1. Coupled Pendula

2. Equation of motion
SHM Coupling
term term
With assumption

3. Let
Natural freq. of each pendulum
Adding
Subtracting

4. Normal Co-ordinates
Normal modes

5. Normal frequencies

6. Normal mode amplitudes : q10 and q20

7. In-phase vibration

8. Out-of-phase vibration

9. Coupled Oscillations
Initial conditions

10. pendulum displacements

11. Behavior with time for individual pendulum

12. Condition for complete energy exchange

13. Resonance q1 q2

14. Normal mode frequencies

16. Stiff coupling
Slow oscillation will be missing

17. For coupled oscillators

18. Modify the normal coordinates to read

19. K.E
P.E

20. Damped forced coupled oscillations
Problem:
Damped forced coupled oscillations
A force F0 Cos ωt is applied on one
of the masses. Write down the
equations of motion try to solve the
system qualitatively using your
knowledge of forced oscillations and
coupled oscillations.
© SPK

21. Amplitude resonances for normal modesω
= 5 ω
= 7 1 2 β
= 0.1

22. 2
Here,

23. Amplitude ratio: |A2/A1|
Filter

24. Wilberforce Pendulum © Saint Mary’s University

25. Summary Coupled system Normal Co-ordinates Normal modes of vibration
Normal frequencies

26. Reference 1. THE PHYSICS OF VIBRATIONS AND WAVES AUTHOR :H.J. PAIN
IIT Central Library
Class no. : PAI/P
PAI/P