Introduction To Mechanism And Kinematics

Introduction to mechanism and kinematics Objectives Up on completion of this chapter, the student will be able to 1. Explain the need for kinematic analysis of mechanism. 2. Define the basic components that comprise a mechanism. 3. Draw the kinematic diagram from a view of a complex mechanism. 4. Compute the number of degrees of freedom of a mechanism. 5. Identify a four bar mechanism and classify it according to its possible motion. 6. Identify a slider crank mechanism. Mechanism and machines Introduction The subject theory of machine deals with the study of motion and forces in machinery devices that provide a wide variety of functions. The subject matter treated inhere includes synthesis and analysis of machinery. The study of a mechanism involves its analysis as well as synthesis Analysis is the study of motions and forces concerning different parts of an existing mechanism, where as synthesis involves the design of its different parts In a reciprocating engine, the displacement of the piston depends up on the length of the connecting rod and the crank. It is independent of the bearing strength of the parts or whether they are able to withstand the forces or not. Thus, for the study of motion, it is immaterial if a machine part is made of mild steel, cast iron or wood. It is not necessary to know the actual shape and area of cross section of the part. The study of mechanisms, therefore, can be divided in to the following disciplines. Kinematics It deals with the relative motions of different parts of a mechanism without taking in to consideration forces producing the motions. It also deals with the way things move. Formally defined, it is the study of the geometry of motion. Kinematic analysis involves -

Determination of position

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Determination of displacement, rotation, speed, velocity and acceleration.

Dynamics It involves the calculations of forces impressed up on different parts of a mechanism. The forces can be either static or dynamic. Another major task in designing machinery is determining the effect of the forces acting in the machine Terminologies (basic definition) Mechanism: - is the mechanical portion of a machine that has the function of transferring motion and from a power source to an output. If a number of bodies are assembled in such a way that the motion of one causes constrained and predictable motion to the others, it is known as a mechanism. A mechanism transmits and modifies a motion. Machines: - are devices used to alter transmit and direct forces to accomplish a specific objective. •

A machine is a mechanism or combination of mechanisms which, Apart from imparting definite motions to the parts, also transmits and modifies the available mechanical energy in to some kind of desired work.

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A machine may also be defined to be a combination of resistant bodies whose relative motions are completely constrained, by means of which the natural energies at our disposal may be transformed in to any special from of works.

In the first place, then, a machine is a combination of bodies- a single body cannot constitute a machine. In each of what are often called the ‘simple machine’ for example the lever, wheel and axle etc… there are at least two bodies, in some more than two. The mare bar which we call a lever does not in itself constitute a machine either simple or otherwise.

n. 1. a. A device consisting of fixed and moving parts that modifies mechanical energy and transmits it in a more useful form. b. A simple device, such as a lever, a pulley, or an inclined plane, that alters the magnitude or direction, or both, of an applied force; a simple machine. 2. A system or device for doing work, as an automobile or a jackhammer, together with its power source and auxiliary equipment. 3. A system or device, such as a computer, that performs or assists in the performance of a human task: The machine is down.

4. An intricate natural system or organism, such as the human body. 5. A person who acts in a rigid, mechanical, or unconscious manner. 6. An organized group of people whose members are or appear to be under the control of one or more leaders: a political machine. 7. a. A device used to produce a stage effect, especially a mechanical means of lowering an actor onto the stage. b. A literary device used to produce an effect, especially the introduction of a supernatural being to resolve a plot. 8. An answering machine: Leave a message on my machine if I'm not home. adj. Of, relating to, or felt to resemble a machine: machine repairs; machine politics. v., -chined, -chin·ing, -chines. v.tr. To cut, shape, or finish by machine. v.intr. To be cut, shaped, or finished by machine: This metal machines easily.

Planar mechanism: - a mechanism its motion is limited to two dimensional spaces. Plane motion when a body moves so that any one selection parallel to the first moves also in its own plane, and the motion of the body is said to be ‘co-plane’ ‘coplanar’ or we may call it simply ‘plane’.

Mechanism terminology

Linkage: - is a mechanism where all parts are connected together to form a closed chain Links: - are individual parts of a mechanism. They are considered rigid body (bodies) and are connected with other links to transmit motion and forces. Link also defined as resistant body or a group of resistant bodies with rigid connections preventing their relative movements.

