Class 5 - 6 Analogous Systems
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Transcript of 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: [email protected], [email protected]
Blog: www.godsfavouritechild.wordpress.comWeb address: http://www.esnips.com/web/SystemModelingClassNotes
Class 5-6 : Analogous Systems
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 2
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 3
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.
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 5
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
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 6
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
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 7
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
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 8
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
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 12
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.
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 13
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.
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 16
Mechanical (Force-Current) to Electrical Conversion
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Label nodes and electrical elements as they were in the original mechanical system.
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 17
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…
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 22
And, before we break…
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The unexamined life is not worth living–
Socrates
Thanks for listening…