44

2

4

4

2

3

2

4

5

3

5

1

1

45

2

4

5

3

4

3

5

1

4

1

2

2

46

2

5

1

5

2

2

5

3

4

4

3

1

47

2

5

2

1

3

3

1

4

5

5

4

2

48

2

5

3

2

4

4

2

5

1

1

5

3

49

2

5

4

3

5

5

3

1

2

2

1

4

50

2

5

5

4

1

1

4

2

3

3

2

5

Table 7.26: L50 (2 × 5 ) Orthogonal Array. 1

11

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

CHAPTER

8

Robust Design

8.1

Quality through design

The main philosophy of quality engineering is that quality can be built into a product during its design phase rather than controlled during its manufacturing process. The fundamental stone behind this philosophy is Robust Design. Robust Design is an engineering methodology whose objective is to create high-quality, cost-effective products that perform well during its useful life independently of how and under which circumstances are used. These external circumstances that are outside the control of the design engineering are called noise. Robust design increases the quality of products minimizing the effect of noise on the performance of the product. The robust design methodology relies on two powerful tools: orthogonal arrays to carry out designed experiments, and signal-to-noise ratios to measure quality. Since robust design is a tool inteded to increase quality of products, it is important to discuss what quality means here. To give a clearer idea of how quality should be measure, consider the next case. At the end of the ’70s, consumers in the United States showed a preference for televisions sets made by Sony Japan over those ones made by Sony America in San Diego. The preference seemed strange at first since both plants worked with exactly the same designs and exactly the same tolerance. In 1979, the Asahi newspaper showed a study of this problem. In the study, the newspaper showed a distribution plot of the c Copyright 2006 Dr. Jos´ e CarlosMiranda. Todos los derechos reservados.

201

8.1 Quality through design

8.2

Sony USA

m−5

m

Color Density

m+5

Grade C

B

A

B

202

on meeting the target seems to be a better alternative to measure quality of products. In order to do so effectively, the concept of quality loss must be studied.

Sony Japan

D

8.2 Quality loss function

C

D

Figure 8.1: Distribution of color density in television sets. (From the Asahi, April 17, 1979).

level of color density of sets made by Sony Japan and Sony America. Color density was chosen as the objective function as it is commonly used to quantify the quality of TV sets. The plot, reproduced in figure 8.1, showed that the color density distribution of TV sets were very different. Sony Japan had an almost normal distribution around the target value m with approximately 0.3 percent of TV sets out of tolerance limits. On the other hand, Sony America had a uniform distribution around tolerance limits with no TV sets out of tolerance limits. So, if Sony America had no units outside tolerance limits, while Sony Japan had 0.3 percent of shipped outside them, how it was then possible for the consumers to prefer sets from Sony Japan? The response to this question lays on how quality is measured. Depending on the deviation from the target m, TV sets are ranked as grade A, if color densitiy is within m ± 1, and ranked B, C and D as color density deviated progressively from target m. From figure 8.1 it is clear that Sony Japan shipped much more grade A TV sets and far fewer grade C sets than Sony America. Hence, in average, the quality of TV sets from Sony Japan was better, and therefore, customers had a preference for sets from Sony Japan.

Quality loss function

From the previous example it is clear that when the objective characteristic from a given product deviates from a target value m, it will loss some performance. Consider for example a connecting rod which objective function is to have a nominal diameter. It is clear that if the diameter is too small compared to its target, the rod will be loose and may not work. On the other hand, if the diameter is too large, then the rod will not fit. Also, as the diameter of the rod deviates from its target value, it may require a larger effort to make it work when assembled. Hence, everytime a the objective characteristic y of a product deviates from its target value, some financial loss L(y) will occur. As engineering specifications are always written as m ± Δ0 , one may be tempted to feel that while the objective characteristic is between the range (m − Δ0 ) and (m + Δ0 ), there is no financial loss as the product is equally good for the customer, independently on the deviation from the target m. This representation of quality loss can be represented as a step function as shown in figure 8.2, 0 if |y − m| ≤ Δ0 (8.1) L(y) = A0 otherwise where A0 is the cost of replacement or repair. The step function is unable to quantify the quality loss when an objective characteristic of a product deviates from its target value but is still within tolerance limits. As shown in the example of Sony TVs, this model is faulty and should be avoided. Consider now the following approach suggested by Taguchi. When y meets the target value m, the loss L(y) will be at minimum. Under ideal conditions, the financial loss can be assumed to be zero under this circumstance

The above example shows the difference between being focused on meeting tolerances rather than being focused on meeting the target. Cleary, being focused

L(m) = 0

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(8.2)

203

8.2 Quality loss function

A0

m −Δ 0

m

Now, it is only necessary to define the constant k. Taguchi suggests to compute it in terms of the replacement or repair cost, A0 , and the magnitude of the deviation from the target value Δ0 , as A0 k= (8.7) Δ0

y m + Δ0 Quality Loss ($)

L(y)

Consider once more the example of the transmission shaft where the target length is 300mm. If the shaft is longer or shorter by 2mm, then it has to be reworked for a cost of $12. For this case, the quality loss function is given by 12 (8.8) L(y) = (y − 300)2 = $ 6(y − 300)2 2

A0

m −Δ 0

m

m + Δ0

204

The above view of quality loss is shown in figure 8.2. As the objective function deviates from its target, the quality loss increases, independently of if the objective characteristic is within or out of tolerances. Of course, if the objective characteristic is outside tolerance limits, the product should be considered defective.

Quality Loss ($)

L(y)

8.2 Quality loss function

y

Figure 8.2: Models for quality loss functions: step function (top) and quadratic function (bottom). As the financial loss will be at a minimum, then the value of the first derivative of the function should also be zero at this point L (m) = 0

(8.3)

If the loss function is expanded through a Taylor series expansion around the target value m, the following equation is obtained L (m) L (m) (8.4) (y − m) + (y − m)2 + · · · 1! 2! or taking into account that L(m) = 0, L (m) = 0 and neglecting high-order terms L (m) (8.5) L(y) = (y − m)2 2! L(y) = L(m) +

Thus, the loss function can be written as a squared term multiplied by a constant k (8.6) L(y) = k(y − m)2 c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

From the above equation it is clear that even when the shaft has a deviation of ±1, and is within tolerance, a financial loss occurs as L(301) = $ 6. 8.2.1. Other types of loss functions

The quadratic loss function discussed above is called nominal–the–best and is only useful when the quality characteristic y has a finite target value and the quality loss incurred when y deviates from the target m is the same on either side of the target, i.e. the function is symmetrical. In many ocassions the quality loss function must accommodate situations different from the above. In that case, other type of functions different from nominal–the–best are needed. Other commonly used loss functions are explained next and shown in figure 8.3. Smaller–the–better This type of characteristic is useful for those situations when the characteristic function can never take negative values, its ideal value is equal to zero and as its value increases its performance becomes progressively worst. An example of this behavior is the pollution from an automobile or the leak of fluid in a reservoir tank. This type of function is obtained from equation (8.6) substituting m = 0, L(y) = ky 2 c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(8.9)

205

8.2 Quality loss function L(y)

A0

m −Δ 0

8.3 Noise factors

L(y)

cases, two different constants k can be specified and the quality loss function could be approxmated as k1 (y − m)2 , y > m L(y) = (8.12) k2 (y − m)2 , y ≤ m

A0

m

m + Δ0

y

y 0

(a) Nominal − the − best

206

Δ0 (b) Smaller − the − better

L(y)

8.3

L(y)

Noise factors

As mentioned before, the goal of robust design is to create product that perform well under all circumstances, despite external and internal uncontrollable influences, that is, despite noise.

A0

Noise factors can generally be classified in three groups: A0

y Δ0 (c) Larger − the − better

m −Δ 0

m

m + Δ0

y

(d) Asymmetric

Figure 8.3: Different quality loss functions. Larger-the-better Some characteristics, like the durability of a mechanical component, do not take negative values, their worst value is zero and as it becomes larger, its performance is progressively better. Its ideal value is zero and reach the point of zero loss at infinity. This behavior is inversely proportional to the smaller–the–better type. Hence, the quality loss function for this type of problems is obtained substituting y by 1/y in function 8.9

 1 (8.10) L(y) = k y2 In this type of problems, Δ0 is taken as the limit below which the product will fail and A0 is the repair or replacement cost, then the constant k, is determined by (8.11) k = A0 Δ20 Asymmetric loss functions Sometimes the deviation of a quality characteristic on one direction makes more harm than in the other. In those c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Outer noise or external noise, groups all environmental factors such as humidity, temperature, pressure, dust, magnetism, vibration, supply voltage, electromagnetic interference and human error during operation of the product. Inner noise or deterioration, refers to changes within the product during its useful life due to wear, tear, etc. Product noise or unit-to-unit variation refers to the inevitable variation that exists between one unit and the next when manufactured. In table 8.1, the strategies to deal with the three different class of noise at different levels of the organization is shown. The table indicates, for example, that the effect of all types of noises can be reduced during the system design but only manufacturing imperfection can be reduced by the manufacturing department. It is important to notice that not only components or products must be robust to withstand noise. Manufacturing processes should also be robustly designed to make them resistant to external noise, inner noise (tool wear) and product noise.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Table 8.1: Sources of noise and the corresponding strategies for managing the noise in each department. (After Taguchi, 1993).

R: The effects of this type of noise can be reduced in this department using the strategy indicated. N: Although the effects of this type of noise can be reduced, it is not recommended to do so at this stage. X: The effects of this type of noise cannot be reduced in this department using the strategy indicated.

X 1. After-sale service

X

X

R X X 2. Product management

Modify:

X 1. Process management

Modify: Manufacturing

Marketing

R X

R X X 3. Tolerance design On-line departments

R

R X

X X

X 1. System design

2. Parameter design

R R N 1. Tolerance design

Modify: Production Technology

R

R 1. System design

R

R Technology departments

Development and design Modify:

R

(deteriorative effects) (environmental effects) Strategy Department

2. Parameter design

imperfections

Inner Outer

Noises

R

8.3 Noise factors Manufacturing

207

8.4 Signal–to–noise ratios

8.4

208

Signal–to–noise ratios

In the field of communication engineering, a common performance ratio is the signal-to-noise ratio. A ignal–to–noise ratio, or S/N ratio, combines a performance characteristic with its sensitivity to noise factors to measure the quality of a design. Because of this characteristic, signal–to–noise rations are ideal to measure quality loss when noise factors are to be taken into account. One of the main contributions of Taguchi was to extend the use of signal– to–noise ratios to non-communication engineering. As it will be shown next, all signal–to–noise ratios involve the use of logarithms, as logarithms allow the transformation of a multiplicative relationship into an additive one which smooths out non-linearities and interactions. As with quality loss functions, three diferent S/N ratios have been developed to use depending on its intended behavior: nominal–the–best, smaller–the– better and larger–the–better. These three S/N ratios are computed from the following formulas:

Nominal–the–best S/N = 10 log

μ2 σ2

 (8.13)

  where μ = (yi /n), σ 2 = 1/(n − 1) (yi − μ)2 and n is the number of observations. Smaller–the–better

  1 yi2 S/N = −10 log n

(8.14)

  1 1 n yi2

(8.15)

Larger–the–better S/N = −10 log

The key characteristic of the S/N ratios is that maximizing them will in all cases minimize the quality loss function L(y).

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

209

8.5 Parameter design Factors Experiment A

Replicates

8.5 Parameter design

Input

S/N

B

AB

···

1

2

···

n

η

1

1

1

1

···

y11

y12

···

y1n

η1

2

1

2

2

···

y21

y22

···

y2n

η2

3

1

3

3

···

y31

y32

···

y3n

η3

4 .. .

2

1

1

··· .. .

y41

y42

··· .. .

y4n .. .

η4 .. .

N

4

4

1

···

yN 1

yN 2

···

yN n

ηN

210

Design Factors (controllable)

Product/System

8.5

Parameter design

Robust design involves ensuring that a given design will perform as expected regardless of the noise factors that may affect it. One answer to this problem may be to isolate the product from noise. This option may be either expensive or even impossible as isolating some products like cars or other types of vehicles. Other option may be to compensate for the noise, for example, including in the product control systems. Although this option is more real, is still expensive and not applicable to all cases. One third option, the one that Taguchi suggest, is to minimize the effect of noise in the product. The question now remains how to include those noise factors in the design. Taguchi suggested that noise factors can be managed similarly to design factors, including their effect in the experimental setup through an orthogonal array. This technique, called Taguchi robust parameter design allows to explore the effect of noises in the design in a efficient and simple way. A graphical c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(uncontrollable)

Quality Characteristics

Table 8.2: An experimental setup using S/N ratios. Consider for example the experimental setup shown in table 8.2. In this example the experiments have been replicated and several different responses yN n have been recorded. Instead of using the average response as a measure of the output of the experiments, a suitable signal–to–noise ratio may be selected. Once more, remembering that maximizing the S/N ratio will minimize the corresponding quality loss function.

