Class 11 - Mathematical Modeling Of Pneumatic System
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System Modeling Coursework
Class 11: Modeling of Pneumatic systems
P.R. VENKATESWARAN Faculty, Instrumentation and Control Engineering, Manipal Institute of Technology, Manipal Karnataka 576 104 INDIA Ph: 0820 2925154, 2925152 Fax: 0820 2571071 Email: [email protected], [email protected] Web address: http://www.esnips.com/web/SystemModelingClassNotes
WARNING! • I claim no originality in all these notes. These are the compilation from various sources for the purpose of delivering lectures. I humbly acknowledge the wonderful help provided by the original sources in this compilation. • For best results, it is always suggested you read the source material. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
2
Contents • Description of a Pneumatic system • Model of the Pneumatic system • Some questions.
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
3
What is a pneumatic system • Pneumatic system uses compressible fluid as working medium and it is usually air. • In pneumatic systems, compressibility effects of gas cannot be neglected and hence dynamic equations are obtained using conservation of mass. • In pneumatic systems, change in fluid inertia energy and the fluid’s internal thermal energy are assumed negligible. • In pneumatic system, the mass and volume flow rates are not readily interchangeable July – December 2008
prv/System Modeling Coursework/MIT-Manipal
4
Description of Pneumatic system • Pneumatic devices involve the flow of gas or air, through connected pipe lines and pressure vessels. • Hence, the variables of pneumatic system are mass flow rate, qm, and pressure P. • The mass flow rate is a through variable and it is analogous to current. The pressure variable is across variable and is analogous to voltage. • The two basic elements of a pneumatic system are the resistance and capacitance. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
5
Definition for Pneumatic Resistance • The gas flow resistance, R is defined as the rate of change in gas pressure difference for a change in gas flow rate. Change in gas pressure difference, N / m 2 R= Change in gas flow rate, Kg / sec
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
6
Definition for pneumatic capacitance • Pneumatic capacitance is defined for a pressure vessel and depends on the type of expansion process involved. The capacitance of a pressure vessel may be defined as the ratio of change in gas stored for a change in gas pressure.
Change in gas stored , Kg C= Change in gas pressure, N / m 2
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
7
Pros and Cons of Pneumatic systems • Advantages – The air or gas used is non inflammable and so it offers safety from fire hazards. – The air or gas has negligible viscosity, compared to high viscosity of hydraulic fluids. – No return pipelines are required and since air can be let out at the end of work cycle.
• Disadvantage – The response is slower than that of hydraulic systems because of the compressibility of the working fluid. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
8
Applications of Pneumatic systems • • • •
Guided Missiles Aircraft systems Automation of production machines Automatic controllers …and many more
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
9
Pneumatic system
Pi = air pressure of the source at steady state (newton/m2) P0 = air pressure in the vessel at steady state (newton/m2) ∆ Pi = small change in air pressure of the source from its steady state ∆ P0 = small change in air pressure of the vessel from its steady state
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
10
System dynamics • Rate of gas storage in vessel = rate of gas inflow C
d ΔP0 ΔP ΔPi − ΔP0 = = dt R R
• The mathematical model of a simple pneumatic system shown in figure is given by d ΔP0 ΔP ΔPi − ΔP0 = = C dt R R
• Applying Laplace and rearranging the terms, we get ΔP0 ( s) 1 = ΔPi ( s) ( RCs + 1) July – December 2008
prv/System Modeling Coursework/MIT-Manipal
11
Try for this system
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
12
Description of the system
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
13
Solution – Part I
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
14
Solution – Part II
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
15
Summary • The transfer function of the system presents the same order as that of the level system with respect to the assumed dynamics and structure. • The common thread however will be with respect to the capacitance and resistance of the system
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
16
References 1. Advanced Control Systems Engineering, Ronald Burns 2. Modern Control Engineering, Ogata 3. Control Systems, Nagoor Kani …amongst others
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
17
And, before we break… • Comparisons give us the cancer of the soul – G. Jampolsky
Thanks for listening…
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
18
Class 11: Modeling of Pneumatic systems
P.R. VENKATESWARAN Faculty, Instrumentation and Control Engineering, Manipal Institute of Technology, Manipal Karnataka 576 104 INDIA Ph: 0820 2925154, 2925152 Fax: 0820 2571071 Email: [email protected], [email protected] Web address: http://www.esnips.com/web/SystemModelingClassNotes
WARNING! • I claim no originality in all these notes. These are the compilation from various sources for the purpose of delivering lectures. I humbly acknowledge the wonderful help provided by the original sources in this compilation. • For best results, it is always suggested you read the source material. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
2
Contents • Description of a Pneumatic system • Model of the Pneumatic system • Some questions.
