Coupled Oscillations 1

Shaking Towers

Coupled oscillations Waves 4

Coupled oscillators 5

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

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

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

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

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

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

Normal modes of Vibration of molecule http://chemwiki.ucdavis.edu/Physical_Chemistry/Spectroscopy/Vibrational_Spectroscopy/Vibrational_Modes

Normal frequencies Slow mode Fast mode

Normal mode amplitudes : q10 and q20

In-phase vibration (Pendulum mode)

Out-of-phase vibration (Breathing mode)

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

Pendulum displacements

Behavior with time for individual pendulum

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

Resonance + q1 q2

Normal mode frequencies

Stiff coupling Slow oscillation will be missing

Coupled oscillators

Equations of motion

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

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

(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

The normal modes have solutions

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

Resonance +

. .

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