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Coupled Pendulums and Normal Modes

What you are seeing: two identical pendulums of length LL and mass mm, coupled by a spring of constant kk attached at distance dd from the pivot. The small-angle EOM has two normal modes: symmetric (both swing in phase) at ω+=g/L\omega_+ = \sqrt{g/L}, and antisymmetric (opposite phase) at ω−=g/L+2kd2/(mL2)\omega_- = \sqrt{g/L + 2 k d^2 / (m L^2)}.

Start with only pendulum 1 displaced (θ2=0\theta_2 = 0) and watch the energy slosh back and forth: pendulum 1 slows while pendulum 2 grows, until after half a beat period almost all the energy lives on pendulum 2. Beat period Tbeat=2π/(ω−−ω+)T_\text{beat} = 2 \pi / (\omega_- - \omega_+). Symmetric or antisymmetric initial conditions stay locked in their respective modes forever.

Figure 1. Two coupled pendulums with symmetric and antisymmetric normal modes. Method: linearized EOM, RK4 integration.
k (N/m)10.0
d / L0.50

WHAT TO TRY

  • Hit "Asymmetric IC" (swing only one): watch the energy slosh fully onto the other pendulum and back. The energy panel makes the beat obvious.
  • Try "Symmetric IC" (both same way) or "Antisym IC" (opposite): these are the pure normal modes, they swing forever without sloshing.
  • Raise the spring stiffness $k$ or move the attachment $d/L$ outward: the coupling grows and the beat gets faster.