Spring-Mass-Damper System

The equation of motion for a spring-mass-damper system is

The damping ratio $\zeta = \frac{c}{2\sqrt{mk}}$ governs behavior. When $\zeta < 1$ the system is underdamped and oscillates with an exponentially decaying envelope. When $\zeta = 1$ it is critically damped and returns to rest fastest without overshooting. When $\zeta > 1$ it is overdamped.

The damped natural frequency is $\omega_d = \omega_0\sqrt{1-\zeta^2}$.

With a harmonic driving force the equation becomes

The steady-state amplitude is

and the phase lag of the response behind the force is $\varphi = \arctan\!\left(\frac{c\omega}{k - m\omega^2}\right)$. At low frequencies the mass follows the force in phase; at high frequencies the mass barely moves and lags by $\pi$ rad.

Amplitude peaks at the resonance frequency

which is slightly below the natural frequency $\omega_0 = \sqrt{k/m}$. For lightly damped systems $(\zeta \ll 1)$, $\omega_r \approx \omega_0$ and the peak amplitude approaches $F_0/(c\omega_0)$. When $\zeta \ge 1/\sqrt{2}$, the resonance peak disappears and the amplitude decreases monotonically from DC.

At resonance the phase lag is exactly $\varphi = \pi/2$ (90 degrees).

The quality factor Q measures how lightly damped a resonator is

A high Q means a sharp, tall resonance peak with slow energy decay. A low Q means a broad, short peak. The bandwidth at the half-power points is $\Delta\omega = \omega_0/Q$.

• Car suspension targets $Q \approx 0.5$ to suppress road bumps without oscillating.
• Tuned mass dampers in tall buildings use $\zeta \approx 0.05-0.1$ to absorb earthquake and wind energy.
• Musical instruments and sensors can have $Q > 100$.