Noise Factors

Smaller−the−better Larger−the−better Nominal−the−best

Output Figure 8.4: The Taguchi robust parameter design.

description of the Taguchi method is shown in figure 8.4. The first step in the robust parameter design technique is to select the factors and/or interactions to be included in the designed experiment and to select an appropriate orthogonal array in exactly the same way as it has been done until now. This orthogonal array of design factors receives the name of inner or design array. Next, the technique suggests to select those noise factors that are believed to affect the design the most and assign numeric values in a range as it is done with design (controllable) factors. Depending on the number of noise factors and the levels of interest, a suitable orthogonal array has to be selected as if the experiment consisted only of noise factors. This orthogonal array receives the name of outer or noise array. Once the inner and outer orthogonal arrays have been selected, both arrays are coupled in a single designed experiment as shown in table 8.4. Each replica of the experiment will involve including certain levels of each one of the noise c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

211

8.5 Parameter design

8.5 Parameter design

212

Outer (Noise) Array

Outer L4 Array

Experiment

Inner (factors) Array

Experiment

1

2

···

M

Noise

1

2

3

4

Noise

1

1

···

2

1

1

2

2

E

1

2

···

2

P .. .

1

2

1

2

F

1

2

···

1

R

1

2

2

1

G

Inner L9 Array Control Factors

Control Factors Experiment A

B

AB

···

S/N

Experiment A

B

C

D

S/N

1

1

1

1

···

y11

y12

···

y1m

η1

1

1

1

1

1

y11

y12

y13

y14

η1

2

1

2

2

···

y21

y22

···

y2m

η2

2

1

2

2

2

y21

y22

y23

y24

η2

1

3

3

3

y31

y32

y33

y34

η3

3

1

3

3

···

y31

y32

···

y3m

η3

3

4 .. .

2

1

1

··· .. .

y41

y42

··· .. .

y4m .. .

η4 .. .

4

2

1

2

3

y41

y42

y43

y44

η4

5

2

2

3

1

y51

y52

y53

y54

η5

N

4

···

yN 1

···

yN m

ηN

6

2

3

1

2

y61

y62

y63

y64

η6

7

3

1

3

2

y71

y72

y73

y74

η7

8

3

2

1

3

y81

y82

y83

y84

η8

9

3

3

2

1

y91

y92

y93

y94

η9

4

1

yN 2

Table 8.3: Experimental setup using inner and outer arrays.

factors selected as specified by the inner array. Overall, if there are N design factors and M noise factors, then M × N experiments must be run.

Table 8.4: Experimental setup using a L9 inner array and a L4 outer array.

Consider for example a designed experiment involving 4 independent factors, namely A, B, C and D, each one with three levels. Furthermore, consider that the design is affected by mainly by three noise factors, E, F and G, and that 2 levels are to be considered on each.

factors. One advantage of measuring the response through S/N ratios is that since the S/N ratio is computed for each experiment of the inner array, a higher S/N ratio will indicate less sensitivity to the effects of noise factors.

From the setup of the experiment, as each experiment of the inner array is carried out multiple times, each one with a different combination of factors, the variation among the responses yn1 to yn4 must be caused by the noise

Once the experiments have been carried out, the selection of optimal values for the design factors can be carried out as done before but remembering that if signal–to–noise ratios are used, then optimal values will be found maximizing the response independently of the S/N formula used. The selection of factor levels that maximize the response can be done as usual from main-effects plots. Nevertheless, it is of extreme importance to select the S/N ratio formula that fits the response of the problem at hand, either nominal–the–best, smaller– the–better or larger–the–better.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

As there are 4 design factors, each one with three levels, a L9 array seems appropriate. On the other hand, for the three two-level noise factors, an L4 array is sufficient. The resulting setup for the experiment with these two arrays is shown in table 8.4.

213

8.5 Parameter design

(a) m

(b) T

Although the above methodology provides some insight into the optimization problem, it requires the finding of a factor that has little or no effect on the S/N but has at the same time a significant effect on the mean. Sometime the problem at hand has more than one response variable of interest. For those cases the above strategy cannot be applied verbatim. Phadke et al. (1983) have suggested the following two steps strategy:

Shift mean to target

m

214

4. For factor that have no or little effect on S/N and the mean function, choose any level that is more convenient from the point of view of other considerations, such as other quality characteristics and cost.

Reduce Variation

m

8.6 Examples

m=T

1. Separately determine the control factors and their optimum levels corresponding to each response variable. if there is a conflict between the optimum levels suggested by the different repsonse variables, use engineering judgment to solve the conflict.

Figure 8.5: Steps to reduce the quality loss. As of now, the discussion about robust design have been centered in minimizing the effects of noise in the design. For that purpose, the S/N ratio has been selected as a convenient choice to minimize the quality loss. From the concept of quality loss, this reduction means, in fact, to reduce product variation in order to keep most products around a given target. This idea is shown graphically in figure 8.5a. But what can be done when the reduction in variation is done around the wrong target? This case, shown in figure 8.5b, calls for a different strategy. In order to reduce variability in the product and keep the mean in target, Phadke (1989) suggest the following strategy: 1. Evaluate the effects of the control factors under consideration on average response and S/N. 2. For factors that have a significant effect on S/N, select the levels that maximize S/N. 3. Select any factor that has no or little effect on S/N but has a significant effect on the mean function as an adjustment factor. Use the adjustment factor to bring the mean function on target. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

2. Select a factor that has the smallest effect or no effect on the S/N ratio for all response variables but has a significant effect on the mean levels. This is called mean adjustment/signal factor. Then, set the level of adjustment factor so that the mean responses are on target. As it can be observed the above strategy is not very different from the oneresponse optimization strategy. Nevertheless, it requires much more careful application.

8.6

Examples

8.6.1. Use of plastics in (Antony et al., 2001) A company wants to investibraking systems gate the possibility of using lightweight plastics in a modern braking system and decided to carry out a Taguchi’s experiment. The production process consists of a heated die, which is then forced down by air pressure onto a valve body forming a plastic lip into which a retaining ring is inserted. A schematic view of the process is shown in figure 8.6. The purpose of the experiment was to obtain the parameters of the process that maximized the pull-out strenght. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

215

8.6 Examples Pre−heated die Metal insert

Thermoplastic moulding

11111 00000 00000 11111 00 11 00 11 00 11 00 00 11 11 00 11

Force

Figure 8.6: Schematic view of the industrial process.

From the many variables affecting the process, five were selected as critical. The choice of control factors and levels is described in table 8.5. As the objetive of the experiment was to obtain the factors levels that maximized the pull-out streght, this quantity, it was natural to select this quantity as response variable. As temperature was the most critical factor, the engineering team working on the problem decided to use four levels for this factor. As the other four factors had only two levels, the orthogonal array that was closer to fit the experiment needs was the L16 . Nevertheless, it had to be modified to fit a four-levels factor. A common technique to fit a column with more levels is to merge two or more columns in the array.

Level Units

Die temperature (A)

◦

Hold time (B)

sec

C

Force application rate (E)

2

3

4

180 200 220 240 5

15

–

–

1

2

–

–

kN

6

7

–

–

kN/sec

5

1

–

–

Batch no. (C) Maximum force (D)

1

216

Run B A

Servo valve body

Control factor

8.6 Examples C

D E B×E

y1

y2

y3

y

1

1

1

1

1

1

1

2.18 2.10 2.14 2.14

6.61

2

1

1

2

2

2

2

2.68 2.65 2.67 2.67

8.52

3

1

2

1

1

1

1

2.46 2.57 2.52 2.52

8.01

4

1

2

2

2

2

2

2.92 2.59 2.76 2.76

8.78

5

1

3

1

1

2

2

2.83 2.74 2.79 2.79

8.90

6

1

3

2

2

1

1

3.61 3.22 3.42 3.42 10.64

7

1

4

1

1

2

2

3.31 3.40 3.36 3.36 10.52

8

1

4

2

2

1

1

4.02 3.98 4.00 4.00 12.04

9

2

1

1

2

1

2

3.08 3.14 3.11 3.11

9.85

10

2

1

2

1

2

1

3.07 2.97 3.02 3.02

9.60

11

2

2

1

2

1

2

3.35 3.15 3.25 3.25 10.23

12

2

2

2

1

2

1

3.46 3.21 3.34 3.34 10.45

13

2

3

1

2

2

1

3.42 3.81 3.62 3.62 11.14

14

2

3

2

1

1

2

3.56 3.70 3.63 3.63 11.19

15

2

4

1

2

2

1

4.33 4.90 4.62 4.62 13.25

16

2

4

2

1

1

2

4.77 4.70 4.74 4.74 13.51

Table 8.6: Experimental layout for the study. The first step in this technique is to identify the degrees of freedom associated with each level. As the degrees of freedom for each factor is the number of levels it has minus one, then, a two-levels factor has one degree of freedom and a four-levels factors has three degrees of freedom. In order to maintain the balance in the array, to fit a four-levels column, three two-levels columns have to be merged. In that way, three one-degree-of-freedom columns are merged to fit one three-degrees-of-freedom column. For the problem under study, it was decided to merge columns 2, 4 and 6. If these columns are isolated, each row will fall in one of the following combinations: 1-1-1, 1-2-2, 2-1-2 or 2-2-1. Each combination can be now assigned to one of four levels of the new factor. For example, 1-1-1 ⇒ 1, 1-2-2 ⇒ 2, 2-1-2 ⇒ 3 and 2-2-1 ⇒ 4.

Table 8.5: Design factors for the Taguchi experiment. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

S/N

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

217

8.6 Examples

8.6 Examples

218

Influence (%)

60

Signal−to−noise ratio

12.5 12 11.5

61.01

50 40 28.59

30 20

11

10

10.5

0

10 9.5

A

B

4.78

3.97

C

D

0.11

0.01

1.53

E

BE

Error

Factor

9

Figure 8.9: Percent influence of the design factors.

8.5 A1 A2 A3 A4 B1 B2 C1 C2 D1 D2 E1 E2 Factors

Figure 8.7: Factor effects on the signal–to–noise ratio.

Average pull−out strength (kN)

The experimental setup from the modified orthogonal array and its results are shown in table 8.6. Notice that three repetitions of each run were carried out and that the pull-out strength is measured in kN. To determine the optimum conditions, the level of each factor has to be chosen in such a way that the maximum pull-out strength is achieved together with the minimum variation. Hence, optimal condition is achieved selecting the levels of each factor that yields the highest S/N ratio. Figure 8.7 shows the response in the S/N ratio for the different factors. From the plot, it is clear that factors A and B have the most influence in the S/N response and that optimal levels based on S/N are obtained with A4 , B2 , C2 , D2 , and E1 .

4.2 4 3.8 3.6 3.4 3.2

For this problem, the same conclusions can be drawn from plots of main effects in average response, shown in figure 8.8, as the levels that maximize the average pull-out strength, remains A4 , B2 , C2 , D2 , and E1 .

3 2.8 2.6 A1 A2 A3 A4 B1 B2 C1 C2 D1 D2 E1 E2 Factors

Figure 8.8: Factor effects on the average response.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

To complete the analysis is always recommendable to check the percent influence of factors. For this problem, the percent influence in the singal-to-noise ratio is shown in figure 8.9. As expected, factors A and B has the most influence with 61% and 29% percent, respectively. Factors C and D have a very small influence with 4.8% and 4%, and factor E and the interaction B×E have a negligible influence as their contribution is respectively 0.11 and 0.01%. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

219

8.6 Examples

(Modified from Reddy et al., 1998) A company manufactures a large range of plastic mouldings from household to large industrial products. The company has experimented some complaints from customers regarding an agitator used in washing machines. The agitator, depicted schematically in figure 8.10, is respnsible for the movement of clothes inside the washing tub. The product is moulded in polypropylene and is fitted on to a serrated shaft spline and locked in position with a screw.

8.6 Examples

220

8.6.2. An injection moulding process

After some initial investigation, it was observed that the problem was mainly due to lack of keeping dimensions, specifically in the outer-diameter and pullout strength. It was decided to use Taguchi’s parameter design methodology to bring the process on target. To ensure success, three response variables were taken into account: outer diameter, height and pull-out strength. The engineering team dealing with the problem identified seven control factors relevent to the investigation. These seven factors are: mould temperature (A), injection pressure (B), hold-on pressure (C), injection time (D), holding time (E), cooling time (F) and fill time (G). In order to keep the experiment of manageable size, it was decided to use only two levels per factor. The choice of levels of each factor is shown in figure 8.7. Also, the team decided to neglect interactions for the sake of simplicity and use instead confirmation or verification experiments to avoid misleading conclusions.

11 00

washer tub

1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 00 11 0000 1111 00 11 0000 1111 00 11

height (114 mm)

1 0 00 11 0 1 00 11

outer−diameter

Level Control factor

Units ◦

Mould temperature (A)

C

1

2

35

50

Injection pressure (B)

kg/cm2

110 150

Hold-on pressure (C)

kg/cm2

70

120

Injection time (D)

sec

30

50

Hold-on time (E)

sec

23

33

Cooling time (F)

sec

50

100

Fill time (G)

sec

7

17

Table 8.7: Control factors and their levels for the agitator experiment.

Outer diameter

Height

Pull-out force

Exp.

Mean

S/N

Mean

S/N

Mean

1

329.30

65.88

113.21 52.99

3.00

2

329.41

65.08

114.07 58.83

1.66

3

329.45

64.02

113.20 43.29

1.69

4

329.48

66.89

113.54 45.17

2.12

5

329.48

60.29

113.88 47.79

2.77

6

329.45

67.89

114.05 50.21

1.48

7

329.43

72.12

113.85 52.21

2.13

8

329.60

69.57

113.72 52.75

2.58

Table 8.8: Summary of responses for the agitator experiment.

