Mechanical and electrical vibrations
Springs, pendulums, and circuits all obey the same second-order equation. Its solutions explain damping, resonance, and why bridges can sway dangerously.
The mass-spring model
A mass \(m\) on a spring with stiffness \(k\) and damping \(c\) satisfies \(my'' + cy' + ky = F(t)\), from Newton's second law: inertia plus damping plus spring force equals applied force. Without damping or forcing, \(my'' + ky = 0\) gives simple harmonic motion with angular frequency \(\omega_0 = \sqrt{k/m}\).
my'' + cy' + ky = F(t)Simple harmonic motion
The solution \(y = A\cos(\omega_0 t - \phi)\) oscillates forever with amplitude \(A\) and period \(\frac{2\pi}{\omega_0}\). A stiffer spring (larger \(k\)) or lighter mass oscillates faster. Clocks, tuning forks, and molecular vibrations follow this pattern.
Damping regimes
The discriminant \(c^2 - 4mk\) decides the behaviour. Underdamped (\(c^2 < 4mk\)): decaying oscillations, like a car bouncing on weak shock absorbers. Critically damped (\(c^2 = 4mk\)): fastest return to rest without oscillating, the design goal for door closers. Overdamped (\(c^2 > 4mk\)): slow return without oscillation.
Worked example
With \(m = 1\), \(c = 2\), \(k = 10\): \(r^2 + 2r + 10 = 0\) gives \(r = -1 \pm 3i\), so \(y = e^{-t}(c_1\cos 3t + c_2\sin 3t)\): an underdamped oscillation at frequency 3 that shrinks by a factor of \(e\) every second.
Forced oscillations
With forcing \(F_0\cos\omega t\), the steady-state response oscillates at the driving frequency \(\omega\), not the natural one. After the transient \(y_h\) dies out (with damping), only this steady state remains. Its amplitude depends on how close \(\omega\) is to \(\omega_0\).
Resonance
Without damping, driving at exactly the natural frequency gives \(y_p = \frac{F_0}{2m\omega_0}t\sin\omega_0 t\), an oscillation whose amplitude grows without bound. With light damping, the amplitude peaks sharply near \(\omega_0\). Resonance lets a radio tune to one station, but it also explains why soldiers break step when crossing bridges.
y_p = \frac{F_0}{2m\omega_0}\,t\sin\omega_0 tThe electrical analogy
An RLC circuit obeys \(Lq'' + Rq' + \frac{1}{C}q = E(t)\), the same equation with inductance as mass, resistance as damping, and \(\frac{1}{C}\) as stiffness. Engineers use this analogy to study mechanical systems with circuits, and the tuning circuit in a radio is a resonant RLC circuit.
Lq'' + Rq' + \tfrac{1}{C}q = E(t)Beats
When the driving frequency is close to, but not equal to, the natural frequency, the response shows beats: a rapid oscillation whose amplitude slowly swells and fades. Musicians tune instruments by listening for beats to disappear.
Key takeaways
\(my'' + cy' + ky = F\) models vibrations. Undamped motion oscillates at \(\sqrt{k/m}\). Damping is under, critical, or over depending on \(c^2 - 4mk\). Forced systems settle to the driving frequency; driving near the natural frequency causes resonance or beats. RLC circuits obey the same mathematics.
Flashcard
Mass-spring-damper equation
Check your understanding
A spring system has m = 1, c = 4, k = 4. It is: