Physics · Oscillations and waves · Year 2

Nonlinear oscillations

How does a child make a swing go without pushing, and how can a pendulum stand upside down?

The situation

In the school yard your little brother is standing on a swing. Nobody is pushing him, but by squatting and standing up on every swing he sends it higher and higher. At a large angle the swing oscillates more slowly — the textbook $T = 2 π \sqrt{l / g}$ no longer works. And in the physics room the teacher turns a pendulum upside down with its pivot vibrating — and it does not fall over! What happens when an oscillation is nonlinear, or when its parameter changes periodically?

Topic

Nonlinear oscillations, parametric resonance. Pumping a swing by standing and crouching, and the Kapitza pendulum

Key ideas

Anharmonic pendulum
Parametric resonance
Kapitza pendulum

Formulas

T = 4 \sqrt{\frac{l}{g}} K (\sin \frac{θ_{0}}{2}) \approx T_{0} (1 + \frac{θ_{0}^{2}}{16} + \frac{11 θ_{0}^{4}}{3072})

Period of a pendulum with a large amplitude: exact (elliptic integral $K$ ) and the series approximation.

\ddot{x} + 2 γ \dot{x} + ω_{0}^{2} x + β x^{3} = f \cos Ω t

The Duffing equation describes an anharmonic oscillator. For amplitude $A$ the frequency is $ω \approx ω_{0} + \frac{3 β A^{2}}{8 ω_{0}}$ .

\ddot{θ} + ω_{0}^{2} [1 + h \cos Ω t] θ = 0, | Ω - 2 ω_{0} | < \frac{1}{2} h ω_{0}

Parametric resonance: if the parameter varies with frequency $Ω \approx 2 ω_{0} / n$ , the oscillation grows. The first zone is $Ω \approx 2 ω_{0}$ .

U_{e f} (θ) = - m g l \cos θ + \frac{m A^{2} Ω^{2}}{4} \sin^{2} θ \Rightarrow A^{2} Ω^{2} > 2 g l

The Kapitza pendulum: fast vibration creates an effective potential; the inverted position is stable under this condition.

y^{''} + (a - 2 q \cos 2 τ) y = 0, a = \frac{4 ω_{0}^{2}}{Ω^{2}}, q = \frac{2 A}{l}

The Mathieu equation: a pendulum whose pivot vibrates as $A \cos Ω t$ ( $τ = Ω t / 2$ ; for the inverted position $a < 0$ ).

Theory

Why does the period get longer at a large angle?

The restoring force is $m g \sin θ$ — smaller than $m g θ$ : at a large displacement the pendulum is “softer” and returns more slowly. The exact period is expressed through an elliptic integral: at $θ_{0} = 90 °$ the period $T$ is 18 % longer than for small oscillations, and as $θ_{0} \to 180 °$ it tends to infinity — the top point is an unstable equilibrium. The isochronism discovered by Galileo holds only for small amplitudes. On the phase portrait ( $θ$ , $ω$ ) closed curves are oscillations, and the separatrix is the boundary between oscillation and rotation.

A method for calculating nonlinear oscillations

The general method is successive approximation (Lindstedt–Poincaré): the solution and the frequency are expanded as a series in the amplitude, and at each order the “resonant” terms are eliminated. The result: the frequency depends on the amplitude, $ω = ω_{0} (1 - θ_{0}^{2} / 16 + \dots)$ , and the harmonics $3 ω$ , $5 ω$ appear in the oscillation. The graph compares the series with the exact curve: the series is good up to 90° and gives a sharp error beyond 150°. For a damped pendulum (switch on friction) the measured period approaches $T_{0}$ as the amplitude decreases.

The anharmonic oscillator and the jump

When a load between two taut springs is displaced sideways, the restoring force has the form $- ω_{0}^{2} x - β x^{3}$ — this is the Duffing oscillator (a hardening spring). The resonance curve leans to the right: at one frequency there are three solutions, and the middle one is unstable. If you raise the frequency slowly, the amplitude grows along the upper branch and then jumps down abruptly; if you lower it, it jumps up at a different frequency — hysteresis in frequency.

Parametric resonance: pumping a swing by squatting and standing

The child does not push the swing — they change the system's parameter, the distance $l$ from the pivot to the centre of mass. Standing at the lowest point (where the centripetal force is greatest) and squatting at the extremes adds energy on every cycle; this happens twice per period, so $Ω = 2 ω_{0}$ . If the child squats and stands to a metronome (regardless of phase), the oscillation grows exponentially only when $Ω$ falls inside the instability zone; once the amplitude is large, the frequency drops and the swing leaves resonance. A swing at rest cannot be pumped parametrically — a small initial oscillation is needed. For length modulation the effective depth is $h_{e f f} \approx 3 h$ .

Kapitza pendulum

If the pivot vibrates quickly ( $Ω ≫ ω_{0}$ ), an additional effective potential $\frac{1}{4} m A^{2} Ω^{2} \sin^{2} θ$ appears for the slow motion of the pendulum: it is a minimum at $θ = 0$ and at $θ = π$ . If $A^{2} Ω^{2} > 2 g l$ , the inverted position becomes stable — the pendulum oscillates slowly at the top. This idea of stability in a rapidly varying field also works in the Paul trap, which holds ions.

The Mathieu equation and the stability map

Small oscillations of a pendulum with a vibrating pivot are described by the Mathieu equation. By calculating the solution over one period (Floquet theory), we find the regions where the growth multiplier is $| λ | > 1$ — these are the red “tongues” on the map. They start at $Ω = 2 ω_{0} / n$ , and friction lifts them upwards (a minimum depth is required). Both maps in the experiment are calculated in exactly this way; move the point and compare the theory with the motion.

Did you know?

The giant incense burner in the cathedral of Santiago de Compostela (Spain) — the Botafumeiro — is swung by a group of people rhythmically pulling and releasing the rope. It is a centuries-old example of parametric resonance in use.
The inverted pendulum with a vibrating pivot was described by A. Stephenson in 1908, and P. L. Kapitza gave its theory in 1951. W. Paul received the Nobel Prize in 1989 for an ion trap that works on a principle close to this idea.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Sound: speed, pitch, ultrasound — Grade 10
The dutar string and sound — Grade 11
Waves and the Doppler effect — Grade 11
Bridge resonance — Grade 11
Coupled oscillations — Grade 11
Fourier: the spectrum of sound — Year 3