A link is also known as kinematic link or element Link can be classified in to binary, ternary, quaternary etc, depending up on their ends on which revolute or turning pairs can be placed Simple link: - is a rigid body that contains only two joins, which connect it to other links. Complex link: - a rigid body that contains more than two joints. Frame: - it serves as the frame of reference for the motion of all other parts. The frame is typically a part that exhibits no motion. Joint: - is a movable connection between links and allows relative motion between the links. The two primary joint also called full joint, are the revolute and sliding joint •

The revolute also called a pin or a hinge joint.

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The sliding joint is also called a piston or a prism joint.

Higher order joint: - permits for both rotation and sliding between the two links that it connects [Also called half joint]. Cam and gear connections are higher order joint. Crank: - is a simple link that is able to complete a full rotation about a fixed center. Rocker: - is a simple link that oscillates through an angle, reversing its direction at certain intervals. A rocker arm: - is a complex link, containing three joints, that is pivoted near its center. A bell crank: - is similar to a rocker arm, but is bent in the center. A point of interest: - is a point on a link where the motion is special interest. Once kinematic analysis is performed, the displacement, velocity, and accelerations of that point are determined. Rigid and resistant bodies A body is said to be rigid if under the action of forces, it does not suffer any distortion or the distance between any two points it remain constant Resistant bodies are those which are some semi rigid bodies which are normally flexible, but under certain loading conditions acts as rigid bodies for the limited purpose.

Actuator: - is the component that drives the mechanism. Common actuator includes •

Motor(electric and hydraulic)

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Engine, cylinder(hydraulic and pneumatic)

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Solenoid.

Kinematic pair Kinematic pair or simply a pair is a joint of two links having relative motion between them Types of kinematic pairs Kinematic pairs can be classified according to

i.

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Nature of contact

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Nature of mechanical constraint

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Nature of relative motion Kinematic pair according to nature of contact

a. Lower pair: - a pair of links having surface or area contact between the members. The contact surfaces of the two links are similar. E.g. nut turning on a screw b. Higher pair:- when a pair has a point or a line contact between the links. The contact surfaces of the two links are dissimilar E.g. wheel rolling on a surface, ball and roller bearing II.

Kinematic pairs according to nature of mechanical constraint a. Closed pair When the elements of a pair are held together mechanically all the lower pairs and some of the higher pairs are closed pairs. b. Unclosed pair When two links of a pair are in contact either due to force of gravity or some spring action, they constitute an unclosed pairs.

iii.

Kinematic pairs according to nature of relative motion

a. Sliding pair If two links have a sliding motion relative to each other, they form a sliding pair b. Turning pair When one link has a turning or revolving motion relative to the other, they constitute a turning or revolving pair c. Rolling pair When the links of a pair having a rolling motion relative to each other, they form a rolling pair. E.g. ball bearing d. Screw pair(helical pair) If two mating links have a turning as well as sliding motion between them, they form a screw joint. e. Spherical pair When one link in the form of a sphere turns inside a fixed link, it is a spherical pair. E.g. the ball and socket joint

Pin joint

Sliding joint

Complex link Simple link

Simple link with two points of interest and their schematic representation

Kinematic diagram: - it shows the schematic diagram of a mechanism. Kinematic diagram should be drawn to scale proportional to the actual mechanism. Kinematic inversion: utilizing alternate link to serve as the fixed link is termed as kinematic inversion The relative motion of the links doesn’t change with the selection of a frame, the choice of a frame link is often not important.

Inversion also defined as, if in a mechanism the link which was fixed is allowed to move and another link becomes fixed, the mechanism is said to be inverted. The inversion of a mechanism does not change the motion of its links relative to each other, but does change their absolute motion. The number of possible kinematic inversions is equal to the number of links in a mechanism. Transmission of motion Motion is transmitted from one member to another in three ways: a. By direct contact between two members; b. Through an intermediate link or a connecting rod; c. By flexible connector such as belt or chain. In transmission of motion one element of the mechanism must be a driver and another element must be a driven element or a follower. Kinematic chain It is an assembly of links in which the relative motions of the links is possible and the motion of each relative to the other is definite. A redundant chain doesn’t allow any motion of a link relative to the other Linkage, mechanism and structure A linkage is obtained if one of the links of a kinematic chain is fixed to the ground. Mobility The number of degrees of freedom of a mechanism is also called the mobility An important property in mechanism analysis is the number of degree of freedom of the mechanism. The degree of freedom: - is the number independent inputs required to precisely position all links of the mechanism with respect to the ground. It can also be defined as the number of actuators needed to operate the mechanism. An unconstrained rigid body moving in space can describe the following independent motion 1. Translational motion along any other mutually perpendicular axes x, y, z 2. Rotational motion about three axes

