1. Coupled Oscillations

2. A single spring-mass system

3. One mass+ Two springs
One mass+ Many springs k3 k4 m

4. Coupled Oscillators
SHM term
Coupling term

5. Let
Natural freq. of each pendulum
Adding
Subtracting

6. Normal Co-ordinates
Normal modes

7. Normal Co-ordinates
Normal modes of vibration
Normal frequencies

8. Normal mode frequencies

9. In-phase vibration x1 x2 t

10. Out-of-phase vibrationx1 x2 t

11. Normal mode amplitudes : q10 and q20
Normal mode frequencies:

12. Initial conditions

13. Mass displacements

14. Behavior with time for individual pendulumTc x1
Behavior with time for individual pendulum
TB/2 x2 t
x1(0) = 0 and x2(0) = 1

15. Condition for complete energy exchange
(Resonance)
x2=0 x
x1=0 x

17. Stiff coupling
Slow oscillation will be missing

18. Compare model Coupled Pendula and Experiment
Equation of motion
SHM Coupling
term term
With assumption

19. Check two pendula are identical
Determine inphase and out of phase mode
Check whether frequency of inphase mode (w1) is less than that
of out of phase mode (w2)

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

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