Damped Harmonic Oscillators (damping proportional to velocity)

• F = ma : m x double dot = - k x - b x dot, or
• m x double dot + b x dot +k x = 0
• try x = A exp (alpha t) and find
• alpha^2 + b/m alpha +k/m = 0
• omega_o = sqrt(k/m), the undamped frequency
• alpha = -b/(2m) +/- sqrt( (b/2m)^2 - k/m) or
• alpha = -b/(2m) +/- i sqrt( k/m - (b/2m)^2 ).
• let beta = - b/(2m), and
• omega = sqrt( k/m - (b/2m)^2 ) = sqrt (k/m - beta^2)

• Assume omega real ( or k/m > beta^2). This is underdamped motion.
• x(t) = exp(-beta t) A exp( i omega t) + exp(-beta t) B exp(- i omega t)
• Rearranging: x(t) = exp(-beta t) [ A exp(i omega t) + B exp(- i omega t) ]
• More rearranging gives x(t) = exp(- beta t) [ C sin (omega t) + D cos (omega t) ]
• Or we could also write x(t) = exp(-beta t) E cos ( omega t + delta)

Using the sine and cosine form, we can fit the initial conditions for an underdamped harmonic oscillator (omega is real)

• x(t) = exp(- beta t) [ C sin (omega t) + D cos (omega t) ]
• x(0) = xo = D
• x dot (t) = -beta exp(-beta t) [ C sin (omega t) + D cos (omega t) ]+omega exp(-beta t)[C cos(omega t) -D sin(omega t)]
• xdot_o = x dot (0) = -beta D + omega C
• We conclude that D = x(0) and C = [x dot (0) + beta x(0) ]/omega
• x(t) = exp(- beta t) [ [x dot (0) + beta x(0) ]/omega sin (omega t) + x(0) cos (omega t)]

We now have the equation of motion in terms of initial conditions.

• When beta is very small we have underdamped motion
• When beta is very large we have overdamped motion
• What happens as omega ->0 ( beta^2 -> k/m) ?
• x(t) = exp(- beta t) [ [x dot (0) + beta x(0) ]/omega sin (omega t) + x(0) cos (omega t)]
• take the limit as omega ->0 ( beta ->omega_o)
• x(t) = lim omega->0 exp(-omega_o t) [[x dot (0) + beta x(0) ]/ sin (omega t)/omega +x(0) ]
• x(t) = ? . This is 'critical damping'.