Figure 8.10: Washing machine agitator. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

8.6 Examples

From the choice of factors and levels, a L8 orthogonal array was selected. To make the experiment more robust, it was conducted in the presence of three noise factors, namely two different operators, two shifts, and raw materials from two different vendors. As all noise factors had two levels, a L4 array was appropriate as noise array. The recorded response from the experiments is described in table 8.8. As the objective of the experiment was to minimize variance and bring the process mean on target for the outer diameter and height responses, the nominal-thebest case was selected to compute the S/N ratio for these quality characteristics. Although it was desirable to minimize the pull-out force, it was decided to deal with this response after the previous two had been optimized.

8.6 Examples

222

40 Influence in S/N (%)

221

35 25 20

20.39

B

C

22.18

10 5

3.98

A

G

Error

Factor

Figure 8.11: Percent of influence of control factors in the S/N ratio for the outer diameter. Factors D, E and F were pooled.

68.5 Signal−to−noise ratio

The main effects plots for the outer diameter in terms of the S/N ratio and the average response are shown in figures 8.12 and 8.14, respectively. For the height, the main effects plot in terms of the S/N ratio is shown in figure 8.16 and in terms of the average response is shown in figure 8.18. Finally, the main effects plot for the average pull-out force is shown in figure 8.20.

20.81

15

0

From the results obtained, the percent influence or percent contribution for each factor on each response, either average or in terms of the S/N ratio, were calculated. The results are presented in figures 8.11 and 8.13 for the outer diameter. For the height, the results are presented in figures 8.15 and 8.17 for the S/N ratio and the average response, respectively. For the pull-out force, the percent influence of factors in the average response is shown in figure 8.19. In all cases all those factors with a small contribution were pooled into error to avoid misleading results with the ANOVA.

32.69

30

68 67.5 67

From the results for the outer diameter it can be concluded that for the S/N ratio, factors B and C have a moderate effect and contributes around 20% each. Factor G, fill time, has the most influence with a 22.18%. From the main effects plot for the S/N ratio, factor G has again the most influence. The optimum levels for the most significant factors are B2 , C1 and G1 . For the average response, factor A, mould temperature, and the fill time have the most significant effect with a contribution of around 30% each. Factors C and F have a moderate effect and factor E has the least influence on the response. From the main effects plot, factors A, B, D and G cause all a relatively large change in the average response.

Figure 8.12: Plots of main effects for the outer diameter using S/N.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

66.5 66 65.5 65 64.5 A1 A2 B1 B2 C1 C2 D1 D2 E1 E2 F1 F2 G1 G2 Factors

223

8.6 Examples

Influence (%)

34.38 29.27

30 25 20 15

12.50

12.50

10 0

7.18

4.17

5 A

C

E

F

G

224

From the above results, factors B, C and G can be treated as control factors as they have a relatively large influence in the S/N response. Factors A and F are the best candidates as adjustment or signal factors since they cause an important effect on the average response but have a little effect on the S/N ratio.

40 35

8.6 Examples

Error

Factor

Figure 8.13: Percent of influence of control factors in the average response for the outer diameter. Factors B and D were pooled.

Analyzing the results for the height, it can be observed that for the S/N ratio, factor C, hold-on pressure, has by far the largest percent contribution with almost 67%, follow by factor B, injection pressure with almost 20% and factor D, injection time, with almost 8%. All other factors have a very small contribution. From the main effects plot for the S/N ratio, factors B and C have most influence in the S/N ratio and have both level 1 as their optimal. Regarding the average response for the height, factor A has a relatively high influence with 32.5%. Factors D and E have a moderate influence of around 20%. Factors B and F have a small influence of around 10%. From the main effects plot, factor A has the most influence and factor C the least.

Average outer diameter (mm)

From the above results, factors C, B and D are the control factors for the height and factors D, E and F are the best candidates to be adjustment factors for the same response. For the pull-out force, results indicate that factor F, cooling time, has the largest influence with almost 70%. Factors C and D have a relatively small contribution of 10 and 17%. All other factors have almost no contribution. From the main effects plot it can be observed that a change in the cooling time causes a large change in the average force required to remove the agitator from the shaft.

329.5 329.48 329.46 329.44 329.42 329.4 A1 A2 B1 B2 C1 C2 D1 D2 E1 E2 F1 F2 G1 G2

The above analyses show that factors B, C and G can be used as control factors while factors A and F can be used as adjustment factors for the problem considering that all three responses, outer diameter, height and pull-out force have to be considered. The final selection of levels have to be carried out taking into account the above results and considering the nature of the industrial process when contradictory results are at hand.

Factors

Figure 8.14: Plots of main effects for the outer diameter using the average response.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Consider for example the injection pressure (factor B). In terms of the S/N ratio, factor B has to be at level 2 for optimal response in the outer diameter. Nevertheless, from the height S/N results, factor B has to be at level 1. Hence, it is necessary to consider more information in order to decide which level is best for the process. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

225

8.6 Examples

8.6 Examples

226

40 66.62

70

35

60

Influence (%)

Influence in S/N (%)

80

50 40 30 20

19.22 7.95

10 0

B

C

D

1.25

2.03

2.93

E

F

Error

32.53

30 25 15

0

Average height (mm)

Signal−to−noise ratio

49 48

A

B

D

E

F

Error

Figure 8.17: Percent of influence of control factors in the average response for the height. Factors C and G were pooled.

113.85

50

4.23

Factor

113.9

51

9.67

5

54

52

11.37

10

55

53

19.47

20

Factor

Figure 8.15: Percent of influence of control factors in the S/N ratio for the height. Factors A and G were pooled.

22.73

113.8 113.75 113.7 113.65 113.6 113.55

47 46 A1 A2 B1 B2 C1 C2 D1 D2 E1 E2 F1 F2 G1 G2 Factors

113.5 A1 A2 B1 B2 C1 C2 D1 D2 E1 E2 F1 F2 G1 G2 Factors

Figure 8.16: Plots of main effects for the height using S/N.

Figure 8.18: Plots of main effects for the height using the average response.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

227

8.6 Examples

Influence (%)

228

In this case, the engineering team concluded the following: “ Higher injection pressure is the main cause of built-in-stress in the end product. Such stresses may increase the vulnerability of the parts to the wear and tear of daily use. If high pressure is exerted on the material and released too soon, some of the material will move out of the mould into the runners and sprue, owing to relief from the state of compression resulting in still greater final shrinkage”. As shrinkage will increase the necessary force to retire the agitator, it was decided to set the injection pressure at level 1.

80 69.86

70

8.6 Examples

60 50 40 30 10 0

From similar analyses the engineering team found the following final results:

16.97

20

9.86 2.21

1.10

A

C

D

F

Error

Factor

Figure 8.19: Percent of influence of control factors in the average response for the pull-out force. Factors B, E and G were pooled.

• Mould temperature (A) = 50◦ C (level 2). • Injection pressure (B) = 110 kg/cm2 (level 1). • Hold-on pressure (C) = 70 kg/cm2 (level 1). • Injection time (D) = 30 sec (level 2). • Hold time (E) = 33 sec (level 2). • Cooling time (F) = 100 sec (level 2).

Average pull−out force (kg/cm²)

• Fill time (G) = 7 sec (level 1). 2.7 2.6

The above combination of factors is not in the array of experimental runs, so verification experiments were carried out yielding good results both in terms of getting target values and reducing variability. The interested reader is encouraged to read the full paper for more details.

2.5 2.4 2.3 2.2 2.1 2

References

1.9 1.8 1.7 A1 A2 B1 B2 C1 C2 D1 D2 E1 E2 F1 F2 G1 G2 Factors

Figure 8.20: Plots of main effects for the pull-out force using the average response.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

1. Antony, J., Warwood, S., Fernandes, K. & Rowlands, H. (2001) Process optimisation using Taguchi methods of experimental design. Work Study, 50, pp. 51-57. 2. Fowlkes, W.Y. & Creveling, C.M. (1995) Engineering methods for robust product design. Using Taguchi Methods in Technology and Product Development. Addison-Wesley. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

229

8.6 Examples

3. Phadke, M.S., Kackar, R.N., Speeney, D.V. & Grieco, M.J. (1983) Off-line quality control integrated circuit fabrication using experimental design. The Bell System Technical Journal, 62, pp. 1273-309. 4. Phadke, M.S. (1989) Quality engineering using robust design. Prentice-Hall. 5. Reddy, P.B.S., Nishina, K. & Subash Babu, A. (1998) Taguchi’s methodology for multi-response optimization. A case study in the Indian plastics industry. International Journal of Quality & Reliability Management, 15, pp. 646-68. 6. Roy, R.K. (2001) Design of experiments using the Taguchi approach. 16 steps to product and process improvement. John Wiley & Sons.

CHAPTER

9

Response Surface Method

7. Taguchi, G. (1993) Taguchi on robust technology development: bringing quality engieering upstream. ASME Press. 8. Yang, L. & El-Haik, B. (2003) Design for Six Sigma. A roadmap for Product Development. McGraw-Hill.

9.1

Overview of the method

From the previous chapters, Taguchi methodology presents itself as a very attractive methodology to effectively and conveniently design robust products and processes. Nevertheless, Taguchi methodology is not infallible as it is possible to miss an optimal setup as the methodology usually involves large changes in the design factors and therefore, it looks for optimal points in a discrete space rather than in a continuous space. To clarify the idea, imagine that an engineer/scientist has setup a single factor experiment. He or she carry out the experiment setting values for the control factor within a pre-established range selecting the minimum (1), middle (2) and maximum (3) points in the range. After the values for the response variable are registered, the engineer/scientist finds that the factor level that maximizes the response is 3 (see figure 9.1a). To be sure, he or she finds a interpolated function by means of a least-squares regression and finds that effectively, the optimum level for the control factor is 3, as shown in the figure 9.1b. As there is a feeling that the optimal point found does not provide the best response, the engineer/scientist decides to carry one more experiment, this time using values for the control factors near the optimum found. After the experiments are run, the engineer/scientist realized that the previous experiment missed the optimal value and that the real optimum is on the new level 2 (see figure c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e CarlosMiranda. Todos los derechos reservados.

231

9.2 Steepest ascent method

y

y

False optimum

y

False optimum

x2 (a)

x3

x

x1

x2

232

True optimum False optimum

interpolated response x1

9.2 Steepest ascent method

x3

(b)

x

x1

x2

x3

(c)

x 0.75 0.5

Figure 9.1: Missing an optimal point in a single factor experiment.

0.25 0

9.1c. The engineer/scientist goes home happy thinking that he/she found the true optimal, not realizing that the true optimal has not been found yet. The above story shows one disadvantage of using discrete values for design factors: the true optimum may be missed. To solve this problem, one option is to try to use continuous values for the design factor. Albeit not as efficient as using discrete values, using continous values would assure that the true optimum would be found in most cases. The Response Surface Method (RSM) aims at solving the above problem in the most efficient possible way. The name response surface comes from the fact that it is possible to represent responses as surfaces, where then a maximum or minimum can be looked for. In many ocassions, analysis is carried out using two factors at a time, in such a way a three-dimensional surface can be plotted or represented by a contour map (see figure 9.2).

−2 −1.5 −1 −0.5

0

0.5

1

1.5

−0.5 −1 −1.5 2−2

0

0.5

1

1.5

2

2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1.5 1 0.5 0

9.2

Steepest ascent method

−0.5 −1 −1.5

9.2.1. Single factor problem

The process of searching for an optimal solution can be illustrated using a single factor, single response problem. In RMS, it is customary to designate factors by the letter X with a subindex 1, 2, etc., to stress that now factors are treated as continuous variables rather than discrete ones. Using this notation, in the following example the response y will be a function of a single factor X1 .

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

−2 −2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Figure 9.2: Three-dimensional response surface and corresponding contour map.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

233

9.2 Steepest ascent method

9.2 Steepest ascent method

234

y

maximum y

X

X

steepest

1−0

ascent

Figure 9.3: A response curve with an initial condition X1−0 .

X y

1−0

X 1-n

X

Figure 9.5: Looking for a maximum point in the steepest ascent direction.

slope intercept

X X 1−0

Figure 9.4: Tangent line at initial condition X1−0 . Since one of advantages of the Response Surface Methodology is that it can be applied to a running system without having to stop production, consider that the system is running at some initial condition X1−0 as shown in figure 9.3. The objective of the procedure is to find the point of optimal response which in this case is the maximum response. The standard approach in RMS is to find for points of maximum response. When the point of interest is a minimum, then the response is simply multiplied by −1.

If the slope at condition X1−0 is positive, that means that the maximum is at the right, since the response increases in that direction. This direction that points to the condition of maximum response is called the direction of steepest ascent. When the slope is negative, then the condition of maximum response will be at the left. In general, this slope will only be accurate in the vecinity of the condition X1−0 , but is good enough to show which direction to follow. Once the direction of steepest descent has been found, the following step is to run an experiment with a new condition X1−1 = X1−0 ± ΔX1 , where the sign depends on the direction to follow. The above procedure is repeated several times until the response stops increasing as shown in figure 9.5. If the response stops increasing, it is because we are at the optimum, or near it. If the new computed slope differs in sign from the previous ones, then the point of maximum response is between the last two chosen conditions. In this case, small increments of X1 can used to hunt for the point of maximum response.

Starting from the initial condition X1−0 , in this simple problem the maximum is either at the right or left. To find which side to go, consider that the process is varied slightly to find the values of the responses in the vecinity of X1−0 . These additional points, represented by crosses in figure 9.4, allows the calculation through linear regression analysis of the slope at point X1−0 . It is important to remember that the response curve is not known in advance and that several observation may have to be carried if there is scatter in the data.

