Harmonic, damped and forced oscillations

Small pushes can build a large swing, while a badly timed push can slow it down. How does the rhythm of the drive change energy transfer, and why do real oscillations weaken without further pushes?

An ideal mass on a spring slows smoothly near the extremes and crosses the middle quickly. A sine or cosine describes this motion. Oscillations with constant amplitude and frequency whose displacement follows this law are called harmonic. Ideal springs and small-angle pendulums behave approximately this way when losses are negligible. Periodic motion does not always have this shape.

Two identical pendulums may share a period while one is on the right and the other on the left. Predicting position also requires phase φ, the stage within the cycle. Phase is measured in radians; one full cycle changes it by 2π. It describes both displacement and motion direction within the cycle. Adding 2π repeats the state, like a clock hand after a full turn.

Phase increases by 2π during period T. Its rate of change is therefore ω = 2π/T = 2πν, the angular frequency in rad/s. Frequency ν counts cycles per second; ω counts phase radians per second. Doubling ν doubles ω and halves T. Their numerical values differ by 2π, so never substitute hertz for radians per second without checking which symbol a formula uses.

Combine the description: x = A cos(ωt + φ₀). Cosine repeats between −1 and +1, and A scales this to displacements from −A to +A in metres. ω sets the repetition rate, t is in seconds and φ₀ sets the initial state. The current phase is ωt + φ₀, not φ₀ alone. Increasing A stretches the graph vertically; increasing ω compresses it in time.

Check a familiar release: pull the mass to +A and let go without a push. Then φ₀ = 0 because cos 0 = 1. After T/4, phase is π/2 and x = 0; after T/2, phase is π and x = −A. The same x = 0 occurs twice per cycle with opposite motion directions. Choose initial phase from the full starting state, not just a coordinate value.

Real swings eventually stop without deliberate braking. Resistance does negative work, turning some mechanical energy into internal energy of objects and air. Free oscillations started by one displacement or push become damped: amplitude decreases. Removing losses in thought would stop this decrease. A formula with one constant A cannot exactly describe the whole damped motion.

To keep a swing moving, supply energy periodically. This produces forced oscillations. Natural frequency characterises the system's free motion, while the source chooses the driving frequency. For a linear system driven sinusoidally, once the initial transient oscillations decay, steady motion has the force's frequency. The two frequencies can differ, and the response depends on how close they are.

Try pushing a swing at different rhythms. With weak losses, amplitude becomes especially large when the driving frequency is close to the natural frequency: energy transfer is efficient from cycle to cycle. This enhanced response is resonance. Frequencies do not create energy; the person pushing does work. An ideal lossless model permits growth, but in a real system growth changes the balance of energy supplied and lost.

Why can a real swing not reach infinite amplitude? As motion grows, friction and resistance limit the response, and the linear model may cease to apply. A damper dissipates mechanical energy and reduces resonant vibration in machines. In a musical instrument, enhanced response in a useful frequency range is desirable. The same mechanism has different consequences depending on system parameters, driving frequency and losses.

A mass oscillates according to x = 0,04 cos(πt), with x in metres and t in seconds. Find amplitude, period and x after 1 s.

Compare x = 0,04 cos(πt) with x = A cos(ωt + φ₀). The cosine coefficient gives A = 0,04 м = 4 см; ω = π рад/с, φ₀ = 0.

A full-cycle phase increase of 2π takes T = 2π/ω = 2 с. Thus 1 s is half a period.

x(1) = 0,04 cos π = −0,04 м = −4 см.

The mass starts at the right extreme +A and reaches the left extreme −A after half a cycle. This agrees with the calculated coordinate, whose magnitude does not exceed amplitude.

A = 4 см, T = 2 с, x(1 с) = −4 см. In one second the mass travels from the positive extreme to the opposite extreme.

The harmonic law separates amplitude A, rate ω and initial state φ₀; current phase changes with time.

Damping transfers mechanical energy to other forms; maintaining motion requires an energy supply.

Resonance enhances the response to an external force at a suitable frequency; the energy source remains external.

When oscillations are passed to neighbours, a wave forms.