Speaker
Description
Standard physics textbooks simplify pendulum motion by assuming the swing angle is small ($\sin\theta \approx \theta$), which only works up to about $23^\circ$ (Crawford, \textit{Waves}). Because of this shortcut, textbooks claim a pendulum's swing time never changes with amplitude, its motion is purely sinusoidal, and two connected pendulums always trade energy completely back and forth.
In this work, we study large angles in single and coupled pendulums alongside energy trapping by solving the nonlinear equations numerically using the fourth-order Runge--Kutta method:
\begin{equation}
\ddot{\theta}_i + \omega_0^2 \sin\theta_i + \kappa\,(\theta_i - \theta_j) = 0, \qquad i, j \in {1, 2}
\end{equation}
Our simulated periods match the exact elliptic-integral values within $10^{-6}\%$ at $10^\circ$, $90^\circ$, and $150^\circ$, while conserving energy within $10^{-5}\%$ over $\sim 20$ oscillations. The actual period is $18\%$ longer at $90^\circ$ and $76\%$ longer at $150^\circ$ than the textbook formula predicts. Furthermore, coupled pendulums exchange energy completely below about $76^\circ$, but above this threshold, energy remains trapped in the released pendulum. Our analytical estimate $\theta_c \approx \sqrt{32\kappa}$ predicts this cutoff within $5\%$ (for $\kappa = 0.05\,\omega_0^2$), pinpointing exactly where textbook linear approximations break down.
Other Fields of Research (Track) for Abstract
Computational mechanics and non-linear dynamics
| Nature of work | Computational |
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