Notes on coupled oscillators.

• We deal with the oscillators on page 460 of Marion and Thornton
• equation of motion #1 mx1-dot-dot +(k+k12)x1 - k12 x2 = 0
• equation of motion #2 mx2-dot-dot +(k+k12)x2 - k12 x1 = 0
• substituting sinusoids in for x1 = B1 exp(i omega t) and x2 = B2 exp( i omega t)
• (-m omega^2 + k + k12)B1 - k12 B2 = 0
• this gives B1/B2 = k12/[-m omega^2 + k + k12)]
• -k12 B1 + (k + k12 - m omega^2) B1 = 0
• this gives B1/B2 = (k + k12 - m omega^2)/k12
• since the B1/B2 ratio must be the same from each we have
• k + k12 - m omega^2 = plus or minus k12
• omega1 = sqrt( (k + k12)/m), omega2 = sqrt(k/m)
• when we substitute omega1 in B1/B2 we find B1/B2 = -1
• when we substitute omega2 in B1/B2 we find B1/B2 = +1
• with mode 1 active (omega1) we have
• x1(t) = C1 cos(omega1 t +delta1)
• x2(t) =-C1 cos(omega1 t +delta1)
• with mode 2 active (omega2) we have
• x1(t) = C2 cos(omega2 t +delta2)
• x2(t) = C2 cos(omega2 t +delta2)
• If mode 1 is active and mode 2 is zero then x1 +x2 = 0 (since B1/B2 = -1)
• This tells us that eta2 = x1 + x2, since eta2 is zero when mode 1 only is active
• Likewise if mode 2 only is active, then B1/B2 = 1 and eta1 = x1-x2 = 0.
• In general we have both modes active:
• x1(t) = C1 cos(omega1 t + delta1) + C2 cos (omega2 t + delta2)
• x2(t) = -C1cos(omega1 t + delta1) + C2 cos (omega2 t + delta2)
• there are four initial conditions x1o, x2o, x1-dot-o and x2-dot-o
• these determine the four constants C1, C2, delta1 and delta 2

For two coupled oscillators there are two distinct frequencies in the motion. These are a function of the masses and spring constants or other factors. How much of each mode shows up depends on how the system is 'launched' (the initial conditions)

It is possible to launch a system so that only one mass is initially moving, then gradually the first one stops and the second one is moving. In this situation, the energy shifts back and forth between the masses.