The Elastic Pendulum: When Bounce and Swing Trade Places

One extra degree of freedom changes everything

A simple pendulum has one moving part in the sense that matters: the swing angle. Swap the rigid rod for a spring and you add a second one, the stretch, and the two are no longer independent. Gravity pulls the mass down and sideways at the same time, and the spring's restoring force depends on how far it is stretched and on the angle it hangs at, so a small nonlinear term couples the vertical bounce to the sideways swing. That single coupling term is responsible for everything interesting about this system.

The two natural frequencies

Ignore the coupling for a moment and the system has two independent natural frequencies: the spring's own bounce frequency omega_r, set by its stiffness k and the hanging mass m, and the pendulum's swing frequency omega_theta, set by the equilibrium length of the spring L and gravity g.

omega_r     = sqrt(k / m)       // vertical bounce, from the spring constant
omega_theta = sqrt(g / L)       // sideways swing, from the pendulum length

For most values of k, m, L and g those two modes barely notice each other: give the system a pure vertical bounce and it stays a pure vertical bounce, and a pure swing stays a pure swing, aside from a very slow, weak leakage.

Autoparametric 2:1 resonance

Everything changes near one special ratio: omega_r ≈ 2 · omega_theta, the spring bounces at roughly twice the frequency the pendulum swings at. This is called autoparametric resonance because the bounce mode effectively acts as a periodic modulation of the pendulum's own restoring force, the same mechanism that lets you pump a swing higher by bobbing your body twice per swing cycle. Start the system in a pure vertical bounce at this ratio and it will not stay vertical: the swing mode is driven parametrically, grows from numerical noise or the tiniest initial angle, and steals energy from the bounce. After it grows large enough the swing gives energy back to the bounce, and the cycle repeats - a slow, striking beat where the pendulum alternates between bouncing straight up and down and swinging in wide arcs, first identified analytically in the 1930s and a textbook example of nonlinear mode coupling ever since.

From a beat pattern to chaos

At small amplitudes the energy exchange near the 2:1 resonance is close to periodic - a smooth, repeating beat. Push the amplitude higher and the nonlinear coupling terms that were small corrections become dominant, and the elastic pendulum turns into a genuinely chaotic system: its trajectory in phase space becomes sensitive to the initial angle and stretch to the point where two nearly identical starting states diverge completely within a handful of oscillations. Unlike the double pendulum, whose chaos comes from having two swinging arms, the elastic pendulum's chaos comes purely from the interplay of one linear stretching mode and one nonlinear swinging mode - a reminder that you do not need many parts to lose predictability, just the right resonance and enough energy.

Simulating it correctly

Because energy sloshes on a slow timescale relative to either natural frequency, an integrator that leaks or gains energy will wreck the beat pattern within a few cycles even if each individual oscillation looks fine. This simulation integrates the coupled Lagrangian equations of motion with RK4 at a fixed, small time step precisely so the slow energy exchange between modes is not swamped by numerical drift - the same lesson that applies to the double pendulum and to N-body orbits.

Frequently asked questions

Try it live

Everything above runs in your browser - open Elastic Pendulum and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.