Resonance and the effects of damping on the sharpness of resonance.

The idea

Resonance occurs when a system is driven at a frequency matching one of its natural frequencies, so energy transfer from the driver is maximised and amplitude peaks. The driver supplies energy each cycle; near resonance the system absorbs it efficiently, while away from resonance the response falls. Damping removes energy each cycle, so the amplitude peak is lower and broader. Light damping gives a tall, narrow resonance curve: the response is sharply tuned to the driving frequency. Heavy damping gives a low, wide curve, so the system responds over a wider frequency range but with smaller amplitude. The sharpness of resonance is therefore a measure of how narrow the peak is, often compared using the width of the curve at a chosen fraction of the peak amplitude. A useful method is to plot amplitude against driving frequency for several damping values and compare peak heights and widths.

What earns the marks

• Resonance is the condition where the driving frequency equals a natural frequency of the system, giving maximum energy transfer per cycle and a maximum steady-state amplitude.
• At resonance the phase relationship between driver and oscillator is such that the driver does work on the system throughout the cycle, so energy input is greatest.
• Damping dissipates mechanical energy, usually as thermal energy, so the resonant amplitude is reduced compared with the undamped or lightly damped case.
• Increasing damping broadens the resonance curve, reducing the sharpness of resonance; decreasing damping narrows the curve and raises the peak.
• The sharpness of resonance can be compared by measuring the width of the resonance peak at a stated level, for example at half the peak amplitude, and relating a narrower width to sharper tuning.
• Practical examples include a mass on a spring driven by a vibrating support, a pendulum driven by a motor, or a loudspeaker cone near its natural frequency.

Common mistakes and how to put them right

• Thinking resonance creates energy.

  Put it right by stating that the driver supplies the energy and resonance maximises the energy transferred per cycle.

• Confusing amplitude with frequency.

  Put it right by keeping the axes distinct, with driving frequency on the horizontal axis and steady-state amplitude on the vertical axis.

• Assuming damping always destroys resonance.

  Put it right by explaining that damping lowers and broadens the peak but the driven system still shows a maximum near the natural frequency.

• Treating the natural frequency as changing with damping.

  Put it right by noting that damping mainly changes peak height and width, while the natural frequency is set by the system's inertia and stiffness.

Exam tips

• Sketch two resonance curves on the same axes, one lightly damped and one heavily damped, and label peak height and width clearly.
• Use the phrase sharpness of resonance only after defining what is being compared, such as the width of the peak at a stated amplitude level.
• When explaining energy transfer, refer to the driver doing work on the oscillating system each cycle rather than saying energy is created.
• Link any practical example back to the driving frequency and the natural frequency of the system, not just to large vibrations.