9.2.2. Two factors problem The above example, although simple in nature, presents how RMS works toward finding an optimum point and the procedure followed can be applied regardless of the number of factors. Consider now a more complicated problem with two factors. In this case, the response is not a curve but a surface.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

235

9.2 Steepest ascent method

y

9.2 Steepest ascent method

1

1

0.5

0.5

0

0

X2−0

X2−0

-0.5

-0.5

-1 -1

-0.5

0

X1−0 0.5

236

Steepest ascent

-1 1

x

Figure 9.6: A response surface with an initial condition (X1−0 , X2−0 ). As in the single factor problem, starting point of the procedure is an initial condition, in this case, (X1−0 , X2−0 ) since two factors namely, X1 and X2 , are considered. Figure 9.6 shows this starting point representing the unknown response surface as a contour map. Next, it is necessary to find the steepest ascent direction, which in this case is a vector. Similarly to the procedure followed in the single factor problem, the steepest ascent direction can be found by observing the response in the vecinity of (X1−0 , X2−0 ) as shown in figure 9.7, and computing a multiple-variable linear regression analysis. Since this case involves a two-factor problem, at least three additional points are required for the linear regression analysis to work. From the linear regression, two slopes will be found, one with respect to factor X1 and one with respect to parameter X2 . The direction of steepest ascent can be found from the slopes as follows: • If the slope with respect to factor X1 is positive, then the direction of steepest ascent is to the right. • If the slope with respect to factor X1 is negative, then the direction of steepest ascent is to the left. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

-1

-0.5

0

X1−0 0.5

1

x

Figure 9.7: Additional sample points at the vecinity of condition (X1−0 , X2−0 ) and steepest ascent vector. • If the slope with respect to factor X2 is positive, then the direction of steepest ascent is to the top. • If the slope with respect to factor X2 is negative, then the direction of steepest ascent is to the left. Since the two slopes with respect to X1 and X2 are available, then it is possible to find not only the direction of steepest ascent but the vector of steepest ascent. This vector will point in which direction the maximum appears to be. Naturally, this direction will be accurate only in the vecinity of the of (X1−0 , X2−0 ). From the direction found, a new set of conditions (X1−i , X2−i ) must be found until the response stops increasing as sketched in figure 9.8. When two or more factors are taken into account, the direction of the increments taken to generate the new condition, in this case (X1−1 , X2−1 ), has to agree with the vector of steepest ascent. In order to do that, increments for the factors must be chosen accordingly.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

237

9.2 Steepest ascent method

9.2 Steepest ascent method

238

1 75 70

y

0.5 65 X1− , X 2− n n

60

0

X2−1 X2−0

55

-0.5

50 45 -1

-1 -1

-0.5

X1−1 0

X1−0 0.5

-0.5

0

X1

1

0.5

1 -1

-0.5

0 X2

0.5

1 y

x

x

Figure 9.8: Tangent line in the direction of steepest ascent and the new condition (X1−1 , X2−1 ) found.

In order to obtain the increments for the factors, typically the factor with the largest magnitude of slope is chosen first and a desired value for the increment is selected. Then, the increment in the other factor is calculated so the direction of the vector of increment agrees with the direction of the vector of steepest ascent. Suppose for example, that the largest slope is for factor X1 . Then, an increment ΔX1 will be chosen taking into account the characteristics of the process and the corresponding increment ΔX2 will be calculated. In this fashion, new conditions (X1−i , X2−i ) can be generated. Once the response stops increasing, then it is necessary to find a new vector of steepest ascent and repeat the procedure to hunt of the optimum as shown in figure 9.9.

1

y

Steepest ascent

0.5

Figure 9.10: The climber analogy.

X 2−n

0

-0.5

9.2.3. The climber analogy

-1 -1

X1−n

-0.5

0

0.5

1

x

Figure 9.9: Steepest ascent vector at condition (X1−1 , X2−1 ).

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

In many occasions, the Response Surface Method of finding maximums is compared to a climber that wants to reach the top of a mountain that is covered by dense fog. As the climber does not know exactly where the top is, he or she needs to rely in his/her instruments to find the way. If the climber has an altimeter and a compass, he/she measure the altitute in some points around his/her current position, and determines in which direction the altitude increases the most. With the help of the compass, the climber proceeds to climb in the same dic Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

239

9.2 Steepest ascent method

rection until he/she notices that altitude is not gained anymore. With the help of the altimeter, the climber takes several measurements of altitude in points nearby. With this information, the climber finds once more the direction where the altitude increases the most and starts to climb again following that direction until he/she notices that altitude fails to increase. Repeating the procedure, the climber will eventually reach the top. Although one may argue that the procedure is not very efficient, it is important to notice that surely is faster than trial and error and somehow guarantees that a peak (maximum) will be found. At this point is important to mention that as an expert climber will observe, to find a peak does not necessarily means that one has reached the top of the mountain as another higher peak may be close by. 9.2.4. Coded variables

In the procedure described above, it is necessary to carry out a regression analysis to find the direction of steepest ascent. As explained above, this regression is performed on selected points laying on the vecinity of a condition (X1−i , X2−i ). Unfortunately, the regression may contain factors of very different units and ranges, reason for which it is not convenient to perform the regression in the raw data. Instead, all data must be normalized before the regression analysis is done. It is convenient to normalize data in the range −1 to +1 so all factors affect the response in the same way. To convert any factor X into a coded variable ˆ the simple following formulas may be used: X,

ˆ =2 X

X − Xmid-value Xmax − Xmin

 (9.1)

where Xmid-value = (Xmax − Xmin )/2 and Xmax and Xmin is the maximum and minimum values for factor X to consider. As it has been discussed, once the regression has been carried out and the vector of steepest distance has been found, it is necessary to select an increment in the factors. This increment can be selected in terms of the coded variables. Nevertheless, it will be necessary to transform the increment in terms of coded variables into an increment in terms of the original physical units in order to setup the next experiment. This backwards transformation is given by ΔX =

Xmax − Xmin ˆ ΔX 2

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(9.2)

9.2 Steepest ascent method

240

ˆ is the selected increment where ΔX is the increment in physical units and ΔX in coded units. 9.2.5. Finding the vector As explained above, two of the key points of the and increments of steepest Response Surface Method are to find the vector ascent of steepest ascent and to obtain the chosen increment in the same direction. Fortunately, once the results of the regression analysis are available, these two key points are easy to carry out. Considering a two-factors problem, after the regression analysis is performed, an equation of the form ˆ 1 + β2 X ˆ2 yˆ = β0 + β1 X

(9.3)

is available. Mathematically speaking, the β1 and β2 factors are the slopes ˆ2, ˆ 1 and X with respect to X ∂y = β1 ∂X1

∂y = β2 ∂X2

and therefore the vector of steepest ascent would be described by

 ∂y ∂y , ˆ1 ∂ X ˆ2 ∂X

(9.4)

(9.5)

Now, it is necessary to choose the desired increment based on both experience and the process itself. Assuming that the largest slope is in factor X1 , then it ˆ as base increment. To keep the increment is convenient to select a given ΔX1 vector in the same direction as the vector as steepest ascent, the increment in any other factor Xn can be calculated from ˆ ˆ 1 (∂y/∂ Xn) ˆ n = ΔX ΔX ˆ 1) (∂y/∂ X

(9.6)

Since these increments are in coded variables, it is necessary to transform them into increments in physical variables. This transformation can be carried out accordingly using equation (9.2). c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

241

9.3 Local exploration method

9.3 Local exploration method Region found by the steepest ascent method

As sometimes the selection of the magnitude of the increments is not easy, some authors suggest to follow the next strategy. Suppose that from linear regression the following linear model can be established: y = β0 + β1 X1 + β2 X2 + · · · + βk Xk

242

y

(9.7)

From the above model, the next coordinate to analyze can be obtained as βi Xinew = Xiold + ρ   k  2  β

(9.8) X

k

j=1

Figure 9.11: Region of the response where the maximum lays.

where Xiold is the current value for the Xi factor, Xinew is the next value for Xi after one increment and ρ is the step length. When no better choice for the step length is available, ρ = 1 is a common selection. From the above formula,  k 2 β it is clear that the increment is given by βi / j=1 k . It is left to the reader to study the equivalence of this procedure to the one shown earlier.

In general, three different types of designed experiments are preferred for local exploration: Central Composites Designs (CCD), Box-Behnken designs and D-Optimal designs. In some occasions, Taguchi arrays may also be used due to their simplicity. In what follows, CCD and Box-Behnken designs will be discussed.

9.3

9.3.1. Central Composite Designs

Local exploration method

After a search for the point of maximum point using the steepest ascent method has been carried out, and the engineer/scientist feels that the optimum point should be close by, then the method of local exploration may be a better choice. Consider figure 9.11. In the case depicted, the steepst ascent method would require to obtain a new direction of steepest ascent, to select an increment, and advance looking for the point where the response starts to diminish. This procedure can become both tedious and long, specially for problems with multiple factors. When the maximum point is near, the method of local exploration is better suited to find it in a quick and effective way. The method of local exploration requires to carry out a designed experiment to fit a non-linear regression model into the reponse to look for the point of maximum yield. Once the non-linear model is obtained, standard exploration and calculus techniques may be used to find the optimum point. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Central Composites Designs (CCD) are a combination of factorial or fractional factorial designs with a group of centerpoints and a group of secondary points that allows the estimation of second order effects. The points belonging to the to the secondary group receive the name of axial points. If the factorial design points are usually at a distance ±1 unit from the center of the design, axial points are at a distance ±α where |α| < 1. The exact value of α depends on the specific factorial design choosen. The layout of a CCD experiment is shown in figure 9.1. The first part of the layout consists of N F factorial design points. Here, full factorial or fractional factorial designs may be used. In the second part, N C centerpoints are specified. The number of centerpoints in the design depends on the number of repetitions required to obtain a predition of the error in the model. In general, the more centerpoints are selected, the lower the prediction error will be. Finally, the number of axial points, N A, is always twice the number of factors. The CCD layout may be represented graphically as depicted in figure 9.12 for a two factors design. As shown, the CCD layout is a combination of factorial c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

243

9.3 Local exploration method

9.3 Local exploration method

244

Factors Point type

Experimental Point

X1

X2

Factorial

1

−1

−1

Design

2 .. .

1

−1

−1

1

NF

1

1

1

0

0

0

2 .. .

0 .. .

0 .. .

−1

NC

0

0

Points

Centerpoints

Axial

1

−α

0

Points

2 .. .

α

0

NA

0

X2

X2 +α +1

0

−α X1

+1

−α

Factorial design with centerpoints

0 −α

0 Axial points

α X2

Table 9.1: Layout of Central Composite Designs. +α +1

points, centerpoints and axial points. The value of α used for the axial points is chosen so the experimental design has a rotatable property. When a design is rotatable, then the variance of the predicted value of the response is only function of the distance between the point of interest and the center of the design, that is, there is no preferential direction in the experiment’s prediction. Since there is no knowledge on in what direction the maximum point may be, it is desirable not to set by design a direction of preference. Rotatables design, are therefore, a better choice. To have a rotatable central composite design, the value of α is a function of the number of experimental runs in the factorial portion of the design, 1 α = (N F ) 4

0

−1 −α −α −1

0

+1 +α

X1

Central Composite Design

Figure 9.12: Central Composite Design (CCD). (9.9)

Values for α for designs including two to six factors are shown in table 9.2. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

+α

X1

245

9.3 Local exploration method Number of factors Factorial design

α

2

2

1.414

3

2

3

1.682

4

24

2.000

5

25−1

2.000

5

25

2.378

6

26−1

2.378

6

26

2.828

2

Table 9.2: Values for α for different factorial designs. One disadvantage of rotatable central designs is that they require to consider five levels for each factor: −α, −1, 0, +1, +α. In many cases, this requirement makes CCD designs difficult to implement or carry out. One alternative is to set α = 1 to reduce the number of levels involved in the experimental design. These designs are called face-centered central composite designs. Although this type of setup requires only three levels for each factor, it looses the CCD rotatable property. For these cases, Box-Benhken designs provide another convenient alternative. 9.3.2. Box-Behnken Box-Behnken designs are quadratic independent designs Designs that do not contain an embedded full or fractional factorial design. Hence, in this design, experimental runs are at the center and at midpoints of edges of the design space. Figure 9.13 shows graphically a Box-Behnken design for three factors. The levels for each one of the factors are specified in table 9.3. Apart from the advantage that experimental designs require only three levels for each factor, another one is that Box-Behnken designs are rotatable or nearly rotatable.

9.3 Local exploration method

246

X1

X2

X3

−1

−1

0

−1

1

0

1

−1

0

1

1

0

−1

0

−1

−1

0

1

1

0

−1

1

0

1

0

−1

−1

0

−1

1

−1

1

0

0

1

−1

0

1

1

0 .. .

0 .. .

0 .. .

0

0

0

Table 9.3: Box-Behnken design for three factors.

Table 9.4 presents the Box-Behnken design for four factors. It is important to notice that in this type of designs centerpoints are added as usual to lower the prediction error.

Figure 9.13: Graphical representation of a Box-Behnken design for three factors. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

247

9.3 Local exploration method

X1

X2

X3

X4

−1

−1

0

0

−1

1

0

0

1

−1

0

0

1

1

0

0

−1

0

−1

0

−1

0

1

0

1

0

−1

0

1

0

1

0

−1

0

0

−1

−1

0

0

1

1

0

0

−1

1

0

0

1

0

−1

−1

0

0

−1

1

0

0

1

−1

0

0

1

1

0

0

−1

0

−1

0

−1

0

1

0

1

0

−1

0

1

0

1

0

0

−1

−1

0

0

−1

1

0

0

1

−1

0

0

1

1

0 .. .

