Spring-Mass-Damper System

Analyze forced oscillations with viscous damping. Sweep the driving frequency to trace the full resonance curve, watch the transient decay, and compare underdamped, critically damped, and overdamped behavior. All computation runs client-side.

Parameters

Presets

System Properties

Mass (m)1.00 kg

Stiffness (k)100 N/m

Damping (c)2.00 N·s/m

Forcing

Force Amplitude (F\u2080)5.00 N

Driving Frequency (\u03C9)8.00 rad/s

Initial Conditions

Initial Displacement (x\u2080)0.10 m

Initial Velocity (v\u2080)0.00 m/s

Results

1\ddot{x} + 2\dot{x} + 100x = 5\cos(8.0t)

Underdamped (\u03B6 = 0.100)

Natural Frequency \u03C9\u208010.000 rad/s

Natural Frequency f\u20801.592 Hz

Damping Ratio \u03B60.1000

Damped Frequency \u03C9\u20919.950 rad/s

Resonance Frequency \u03C9\u1D639.899 rad/s

Quality Factor Q5.000

Steady-State Amplitude0.1269 m

Phase Lag \u03C624.0 \u00B0

Damped Period0.631 s

Show formulas

\omega_0 = \sqrt{\frac{k}{m}} = \sqrt{\frac{100}{1}} = 10.000\,\text{rad/s}

\zeta = \frac{c}{2\sqrt{mk}} = \frac{2}{2\sqrt{1\cdot100}} = 0.1000

\omega_r = \omega_0\sqrt{1-2\zeta^2} = 9.899\,\text{rad/s}

A = \frac{F_0}{\sqrt{(k-m\omega^2)^2+(c\omega)^2}} = 0.1269\,\text{m}

Resonance Curve (Amplitude vs Driving Frequency)

Orange dashed line marks natural frequency \u03C9\u2080. Red dot is the current operating point.

Displacement vs Time

Red dashed lines show steady-state amplitude envelope. Transient decays over time.

Reference Guide

Damped Oscillations

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

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

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 $\omega_d = \omega_0\sqrt{1-\zeta^2}$ .

Forced Vibrations

With a harmonic driving force the equation becomes

m\ddot{x} + c\dot{x} + kx = F_0\cos(\omega t)

The steady-state amplitude is

A = \frac{F_0}{\sqrt{(k-m\omega^2)^2 + (c\omega)^2}}

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.

Resonance

Amplitude peaks at the resonance frequency

\omega_r = \omega_0\sqrt{1-2\zeta^2}

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).

Quality Factor

The quality factor Q measures how lightly damped a resonator is

Q = \frac{\sqrt{mk}}{c} = \frac{\omega_0}{2\zeta}

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 $Q > 100$ .