Coupled Pendula

Equation of motion SHM Coupling term term With assumption

Let Natural freq. of each pendulum Adding Subtracting

Normal Co-ordinates Normal modes

Normal frequencies

Normal mode amplitudes : q10 and q20

In-phase vibration

Out-of-phase vibration

Coupled Oscillations Initial conditions

pendulum displacements

Behavior with time for individual pendulum

Condition for complete energy exchange

Resonance q1 q2

Normal mode frequencies

Stiff coupling Slow oscillation will be missing

For coupled oscillators

Modify the normal coordinates to read

K.E P.E

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

Amplitude resonances for normal modes ω = 5 ω = 7 1 2 β = 0.1

2 Here,

Amplitude ratio: |A2/A1| Filter

Wilberforce Pendulum © Saint Mary’s University http://www.ap.smu.ca/demos/content/osc_and_waves/wilberforce_pendulum/wilberforce_pendulum.html

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

Reference 1. THE PHYSICS OF VIBRATIONS AND WAVES AUTHOR :H.J. PAIN IIT Central Library Class no. : 530.124 PAI/P 531.1133 PAI/P