Damped Oscillations

The Damped Harmonic Oscillator

$m\ddot{x} + c\dot{x} + kx = 0$

Or in standard form:

$\ddot{x} + 2\gamma\dot{x} + \omega_0^2 x = 0$

where:

• $\gamma = \frac{c}{2m}$ (damping coefficient, per unit mass)
• $\omega_0 = \sqrt{\frac{k}{m}}$ (natural frequency, undamped)

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Three Damping Cases

1. Overdamped ($\gamma > \omega_0$)

Large damping, no oscillation.

Characteristic equation roots are real, distinct, and negative:

$\lambda_{\pm} = -\gamma \pm \sqrt{\gamma^2 - \omega_0^2} < 0$

Solution:

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

Decays monotonically to zero. The system returns to equilibrium without oscillating.

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2. Critically Damped ($\gamma = \omega_0$)

Boundary case: fastest return to equilibrium without oscillation.

Repeated root: $\lambda = -\gamma$

Solution:

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

Initially, the linear term may dominate if $B \neq 0$, but exponential decay eventually wins.

Application: Car suspension systems are designed to be critically damped.

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3. Underdamped ($\gamma < \omega_0$)

Small damping: oscillation with decaying amplitude.

Complex conjugate roots: $-\gamma \pm i\Omega$ where $\Omega = \sqrt{\omega_0^2 - \gamma^2}$

Solution:

$x(t) = e^{-\gamma t}(A\cos\Omega t + B\sin\Omega t)$

Or equivalently:

$x(t) = Ce^{-\gamma t}\cos(\Omega t - \phi)$

Properties:

• Oscillates with frequency $\Omega$ (less than $\omega_0$)
• Amplitude decays as $e^{-\gamma t}$
• Period $T = \frac{2\pi}{\Omega}$

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Energy Dissipation

For undamped oscillator: total energy $E = \frac{1}{2}m\dot{x}^2 + \frac{1}{2}kx^2$ is conserved.

With damping, multiply equation by $\dot{x}$:

$m\ddot{x}\dot{x} + c\dot{x}^2 + kx\dot{x} = 0$

$\frac{d}{dt}\left(\frac{1}{2}m\dot{x}^2 + \frac{1}{2}kx^2\right) = -c\dot{x}^2$

Energy decays at rate $c\dot{x}^2$ (power dissipated by friction).

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Quality Factor (Q-factor)

The Q-factor measures how underdamped the system is:

$Q = \frac{\omega_0}{2\gamma} = \frac{\sqrt{km}}{c}$

Interpretation:

• $Q \gg 1$: lightly damped, many oscillations
• $Q \approx \frac{1}{2}$: critically damped
• $Q$ equals the number of oscillations for amplitude to decay by factor $e^{-\pi}$

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Forced Oscillations

Add external forcing:

$\ddot{x} + 2\gamma\dot{x} + \omega_0^2 x = f_0\cos\omega t$

Particular Integral

Try $x_p = a\cos\omega t + b\sin\omega t$.

Substituting and solving:

$a = \frac{f_0(\omega_0^2 - \omega^2)}{(\omega_0^2 - \omega^2)^2 + 4\gamma^2\omega^2}$

$b = \frac{2\gamma\omega f_0}{(\omega_0^2 - \omega^2)^2 + 4\gamma^2\omega^2}$

Amplitude and Phase

Write $x_p = \mu f_0\cos(\omega t - \phi)$ where:

$\mu = \frac{1}{\sqrt{(\omega_0^2 - \omega^2)^2 + 4\gamma^2\omega^2}}$

$\tan\phi = \frac{2\gamma\omega}{\omega_0^2 - \omega^2}$

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Resonance

Resonance occurs when the amplitude $\mu$ is maximum.

For small $\gamma$: maximum near $\omega = \omega_0$.

Resonant frequency (exact):

$\omega_{\text{res}} = \sqrt{\omega_0^2 - 2\gamma^2}$

At resonance:

• Amplitude is large (inversely proportional to $\gamma$)
• Phase shift is $\pi/2$: response lags forcing by $90°$

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Transients

The complete solution is:

$x(t) = x_c(t) + x_p(t)$

where $x_c$ (complementary function) decays as $e^{-\gamma t}$.

Transient behaviour: Initially dominated by $x_c$, which depends on initial conditions.

Steady state: At large $t$, only $x_p$ remains (forced oscillation).

Physical interpretation: The system “forgets” its initial state and responds only to the forcing.