1. Coupled Oscillations 1

2. Shaking Towers

4. Coupled oscillations
Waves 4

5. Coupled oscillators 5

6. Coupled pendula θ1 θo l xo x1 k
Mass-0
Mass-1

7. Equation of motion k SHM Coupling term term
Considering small-angle approximation
Equation of motion
Total force on Mass-1 θ1 θo l
Total force on Mass-0 xo x1 k
Mass-0
Mass-1
SHM Coupling
term term 7

8. Natural freq. of each pendulum
Let
Natural freq. of each pendulum
Adding
Subtracting 8

9. In this case, we can easily identify 2 such variables (q1 and q2):
These are coupled differential equations. We have to identify linear combinations of x0 and x1 for which the equations become decoupled.
In this case, we can easily identify 2 such variables (q1 and q2): 9

10. Normal Co-ordinates Normal modes
Normal coordinates are coordinates in which the equations of motion take the form of a set of linear differential equations with constant coefficients in which each equation contains only one dependent variable.
(Here, simple harmonic equations are in q1 and q2 only)
Normal Co-ordinates
A vibration involving only one dependent variable is called a normal mode of vibration and has its own normal frequency.
The importance of the normal modes of vibration is that they are entirely independent of each other.
The energy associated with a normal mode is never exchanged with another mode; this is why we can add the energies of the separate modes to give the total energy.
Normal modes

11. Normal mode: A way in which the system can move in a steady state, in which all parts of the system move with the same frequency. The parts may have different (zero or negative) amplitudes.
Breathing mode
C-H Bending mode
C-C Stretching mode

12. Normal modes of Vibration of molecule

13. Normal frequencies
Slow mode
Fast mode

14. Normal mode amplitudes : q10 and q20

15. In-phase vibration (Pendulum mode)

16. Out-of-phase vibration (Breathing mode)

17. Coupled Oscillations
To simplify discussion, let us choose:
Initial conditions

18. Pendulum displacements

19. Behavior with time for individual pendulum

20. xo=0 Condition for complete energy exchange
The masses, M0 and M1, have to be equal.
And,
xo=0
For
Else, neither of the two pendulums will ever be stationary

22. Resonance + q1 q2

23. Normal mode frequencies

25. Stiff coupling
Slow oscillation will be missing

26. Coupled oscillators

27. Equations of motion

28. Normal modes
The two normal modes execute SHO with respective angular frequencies

29. qo represents Centre of mass
Normal modes: Physical Interpretation
qo represents Centre of mass
Slow mode

30. (relative motion of the two masses with the centre of mass unchanged)
q1 represents Relative coordinate
(relative motion of the two masses with the centre of mass unchanged)
Fast mode

33. The normal modes have solutions

35. i.e., |Ao| sin ψ = 0 which gives initial phase ψ = 0
Remember

36. Resonance +

37. . .

38. x0 (t) x1 (t) t Coupled oscillations (Resonance) t TB/2
TB=Beats Period
x0 (t) t
Coupled
oscillations
(Resonance) TC
x1 (t) t