0 .. .

0 .. .

0 .. .

0

0

0

0

Table 9.4: Box-Behnken design for four factors. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

9.4 Examples

248

9.3.3. Single response model creation

Once the experimental design layout has been choosen, and the experimental runs have been carried out, it is time to create a quadratic model that can be analyzed in order to find the optimum response point. In general, a quadratic model can be described as: y = β0 +

k  i=1

βi xi +

k 

βii x2i +



i=1

βij xi xj + ε

(9.10)

i<j

The above equation can be rewritten in matrix form as: y = Xβ + ε

(9.11)

where y is the response vector, X is the factor or information matrix and β is the coefficent vector. In order to find the coefficient vector β, a least-squares fit for multiple linear regression can be obtained from ˆ = (XX)−1 X y β

(9.12)

Once coefficients are found, the model is complente and can be analyzed using standard calculus techniques in order to find the point of maximum yield.

9.4

Examples

9.4.1. Chemical process response

(Yang & El-Haik, 2003). In a given chemical process, the two most important factors are temperature and reaction time. The response surface method is selected to obtain the point that maximizes yield. To carry out experiments, a central composite layout is used where α = 1.414. The summary of experimental runs is shown in table 9.5. ˆ From the experimental results, the information matrix in coded variables X and the response vector y are

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

249

9.4 Examples

9.4 Examples ⎡

Coded variables

Natural variables

X1

X2

Temperature, C Reaction Time, min Yield

−1

−1

170

300

64.33

+1

−1

230

300

51.78

−1

+1

170

400

77.30

+1

+1

230

400

45.37

0

0

200

350

62.08

0

0

200

350

79.36

0

0

200

350

75.29

0

0

200

350

73.81

0

0

200

350

69.45

−1.414 0

157.58

350

72.58

+1.414 0

242.42

350

37.42

0

−1.414

200

279.3

54.63

0

+1.414

200

420.7

54.18

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ˆ =⎢ X ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

250

1

−1

−1

1

1

1

1

−1

1

1

1

−1

1

1

1

1

1

1

1

1

1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

−1.414

0

2

0

1

1.414

0

2

0

1

0

−1.414

0

2

1

0

1.414

0

2

1

⎤

⎥ −1 ⎥ ⎥ ⎥ −1 ⎥ ⎥ 1 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎦

⎤ ⎡ 64.33 ⎥ ⎢ ⎢51.78⎥ ⎥ ⎢ ⎥ ⎢ ⎢77.30⎥ ⎥ ⎢ ⎢45.37⎥ ⎥ ⎢ ⎥ ⎢ ⎢62.08⎥ ⎥ ⎢ ⎢79.36⎥ ⎥ ⎢ ⎥ ⎢ y = ⎢75.29⎥ ⎥ ⎢ ⎢73.81⎥ ⎥ ⎢ ⎥ ⎢ ⎢69.45⎥ ⎥ ⎢ ⎥ ⎢ ⎢72.58⎥ ⎥ ⎢ ⎢37.52⎥ ⎥ ⎢ ⎥ ⎢ ⎣54.63⎦

(9.13)

54.18

0

where the information matrix is given by the column vectors   2 2 ˆ = 1 ˆ ˆ ˆ ˆ ˆ ˆ X X1 X2 X1 X2 X1 X2

(9.14)

ˆ −1 X ˆ  y, the coefficient vector βˆ is ˆ  X) Solving the system (X ˆ = [72.0 β

− 11.78 0.74

− 7.25

− 7.55

− 4.85]T

(9.15)

Hence, the prediction model is described by the quadratic equation ˆ2 ˆ 2 − 7.25X ˆ 2 − 7.55X ˆ 2 − 4.85X ˆ 1X ˆ 1 + 0.74X yˆ = 72.0 − 11.78X 1 2

(9.16)

Table 9.5: Experimental layout for the chemical process example. The response surface predicted by the model is shown in figure 9.14. In order to find the point that either maximizes or minimizes the predicted ˆk response, it is necessary to find the point where all partial derivatives ∂ yˆ/∂ X are equal to zero. The first partial derivative is ∂ yˆ ˆ 1 − 4.85X ˆ 2 − 11.78 = −14.5X ˆ1 ∂X c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(9.17)

251

9.4 Examples

9.4 Examples

252

The final step is to convert the coded temperature and reaction time to physical variables, X1 high + X1 low ˆ1 X1 high − X1 low +X 2 2 230 + 170 230 − 170 = + (−0.9285) = 172.14 2 2

X1 = 80 70 60 50 40 30 20 10

X2 high + X2 low ˆ2 X2 high − X2 low +X 2 2 300 + 400 400 − 300 = + (0.3472) = 367.36 2 2

(9.22)

X2 =

−1.5

−1

−0.5

0

0.5

−0.5 −1 −1.5 1.5

1

0

0.5

1

1.5

Hence, the conditions that maximize the yield are a temperature of 172.14 C and 367.36 minutes of reaction time.

Figure 9.14: Predicted response surface for the chemical reaction problem.

(9.18)

Hence, the system of equations to solve is: ˆ 1 − 4.85X ˆ 2 − 11.78 = 0 −14.5X ˆ ˆ1 = 0 0.74 − 15.1X2 − 4.85X

(9.19)

ˆ 2 = 0.3472 X

2. Montgomery, D.C. (2000) Design and Analysis of Experiments. 5th Edition. Wiley Text Books, New York. 3. Myers, R.H. & Montgomery D.C. (1995). Respose Surface Methodology. Process and product optimization using designed experiments. Wiley series in Probability and Statistics. 4. Yang, L. & El-Haik, B. (2003) Design for Six Sigma. A roadmap for Product Development. McGraw-Hill.

ˆ 2 yields ˆ1 and X Solving for X ˆ1 = −0.9285 X

References 1. Berger, P.D. and Maurer, R.E. (2002) Experimental Design with applications in management, engineering and the Sciences. Duxbury Thomson-Learning.

The second partial derivative is ∂ yˆ ˆ 2 − 4.85X ˆ1 = 0.74 − 15.1X ˆ2 ∂X

(9.20)

Subsituting these values into the predicted response model yields the maximum response: yˆ = 72.0 − 11.78 × (−0.9285) + 0.74 × (0.3472) − 7.25 × (−0.9285)2 − 7.55 × (0.3472)2 − 4.85 × (−0.9285) × (0.3472) = 77.597 c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(9.23)

(9.21) c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

10.1 Optimum design

254

Good design

Optimal design work as intended

CHAPTER

10

work as intended

vs minimizes cost maximizes performance

Optimum design

In many occasions, a design team is faced with the task to provide a design that is not only good but also optimal. Here, optimal may be understood as the condition where a certain characteristic or group of characteristics are at its best. As it has been discussed previously, a good design is a design that will work as intended, fulfilling customer’s expectations. But a design that satisfy the needs of the customer may not necessarily be one that minimizes cost or maximizes certain desired characteristic. Such a design is called optimal design. The difference between these two types of designs is shown graphically in figure 10.1. Optimization is a branch of applied mathematics and in this chapter, some optimization techniques will be applied to maximize/minimize desired characteristics in engineering designs.

10.1

Figure 10.1: Difference between a good design and an optimal design.

Design variables Problem Formulation

Objective/Cost function Design constraints

equalities inequalities domain of solution

Standard model/ Mathematical model

constrained unconstrained

Optimum design Graphical method

In order to optimize a given problem, three steps are needed. First, the problem must be thoroughly understood. Second, it is necessary to formulate the problem in order to create a model that can be subject to optimization. Third, using an optimization technique, the selected characteristic(s) are maximized or minimized.

c Copyright 2006 Dr. Jos´ e CarlosMiranda. Todos los derechos reservados.

Optimization

Analytical methods Numerical methods

Figure 10.2: Path to optimization.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

255

10.2 Problem formulation

10.2 Problem formulation

256 h

In figure 10.2, a brief conceptual map of the path to optimization is presented. First during the problem formulation, design variables, those variables that will be changed in order to maximize/minimize the desired characteristic are identified. Then, the characteristic to be optimized is expressed in mathematical form. The resultant mathematical expression is called objective function or design function. In this step, design constraints are also identified. A design constraint is a condition that limit the possible values that design variables may take in order to consider physical or technical limitations. Design constraints may come in the form of solution domain limitation, equalities or inequalities.

l s/2

α α

s/2

l W

Figure 10.3: A truss problem.

The second step is to use all the above equations to create a model that can be optimized. This model is usually constructed as the union of design variables, objective functions and constraints.

F1 α

The final step is to optimize the problem using a given technique. Simple problems involving one, two or even three variables can be optimized using a graphical method. More complicated models are optimized using numerical techniques such as Newton-Rhapson, Simplex or Steepest descent methods.

α F2 W

Figure 10.4: Body forces diagram for the truss problem.

10.2

Problem formulation From the figure sin α =

In order to show how problems may be formulated, four different examples will be shown. The examples include simple problems of structural optimization.

s/2 l

cos α =

h l

(10.3)

Solving equations (10.1) for F1 and F2 yields 10.2.1. Truss optimization

Consider the problem of a truss structure with two members shown in figure 10.6. The optimization problem is to design the structure in order to minimize its weight. The structure must avoid yield and buckling. As with any truss problem, the first step is to sketch the free-body diagram and perform summation of forces in both x and y directions. From the diagram shown in figure 10.7, summation in forces become F1 cos α + F2 cos α = 0 −F1 sin α + F2 sin α = −W

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(10.1) (10.2)

F1 = where l =



W Wl = 2 sin α 2s

F2 = −

W Wl =− 2 sin α 2s

(10.4)

h2 + (s/2)2.

To specify the design, it is necessary to choose a cross-sectional shape for each member. Although the choice of a standard shape like the ones shown in figure 10.8, limits the generality of the solution, it helps to maintain the complexity of the problem to a minimum. Suppose that cross-section (3) is selected for both members. Since weight must be minimized, then the mass of the structure can be specified as the objective function which value must be at a minimum. Hence, the objective function for c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

257

10.2 Problem formulation

10.2 Problem formulation

258

Considering the calculation of areas, the above restrictions become t

d

d

di do

Wl (b1 t1 − 2d1 t1 + t21 ) < σYt 2s Wl (b2 t2 − 2d2 t2 + t22 ) < σYc − 2s

(10.8)

b (2)

(1)

It is necessary to state that all dimensions defining the cross sectional areas cannot be zero or negative. Hence, the next restrictions arise

(3)

b

bi > 0 di > 0 ti > 0 and ti < b/2 bi ≥ di

b t1

d

t1 d

b

t2

(4)

d

t2 (5)

(10.9) (10.10) (10.11) (10.12) (10.13)

As the lower member is subject to compression, it is necessary to check for buckling. The critical load for a column with pinned ends is

(6)

Figure 10.5: Some cross-sectional areas for structural members.

Pcr =

π 2 EI L2

(10.14)

where E is the Young Modulus of the material and I is the moment of inertia.

the problem is mass = density × volume = ρ × A1 × l + ρ × A2 × l  = ρ × (b1 t1 − 2d1 t1 + t21 ) × h2 + (s/2)2  + ρ × (b2 t2 − 2d2 t2 + t22 ) × h2 + (s/2)2

From the critical load formula, the buckling condition restriction becomes (10.5) (10.6)

where ρ is the density of the material, Ai is the cross-sectional area, bi , di and ti are the width, height and thickness of the cross-section for the i-th member.

(10.15)

For the cross-sectional area selected, a rectangular tube, the moment of inertia can be computed as follows I=

bd3 − (b − 2t)(d − 2t)3 12 12 

π2E (10.7)

where σYt is the yield stress at tension and σYc is the yield stress at compression. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

π 2 EI L2

(10.16)

Hence, the restriction becomes

As the structure must withstand yield, then F1 A1 < σYt F2 A2 < σYc

F2 <

F2 <

bd3 (b − 2t)(d − 2t)3 − 12 12   s 2  h2 + 2



c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(10.17)

259

10.2 Problem formulation

Minimize:

+ ρ × (b2 t2 − 2d2 t2 + t22 ) ×

260

The can capacity should be 400ml (400 cm3 ). Do not consider the top of the can.

