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 relationship between the two frequencies determines how large the response is.

Resonance

When the driving frequency equals the natural frequency, every push arrives in step with the velocity, pushing along the motion, so energy transfers in at the most efficient rate and the amplitude climbs to a maximum. That is resonance, stated for a lightly damped system. 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.

Describe the drawing

The driver sweeps slowly from below the natural frequency to above it. Far from f₀ the bob barely answers; as the frequencies meet it blooms, then shrinks again, and the cyan curve is the record of that sweep being drawn as it happens. The second pass reruns the sweep with heavier damping: the amber response is lower, broader, and peaks slightly early. Watch the lag too: in step below resonance, a quarter cycle behind at it, opposite above.

Damping sets the sharpness

The shape of the amplitude-frequency curve is set by the damping. Light damping gives a tall, narrow peak at the natural frequency, so the system responds strongly only very close to that frequency and weakly at all others. Heavier damping gives a lower, broader peak, with its maximum slightly below the natural frequency. Sketch the family of curves on one set of axes.

The phase relationship between driver and oscillator 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.

The maximum of the amplitude curve moves further below the natural frequency as the damping increases, and beyond a certain amount of damping there is no peak at all: the response falls away as the driving frequency rises. The quarter-cycle phase lag at the natural frequency does not move. Elsewhere on this page, resonance refers to the lightly damped case, in which the two frequencies are close together.

Engineering uses both cases. 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 in stationary waves

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. The large stationary wave corresponds to the peak of the resonance curve, and the harmonics of a stretched string are its set of resonant frequencies.

How damping removes energy from an oscillation

Exam-board note, Edexcel only. Edexcel asks by what mechanism damping reduces the amplitude, as well as what it does to the resonance curve. Damping reduces an amplitude 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 between the loading line and the unloading line, which is the energy that is not recovered on unloading.

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.