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Physics 1

Oscillations: simple harmonic motion, the pendulum, damping and resonance

The mass-spring system and its solution, energy in simple harmonic motion, the simple pendulum, damped and driven oscillations, resonance and the analogy with the RLC circuit.

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A mass on a spring, a pendulum, a guitar string, a watch crystal, an LC circuit: very different systems that all oscillate following the same equation. Simple harmonic motion is perhaps the most useful model in physics, because almost any system near a point of stable equilibrium behaves this way. In this lesson we derive it, then add damping and a driving force, up to resonance.

The mass-spring system

A mass m on a frictionless surface is attached to a spring of constant k. Call x the displacement from equilibrium, where the spring is neither stretched nor compressed.

A mass attached to a spring and a graph of its position over time
Top: the mass displaced by x from equilibrium feels the spring force F = -kx, which pulls it back towards zero. Bottom: the position oscillates like a cosine between +A and -A, with period T.

Proof: the equation of motion and its solution

The only horizontal force is the spring’s, so m\,\ddot x = -k\,x, that is

    \[ \ddot x + \omega^2 x = 0,\qquad \omega = \sqrt{\frac{k}{m}} \]

We need a function whose second derivative is proportional to itself with a minus sign: sines and cosines. Try x(t) = A\cos(\omega t + \varphi). Differentiating twice:

    \[ \dot x = -A\omega\sin(\omega t + \varphi),\qquad \ddot x = -A\omega^2\cos(\omega t + \varphi) = -\omega^2 x \]

The equation is satisfied for any A and \varphi, which follow from the initial conditions (position and velocity at time zero). Since the equation is second order, this is the general solution.

  • A is the amplitude, the maximum displacement;
  • \omega is the angular frequency (rad/s), and the period is T = 2\pi/\omega = 2\pi\sqrt{m/k};
  • \varphi is the initial phase.

The period does not depend on the amplitude: a large swing and a small one take the same time. This property, isochronism, is what makes oscillations suitable for measuring time.

Energy

With x = A\cos(\omega t + \varphi) and \omega^2 = k/m:

    \[ E = \tfrac12 m\dot x^2 + \tfrac12 k x^2 = \tfrac12 k A^2\sin^2(\omega t + \varphi) + \tfrac12 k A^2\cos^2(\omega t + \varphi) = \tfrac12\,k\,A^2 \]

Energy keeps moving from potential (at the ends) to kinetic (in the middle), but the total is constant and proportional to the square of the amplitude.

The simple pendulum

A mass m hanging from a string of length l is displaced by an angle \theta. The component of the weight along the path is -mg\sin\theta and the displacement along the arc is s = l\theta, so

    \[ m\,l\,\ddot\theta = -m\,g\sin\theta \]

Proof: the small-angle approximation

For small angles \sin\theta \approx \theta (with \theta in radians; at 10° the error is 0.5%). The equation becomes \ddot\theta + (g/l)\,\theta = 0, the same as the mass-spring system with \omega^2 = g/l:

    \[ T = 2\pi\sqrt{\frac{l}{g}} \]

This is the 2\pi that dimensional analysis could not give. A 1 m pendulum ticks a second every half period: T \approx 2.006 s. For large amplitudes the period grows slightly: at 30° it is about 1.7% longer.

Damped oscillations

In reality there is always friction. If it is proportional to velocity, F = -b\,\dot x (viscous friction, a shock absorber), the equation becomes

    \[ \ddot x + 2\gamma\,\dot x + \omega_0^2\,x = 0,\qquad \gamma = \frac{b}{2m},\quad \omega_0 = \sqrt{\frac{k}{m}} \]

Looking for solutions of the form e^{\lambda t} gives the characteristic equation \lambda^2 + 2\gamma\lambda + \omega_0^2 = 0, with roots \lambda = -\gamma \pm \sqrt{\gamma^2 - \omega_0^2}. Three behaviours:

  • underdamped (\gamma \lt \omega_0): it oscillates with an amplitude decaying as e^{-\gamma t}, at angular frequency \omega_d = \sqrt{\omega_0^2 - \gamma^2};
  • critically damped (\gamma = \omega_0): it returns to equilibrium in the shortest possible time without oscillating. It is the ideal tuning for a car’s suspension and for the needle of a measuring instrument;
  • overdamped (\gamma \gt \omega_0): it creeps back to equilibrium without oscillating.

Driven oscillations and resonance

If a periodic external force F_0\cos(\omega t) drives the system, after a transient it oscillates at the driving frequency, with amplitude

    \[ A(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + 4\gamma^2\omega^2}} \]

When \omega approaches \omega_0 the first term under the root vanishes and the amplitude becomes very large, the more so the smaller the damping: this is resonance. A swing pushed at the right moment, a glass shattered by a high note, a bridge vibrating in step with pedestrians (it happened to London’s Millennium Bridge in 2000). The quality factor Q = \omega_0/(2\gamma) measures how narrow and tall the peak is.

The same model in circuits

A circuit with inductance L, resistance R and capacitor C in series obeys, for the charge q on the capacitor, the equation

    \[ L\,\ddot q + R\,\dot q + \frac{q}{C} = 0 \]

It is identical to the damped mass with the correspondences m \to L, b \to R, k \to 1/C. The resonant frequency follows at once:

    \[ \omega_0 = \frac{1}{\sqrt{L\,C}} \]

and everything we said about damping and resonance applies to filters, oscillators and the tuning circuits of a radio. A watch crystal is literally a mechanical oscillator: a tiny quartz element vibrating at 32,768 Hz with a quality factor in the tens of thousands, which is why it drifts by only a few seconds a month.

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