Thus a rigid body possesses six degree of freedom. Degree of freedom of a pair is defined as the number of independent relative motions, both translational and rotational, a pair can have Degree of freedom=6-number of restraint Degree of freedom of a mechanism in space can be determined as follow n= number of links in a mechanism F= degree of freedom P1= number of pairs having one degree of freedom P2= number of pairs having two degree of freedom In a mechanism, one link is fixed. Therefore Number of movable links=n-1 Number of degree of freedom of (n-1) movable links=6(n-1) Each pair having one degree of freedom imposes 5 restraints on the mechanism reducing its degree of freedom by 5P1 Each pair having two degree of freedom imposes 4 restraints on the mechanism reducing its degree of freedom by 4P2 Similarly other pairs having 3, 4, and 5 degrees of freedom reduce the degree of freedom of the mechanism. Thus F=6(n-1)-5P1-4 P2-3 P3-2P4-1P5 For planar mechanism the degree of freedom F=3(n-1)-2P1-P2 This is known as Grubler equation Most of the linkages are expected to have one degree of freedom so that one input to any of the links, a constrained motion of other is obtained 1=3(n-1)-2P1 P1=1.5*n-2 If n and P1 are whole number, the relation can be satisfied only if n is even For turning pair the following relations are valid

F=n-(2L+1) P1=n+ (L-1) Where L- number of loops in a linkage L 1 2 3 4 5

F n-3 n-5 n-7 n-9 n-11

P1 n n+1 n+2 n+3 n+4

For example, if a linkage, there are 4 loops and 11 links, its degree of freedom will be 2 and number of joints 14 If a linkage has 3 loops, it will require 8 links to have one degree of freedom, 9 links to have 2 degree of freedom Grubler’s equation Degree of freedom for planar mechanism joined with common joints can be calculated through Grubler’s equation F=degree of freedom=3(n-1)-2*Jp-2Jh Where n=total number of links in the mechanism Jp= total number of primary joints (pins or sliding joints) Jh= total number of higher joints (cam or gear joint)

The four bar linkage •

Four bar linkage is the simplest and most common linkage

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It is a combination of four links, one being designated as frame, and connected by four pin joint.

The mobility of a four bar mechanism consists of n=4,

Jp=4

F=3(4-1)-2*4

Jh=0 F=1

Because the four bar mechanism has one degree of freedom, it is constrained or fully operated with one driver The pivoted link that is connected to the driver or power source called the input link. The other pivoted link that is attached to the frame is designated the output link or follower. The coupler or connecting arm “couple” the motion of the input link to the output link. Grashof’s criterion The following nomenclature is used to describe the length of the four links S=length of the shortest link L=length of the longest link P=length of one of the intermediate link q=length of the other intermediate link Grashof’s theorem states that a four bar mechanism has at least one revolving link if

Conversely, the three non-fixed links merely rock if

Four bar mechanism fall in to one of the five categories listed as follow case 1 2 3 4 5

criteri on s+l

shortest link

category

frame

double crank

slide

crank rocker

coupler

double rocker

any

change point

any

triple rocker

1. Double crank or crank-crank It has the shortest link of the four bar mechanism configured as the fixed link or the frame. If one of the pivoted links is rotated continuously, the other pivoted link will also rotate continuously. If double crank mechanism is also called a drag link mechanism 2. Crank rocker It has the shortest link of the four bar mechanism configured adjacent to the frame. If this shortest link is continuously rotated the output link will oscillate between limits. Thus the shortest link is called the crank; the output link is called the rocker. 3. Double rocker The double rocker or rocker-rocker, it has the link opposite the shortest link of the four bar mechanism configured as frame. In this configuration neither link connected to the frame will be able to complete a full revolution. Thus, both input and out put links are constrained to oscillate between limits and called rocker. However, the coupler is able to complete a full revolution. 4. Change point mechanism

