Forced vibrations and resonance

Free and forced

Displace a system and let go and it performs free vibrations at its own natural frequency, the one set by its mass and stiffness. Push it periodically instead and it performs forced vibrations at the driving frequency, whatever that is. The interesting physics lives in the relationship between the two frequencies.

Resonance

When the driving frequency equals the natural frequency, every push arrives in step with the velocity, pushing along the motion, energy transfers in at the most efficient rate available, and the amplitude climbs to a maximum. That is resonance, stated for a lightly damped system, which is the version A-level answers need. Think of a child on a swing. Push twice as hard, at any frequency, and you get twice the amplitude: a linear oscillator's response is proportional to the driving force. What the timing changes is the factor by which that push is magnified. A mistimed push partly cancels the motion. A well-timed one builds it until the energy lost per cycle matches the energy supplied.

Damping sets the sharpness

The amplitude-frequency curve is the topic's exam centrepiece, and damping controls its shape. Light damping draws a tall narrow spike at the natural frequency, a system that responds strongly only very close to the natural frequency and weakly at all others. Heavier damping gives a lower, broader hump, duller but safer, with the peak nudged slightly below the natural frequency. Sketching that family of curves on one set of axes is a standard mark.

Phase deserves a sentence of its own, since it is examined directly. Drive the system well below its natural frequency and the oscillator moves almost in step with the driver. At the natural frequency it lags by a quarter of a cycle, and by exactly that whatever the damping. Push the driving frequency well above the natural frequency and the lag grows towards half a cycle, leaving driver and oscillator moving in opposite directions.

Two frequencies sit behind the one word, then, and only heavy damping separates them. The tallest point of the amplitude curve slides further below the natural frequency the more damping there is, and past a certain amount of damping there is no peak at all: the response simply falls away as the driving frequency climbs. The quarter-cycle lag never moves. Under light damping the two sit close enough to share the name resonance, which is the reading every mark scheme takes and the one assumed everywhere else on this page.

Engineering runs both directions. Where resonance is a hazard, vehicle suspensions, buildings in wind or earthquakes, bridges under marching feet, designers add damping to flatten the peak. Where it is the aim, a musical instrument's body, radio tuning circuits, damping is kept light so the response stays sharp.

Resonance with a shape

Exam examples include situations involving stationary waves, and the stretched string is the cleanest of them. Drive it at one of its natural frequencies and the reflections reinforce, cycle after cycle, until a large stationary wave stands on the string. That is the resonance curve's tall peak, rendered as a shape you can see. From this vantage the harmonics of the stationary-waves lesson are the string's set of resonant frequencies.

What the damper is actually doing, an Edexcel extension

One specification asks what damping does to the curve and also by what means, so read this section if Edexcel is your board. Damping brings an amplitude down for a single reason. Something takes energy out of the oscillator on every cycle. In a free oscillation that leaves less to swing with each time, so the amplitude decays; in a forced one it settles wherever the energy dissipated per cycle has grown to match the energy the driver puts in per cycle, which is exactly why a heavily damped system peaks so much lower. Two mechanisms are named, and both amount to the same energy loss described in different ways.

The first is a resistive force, air resistance, friction at a pivot, or oil driven through the narrow orifice of a car's dashpot. Such a force opposes the velocity at every instant, so it does negative work all the way round the cycle and never gives any of it back. The energy becomes internal energy, which is why a suspension worked hard on a rough road is warm to the touch afterwards.

The second is the plastic deformation of a ductile material. Take a ductile metal past its yield point and its structure rearranges permanently, so the work done in deforming it is not stored elastically and returned on the way back. It goes into moving dislocations through the metal and warming it, and on a force-extension graph it is the area trapped between the loading line and the unloading line, the energy that is not recovered on unloading, as the materials unit sets out.

Engineers use that dissipation deliberately. Build a structure so that a sacrificial piece of mild steel yields a little on each large swing, and every swing removes energy from the oscillation until the amplitude falls. Steel yielding dampers in tall buildings and the crumpling members of a vehicle both work this way. The difference from a dashpot is that the deformation is permanent, so the part is replaced after a large event rather than simply used again.