Damped Pendulum Simulator

The damped simple pendulum is governed by the nonlinear second-order ODE

where \(L\) is the rod length, \(m\) the bob mass, \(b\) the damping coefficient, and \(g=9.81\,\text{m/s}^2\).

Lagrangian Formulation

The undamped pendulum has kinetic and potential energies \(T = \tfrac{1}{2}mL^2\dot\theta^2\) and \(V = -mgL\cos\theta\), giving the Lagrangian

Damping is a non-conservative force and enters via the Rayleigh dissipation function \(\mathcal{F} = \tfrac{1}{2}b(L\dot\theta)^2 = \tfrac{1}{2}bL^2\dot\theta^2\). The generalised Euler–Lagrange equation including dissipation is

Substituting:

Damping Regimes (linearised, \(\sin\theta\approx\theta\))

Let \(\gamma = b/m\) and \(\omega_n = \sqrt{g/L}\). The characteristic equation \(\lambda^2+\gamma\lambda+\omega_n^2=0\) has discriminant \(\Delta = \gamma^2 - 4\omega_n^2\):

• Underdamped \((\Delta < 0,\;\zeta<1)\): \(\theta(t)=e^{-\gamma t/2}(A\cos\omega_d t+B\sin\omega_d t)\), \(\omega_d=\sqrt{\omega_n^2-(\gamma/2)^2}\) - decaying oscillations, spiral in phase space.
• Critically damped \((\Delta = 0,\;\zeta=1)\): \(\theta(t)=(A+Bt)e^{-\gamma t/2}\) - fastest return without oscillation.
• Overdamped \((\Delta > 0,\;\zeta>1)\): two distinct negative real roots, slow monotone return.

The damping ratio is \(\zeta = \gamma/(2\omega_n) = b/(2m\sqrt{g/L})\).

Phase Space

The phase portrait plots \(\omega = \dot{\theta}\) vs \(\theta\) (wrapped to \((-\pi,\pi]\)). Fixed points: stable centers/spirals at \(\theta = 0\) and unstable saddles at \(\theta = \pm\pi\).

Numerical Method

RK4 (step \(\Delta t = 0.025\,\text{s}\), 4 steps/frame) by default. Switches to backward Euler when \(\Delta = \gamma^2 - 4\omega_n^2 > 0\) (overdamped / stiff). Inputs accept expressions such as pi/4, 2*pi, sqrt(2).