Coupled Pendula with Spring

Simulate two spring-coupled pendulums - observe normal modes, beats, synchronization, and energy transfer governed by \(\ddot{\theta}_i = -\tfrac{g}{L_i}\sin\theta_i - \tfrac{b}{m_i L_i}\dot{\theta}_i \pm \tfrac{k_c}{m_i L_i}\cos\theta_i\,\Delta x\).

Two pendulums are coupled by a spring whose natural length equals the pivot separation. The spring exerts a horizontal restoring force proportional to the difference in bob displacements \(\Delta x = L_2\sin\theta_2 - L_1\sin\theta_1\).

Equations of Motion

\(\ddot{\theta}_1 = -\dfrac{g}{L_1}\sin\theta_1 - \dfrac{b}{m_1 L_1}\dot{\theta}_1 + \dfrac{k_c}{m_1 L_1}\cos\theta_1\cdot\Delta x\)

\(\ddot{\theta}_2 = -\dfrac{g}{L_2}\sin\theta_2 - \dfrac{b}{m_2 L_2}\dot{\theta}_2 - \dfrac{k_c}{m_2 L_2}\cos\theta_2\cdot\Delta x\)

Normal Modes (Identical Pendula)

For \(L_1=L_2=L\), \(m_1=m_2=m\), the two normal mode frequencies are:

In-phase mode: \(\omega_+ = \sqrt{g/L}\) - both bobs swing together, spring unstretched.

Anti-phase mode: \(\omega_- = \sqrt{g/L + 2k_c/m}\) - bobs swing oppositely, spring compressed and extended alternately.

Beats & Energy Transfer

Displacing only one pendulum excites both modes simultaneously. Energy transfers between pendulums periodically at the beat frequency \(\omega_{\text{beat}} = (\omega_- - \omega_+)/2\). Weaker coupling \(\Rightarrow\) slower beats.

Observable Phenomena

Beats: periodic exchange of energy when one pendulum is displaced.
Synchronization: damping drives the system toward the lower-energy in-phase mode.
Symmetry breaking: unequal lengths or masses prevent complete energy transfer.
Large amplitude: nonlinear \(\sin\theta\) terms produce richer dynamics.

Lagrangian Formulation

\(\mathcal{L} = \tfrac{1}{2}m_1 L_1^2\dot{\theta}_1^2 + \tfrac{1}{2}m_2 L_2^2\dot{\theta}_2^2 + m_1 g L_1\cos\theta_1 + m_2 g L_2\cos\theta_2 - \tfrac{k_c}{2}(L_2\sin\theta_2 - L_1\sin\theta_1)^2\)

Numerical Method

The 4D system is integrated with the classical 4th-order Runge-Kutta (RK4) method at fixed step \(\Delta t = 0.005\,\text{s}\), providing excellent accuracy for nonlinear oscillatory dynamics.

\[ \ddot{\theta}_i = -\frac{g}{L_i}\sin\theta_i - \frac{b}{m_i L_i}\dot{\theta}_i \pm \frac{k_c}{m_i L_i}\cos\theta_i\,(L_2\sin\theta_2 - L_1\sin\theta_1), \quad i=1,2 \]

Parameters & Initial Conditions

Pendulum 1

Mass \(m_1\) [kg]

Length \(L_1\) [m]

Pendulum 2

Mass \(m_2\) [kg]

Length \(L_2\) [m]

Coupling & Damping

Spring \(k_c\) [N/m]

Damping \(b\) [kg/s]

Initial Conditions

\(\theta_1(0)\) [rad]

\(\theta_2(0)\) [rad]

End time \(T\) [s]

Simulation speed 0.50×

Load example

Numerical method

Animation

\(\theta(t)\) - Angular Displacement

Phase Plot \(\theta_1\) vs \(\theta_2\)

Numerical accuracy - global error vs the exact solution

In the small-angle limit the two coupled pendula reduce to a linear normal-mode system whose exact solution is the matrix exponential e^AtX₀. Each integrator is run on that linearised system at the tool's own time step and compared against the exact reference; lower is better, and the ordering follows each method's order of accuracy.

Cite this tool

DOI: https://doi.org/10.5281/zenodo.20981189

Kapita, S. (2026). Coupled Pendula with Spring. Math Tools. https://doi.org/10.5281/zenodo.20981189

Kapita, Shelvean. "Coupled Pendula with Spring." Math Tools, 2026, doi.org/10.5281/zenodo.20981189.

@online{kapita2026coupled,
  author       = {Shelvean Kapita},
  title        = {{Coupled Pendula with Spring}},
  year         = {2026},
  organization = {Math Tools},
  doi          = {10.5281/zenodo.20981189},
  url          = {https://doi.org/10.5281/zenodo.20981189}
}