Damped & Driven Oscillation Lab

Set mass, spring constant, and damping coefficient, then watch real-time oscillation. Switch to driven mode to sweep driving frequency and discover resonance peaks, phase lag, and the role of the quality factor Q.

Guided Experiment: Observing Damping Regimes

Hypothesis

Setup

Run Experiment

Analyze

Conclude

How will the motion differ when the damping ratio ζ is less than 1, equal to 1, and greater than 1? Which regime returns the system to equilibrium fastest?

Write your hypothesis in the Lab Report panel, then click Next.

Mass-Spring-Damper System

SpringDamperDisplacement

Displacement vs Time

Controls

Mode

Presets

Mass (m)1.0 kg

Spring Constant (k)10 N/m

Damping Coefficient (b)0.5 N·s/m

Results

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

m = 1,\; k = 10,\; b = 0.5

Natural Freq. ω₀

3.162 rad/s

Damping Ratio ζ

0.0791

Quality Factor Q

6.32

Damped Freq. ω_d

3.152 rad/s

Classification

Underdamped (ζ < 1)

\omega_0 = \sqrt{k/m},\quad \zeta = \frac{b}{2\sqrt{mk}},\quad Q = \frac{1}{2\zeta}

Data Table

(0 rows)

#	Trial	Mode	ω_d(rad/s)	Amplitude(cm)	Phase Lag(°)	Q Factor	ζ	Classification

Hypothesis

0 / 500

Observations

0 / 500

Conclusions

0 / 500

Reference Guide

Free Damped Oscillation

A mass on a spring with damping follows the equation of motion

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

The natural angular frequency is $\omega_0 = \sqrt{k/m}$ and the damping ratio is $\zeta = b/(2\sqrt{mk})$ .

When $\zeta < 1$ (underdamped), the system oscillates with exponentially decaying amplitude. When $\zeta = 1$ (critically damped), it returns to zero fastest without overshoot. When $\zeta > 1$ (overdamped), it returns slowly without oscillating.

Driven Oscillation and Resonance

When an external periodic force is applied, the equation becomes

m\ddot{x} + b\dot{x} + kx = F_0\cos(\omega_d t)

After transients decay, the steady-state amplitude is

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

The amplitude peaks near the natural frequency. The resonance frequency is $\omega_{\text{res}} = \omega_0\sqrt{1 - 2\zeta^2}$ (valid when $\zeta < 1/\sqrt{2}$ ).

Quality Factor Q

The quality factor measures how underdamped the system is

Q = \frac{\sqrt{mk}}{b} = \frac{1}{2\zeta}

A high Q means the system rings for many cycles before damping out. At resonance, the peak amplitude is approximately

A_{\max} \approx \frac{QF_0}{k}

The width of the resonance peak (half-power bandwidth) is inversely proportional to Q.

Phase Lag

The response of a driven oscillator lags behind the driving force by a phase angle

\varphi = \arctan\!\left(\frac{b\omega_d}{k - m\omega_d^2}\right)

At low frequency the displacement is nearly in phase with the force (φ near 0). At resonance the phase lag is 90°. Well above resonance the displacement is nearly 180° out of phase, meaning the mass moves opposite to the applied force.