The damped harmonic oscillator is a classic model in physics used to describe systems where an oscillating mass experiences a resistive force proportional to its velocity. This applies to everything from car shock absorbers to RLC electrical circuits.
The Equation of Motion
According to Newton's second law, the sum of the forces on the mass is equal to its mass times acceleration. In a damped system, we have the restoring force of the spring (Hooke's Law: \(-kx\)) and a damping force (\(-cv\), where \(c\) is the damping coefficient and \(v\) is velocity).
\[m \frac{d^2x}{dt^2} = -kx - c\frac{dx}{dt}\]Rearranging this gives us the canonical second-order linear homogeneous differential equation:
\[\frac{d^2x}{dt^2} + 2\zeta\omega_n\frac{dx}{dt} + \omega_n^2x = 0\]Where:
- \(\omega_n = \sqrt{k/m}\) is the natural frequency.
- \(\zeta = \frac{c}{2\sqrt{mk}}\) is the damping ratio.
Solutions based on Damping Ratio \(\zeta\)
Depending on the value of \(\zeta\), the system behaves in one of three distinct ways:
1. Underdamped ( \(0 \le \zeta < 1\) )
The system oscillates with gradually decreasing amplitude.
\[x(t) = A e^{-\zeta\omega_n t} \cos(\omega_d t - \phi)\]2. Critically Damped ( \(\zeta = 1\) )
The system returns to equilibrium as fast as possible without oscillating. This is mathematically the boundary condition and extremely useful in engineering (e.g., door closers).
\[x(t) = (A + Bt)e^{-\omega_n t}\]3. Overdamped ( \(\zeta > 1\) )
The system returns to equilibrium without oscillating, but slower than in the critically damped case due to intense resistance.
\[x(t) = A e^{s_1 t} + B e^{s_2 t}\]Interactive Physics Simulation
Try interacting with the native physics simulator below. Notice how tweaking the damping parameter drastically changes the envelope shape, acting completely predictably within the mathematical bounds established above! We use a pure React-rendered SVG to interpolate the math without heavy external charting libraries.
This visualization interpolates directly through raw calculus.