Pendulum simulations: exploring classic and chaotic motion
In this blog post, I present interactive simulations of several pendulum systems that illustrate non-linear dynamics and classical physics concepts. Pendulums are an ideal testbed for understanding periodic and chaotic motion, and their mathematical models can range from the simple to those requiring numerical integration of non-linear equations.
Simple Pendulum
The simple pendulum, composed of a mass suspended by a massless string, behaves predictably for small angles but exhibits richer dynamics at larger amplitudes. Its motion is governed by
\frac{d^2\theta}{dt^2} = -\frac{g}{L} \sin(\theta)
where L is the string length and g the gravitational acceleration. I use the non-linear equation and solve it numerically, so the simulation remains accurate even when the small angle approximation is not valid.
Driven Damped Pendulum
By introducing damping and a periodic driving force, the same basic system becomes a rich source of phenomena including resonance and chaos. The motion is modeled by
\frac{d^2\theta}{dt^2} = -\frac{g}{L} \sin(\theta) - b \frac{d\theta}{dt} + F_d \cos(\omega_d t)
I numerically integrate this equation to display how varying the damping and driving changes the resulting motion, from steady oscillations to complex and irregular behavior.
Cycloidal Pendulum
Unlike the ordinary pendulum, the cycloidal pendulum achieves perfect isochronism for all amplitudes due to the path traced by the bob. Its angular parameter \psi satisfies
\psi(t) = \psi_0 \cos\left(\sqrt{\frac{g}{L}} t\right)
The bob’s position is determined parametrically:
\begin{aligned} x_B &= O_x + r(\psi + \sin\psi) \\ y_B &= O_y + r(3 + \cos\psi) \end{aligned}
Here, L = 4r is the pendulum length, and the geometry ensures equal-period swings for any initial angle.
Elastic (Spring) Pendulum
The elastic, or spring, pendulum couples angular and radial motion, leading to energy exchanges between swinging and stretching modes. Its coupled dynamics are
\begin{aligned} \dot{\theta} &= \omega \\ \dot{\omega} &= \frac{-g \sin\theta - 2 \omega v_x}{r_0 + x} \\ \dot{x} &= v_x \\ \dot{v_x} &= (r_0 + x)\omega^2 + g \cos\theta - \frac{k}{m}x \end{aligned}
This system displays interesting mode coupling and can exhibit transition to chaotic motion depending on parameter values.
Double Pendulum
The double pendulum, with two rods and masses linked in series, is a well-studied chaotic system. I numerically solve the Lagrange-derived equations to capture its sensitive dependence on initial conditions and its unpredictable long-term behavior. Even with rigid, massless rods and no external damping or forcing, this system reveals how classical mechanics transitions from regular motion to chaos.
For more insights into this topic, you can find the details here.