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
3
What is a pneumatic system • Pneumatic system uses compressible fluid as working medium and it is usually air. • In pneumatic systems, compressibility effects of gas cannot be neglected and hence dynamic equations are obtained using conservation of mass. • In pneumatic systems, change in fluid inertia energy and the fluid’s internal thermal energy are assumed negligible. • In pneumatic system, the mass and volume flow rates are not readily interchangeable July – December 2008
prv/System Modeling Coursework/MIT-Manipal
4
Description of Pneumatic system • Pneumatic devices involve the flow of gas or air, through connected pipe lines and pressure vessels. • Hence, the variables of pneumatic system are mass flow rate, qm, and pressure P. • The mass flow rate is a through variable and it is analogous to current. The pressure variable is across variable and is analogous to voltage. • The two basic elements of a pneumatic system are the resistance and capacitance. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
5
Definition for Pneumatic Resistance • The gas flow resistance, R is defined as the rate of change in gas pressure difference for a change in gas flow rate. Change in gas pressure difference, N / m 2 R= Change in gas flow rate, Kg / sec
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
6
Definition for pneumatic capacitance • Pneumatic capacitance is defined for a pressure vessel and depends on the type of expansion process involved. The capacitance of a pressure vessel may be defined as the ratio of change in gas stored for a change in gas pressure.
Change in gas stored , Kg C= Change in gas pressure, N / m 2
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
7
Pros and Cons of Pneumatic systems • Advantages – The air or gas used is non inflammable and so it offers safety from fire hazards. – The air or gas has negligible viscosity, compared to high viscosity of hydraulic fluids. – No return pipelines are required and since air can be let out at the end of work cycle.
• Disadvantage – The response is slower than that of hydraulic systems because of the compressibility of the working fluid. July – December 2008
prv/System Modeling Coursework/MIT-Manipal
8
Applications of Pneumatic systems • • • •
Guided Missiles Aircraft systems Automation of production machines Automatic controllers …and many more
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
9
Pneumatic system
Pi = air pressure of the source at steady state (newton/m2) P0 = air pressure in the vessel at steady state (newton/m2) ∆ Pi = small change in air pressure of the source from its steady state ∆ P0 = small change in air pressure of the vessel from its steady state
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
10
System dynamics • Rate of gas storage in vessel = rate of gas inflow C
d ΔP0 ΔP ΔPi − ΔP0 = = dt R R
• The mathematical model of a simple pneumatic system shown in figure is given by d ΔP0 ΔP ΔPi − ΔP0 = = C dt R R
• Applying Laplace and rearranging the terms, we get ΔP0 ( s) 1 = ΔPi ( s) ( RCs + 1) July – December 2008
prv/System Modeling Coursework/MIT-Manipal
11
Try for this system
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
12
Description of the system
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
13
Solution – Part I
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
14
Solution – Part II
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
15
Summary • The transfer function of the system presents the same order as that of the level system with respect to the assumed dynamics and structure. • The common thread however will be with respect to the capacitance and resistance of the system
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
16
References 1. Advanced Control Systems Engineering, Ronald Burns 2. Modern Control Engineering, Ogata 3. Control Systems, Nagoor Kani …amongst others
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
17
And, before we break… • Comparisons give us the cancer of the soul – G. Jampolsky
Thanks for listening…
July – December 2008
prv/System Modeling Coursework/MIT-Manipal
18
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