5.3 Second-order linear ODEs

Add one more derivative and the world changes. With $\ddot x$ in play the system has inertia — once it starts moving it tends to keep moving — and inertia is what lets a system oscillate. Almost every system in this bookshelf that vibrates, rings, resonates, or oscillates is described by a second-order linear ODE. Mass on a spring, pendulum, tuning fork, LC circuit, every acoustic mode, the cochlear filter at each place on the basilar membrane — the same equation, different labels.

The characteristic-equation trick from 5.2 carries through with one extra subtlety: the polynomial is now quadratic, so there are two roots, and the interplay between them sets the character of the motion.

The general form

A second-order linear ODE with constant coefficients looks like

with constants $a, b, c$ and an external forcing $f(t)$. In physical problems each of the three constants always means something — inertia, friction, restoring stiffness — and we’ll see them under several different names as the lesson goes on.

We’ll build up the story in three stages:

1. Undamped ($b = 0$, $f = 0$) — pure sinusoidal motion forever. Simple harmonic motion.
2. Damped ($b > 0$, $f = 0$) — decay and four distinct regimes.
3. Forced ($f(t)$ a sinusoid) — resonance: dramatic amplification when the drive matches the natural frequency.

Each stage adds one feature at a time. By the end you’ll have the full damped-driven-oscillator equation, which is one of the most useful equations in all of physics and engineering.

Stage 1 — Simple harmonic motion

Set $b = 0$ (no friction) and $f = 0$ (no forcing). Rename constants to their physical meanings: $a = m$ (mass), $c = k$ (spring stiffness). Newton’s second law for a mass on a frictionless spring is

with the natural frequency $\omega_0 \equiv \sqrt{k/m}$. Try the trick. Substitute $x = e^{\lambda t}$:

The two characteristic roots are pure imaginary. By Euler’s formula (refresher →), $e^{i \omega_0 t} = \cos(\omega_0 t) + i \sin(\omega_0 t)$. The real-valued general solution is the linear combination

with constants $A$ and $B$ fixed by the initial position $x(0)$ and initial velocity $\dot x(0)$. The motion is purely sinusoidal at $\omega_0$ — no decay, no drift. The oscillator runs forever.

On the left, a mass slides back and forth on a spring; on the right, the same motion appears as a sine curve in time. Slide $\omega_0$ and the period $T = 2\pi/\omega_0$ shrinks. Set $x_0$ to start the mass displaced; set $v_0$ to give it an initial kick. The two pieces of initial data uniquely fix the curve.

The mass and the curve are the same thing seen two ways: the mass’s position at time $t$ is the value of $x(t)$ on the right.

Stage 2 — Add damping

Real oscillators lose energy. Air viscosity drags on a tuning fork; mechanical friction warms a shock absorber; electrical resistance dissipates current in an RLC circuit. We model the loss with a force proportional to velocity, opposing the motion:

The coefficient $\gamma > 0$ is the damping rate. (The factor of 2 makes downstream formulas tidier.) Apply the same trick — substitute $x = e^{\lambda t}$:

This is just the quadratic formula. Now look at what happens as we crank $\gamma$ up from zero. The character of $\lambda$ — and hence of $x(t)$ — depends on the sign of the discriminant $\gamma^2 - \omega_0^2$. Four regimes:

• $\gamma = 0$ — undamped. $\lambda = \pm i\omega_0$. Pure sinusoid; the previous stage.

• $0 < \gamma < \omega_0$ — underdamped. $\lambda = -\gamma \pm i\,\omega_d$ with $\omega_d = \sqrt{\omega_0^2 - \gamma^2}$. The roots have both a real part (decay) and an imaginary part (oscillation). The solution is a sinusoid inside a decaying envelope:

  $$x(t) \;=\; e^{-\gamma t}\, \bigl[\,A\cos(\omega_d t) + B\sin(\omega_d t)\,\bigr].$$

• $\gamma = \omega_0$ — critically damped. The two roots collide at the single real value $-\gamma$. The repeated-root case from 5.2 kicks in: $x(t) = (A + Bt)\,e^{-\gamma t}$. The system returns to equilibrium as fast as possible without oscillating.

• $\gamma > \omega_0$ — overdamped. Two distinct real negative roots. Sluggish, monotone return to equilibrium with no overshoot.

Slide $\gamma$ from 0 up through and beyond $\omega_0$. At low damping you get slow ringdown — a sinusoid inside a slowly-shrinking envelope (the dashed lines trace $\pm e^{-\gamma t}$). At critical damping the oscillation just barely doesn’t happen. At high damping the curve sags back to zero without crossing it.

You can also vary $x_0$ and $v_0$ — the two pieces of data a second-order ODE always demands.

Why each regime matters

Each regime is the right answer for a different engineering problem. Engineers spend a lot of time deliberately choosing damping to land in one of these zones.

• Underdamped is what makes things ring. A tuning fork, a wine glass tapped with a knuckle, a struck bell, the strings of an instrument, the cochlear filters in your ear — all are slightly underdamped, so they sustain audible oscillation while losing a small fraction of their energy per cycle. The number of cycles before they fade is the quality factor $Q \sim \omega_0/(2\gamma)$.
• Critical damping is the optimum for systems that should settle quickly without bouncing: a car shock absorber, a swinging door with a closer, the read-arm of a hard drive, the auto-focus actuator in a camera lens, an analog meter needle. Engineers tune for $\gamma \approx \omega_0$ deliberately so the response is fast but free of overshoot.
• Overdamped is what you get when friction dominates: a heavy door swinging through air thick with friction, a galvanometer in a viscous fluid bath, a slow chemical reaction settling toward equilibrium.

