Damped and Driven Oscillations

Damped Oscillations

With damping force $F_d = -bv$:

$m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0$

Divide by $m$:

$\frac{d^2x}{dt^2} + 2\gamma\frac{dx}{dt} + \omega_0^2x = 0$

where:

• $\gamma = b/(2m)$ (damping coefficient)
• $\omega_0 = \sqrt{k/m}$ (natural frequency)

Three Cases of Damping

Underdamped ($\gamma < \omega_0$):

$x(t) = Ae^{-\gamma t}\cos(\omega' t + \phi)$

where $\omega' = \sqrt{\omega_0^2 - \gamma^2}$ (damped frequency)

Oscillates with decreasing amplitude.

Critically damped ($\gamma = \omega_0$):

$x(t) = (A + Bt)e^{-\gamma t}$

Returns to equilibrium fastest without oscillating.

Overdamped ($\gamma > \omega_0$):

$x(t) = Ae^{-\alpha t} + Be^{-\beta t}$

where $\alpha, \beta = \gamma \pm \sqrt{\gamma^2 - \omega_0^2}$

Returns to equilibrium slowly without oscillating.

Quality Factor

$Q = \frac{\omega_0}{2\gamma}$

High Q: Light damping, many oscillations before amplitude decays

Low Q: Heavy damping, few oscillations

Energy decay: $E(t) = E_0 e^{-2\gamma t}$

Time constant: $\tau = 1/(2\gamma)$

Driven Oscillations

With external driving force $F(t) = F_0\cos(\omega t)$:

$m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = F_0\cos(\omega t)$

$\frac{d^2x}{dt^2} + 2\gamma\frac{dx}{dt} + \omega_0^2x = \frac{F_0}{m}\cos(\omega t)$

Steady-State Solution

After transients die out:

$x(t) = A(\omega)\cos(\omega t - \delta)$

Amplitude: $A(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2\gamma\omega)^2}}$

Phase lag: $\tan\delta = \frac{2\gamma\omega}{\omega_0^2 - \omega^2}$

Resonance

Amplitude is maximum when denominator is minimum.

Resonant frequency: $\omega_r = \sqrt{\omega_0^2 - 2\gamma^2}$

For light damping ($\gamma \ll \omega_0$): $\omega_r \approx \omega_0$

Maximum amplitude: $A_{max} \approx \frac{F_0}{2m\gamma\omega_0} = \frac{F_0Q}{m\omega_0^2} = \frac{F_0Q}{k}$

Power and Resonance

Average power absorbed: $P_{avg} = \frac{1}{2}b\omega^2 A^2$

Maximum at: $\omega = \omega_0$ (exactly)

Power at resonance: $P_{max} = \frac{F_0^2}{4m\gamma}$

Bandwidth and Q

Half-power points: where $P = P_{max}/2$

Occur at $\omega = \omega_0 \pm \gamma$ (for small $\gamma$)

Bandwidth: $\Delta\omega = 2\gamma$

$Q = \frac{\omega_0}{\Delta\omega} = \frac{\omega_0}{2\gamma}$

Sharp resonance: high Q, narrow bandwidth

Complex Representation

Using $\tilde{x} = x_0 e^{i\omega t}$:

$-\omega^2\tilde{x} + 2i\gamma\omega\tilde{x} + \omega_0^2\tilde{x} = \frac{F_0}{m}$

$\tilde{x} = \frac{F_0/m}{\omega_0^2 - \omega^2 + 2i\gamma\omega}$

Magnitude gives amplitude, argument gives phase.

Applications

Shock absorbers: Critical damping for quick settling

Musical instruments: High Q for pure tones

Bridges: Avoid driving at natural frequency

RLC circuits: Same equations as mechanical oscillators