Damped Harmonic Oscillation Lab

Exploring Restoring Forces, Damping Forces & Dynamic Regimes

Differential Equation of Motion
d²x/dt² + 2β dx/dt + ω₀² x = 0
d²x/dt² + 0.60·(dx/dt) + 4.00·x = 0
Current Analytical Solution x(t)
x(t) = A · e-βt · cos(ωt + φ)
x(t) = 4.00 · e-0.30t · cos(1.98t)
Underdamped Regime β (0.30) < ω₀ (2.00)

Oscillation Waveform & Force Dynamics

Time vs. Displacement x(t), Envelopes, & Opposing Forces
t = 0.00 s
x(t): 0.00 m
v(t): 0.00 m/s
Frestoring: 0.00 N
Fdamping: 0.00 N
Displacement x(t)
Envelopes ±A e-βt
Restoring Force Fk = -kx
Damping Force Fb = -bv

Spring-Mass-Dashpot Apparatus

Simultaneous mechanical motion & opposing forces
Current Time: 0.00 s 16.0 s
→ Restoring Force Fk → Damping Force Fb → Velocity v

Live Oscillator Telemetry

Derived frequencies, coefficients & decay metrics
ζ Damping Ratio
0.150
ζ = β / ω₀ < 1 (Underdamped)
ω Damped Angular Freq
1.977 rad/s
√(ω₀² - β²)
ω₀ Natural Angular Freq
2.000 rad/s
√(k / m)
Td Oscillation Period
3.178 s
2π / ω
τ Decay Time Constant
3.333 s
1 / β (e⁻¹ decay)
bcrit Critical Damping Coeff
4.000 N·s/m
2√(m·k)

Why Underdamping Occurs

Here, the damping constant β is smaller than the natural angular frequency ω₀ (β < ω₀). The restoring force (spring) is strong enough to repeatedly pull the mass past equilibrium, while the damping force progressively extracts kinetic energy, forming decaying sinusoidal oscillations contained within the exponential envelope ±A e-βt.