The characteristic roots in the complex plane

The four regimes correspond to four configurations of $\lambda_\pm$ in the complex plane. This is the most important picture in the chapter — once you see it the four-regime story becomes a single geometric statement.

Two synchronised pictures. On the left, the roots’ positions in the complex plane. On the right, the resulting time-domain $x(t)$ from unit initial conditions. Two coloured rules tie them together:

• Red — the real part of $\lambda$. Sets the decay rate. On the left it’s the horizontal distance from the imaginary axis to each root; on the right it’s the dashed envelope $\pm e^{-\gamma t}$ that the solution lives inside.
• Blue — the imaginary part of $\lambda$. Sets the oscillation rate. On the left it’s the vertical distance from the real axis to each root; on the right it’s the period $T = 2\pi / |\mathrm{Im}\,\lambda|$ marked with vertical dashed lines.

Slide $\gamma$ from 0 upward and watch both panels move together. The roots start on the imaginary axis (undamped — pure oscillation, no envelope), slide together along a semicircle into the left half-plane (underdamped — oscillation inside a shrinking envelope), meet on the real axis at $\gamma = \omega_0$ (critically damped — envelope collapses, no oscillation), and then walk apart along the real axis (overdamped — two decay rates, no oscillation).

Stable systems — ones that decay to equilibrium rather than blow up — live in the left half of the complex plane: $\mathrm{Re}\,\lambda < 0$. Unstable systems live in the right half. The imaginary axis is the stability boundary. The central question of control theory and circuit design is keeping the eigenvalues of a closed-loop system on the stable side of that axis.

Stage 3 — Force it

Now turn on the forcing. The simplest case is a sinusoidal drive at some frequency $\omega$:

The full solution is a sum of two parts: a transient (the homogeneous solution we just worked out, decaying with $\gamma$) plus a particular solution at the driving frequency $\omega$. After the transient dies out, only the steady-state response remains. Using the complex-exponential trick (refresher →), the steady-state amplitude as a function of drive frequency is

The magnitude is maximised when $\omega \approx \omega_0$ — this is resonance. The peak height grows like $1/\gamma$ (so the peak diverges as damping vanishes); the peak width is $\sim \gamma$ (so the peak narrows as damping vanishes). The quality factor $Q = \omega_0/(2\gamma)$ captures both: high $Q$ means tall and narrow.

Slide the drive frequency past the natural frequency and watch the amplitude peak rise and fall. The phasor diagram on the side shows the phase of the response relative to the drive — exactly $90°$ behind at resonance, in phase well below, $180°$ out of phase well above. That $90°$ lag at resonance is a universal signature; it appears in every resonant system ever built.

Why resonance matters

Resonance is what makes acoustics, music, and hearing work — and what occasionally breaks bridges and shatters wine glasses. A small sample of real systems described by exactly the equation above:

• A wine glass has natural frequencies set by its geometry and material. Sing the right note loudly enough at low damping and the amplitude can climb past the glass’s elastic limit. It shatters.
• A child on a swing is a damped pendulum. Push at the natural frequency (and at the right phase) and the amplitude grows; push off-frequency and you do almost no net work. Every parent at a playground is solving the forced-oscillator problem.
• The Tacoma Narrows Bridge collapsed in 1940 because a vibrational mode of the deck happened to match the frequency of wind-driven aerodynamic forcing. The damping was too low to absorb the energy and the amplitude grew until the bridge tore itself apart. Modern long-span bridges are deliberately detuned or fitted with dampers to prevent such matches.
• An MRI machine drives nuclear-spin resonances at $\omega_0 = \gamma_n B_0$ by hitting them with an RF pulse at that frequency. The image is reconstructed from the ring-down of the response after the drive is removed.
• A cochlear hair cell sits on a basilar-membrane segment whose mechanical resonance places it at a characteristic frequency. Sound at that frequency drives that hair cell preferentially — the ear is built as a bank of resonators tuned across the audible range (Hearing Ch 4).
• An AM/FM radio tuner is an LC circuit whose resonance is steered to the carrier frequency of the station you want. The narrow peak rejects all the other broadcasts at adjacent frequencies.
• A vocal-tract formant is a resonance of the air column in your mouth, throat, and nose. By reshaping the geometry you move $\omega_0$ around; the formant frequencies are how vowels are distinguished.
• The body of a violin, the port of a bass-reflex loudspeaker, the soundboard of a piano — each is an acoustic resonator whose response peak shapes the radiated spectrum.
• A dye-laser tuning curve, a mass-spectrometer ion trap, an atomic-clock cavity: physics labs are full of forced oscillators driven near their natural frequencies for precision-measurement reasons.
• A building during an earthquake has natural sway modes; if the seismic excitation overlaps them, the amplitudes climb. Tuned mass dampers (a heavy pendulum tuned to the building’s $\omega_0$, off in phase) are added to skyscrapers to bleed energy away from the structural mode.

Every one of those systems is described by the equation at the top of this section, with different physical meanings for $\omega_0$ and $\gamma$ but identical mathematics.

A pause before lesson 4

We’ve now done the canonical second-order story: SHM → damping → forcing. The same characteristic-equation trick handled all three. In the final lesson — 5.4 — Phase plane and classification — we step back and look at the geometric view, in which the same motion becomes a curve in $(x, \dot x)$ space, and meet the classification of equilibria that organises every 2-D linear system on a single map.