Finally, the problem can be stated as:

S = ρ × (b1 t1 − 2d1 t1 + t21 ) ×

10.2 Problem formulation

 

h2 + (s/2)2 h2 + (s/2)2

subject to: bi di ti ti bi

> > > < ≥

0 0 0 b/2 di

Wl (b1 t1 − 2d1 t1 + t21 ) < σYt 2s Wl (b2 t2 − 2d2 t2 + t22 ) < σYc 2s   3 bd (b − 2t)(d − 2t)3 2 π E − 12 12  < F2  s 2  h2 + 2 −

10.2.2. Beverage can A company want to verify that the aluminum beveroptimization age can that they manufacture is optimal from the cost point of view. Since most of the production cost is associated to the material needed to manufacture the can, it has been decided that an optimal can would have the least material which in turn translate into having the least possible surface area. In order to use current production and handling facilities, the can must fulfill the following restrictions (Arora, 1989): • The diameter of the can should be more than 3.5 and less than 8 cm. • The height of the should be more than 8 cm and less than 18 cm. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

As the surface area should be minimized, it is necessary to consider the surfaces covering the cylinder and bottom of the can. Hence, surface area can be calculated from π S(d, h) = d2 + πdh (10.18) 4 where d is the diameter of the can and h is its height. Now, the volume of the can must also be considered as it has to be equal to 400 cm3 . The volume of a cylinder is given by π (10.19) V (d, h) = d2 h 4 With the above equations and the restrictions stated, the optimization problem can be formulated as: Minimize: S(d, h) =

π 2 d + πdh 4

subject to: d d h h

≥ ≤ ≥ ≤

3.5 8.0 8.0 18.0 π 2 V (d, h) = dh 4 10.2.3. Tubular column

Consider the design of a tubular column completely fixed at its lower end. The design objective is to minimize the weight of the column. To design a column is necessary to consider failure by yielding and failure by buckling. Failure by yielding occurs when the compression stress is larger than the yielding stress of the material. Failure by buckling occurs when the compression force acting on the column is larger than the critical load for the beam. Since the weight of the column has to be minimized, it is possible to set as an objective function to minimize the total mass of the column. The mass of the beam is given by (10.20) mass = ρhπ(Ro2 − Ri2 ) c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

261

10.2 Problem formulation

where ρ is the density of the material, h is the height of the column and Ro and Ri are the outer and inner radius of the column. Since the column must withstand yield, then F < σY π(Ro2 − Ri2 )

(10.21)

where F is the force that the column must support and σY is the yield stress for the material. Now, since the critical load of the column must not be exceeded, it is necessary to fulfill in the design F < Pcr π 2 EI (10.22) 4h2 π  π 2 E (Ro2 − Ri2 ) 4 F < 4h2 where E is the Young Modulus of the material and I is the moment of inertia of the cross-sectional area. F <

Finally, the inner radius Ri must be greater than zero and less than Ro Ri ≥ 0 Ro > Ri Hence, the optimization problem can be formulated as: Minimize: S(Ro , Ri ) = ρhπ(Ro2 − Ri2 )

(10.23)

10.2 Problem formulation

262

10.2.4. Rectangular beam

As another example, consider now that a beam of solid rectangular cross-sectional area subject to shear and bending forces must be optimized (Arora, 1989). The bending and shear stress in the beam can be computed from 6M (10.24) bd2 3V τ= (10.25) 2bd where σ is the bending stress, τ is the shear stress, M and V are the moment and shear force acting on the beam and b and d are the base and depth of the rectangular cross-section. Another restriction is that the depth of the beam must not exceed twice its base. σ=

The above problem can be easily formulated as follows. First, as the beam must have a minimum cross-section, the objective function becomes cross-section = b × d Second, since the beam must withstand a bending stress M , then 6M bd2 where σa is the maximum allowable bending stress for the beam. σa >

3V 2bd where τa is the maximum allowable shear stress for the beam. τa >

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(10.28)

With the above information, the problem can be summarized into minimize: S(b, d) = bd

F < σY π(Ro2 − Ri2 ) π  π 2 E (Ro2 − Ri2 ) 4 > F 4h2

(10.27)

Third, since the beam must also withstand shear stress, then

subject to: Ri ≥ 0 Ro > Ri

(10.26)

subject to: d < 2b 6M σa > bd2 3V τa > 2bd c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

263

10.3 Solution by the graphical method

Solution by the graphical method

One simple yet effective method to solve optimization problems is the so called graphical method. This method relies in plotting the restrictions and objective function to determine regions, curves and points of optimal solution. The method is one of the best ways to gain insight of the problem at hand. Unfortunately, as it relies on plotting all the different functions and restrictions, it is only of practical use with two-variables problems and in some cases threevariable problems. To explain how the graphical method works, the last three optimization problems shown in the last section will be solved by this method. 10.3.1. Beverage can From the previous section, the beverage can problem problem can be formulated as: Minimize: S(d, h) =

π 2 d + πdh 4

264

18

(10.29)

16 height [cm]

10.3

10.3 Solution by the graphical method

Feasible region 14 12 10 8

r5 S=250 cm² S=315 cm² S=400 cm² 3.5

4

4.5

5 5.5 6 6.5 diameter [cm]

7

7.5

8

Figure 10.6: Graphical solution of the beverage can.

subject to restrictions r1 to r5 : plotted as a function of d r1 r2 r3 r4 r5

d ≥ 3.5 d ≤ 8.0 h ≥ 8.0 h ≤ 18.0 π = 400 = d2 h 4 = = = =

(10.30) (10.31) (10.32) (10.33) (10.34)

The first step towards the solution of an optimization problem is to identify the variables that will be used to plot. For this problem, diameter will be used as the abscissas axis (x-axis) and height will be used as the ordinates axis (y-axis). The second step to solve an optimization problem graphically is to plot all the restrictions involved in the problem. In this case, restrictions r1 to r4 are limits to the solution domain. Hence, any possible solution must be in the windows 3.5 ≤ d ≤ 8.0 and 8.0 ≤ h ≤ 18.0. Such solution domain is shown in figure 10.6. The last restriction r5 , which sets a volume restriction of 400cm3 , can be c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

400 h(d) = π d2 4

(10.35)

From figure 10.6 it can be seen that the restriction divides the domain window in two parts. Below the curve r5 lay all the possible solution where volume is less than 400cm3 . Above the same curve, lay all possible solutions where V is larger than 400cm3 . As practically speaking the can may have a volume of more than 400cm3, the feasible region for the solution is the region to the right of the r5 curve. If, on the other hand, the can must have exactly a volume of 400cm3 , then all possible solutions lay on the curve r5 . The third step is to plot different possible solutions that fulfill the objective function S(d, h). In order to plot possible solutions, it is necessary to assume a given final surface area. The objective function 10.29 can be plotted as π 2 d 4 πd

S− h(d) =

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(10.36)

265

10.3 Solution by the graphical method

10.3 Solution by the graphical method

where S is the value chosen as a guess for the final surface.

To verify the selected point, d = 8 and h = 8 can be substituted in the volume equation. Using equation 10.19, it can be verified that the volume gives 402cm3 , which closely satisfy the volume restriction. These two values give a final surface area of 251cm2 . If the proposed solution is not accurate enough, then more accurate values for the diameter and the height can be obtained plotting the area near point (8,8).

outer radius [m]

0.1

In figure 10.6, chosen values of S = 250, S = 315 and S = 400 where chosen as possible solutions. As all these curves represent solutions to the optimization problem, it is necessary to check which curve give values of d and h that are in the feasible region and are closer to the restriction curve r5 . For this problem, it seems like the point d = 8 and h = 8 give a possible solution as it lays on the curve r5 and intersects with the possible solution curve S = 250.

0.08 Feasible region 0.06 0.04 r1 r2 r3 S=10 kg S=20 kg

0.02 0 0

10.3.2. Tubular column problem

Consider the tubular column problem described in the previous section. Furthermore consider that the column must have a height of 5m, support a load of 50kN and that it will be build from steel having the following properties: E = 110GPa, σY = 120MPa and ρ = 2700 kg/m3 . As stated in the previous section, this problem can be stated as: Minimize: S(Ro , Ri ) = ρhπ(Ro2 − Ri2 )

(10.37)

r 0 = Ri ≥ 0 r 1 = Ro > Ri F r2 = < σY 2 π(Ro − Ri2 ) π  π 2 E (Ro2 − Ri2 ) 4 r3 = >F 4h2

(10.38) (10.39)

subject to:

(10.40)

(10.41)

In order to plot restrictions and possible solutions, it is possible to select the inner radius as abscissas and the outer radius as ordinates. The r0 and r1 restrictions limit the solution domain to the first quadrant of the coordinate c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

266

0.02

0.04 0.06 inner radius [m]

0.08

0.1

Figure 10.7: Graphical solution of the tubular column. system. If plotted, the r1 restriction will draw a 45◦ from (0,0) limiting the feasible region to the upper part of the quadrant as shown in figure 10.7. The r2 and r3 restrictions will further limit the solution domain, although the restriction will only be of importance for values of the inner radius between 0 and 0.04 since farther away the three restrictions are very close together. Restrictions r2 and r3 can be plotted respectively as  F + Ri2 (10.42) Ro (Ri ) = πσY  2 4 16F h (10.43) Ro (Ri ) = + Ri4 3 π E Finally, curve lines for solutions S = 10 and S = 20 were plotted to look for points of optimal solution. These two solutions are shown in figure 10.8 where only the area of interest is shown. From the plots it can be observed that the optimal solution must lay between 10 and 20kg since both curves cross the curve of restriction r3 . Hence, the c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

267

10.3 Solution by the graphical method

10.3 Solution by the graphical method

0.03

268

1

r1 r2 r3 S=0.1 m² S=0.2 m²

Feasible region 0.8

0.02

depth [m]

outer radius [m]

0.025

0.015 r1 r2 r3 S = 5 kg S = 10 kg S = 20 kg

0.01 0.005 0 0

0.6 Feasible region 0.4 0.2 0

0.005 0.01 0.015 0.02 0.025 0.03 inner radius [m]

0

0.2

0.4 0.6 width [m]

0.8

1

Figure 10.8: Area of interest for the tubular column problem.

Figure 10.9: Graphical solution of the rectangular beam.

optimal value for the inner radius must be between 0.02 and 0.04 meters and for the outer radius between 0.02 and 0.04 meters. Further plots of S with values for the inner radius slightly smaller than 10kg would help to find the point of optimal solution.

As both base and width must be positive, then the solution domain is restricted to the first quadrant of the coordinate system, which in this case was selected to have the width in the abscissas and depth as the ordinates. The first restriction, d < 2b, draws a straight dividing line limiting the feasible region to the lower part of the quadrant as shown in figure 10.10. The second and third restrictions, plotted as  6M d(b) = (10.48) σa b

10.3.3. Rectangular beam problem

The problem of finding the minimum cross-sectional area of a rectangular beam subject to a bending moment and shear force can be formulated as S(b, d) = bd

(10.44)

and

subject to:

d(b) = r1 = d < 2b 6M r2 = σa > 2 bd 3V r3 = τa > 2bd as it was discussed in the previous section. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(10.45) (10.46) (10.47)

3V τa 2b

(10.49)

further limit the region of feasible solution. Consider now the values of M = 40×103 Nm, V = 150×103N, σa = 10×106Pa and τa = 2 × 106 Pa. After plotting the three restrictions, the feasible solution is bounded first by r1 . Then, it is bounded by r3 until it crosses the curve of r2 where the latter becomes the limit of the feasible region. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

269

10.4 Lagrange multipliers

10.4 Lagrange multipliers

270

For an extremum of the function f (x, y) to exist on g(x, y), the gradients of f and g must line up. The gradient is a vector that shows the direction that the function increases. This vector is horizontal, i.e. it has no z-component. Consider the example shown in figure 10.10. The bold line shows the interception between the surfaces f (x, y) = xy 2 and g(x, y) = x2 + 2y. The extremum of f subject to g can be found finding the directions of the gradients of f and g.

6 0 −6

2 1.5 1

−2

0.5 −1.5

0 −1

−0.5

0

0.5

1

1.5

−0.5 −1 −1.5 −2

Figure 10.10: Intersection between f (x, y) = xy 2 and g(x, y) = x2 + 2y. After plotting possible solutions with S = 0.1m2 and S = 0.2m2, it is clear that all solutions are “parallel” to r3 and hence, the curve describing this restriction holds all possible solutions for the problem between the crossings with r1 and r2 . Any point (b, d) laying in this curve between these two crossing will be a point of optimal solution.

10.4

Lagrange multipliers

Lagrange multipliers or Lagrangian multipliers are a very powerful method to deal analytically with mathematical optimization problems with constraints. Consider that it is necessary to find the local maximum or minimum point of a function of several variables subject to one or more constraints. In order to find this local extrema, the method introduces an unknown scalar variable, called the Lagrange multiplier, for each one of the constrains. These new variables are used to generate a linear system of equations that can be solved to find the local extremum. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

To show how Lagrange multipliers work, consider a simple example in two dimensions. In order to minimize a function f (x, y) it is necessary to set its partial derivatives to zero ∂f =0 ∂x

∂f =0 ∂y

(10.50)

and solve simultaneously the two equations to find the point (x, y) that maximize or minimize f . If there is a constraint g(x, y) = 0, then g(x, y) is a line or curve in the x, y plane. The problem is to find find the maximum/minimum f along g. The point that maximizes/minimizes f subject to g has two properties: • The gradient ∇f is perpendicular to the line of constant g(x, y) since f is stationary here. • The gradient ∇g is also perpendicular to the line of constant g(x, y) since by construction g(x, y) = 0. Hence, if ∇f and ∇g may have different magnitudes but must point in the same direction, therefore it is possible to write ∇f = −λ∇g

(10.51)

where λ is a proportionality constant called Lagrange multiplier. The expression ∇f + λ∇g = 0 is usually expressed as ∇F = 0 since ∇F = 0, then all partial derivatives of F are zero. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(10.52)

271

10.4 Lagrange multipliers

In order to solve the problem it is just necessary to set ∂F ∂x ∂F ∂y ∂F ∂λ

10.4 Lagrange multipliers 10.4.1. Inequalities and Lagrange multipliers

= 0

272 In many occasions, constraints come in the form of an inequality. Lagrange multipliers can also accommodate these common cases when a constraint is

given in the form = 0 = 0

g(x) ≤ 0 (10.53)

and solve simultaneously these three equations.

