In this interactive physics simulation page it is possible to explore the dynamics of various pendulum systems.
Each simulation allows to adjust its parameters and observe the resulting motion in real-time.
From the predictable swings of a simple pendulum to the chaotic motion of a double pendulum and the isochronous nature of the cycloidal path, these examples show fundamental concepts in classical mechanics.
The simple pendulum is an idealized model consisting of a point mass (the bob) suspended by a massless, inextensible string of length L from a fixed pivot point.
When displaced from its equilibrium position, it swings back and forth under the influence of gravity.
The equation of motion for the angle \theta the string makes with the vertical is given by:
\frac{d^2\theta}{dt^2} = -\frac{g}{L} \sin(\theta)
where g is the acceleration due to gravity.
For small angles, \sin(\theta) \approx \theta, and the motion approximates simple harmonic motion with a period T \approx 2\pi\sqrt{L/g}.
This simulation uses the full non-linear equation, solved numerically using the Runge-Kutta method, making it accurate even for large angles of swing.
This simulation extends the simple pendulum by incorporating two additional forces: damping and a periodic driving force.
Damping represents energy loss (e.g., due to air resistance or friction at the pivot) and is typically modeled as a force proportional to the angular velocity.
The driving force is an external periodic push applied to the system.
The equation of motion becomes:
\frac{d^2\theta}{dt^2} = -\frac{g}{L} \sin(\theta) - b \frac{d\theta}{dt} + F_d \cos(\omega_d t)
where b is the damping coefficient, F_d is the amplitude of the driving force (scaled), and \omega_d is its angular frequency.
This system is exhibits a rich variety of behaviors, including resonance, period-doubling bifurcations, and chaotic motion, depending on the parameters.
A cycloidal pendulum is designed such that its bob follows a cycloidal path. This is achieved by having the pendulum string “unwrap” from guide curves (cheeks) that are themselves cycloids (evolutes of the bob’s path).
An inverted cycloid is a tautochrone curve, meaning the time taken for an object to slide down to its lowest point under uniform gravity is independent of its starting point on the curve.
The property of the cycloidal pendulum is that its period of oscillation is independent of its amplitude, unlike a simple pendulum at large angles.
If L = 4r is the total length of the pendulum string (where r is the radius of the generating circle for the cycloids), the motion of the angular parameter \psi (which dictates the point of unwrapping and the bob’s position) is described by simple harmonic motion:
\psi(t) = \psi_0 \cos\left(\sqrt{\frac{g}{L}} t\right)
The period is exactly T = 2\pi\sqrt{L/g}. The coordinates of the bob (x_B, y_B) relative to the pivot (O_x, O_y) are given by:
\begin{aligned} x_B &= O_x + r(\psi + \sin\psi) \\ y_B &= O_y + r(3 + \cos\psi) \end{aligned}
assuming y increases downwards, and the pivot is at the cusp of the downward-curving cheeks, and the bob’s path is an upward-opening cycloid whose lowest point is L below the pivot.
The elastic pendulum, also known as a spring pendulum, consists of a bob of mass m attached to a spring with spring constant k and rest length r_0. The other end of the spring is fixed to a pivot.
This system has two degrees of freedom: the angle \theta the spring makes with the vertical, and the extension x of the spring beyond its rest length (so its instantaneous length is r = r_0 + x).
The motion is governed by a set of coupled non-linear differential equations. If \omega = \dot{\theta} is the angular velocity and v_x = \dot{x} is the radial velocity of the extension, the equations 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}
The interplay between the swinging motion and the spring’s oscillation can lead to complex behaviors, including energy transfer between modes and chaotic motion for certain parameter values. This simulation uses Runge-Kutta to solve these equations.
The double pendulum consists of one pendulum bob attached by a rod to another pendulum bob, which is itself suspended from a fixed pivot. Both rods are typically considered massless and rigid.
Despite its simple construction, the double pendulum is a classic example of a chaotic system. Its motion is described by a set of two coupled, non-linear second-order differential equations for the angles of the two rods, \theta_1 and \theta_2.
The Lagrangian \mathcal{L} = T - V, where T is the kinetic energy and V is the potential energy, can be used to derive these equations:
\begin{aligned} \frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{\theta}_1}\right) - \frac{\partial \mathcal{L}}{\partial \theta_1} &= 0 \\ \frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{\theta}_2}\right) - \frac{\partial \mathcal{L}}{\partial \theta_2} &= 0 \end{aligned}
The resulting equations are solved numerically using Runge-Kutta. The system's behavior is highly sensitive to initial conditions, and shows chaotic motion.