Damped Oscillations

When you push a child on a swing and then stop pushing, the swing does not oscillate forever — it gradually slows down and comes to rest. This happens because energy is continuously being removed from the system by resistive forces such as air resistance and friction at the pivot. This process is called damping, and the resulting motion is called a damped oscillation. Damping is not a flaw; it is a fundamental feature of almost every real oscillating system, from a car's suspension to the strings of a guitar. The mathematics of damped oscillations reveals three distinct regimes depending on how strong the damping force is relative to the restoring force. In light (under) damping the system still oscillates, but with an amplitude that decays exponentially over time. In critical damping the system returns to equilibrium as quickly as possible without oscillating at all. In heavy (over) damping the system returns to equilibrium slowly, again without oscillating. Engineers choose between these regimes deliberately — a car's shock absorber, for example, is designed to be critically or slightly over-damped so the car settles quickly without bouncing. Understanding damped oscillations is essential in physics and engineering because truly undamped systems do not exist in nature. Every oscillator — mechanical, electrical, or acoustic — loses energy to its environment. Knowing the rate of that energy loss, quantified by the damping coefficient and the quality factor Q, allows us to design systems that behave exactly as required, whether we want long-lasting vibrations or a rapid return to rest.

Key takeaways

The Equation of Motion for a Damped Oscillator
Light (Under) Damping — b² < 4mk
Critical Damping — b² = 4mk
Heavy (Over) Damping — b² > 4mk
The Quality Factor Q

The Equation of Motion for a Damped Oscillator

For a mass m on a spring (spring constant k) with a damping force proportional to velocity — F_damping = −bv — Newton's second law gives: m(d²x/dt²) + b(dx/dt) + kx = 0. Here b is the damping coefficient (units: kg s⁻¹). This is a second-order linear differential equation whose solutions depend critically on the relative sizes of b, m, and k.

Light (Under) Damping — b² < 4mk

When damping is weak, the discriminant b² − 4mk is negative and the solution is oscillatory: x(t) = A₀ e^(−bt/2m) cos(ω_d t + φ), where the damped angular frequency ω_d = √(k/m − b²/4m²) < ω₀. The amplitude decays exponentially like a shrinking envelope wrapped around the cosine wave. Think of a plucked guitar string — it vibrates audibly but the sound fades away. The time constant τ = 2m/b tells you how long it takes the amplitude to fall to 1/e (about 37%) of its initial value.

Critical Damping — b² = 4mk

When b = 2√(mk), the system returns to x = 0 in the shortest possible time without overshoot. The solution is x(t) = (A + Bt)e^(−bt/2m), which is no longer oscillatory. Car shock absorbers and the needle on an analogue measuring instrument are designed to be critically damped — the car settles after a bump without bouncing, and the needle reaches its true reading quickly without oscillating past it.

Heavy (Over) Damping — b² > 4mk

When damping is very strong, the system creeps slowly back to equilibrium, never crossing x = 0. The solution involves two decaying exponentials: x(t) = Ae^(r₁t) + Be^(r₂t), where both r₁ and r₂ are negative real numbers. A heavy door closer is deliberately over-damped — the door closes without slamming but takes a long time to shut fully. Although safe, over-damping is inefficient in most engineering contexts.

The Quality Factor Q

Engineers and physicists use the dimensionless quality factor Q = (1/b)√(mk) = ω₀m/b to characterise how well an oscillator sustains oscillations. Q equals 2π times the ratio of energy stored to energy lost per cycle. A lightly damped system (Q ≫ 1) oscillates many times before losing significant energy — a crystal oscillator in a watch has Q ~ 10⁵. Critical damping corresponds to Q = 0.5. A heavily damped system has Q < 0.5. The higher the Q, the sharper and taller the resonance peak when the system is driven.

The Key Equation

x(t) = A_0 \, e^{-\frac{b}{2m}t} \cos\!\left(\omega_d\, t + \varphi\right), \quad \omega_d = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}

x(t) — Displacement (m): Position of the oscillating mass relative to equilibrium at time t.
A_0 — Initial amplitude (m): The amplitude at t = 0 before damping has reduced it.
b — Damping coefficient (kg s⁻¹): Quantifies the strength of the resistive force; larger b means faster energy loss.
m — Mass (kg): The mass of the oscillating object.
\omega_d — Damped angular frequency (rad s⁻¹): The actual oscillation frequency, always less than the natural frequency ω₀.
\varphi — Phase constant (rad): Sets the initial phase of the oscillation, determined by initial conditions.
k — Spring constant (N m⁻¹): The stiffness of the restoring force — larger k gives faster oscillation.

Worked Example

Problem. A 0.50 kg mass is attached to a spring of spring constant k = 50 N m⁻¹ and oscillates in a medium with damping coefficient b = 2.0 kg s⁻¹. (a) Identify the damping regime. (b) Calculate the damped angular frequency ω_d. (c) Find the time for the amplitude to fall to 50% of its initial value.

\omega_d \approx 9.80 \text{ rad s}^{-1}, \quad t_{1/2} = \frac{2m\ln 2}{b} \approx 0.35 \text{ s}

Answer. The system is lightly damped. The damped frequency is approximately 9.80 rad s⁻¹, and the amplitude drops to half its initial value after approximately 0.35 s.

Figure

Diagram

Origins · 1 minLord Rayleigh (John William Strutt) and how this was discovered

Lord Rayleigh (John William Strutt)

1842–1919 · British

Lord Rayleigh provided the rigorous mathematical framework for damped and driven oscillations in his landmark 1877 treatise 'The Theory of Sound'. He introduced the concept of the dissipation function — a quantity analogous to potential energy but describing energy lost to damping — and used it to derive equations of motion for damped mechanical and acoustic systems. His analysis of how damping affects resonance frequency and sharpness of resonance remains foundational to all modern treatments of oscillation theory. Rayleigh discovered the noble gas argon in 1894 (jointly with William Ramsay), earning the 1904 Nobel Prize in Physics — making him one of the very few scientists to win a Nobel Prize for chemistry-related work while being primarily a physicist.

Simple Harmonic Motion — Damped oscillation is built directly on SHM — it is SHM modified by an energy-loss term. Understanding the undamped case is essential before analysing damping.
Resonance — The sharpness and height of a resonance peak are directly controlled by damping via the quality factor Q — a lightly damped system resonates much more dramatically than a heavily damped one.
Forced Oscillations — When an external periodic force drives a damped oscillator, the interplay between driving frequency, natural frequency, and damping determines the amplitude and phase of the steady-state response.
Wave Properties — Waves are oscillations propagating through a medium; wave attenuation (amplitude decrease with distance) is the spatial analogue of temporal damping in oscillators.