For these cases, it is possible to transform an inequality to an equality using what is called a ’slack variable’. Slack variables transform the constraints into gi + s2i = 0

When there is more than one restriction, ∇F becomes ∇f + λ1 ∇g1 + λ2 ∇g2 + · · · + λn ∇gn = 0

(10.54)

(10.60)

(10.61)

where s2 takes care of the fact that s2 must be always non-negative since the constraint is of the form g(x) ≤ 0.

To solve the problem, the derivatives become ∂F = 0 ∂xi ∂F = 0 ∂λj

There is an additional necessary condition for the Lagrange multipliers of the form ”” given as λj  0

As an example, consider that the function f (x1 x2 ) = (x1 − 1.5)2 + (x2 − 2)2

(10.56)

(10.57)

For the above functions, F is given by F = (x1 − 1.5)2 + (x2 − 2)2 + λ(x1 + x2 − 2)

where λj is the Lagrange multiplier for the jth inequality constraint. If the constraint is inactive at the optimum, its associated Lagrange multiplier is zero. If it is active (gi = 0) then the associated λj must be non-negative.. Consider now the function used in the previous example

wants to be minimized subject to the condition g(x1 x2 ) = x1 + x2 − 2 = 0

f (x1 , x2 ) = (x1 − 1.5)2 + (x2 − 1.5)2 In this occasion it is desired to minimize f subject to the inequality g(x1 , x2 ) = x1 + x2 − 2 ≤ 0

(10.64)

Using the slack variable s, the constraint g can be expressed as g(x1 , x2 ) = x1 + x2 − 2 + s2 ≤ 0 (10.59)

The above three equations can be solved simultaneously yielding x1 = 0.75 and x2 = 1.25 with λ = 1.5. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(10.63)

(10.58)

and the partial derivatives are therefore ∂F = λ + 2x1 − 3 = 0 ∂x1 ∂F = λ + 2x2 − 4 = 0 ∂x2 ∂F = x1 + x2 − 2 = 0 ∂λ

(10.62)

(10.55)

(10.65)

and therefore F becomes F (x1 , x2 ) = (x1 − 1.5)2 + (x2 − 1.5)2 + λ(x1 + x2 − 2 + s2 ) c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(10.66)

273

10.4 Lagrange multipliers

10.5

The partial derivatives of F with respect to xi , λ and s are ∂F ∂x1 ∂F ∂x2 ∂F ∂λ ∂F ∂s

10.5 Examples

= λ + 2x1 − 3 = 0

Examples

10.5.1. Beverage can

= λ + 2x2 − 4 = 0

Consider once more the beverage can problem discussed in the previous sections. The problem can be formulated as minimizing: S(d, h) =

= x1 + x2 − 2 + s2 = 0 = 2λs = 0

274

(10.67)

s2 = 0;

x1 = 0.75;

x2 = 1.25

(10.68)

From the above example, the solution procedure can be stated as finding ∂F = 0 ∂xi ∂F = 0 ∂λj ∂F = 0 ∂sn

r1 = d ≥ 3.5

(10.71)

r2 = d ≤ 8.0

(10.72)

r3 = h ≥ 8.0

(10.73)

r4 = h ≤ 18.0 π r5 = 400 = d2 h 4

(10.74)

π 2 d + πdh 4

π 2 dh 4 As F = f + λg, the partial derivatives of F are g := 400 −

and then solve the resultant system of equations. For problems involving inequalities it is important that the Kuhn-Tucker (KT) necessary conditions are fulfilled including that f and gi are functions with continuous first partial derivatives along the curves gi = 0 and ∇gi = 0 at any point on the curve. These conditions ensures that the solution of the Lagragian multipliers is a minimum or maximum.

(10.75)

The problem can be easily solved using Lagrange multipliers. Suppose that at this time that there are no minimum or maximum diameter and height restrictions. For this specific case, f and g become f :=

(10.69)

(10.70)

subject to restrictions r1 to r5 :

Solving the above system of equations yields the following solution λ = 1.5;

π 2 d + πdh 4

(10.76) (10.77)

πd πdhλ + hπ − 2 2

∂F ∂d

=

∂F ∂h

= πd −

∂F ∂λ

= 400 −

πd2 λ 4 πd2 h 4

(10.78)

One very useful property of Lagrange multipliers is that the larger the value of λ, the larger the effect on the objective function optimum.

Solving the above equations for d, h and λ gives d = 10.06cm and h = 5.03cm for a total surface of 238.50cm2. Although this surface is less than the minimum

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

275

10.5 Examples

of 251cm2 found with the graphical method, the value for the diameter is not suitable for a beverage can.

10.5 Examples

276

subject to: r1 = d < 2b

Taking into account the restriction that the diameter of the can cannot be larger than 8 cm, the problem can be re-formulated in terms of two restrictions g1 and g2 π g1 := 400 − d2 h (10.79) 4 g2 := 8 − d +

s22

6M bd2 3V = τa > 2bd

r2 = σa >

(10.85)

r3

(10.86)

(10.80)

where s2 is the slack variable that considers the inequality in restriction r2 . With these restriction, F becomes

Using the above formulation and Lagrange multipliers the problem can be expressed as ∇f + λ1 ∇g1 + λ2 ∇g2 + λ3 ∇g3 = 0

F = f + λ1 g1 + λ2 g2

(10.84)

(10.87)

(10.81) where

and its derivatives are given by ∂F ∂d ∂F ∂h

=

f = bd

πdhλ1 πd − λ2 + hπ − 2 2

= πd −

πd λ1 4

g2

∂F πd2 h = 400 − ∂λ1 4

g3 = 2b − d + s23

∂F = s22 − d + 8 ∂λ2 ∂F = 2λ2 s2 ∂s2

6M + s21 bd2 3V = τa − + s23 2bd

g1 = σa −

2

Hence, the partial derivatives of F become (10.82)

∂F ∂b

= d + 2λ3 +

Solving the above equations yields d = 8 and h = 25/π = 7.96 for a total surface of 250.26cm2 which is the same value found with the graphical method.

∂F ∂d

= b − λ3 +

10.5.2. Rectangular beam problem

Consider now the problem of finding the minimum cross-sectional area of a rectangular beam subject to a bending moment and shear force. As discussed before, the problem can be stated as S(b, d) = bd c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(10.83)

6M λ1 3V λ2 + 2 b2 d2 2b d

12M λ1 3V λ2 + bd3 2bd2

6M ∂F = σa + s21 − 2 ∂λ1 bd 3V ∂F = τa + s22 − ∂λ2 2bd ∂F = 2b − d + s23 ∂λ3 c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(10.88)

277

10.6 Solving problems with Excel

10.6 Solving problems with Excel

278

∂F = 2λ1 s1 ∂s1 ∂F = 2λ2 s2 ∂s2 ∂F = 2λ3 s3 ∂s3

(10.89)

Solving for b and d using the values M = 40 × 103 Nm, V = 150 × 103 N, σa = 10 × 106 Pa and τa = 2 × 106 Pa yields b = 0.2371m and d = 0.4743m for a total cross sectional area of 0.1125m2.

10.6

Solving problems with Excel

Simple optimization problems can be solved quickly and easily using a spreadc In this sheet with a Solver tool as the one integrated to Microsoft Excel . section, the problems of the beverage can and the rectangular beam will be solved using this tool. In Excel, the solver must be activated on the Tools menu under Complements. After the solver has been activated, it will appear directly on the same Tools menu. 10.6.1. Beverage can In order to solve the problem, the first step is to add the data to Excel as it done normally. For clarity it is recommended to define separately the objective function, the variables and the restrictions as shown in figure 10.11. As the objective function for this problem is S(d, h) = π/4d2 + πdh, then the value of cell B4 will be =B7*B8. Note that cells B7 and B8 have at this time initial guesses for b and d. Restrictions R1 to R5 must also be computed from cells B7 and B8 as the solver will check that the values of these cells fulfills their corresponding restriction. Once the spreadsheet has been constructed, the next step is to call the solver tool. When selected, the Solver windows will appear as shown in figure 10.12. The solver requires four pieces of information in order to solve a problem. First, it needs to know which cell contains the objective function, in this case B4. Second, it needs to now if the solver must maximize the value of the objective c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Figure 10.11: Data for the optimization problem of a beverage can.

function, minimize it or make it equal to a given value. Next, the values that the solver can change must be specified. In the problem at hand, that is the cells containing the values of d and h, that is, cells B7 and B8. Finally, the solver must be made aware of the restrictions for the problem. In order to add restrictions to the solver, it is necessary to click the Add button at the right of the restrictions field. When clicked, a figure like the one shown in figure 10.13 will appear. To add a restriction is necessary to specify in what cell is the value that computes the restriction and the cell where the value of the restriction has been entered and the relation between them. For example, for the volume restriction, π/4d2 h = 400, the volume from the values of d and h is computed in cell B11 whereas the value of the restriction, 400, is specified in cell D11. As this restriction is an equality, the sign = must be selected from the drop-down menu between the cell fields. After all restrictions have been entered, the solver can be started clicking on the Solve button at the top right corner of the window. The solver will change the values of d and h until it finds a minimum value for the objective function. In this case the solver finds the values of d = 7.97cm and h = 8cm for a total surface area of 250.53cm. This value is consistent with with the solutions found through the graphical and Lagrange multipliers methods. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

279

10.6 Solving problems with Excel

10.6 Solving problems with Excel

280

Figure 10.14: Data for the optimization problem of a rectangular beam. Figure 10.12: Solver window with all the information needed to solve the problem of the beverage can.

10.6.2. Rectangular beam problem

The optimization of the cross-sectional area of the rectangular beam can be carried out in a similar fashion as the beverage can optimization problem. The spreadsheet for this problem is shown in figure 10.14. For this case, the spreadsheet includes also the data for the problem, that is, the momentum and shear force applied to the beam as well as the yield strength and shear yield of the material. After entering the objective function S(b, d) = bd in cell D4, the problem variables in cells D7 and D8 and the restrictions given in equations (10.84) in cells D13, D14 and D15, the problem is ready to solve. Figure 10.15 shows the solver windows with all the information of the problem. With the above information, Excel finds optimal values of b = 0.2890 and d = 0.3892 for a minimal cross sectional area of 0.1125 cm. Please not that this value is also consistent with the solutions found through the graphical and Lagrange multipliers methods.

Figure 10.13: Adding a restriction for the problem.

References 1. Arora, J.S. (1989) Introduction to optimal design. McGraw-Hill international editions.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

281

10.6 Solving problems with Excel

CHAPTER

11

Fracture Mechanics Figure 10.15: Solver window for the problem of a rectangular beam. 2. Papalambros, P.Y. & Wilde, D.J. (2000) Principles of optimal design. Modeling and computation. 2nd edition. Cambridge University Press. 3. Yang, L. & El-Haik, B. (2003) Design for Six Sigma. A roadmap for Product Development. McGraw-Hill.

Failure by fracture is theoretically limited by the ultimate strength of the material σu . Nevertheless, many products fail by fracture even if the stress part never reaches, by design, yield. The answer to this question is that small cracks in the product due to material, manufacture or use develop, lowering the resistance of the material. In 1921 Griffith established the foundations of fracture mechanics using the solution for the stress field around an elliptical flaw in an infinite plate developed by Inglis in 1913. Consider the elliptical flaw in an elliptical plate shown in Figure 11.1. For this case, the maximum stress occurs at (±a, 0) and is given by  a (11.1) σymax = 1 + 2 σ b If a = b, the ellipse becomes a circle and the maximum stress becomes 3σ. For a fine crack b/a → 0 and σ → ∞. To explain Griffith findings consider Figure 11.2. If a plate of unit thickness and containing a crack such that b/a ≈ 0 is loaded, its load-deformation curve would follow the line OAB. Assuming that when the load reaches Pc the deformation remains fixed and the crack grows such that a becomes a + da, then the cross section of the

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e CarlosMiranda. Todos los derechos reservados.

283

284 plate decreases and hence the load decreases to point D in order to keep the displacement fixed.

y σ

Since the area below the curve is equal to the strain energy in the plate, then the extension of the crack causes a release of strain energy equal to the area OAD.

a

x

Griffith criterion states that the energy released in extending the crack must be sufficient to provide the energy required to create the new surfaces of the propagating crack.

b

If the energy (work) necessary for crack growth is Wf and the strain energy is denoted by u, the condition for crack growth is given by

σ

d dWf (W p + u)  (11.2) da da where the condition of fixed deformation has been removed and Wp denotes the work due to the applied load.

Figure 11.1: Elliptical flaw in an infinite plate

The previous equation is usually expressed as GR

(11.3)

where G is the energy release rate and R is the rate energy for crack growth called crack resistance force.

Applied Load B

It has been shown that crack growth is stable until dG/da = dR/da, where stable means that the crack will grow in a stable, slow manner.

A

111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 D 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 O 111111111 C 000000000

PC

Unstable crack growth occurs when G  R and dG/da = dR/da. Here unstable means that growth will be fast.

E

From Inglis equations, Griffith found the energy release rate per crack to be G= Deformation

Figure 11.2: Griffith’s diagram. The line OAB represents the deformation– applied curve before deformation. The line ODE represents the curve with a crack extension da.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

πσ 2 a E

(11.4)

Griffith’s theory is in well accordance to experimental work with brittle materials where the energy release upon crack formation exceeds the energy necessary to form the crack surface. For ductile materials, where the energy needed to perform plastic deformation at the crack tip is more critical than the energy needed to create new surfaces, Griffith’s theory needs some corrections. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

285

11.1 Crack Modes y

y

11.2 Stress intensity factor

286 y

y

σ

x

z

x

x

z

z

dxdy r, θ x 2a

Mode I Tension, Opening

Mode II In-Plane Shear, Sliding

Mode III Out-of-plane Shear, Tearing

σ

Figure 11.3: Three fracture modes.

11.1

Crack Modes

There are three modes of crack propagation which depends on the loading and the crack surface displacements. The three cases are:

Figure 11.4: Mode I crack model.

11.2

Stress intensity factor

Consider the Mode I crack model shown in Figure 11.4. It has been shown that the stress field on a dxdy element in the vecinity of the crack tip is given by the following expressions

Mode I or opening mode, when the crack faces are pulled apart. 

Mode II or Sliding mode, when the crack surfaces slide over each other.

σx

Mode III or Tearing mode, when the crack surfaces move parallel to the leading edge of the crack and relative to each other.

σy τxy

The three modes are shown in Figure 11.3. Since the most common mode found in engineering applications is Mode I, this mode will be consider in what follows. Similar developments can be done for modes II and III. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

σz

 θ 3θ 1 − sin sin = σ 2 2 

 θ a θ 3θ 1 + sin sin = σ cos 2r 2 2 2  a θ θ 3θ sin cos cos = σ 2r 2 2 2 ⎧ ⎨ 0 if plane stress = ⎩ ν(σx + σy ) if plane strain a θ cos 2r 2

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(11.5)

287

11.2 Stress intensity factor

The stress near the tip when θ = 0 is therefore given by  a σy |θ=0 = σ 2r

11.3 Crack growth

288 σ

(11.6) b

It is common practice to define a factor KI , called stress intensity factor. This factor defines the magnitude of the local stresses aroud the crack tip and its given by √ (11.7) KI = σ πa √ √ where KI has units of Pa m or kpsi in.

2a

σ

√ KI = C1 σ πa

The stress intensity factor depends on factors such as the loading conditions, the size of the crack, the crack shape and the geometric boundaries of the surrounding material. It is important that the reader do not confuse the stress intensity factor KI with the static stress concentration factor Kt . With the stress intensity factor definition, equations (11.5) can be rewriten as

 θ θ KI 3θ 1 − sin sin cos σx = √ 2 2 2 2πr

 θ θ KI 3θ 1 + sin sin cos σy = √ 2 2 2 2πr θ KI θ 3θ sin cos cos τxy = √ 2 2 2 2πr

C1 = (1 − 0.1η 2 + 0.96η 4 )



sec(πη)

Figure 11.5: Plate in tension with center crack. As an illustration, figures 11.5 to 11.8 show four of the most common Mode I crack cases. Each case state the value of its stress intensity factors. In all cases η = a/b.

(11.8)

11.3

(11.9)

As mentioned before, stress intensity factors depend on the geometry of the component, size and shape of the crack and the loading acting on the part. The Mode I cases shown prove this dependency.

From the above equation it is clear that the Griffith energy release rate G is proportional to the stress intensity factor KI .

From the figures and the values of KI , one can compute the value when the crack growth will become unstable by comparing KI with the critical stress intensity factor KIc .

11.2.1. Some Mode I stress Some mode I stress intensity factors are readintensity factors ily available for a variety of loading conditions and crack geometries. The principle of superposition can be applied to stress intensity factors of the same loading mode. Hence, stress intensity factors of the same mode can be added algrebraically.

The critical stress intensity factor KIc is a material property that depends on the crack mode, material, temperature, loading rate and the state of stress at the crack. Some valies for KIc are given in the table above for different materials at room temperature. The critical stress intensity factor is also called the fracture toughness of the material.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Crack growth

and equation (11.4) can be expressed as K2 G= I E

289

11.3 Crack growth

11.3 Crack growth

290 M

σ b b a

a

a

σ

√ KI = C2 σ πa  tan(πη) C2 = [1 − 0.122 cos4 (πη)] πη Figure 11.6: Plate in tension with double-edge crack.

M √

6M tb2   0.923 + 0.199 [1 − sin (πη/2)]4 2 πη C4 = × tan( ) cos (πη/2) πη 2 KI = C4 σ πa

σ=

Figure 11.8: Plate of thickness t in bending with a single-edge crack. σ

The fracture toughness for plane strain is normally lower than that for the plane stress. For this reason KIc is usually termed ”mode I, plane strain fracture toughness”. As plane stress conditions always exists on the free surface perpendicular to the crack surface, it is always necessary to check if plane stress or plane strain conditions will dominate.

b

The ASTM recommends for plane strain conditions that the thickness, t, is such that 2

KIc (11.10) t  2.5 σY

a

σ C3 =

√

KI = C3 σ πa 0.752 + 2.02η + 0.37 [1 − sin (πη/2)]3 cos (πη/2)



 ×

 πη  2 tan πη 2

Consider for example the values for σY and K√ Ic for the aluminum alloy 2014T6. Considering an average KIc = 24 MPa· m, the thickness required for plane strain conditions to dominate is

t  2.5

Figure 11.7: Plate in tension with single-edge crack.

t  2.5

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

KIc σY

2

√  24 MPa · m = 0.136 m 440 MPa

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

(11.11)

291

11.3 Crack growth Material Carbon steel

√ σY (MPa) KIc (MPa· m) 245

>225

◦

1600

44-66

◦

1400

79-91

4340 (205 temper) 4340 (425 temper) Ti-GAL-4v-25n

800

110

2014-T6

440

18-30

Table 11.1: Typical KIc values at room temperature. or 136 mm. A large value indeed! If the thickness of the material is less than the t found by the previous equation, then plane stress conditions will arise causing yields at the crack tip. If appreciable yielding occurs at the crack tip, then the value for the actual stress intensity factor may be quite large compared to the original KIc . If plane stress conditions are present, the effective fracture toughness can be aproximated by

4 1.4 KIc (11.12) KIceff = KIc 1 + 2 t σY It is also necessary to check that the plastic zone is small comapared to the crack length. The crack size limitation is given by the same requiremnent for the thickness 2

KIc (11.13) a  2.5 σY

11.3 Crack growth

292

Consider first the problem of finding the highest load P . Since inestable crack growth begins when KI = KIc √ KIc = C1 σ πa √ where C1 = (1 − 1.01η 2 + 0.96η 4 ) sec πη and η = a/b. Solving for σ σ=

KIc √ C1 πa

and substituting numerical values 60 √ = 47.90 ksi 0.9993 π0.5 where the value of C1 = 0.9993 was previously calculated. σ=

The highest load can be found from P = σA, P = σ(6 × 0.06) = 17.24 kpi The critical crack length acr can be found from the relationship KIc = KI . Considering a plate in tension with a central crack, it is possible to write √ C1 σ πacr = KIc √ where for this case C1 = [1 − 0.1η 2 + 0.96η 4 ] sec πη and η = acr /b. The equation resultant from substituting C1 into KIc = KI is

  a 2  a 4    a  √ cr cr cr 1 − 0.1 + 0.96 sec π σ πacr KIc = b b b which must be solved for acr .

If this relationship is not fulfilled, then the value of KIc may be larger than the actual value. 11.3.1. Example

A plate of whith b = 6 in and thickness t = 0.06 in is made of aluminum 7075-T651 (σY = 70 ksi and KIc for plane √ stress 60 ksi in). Estimate: a) The highest load P that can be applied without causing unstable crack growth considering a central crack of 2a = 1 in. b) The critical crack length a if σ = 0.75σY . c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

The above equation nannot be solved analytically and must be solved√through iterations supposing a given acr and verifying if KIc is equal to 60 ksi in. The process can be carried out esily with the help of a solver application such the one integrated in most spreadsheets. √ Using the problem’s data σY = 0.75(70) ksi, b = 6 in, and KIc = 60 ksi in, acr = 0.4067 in If the above above result is substituted back into the equation for KIc , a value of 60 k arises, verifying the solution obtained. c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

293

11.4 Plastic zone correction

11.4 Plastic zone correction

294

in terms of principal stresses: σ1

y rp ∗

σ2

θ z

a

x

Free surface Free surface enlarged view

σ3

 θ 1 + sin 2

 θ KI θ 1 − sin = √ cos 2 2 2πr ⎧ ⎪ if plane stress ⎨ 0 = KI θ ⎪ ⎩ 2ν √ cos if plane strain 2 2πr θ KI cos = √ 2 2πr

(11.14)

To find rp ∗ consider first the case of plane stress. Applying von Mises criteria, yield occurs when

 3 0.5 1 + sin2 θ + cos θ = σY 2

KI √ 2πr

Figure 11.9: Plastic zone at crack tip.

(11.15)

Solving for r gives the extent of the plastic zone as a function of θ resulting in rp ∗ (θ) =

11.4

1 4π

KI σY

2  3 1 + sin2 θ + cos θ 2

(11.16)

Plastic zone correction When θ = 0 the above equation simplifies to

As discussed before, equations based on the mode I crack model predict an infinite stress at the crack tip. The infinite stress does never occur in reality as materials will experience plastic deformation once the stress reach the yield point. Consider the plastic zone diagram shown in Figure 11.9. The extent of the plastic zone is given by the vector rp ∗ (θ). Although at the free surfaces a state of plane stress exists, within the plate a state of plane strain dominates if the plate is thick enough. Because of this change of conditions, rp ∗ (θ) will vary along the thickness of the plate.

1 rp ∗ = 2π

KI σY

2 (11.17)

The above procedure can be repeated for plane strain conditions giving rp ∗ (θ) =

1 4π

KI σY

2 

3 2 sin θ (1 − 2ν)2 (1 + cos θ) 2

 (11.18)

In order to find the extent of rp∗ it is convenient to express equations (11.8)

If θ = 0 and assuming ν = 1/3, rp ∗ for plane strain conditions simplifies to simplifies to

2 KI 1 (11.19) rp ∗ = 18π σy

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

295

11.4 Plastic zone correction

The above expression can lead to some interesting results. Using equations (11.14) for plane strain at θ = 0, σ2 becomes equal to σ1 and σ3 = 2νσ1 . Substituting these results and ν = 1/3 into von Mises equation given by  # 1" (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 (11.20) σY = 2 the result 3σY = σ1 (11.21) arises predicting that σ1 will be three times the yields stress at the time of yielding. As plane stress conditions must occur at the free inner surface independently of other conditions, σ1 = σY at θ = 0. Nevertheless, as r increases, the stress rises to 3σY very quickly. As the average stress in the plastic zone will be something between σY and 3σY , it has been suggested to use for plane strain the following expression

2 1 KI (11.22) rp ∗ = 6π σY Once rp∗ has been estimated, the effective crack length used for the determination of stress intensity factors becomes aeff = a + rp∗ 11.4.1. Example

(11.23)

Consider once more the plate of thickness t = 0.06 in and with b = 6 in made of aluminum 7075-T651. Estimate:

a) The highest load P that can be applied without causing unstable crack growth considering a central crack of 2a = 1 in taking into account the plastic zone correction. b) The critical length a if σ = 0.75σY taking into account the plastic zone correction.

11.4 Plastic zone correction

296

Remembering that aeff = a + rp ∗ and since plane strain conditions apply

2 1 KI aeff = a + 6π σy

2 1 60 = 0.5 + 6π 70 = 0.5 + 0.0389 = 0.5389 The stress at which unstable crack growth begins is σ=

KIc √ C1 πaeff

√ where C1 = (1 − 0.1η 2 + 0.96η 4) sec πη with η = aeff /b. Substituting numerical values σ=

60  = 44.32178 ksi 1.0404 π(0.5389)

and from P = σA P = 44.3218(6 × 0.06) = 15.956 kpi The initial crack length for σ = 0.75σY can be found from √ KIc = C1 σ πaeff where aeff indicates that the plastic zone correction has been considered. From the above formula

2 1 KIc aeff = π C1 σ and substituting the values for the problem yields

2 60 1 = 0.4157 aeff = π 0.75 × 70

c) Compare results with those obtained without the plastic zone correction. As in the previous example, unstable crack growth begins with KI = KIc . When plastic deformation is being taken into account, the previous equation transforms into √ KIc = C1 σ πaeff c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

Since aeff = a + rp ∗, then a = aeff − rp∗

2 60 1 a = aeff − 6π 0.75 × 70 = 0.4157 − 0.0693 = 0.3464 c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.

297

11.4 Plastic zone correction

From this result, unstable crack growth will begin when σ = 0.75σY if the crack has a length of 0.3464 in. With these results it is possible to compare the behavior of the crack in brittle and ductile materials. In the case of the highest load that can be applied, in the case of brittle materials this would be of 17.24 kpi whereas for ductile materials it would be of 15.95 kpi or 16.3% less force. For the critical crack lenght if σ = 0.75σY , for brittle materials it would be of 0.4067 in whereas for ductile materials it would be of 0.3464 in or 44.5% shorter. From the above results it can be said that for the same load brittle materials will withstand larger cracks, and consequently, for the same crack size brittle materials will support a larger crack.

References 1. Arora, J.S. (1989) Introduction to optimal design. McGraw-Hill international editions. 2. Papalambros, P.Y. & Wilde, D.J. (2000) Principles of optimal design. Modeling and computation. 2nd edition. Cambridge University Press. 3. Yang, L. & El-Haik, B. (2003) Design for Six Sigma. A roadmap for Product Development. McGraw-Hill.

c Copyright 2006 Dr. Jos´ e Carlos Miranda. Todos los derechos reservados.