The change point mechanism can be positioned such that all the links become collinear. The most familiar type of change point mechanism is a parallelogram linkage. The frame and the coupler are the same length, and the two pivoting links are the same length. In that collinear configuration, the nption becomes indeterminate. The motion may remain in a parallelogram arrangement. Or cross in to anti parallelogram, or butterfly, arrangement, for this reason, the change point is called a singularity configuration. 5. Triple rocker In a triple rocker linkage, none of its links are able to complete a full revolution. All three moving links rock

Deltoid linkage In a deltoid linkage the equal links are adjacent to each other. When any of the shorter link is fixed, a double crank mechanism is obtained in which one revolution of the longer link causes two revolution of the other shorter link.

Slider crank mechanism Another mechanism that is commonly encountered is a slider crank. This mechanism also consists of a combination of four links, with one being designated as the frame. This mechanism, however, is connected by three pin joints and one sliding joints.

The mobility of a slider crank mechanism is represented by the following relation n=4,

Jp= (3pins+1sliding)

Jh=0

F=3(4-1)2*4 F=1 Because the slider crank mechanism has one degree of freedom, it is constrained or fully operated with one driver In general, the pivoted link connected to the frame is called the crank. This link is not always capable of completing a full revolution. The link that translates is called the slider. The coupler or connecting rod “couples” the motion of the crank to the slider. Simple machine A simple machine is a device that enables us to do work more easily. Simple machines help us accomplish a task faster or more conveniently i.

Machine transform energy

ii.

Some machines transfer energy from one place to another

iii.

Multiply force

iv.

Multiply speed

v.

Change direction of force

Type of simple machine •

lever

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Wedge

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Inclines plane

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Pulley

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Wheel and axle

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Screw

Laws of lever FE+LE=FL+LL Theory of simple machine i.

Mechanical advantage MA=Load/Effort

ii.

Velocity ratio VR=velocity of the effort/velocity of the load VR=distance of effort/distance of the load

iii.

Efficiency Efficiency=[used work output/useful work input]*100%

If VR=MA then the ideal machine Equivalent mechanism This can be done according to some set rules so that the new mechanisms also have the same degree of freedom and are kinematically similar

When D lies at infinity so link C moves perpendicular to link4. Spring in place of turning pairs

To find the degree of freedom of such a mechanism, the spring has to be replaced by the binary link Cam pairs In place of turning pair A cam pair has two degree of freedom. For linkage with one degree of freedom, application of grubler’s equation yields F=3(n-1)-2P1-P2

1=3n-3-2P1-1 P1=3n-5/2 This shows that to have one cam pair in a mechanism with one degree of freedom, the number of links and turning pairs should be as ……………………….

Slider crank mechanism When one of the pairs of a four bar chain is replaced by a sliding pair, it becomes a single slider crank chain or simply a slider crank chain. It is also possible to replace two sliding pairs of a four bar chain to get a double slider crank chain In a slider may be passing through the fixed pivoted O or may be displaced. The distance e between the fixed pivot O and the straight line path of the slider is called the off-set and the chain so formed an off-set slider crank chain. Different mechanisms obtained by fixing different links of a kinematic chain are known as its INVERSION

First inversion This inversion is obtained when link 1 is fixed and link 2 and 4 are made the crank and slider respectively Application:- reciprocating engine, reciprocating compressor

Second inversion Fixing of link 2 of a slider crank chain results in the second inversion

Application: - whit worth quick return mechanism, rotary engine

Third inversion

By fixing link three the slider crank mechanism, third inversion is obtained

Application:- oscillating cylinder, crank and slotted lever mechanism

Fourth inversion If link 4 of the slider crank mechanism is fixed, the fourth inversion is obtained

Application: - hand pump

Four bar linkage Transmission angle The angle µ between the output link and the coupler is known as transmission angle If link AB id the input link the force applied to the output link DC is transmitted through the coupler is when µ is 900. If links BC and DC becomes coincident, the transmission angle is zero and the mechanism would lock or jam From cosine law we have