class 5 - 6 analogous systems

System Modeling Coursework

P.R. VENKATESWARANFaculty, Instrumentation and Control Engineering,

Manipal Institute of Technology, ManipalKarnataka 576 104 INDIAPh: 0820 2925154, 2925152

Fax: 0820 2571071Email: pr.venkat@manipal.edu, prv_i@yahoo.com

Blog: www.godsfavouritechild.wordpress.comWeb address: http://www.esnips.com/web/SystemModelingClassNotes

Class 5-6 : Analogous Systems

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WARNING!

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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.

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For best results, it is always suggested you read the source material.

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Contents

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What is meant by analogous systems?

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It is possible to make electrical and mechanical systems using analogs.

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An analogous electrical and mechanical system will have differential equations of the same form.

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Analogous Quantities

Electrical

Quantity

Mechanical

Analog I

(Force-Current)

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

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Analogous quantities in force(Torque)-Voltage Analogy

Mechanical Translational systems

Mechanical rotational Systems Electrical System

Force F Torque, T Voltage, e

Mass, M Moment of inertia J Inductance L

Viscous Friction Coefficient B

Viscous Friction Coefficient B Resistance R

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

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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 Friction Coefficient B

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

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Converting between systems

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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 Mechanical 1

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Procedure for Conversion from Electrical to Mechanical

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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.

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Procedure for Conversion from Electrical to Mechanical

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Start with an electrical circuit.

Label all node voltages

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Procedure for Conversion from Electrical to Mechanical

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Write a node equations for each node voltage

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Procedure for Conversion from Electrical to Mechanical

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

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Procedure for Conversion from Electrical to Mechanical

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Draw the mechanical system that corresponds with the equations.

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Mechanical (Force-Current) to Electrical Conversion

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Start with the mechanical system. Label all

positions.

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Mechanical (Force-Current) to Electrical Conversion

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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.

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Mechanical (Force-Current) to Electrical Conversion

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

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Verification of Mechanical to Electrical Analog

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Verification of Mechanical to Electrical Analog

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Numerical No.1 Convert into Mechanical equivalent

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Summary

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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…

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And, before we break…

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The unexamined life is not worth living–

Socrates

Thanks for listening…