Coupled Harmonic Oscillators

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X1(0) and X2(0) are the initial postions of the two masses and can range from -1...1. Press "Reset" to initialize the graphs. Press "Start" to start the animation and "Stop" to stop it.(The Stop button is only active while the animation is running)

In this coupled system both masses(M) are identical. The outer two springs are identical and have a spring constant (K1) about half the size of the middle spring's spring constant (K2).

The equations that represent the system are:

X1(t) = exp(-Bt)( X1(0) ( cos(w1t) + cos(w2t) )/2 + X2(0) ( cos(w2t) - cos(w1t) )/2 )

X2(t) = exp(-Bt)( X2(0) ( cos(w2t) + cos(w1t) )/2 + X1(0) ( cos(w2t) - cos(w1t) )/2 )

w1 = sqrt( (K1 + 2K2)/M )

w2 = sqrt( K1/M )

Some interesting initial conditions to try are:

X1(0) = -1.0 and X2(0) = 1.0

X1(0) = -1.0 and X2(0) = -1.0

Both of these have a simple period of oscillation.