Electrical analogous of mechanical translational system

Anna University, Regulation 2017,EEE,IC 8451- CONTROL SYSTEMS

1. IC 8451&CONTROL SYSTEMS Department of Electrical and Electronics Engineering Approved by AICTE | Affiliated to Anna University | Accredited by NAAC | Accredited NBA | Recognized by UGC under 2(f) and 12(B) Chennai Main Road, Kumbakonam- 612 501. ARASU ENGINEERING COLLEGE 1

2. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 2 Prepared by Mrs.K. Kalpana .,M.E.,( Ph.D )., Department of Electrical and Electronics Engineering

3. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 3 Electrical Analogous of mechanical translational systems There are two types of electrical analogies of translational mechanical systems. Those are force voltage analogy and force current analogy. Force Voltage Analogy In force voltage analogy, the mathematical equations of translational mechanical system are compared with mesh equations of the electrical system . Translational Mechanical System Electrical System Force(F) Voltage(V) Mass(M) Inductance(L) Frictional Coefficient(B) Resistance(R) Spring Constant(K) Reciprocal of Capacitance ( 1/c ) Displacement(x) Charge(q) Velocity(v) Current( i )

4. Force Current Analogy In force current analogy, the mathematical equations of the translational mechanical system are compared with the nodal equations of the electrical system. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 4 Translational Mechanical System Electrical System Force(F) Current(i) Mass(M) Capacitance(C) Frictional coefficient(B) Reciprocal of Resistance(1/R ) Spring constant(K) Reciprocal of Inductance(1/L ) Displacement(x) Magnetic Flux(ψ) Velocity(v) Voltage(V)

5. Analogous elements in Force voltage analogy ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 5

6. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 6 Analogous elements in Force current analogy

7. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 7 EXAMPLE PROBLEM 1 Write the differential equations governing the mechanical system shown in fig. Draw the force voltage and force current electrical analogous circuits and verify by writing mesh and node equations. The free body diagram of M1 is shown in fig. The opposing forces are f m1 ,f b1 ,f b12 ,f k1

8. The free body diagram of M1 is shown in fig. The opposing forces are f m2 ,f b2 ,f b12 ,f k1 and f k2 ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 8

9. Force voltage analogous circuit The electrical analogous elements for the elements of mechanical systems are f(t) - e(t) , M 1 - L 1, M 2 - L 2, B 1 - R 1, B 2 - R 2 , B 12 - R 12, K 1 -1/C 1 , K 2 -1/C 2 The mesh equations are Force -voltage analogous circuit ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 9

10. Force current analogous circuit The electrical analogous elements for the elements of mechanical systems are f(t) - i (t) , M 1 - C 1, M 2 - C 2, B 1- 1/R 1, B 2- 1/R 2 , B 12- 1/R 12, K 1 -1/L 1 , K 2 -1/L 2 The node equations are ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 10

11. Force current analogous circuit ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 11

12. EXAMPLE PROBLEM 2 Write the differential equations governing the mechanical system shown in fig. Draw the force voltage and force current electrical analogous circuits and verify by writing mesh and node equations. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 12

13. The free body diagram of M1 is shown in fig. The opposing forces are marked as f m 1, fb 1 , and f k 1, f k 2 ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 13

14. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 14 Free body diagram of M2 is shown in fig. The opposing forces are marked as fm 2 , fb 3 , fk 2 & fk 3 .

15. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 15 The free body diagram of M3 is shown in fig. The opposing forces are marked as fm 3 , f b 3 and fk 3

16. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 16 FORCE VOLTAGE ANALOGOUS CIRCUIT MESH EQUATIONS

17. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 17 FORCE VOLTAGE ANALOGOUS CIRCUIT FORCE CURRENT ANALOGOUS CIRCUIT

18. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 18 NODE EQUATIONS FORCE CURRENT ANALOGOUS CIRCUIT

19. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 19 HOME WORK PROBLEM Write differential equations governing mechanical system shown in fig. Draw force voltage and force current electrical analogous circuits and verify by writing mesh and node equations.