Class 5 - 6 Analogous Systems

System Modeling Coursework

Class 5-6 : Analogous 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] Blog: www.godsfavouritechild.wordpress.com 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

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

3

What is meant by analogous systems? • It is possible to make electrical and mechanical systems using analogs. • An analogous electrical and mechanical system will have differential equations of the same form.

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

4

Analogous Quantities Mechanical Analog I (Force-Current)

Electrical Quantity

Mechanical Analog II (Force Voltage)

Voltage, e

Velocity, v

Force, f

Current, i

Force, f

Velocity, v

Resistance, R

Lubricity, 1/B (Inverse friction)

Friction, B

Capacitance, C

Mass, M

Compliance, 1/K (Inverse spring constant)

Inductance, L

Compliance, 1/K (Inverse spring constant)

Mass, M

Transformer, N1:N2

Lever, L1:L2

Lever, L1:L2

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

5

Analogous quantities in force(Torque)-Voltage Analogy Mechanical Mechanical rotational Systems Translational systems

Electrical System

Force F

Torque, T

Voltage, e

Mass, M

Moment of inertia J

Inductance L

Friction Viscous Friction Coefficient B

Resistance R

Viscous Coefficient B Spring stiffness K

Torsional spring stiffness K

Reciprocal of capacitance 1/C

Displacement x

Angular Displacement, θ

Charge q

Velocity, v

Angular Velocity, ω

Charge i

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

6

Analogous quantities in Force (Torque)-Current Analogy

Mechanical Translational systems

Mechanical Rotational Systems

Electrical System

Force F

Torque, T

Current, i

Mass, M

Moment of inertia J

Capacitance C

Viscous Coefficient B

Friction Viscous Friction Coefficient B

Reciprocal of Resistance 1/R

Spring stiffness K

Torsional spring stiffness K

Reciprocal of inductance 1/L

Displacement x

Angular Displacement, θ

Magnetic flux linkage φ

Velocity, v

Angular Velocity, ω

Voltage e

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

7

Converting between systems • The important relationship when converting from a circuit to the Mechanical 1 analog is that between Kirchoff's Current Law and D'Alemberts Law (with inertial forces included).

Electrical

July – December 2008

Mechanical 1

prv/System Modeling Coursework/MIT-Manipal

8

Procedure for Conversion from Electrical to Mechanical

• The conversion from an electrical circuit to a mechanical analog is easily accomplished if capacitors in the circuit are grounded. If they are not, the process results in a mechanical system where positions must be chosen very carefully and the process can be much more difficult.

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

9

Procedure for Conversion from Electrical to Mechanical

• Start with an electrical circuit. Label all node voltages

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

10

Procedure for Conversion from Electrical to Mechanical

• Write a node equations for each node voltage

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

11

Procedure for Conversion from Electrical to Mechanical • Re-write the equations using analogs (make making substitutions from the table of analogous quantities), with each electrical node being replaced by a position.

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

12

Procedure for Conversion from Electrical to Mechanical

• Draw the mechanical system that corresponds with the equations.

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

13

Mechanical (Force-Current) to Electrical Conversion

• Start with the mechanical system. positions.

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

Label all

14

Mechanical (Force-Current) to Electrical Conversion • Draw over circuit, replacing mechanical elements with their analogs; force generators by current sources, input velocities by voltage sources, friction elements by resistors, springs by inductors, and masses by capacitors (which are grounded). Each position becomes a node.

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

15

Mechanical (Force-Current) to Electrical Conversion

• Label nodes and electrical elements as they were in the original mechanical system.

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

16

Verification of Mechanical to Electrical Analog

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

17

Verification of Mechanical to Electrical Analog

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

18

Numerical No.1 Convert into Mechanical equivalent

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

19

Summary

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

20

References • •

http://www.swarthmore.edu/NatSci/echeeve1/A nalogous Electrical and Mechanical Systems http://www.swarthmore.edu/NatSci/echeeve1/R ef/Analogs/ElectricalMechanicalAnalogs.html#Q uantities#Quantities amongst others…

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

21

And, before we break… • The unexamined life is not worth living – Socrates

Thanks for listening…

July – December 2008

prv/System Modeling Coursework/MIT-Manipal

22