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Oscillations overview

Topic 6 of 6

Driving and resonance

A periodic external force can keep an oscillator moving by supplying energy. The size of the response depends on driving frequency, damping and drive strength.

Separate the driving frequency from the natural frequency

The natural frequency is the frequency of free oscillation. For the undamped spring-cart model with m = 0.400 kg and k = 6.40 N/m, ω0 = 4.00 rad/s and f0 = ω0/(2π) = 0.636620 Hz.

The driving frequency is set by the external periodic force. After the initial transient has died away, the steady forced response of the linear oscillator repeats at that driving frequency. Natural and driving frequency are different concepts even when their numerical values coincide.

With damping present, maintaining a steady amplitude requires energy input. Over each steady cycle, energy supplied equals energy dissipated. Constant amplitude does not mean that damping has stopped acting.

Read an amplitude-frequency response

Vary the driving frequency while keeping the driving-force amplitude and the oscillator fixed. At resonance, the response amplitude reaches a maximum when the driving frequency is close to or at the natural frequency.

The vertical axis below is response amplitude, not signed displacement. The horizontal axis is driving frequency in Hz, not time. Each point represents the steady response at one chosen driving frequency.

Response amplitude depends on driving frequency

Keep the mass 0.400 kg, spring stiffness 6.40 N/m and driving-force amplitude 0.0128 N fixed. Each curve predicts the steady response after transients have settled. Only the damping changes.

A: solid, least damping

B: long dash, intermediate damping

C: short dash, greatest damping of these three

Greater damping lowers and broadens the finite response peakResponse amplitude in millimetres is plotted against driving frequency from zero to 1.2 hertz, rather than against time. All three smooth response curves share a nonzero two-millimetre limit at low frequency. Solid A has the least damping and peaks near 0.632532 hertz at 12.5402 millimetres. Long-dashed B peaks near 0.615648 hertz at 5.64780 millimetres. Short-dashed C has the greatest damping of these three and peaks near 0.576484 hertz at 3.49428 millimetres. A vertical dotted reference marks the natural frequency 0.636620 hertz. The true damped maxima are close to but below that reference; the dots mark their actual maxima. All peaks are finite, and no ideal undamped steady-state curve is shown. Letters and line patterns identify the curves independently of colour.0.00.20.40.60.81.01.202468101214ABCAmplitude X / mmf0Driving frequency f / Hz

The dotted reference is f0 = 0.636620 Hz. The peaks shift slightly below it as damping increases. The common 2.00 mm low-frequency limit is the static-deflection scale; it is not zero amplitude, and f = 0 is not an oscillatory cycle.

The curves compare the same mass, stiffness and driving-force amplitude, with increasing damping from A to B to C. The separate f0 reference marks the undamped model's natural frequency; the response peaks need not lie exactly on it.

These supplied model curves use m = 0.400 kg, k = 6.40 N/m and sinusoidal driving-force amplitude 0.0128 N. They are predictions for a controlled comparison, not measured responses.

Curve A, with the least damping of the three, has the highest, sharpest peak: about 12.5 mm near 0.633 Hz. The intermediate curve B peaks at about 5.65 mm near 0.616 Hz. Curve C, with the greatest damping, has a lower, broader peak: about 3.49 mm near 0.576 Hz.

Greater damping reduces the large response and makes the resonance less sharp. In this displacement-response comparison, it also moves the peak farther below the undamped f0. Do not force every damped curve to peak at exactly that reference frequency.

At very low driving frequency, the displacement follows the slowly varying force almost statically. The limiting amplitude here is F0/k = 0.0128/6.40 = 0.00200 m = 2.00 mm, so the response curves do not begin with a fabricated zero-amplitude limit.

The driven amplitude is determined by these drive and damping conditions; it need not equal the 0.0500 m amplitude chosen for the earlier free motion. In a perfectly lossless oscillator driven at exact resonance, a finite steady amplitude would not be established: continued input increases the response. The finite peaks above include damping.

Optional check Frequency-response curves compare the same oscillator and driving-force amplitude with different damping. What does a lower, broader resonance peak indicate?
Frequency-response curves compare the same oscillator and driving-force amplitude with different damping. What does a lower, broader resonance peak indicate?

Useful and unwanted resonance

A swing: the swing is the responding system and the repeated pushes are the drive. Well-timed pushes do positive work each cycle and build its motion when their timing is close to the free-oscillation rhythm. A maintained amplitude requires the pushes to replace losses.

A musical resonator: a driving vibration can produce a strong response near one of the resonator's natural frequencies. Energy passes into its vibration and then into the surrounding air, supporting sound production.

A machine support: a running motor can exert a periodic force. If its driving frequency is near a support's natural frequency, a large response can cause unwanted vibration and stress. Damping can reduce the response, while a suitable design or operating change can separate the driving and natural frequencies.

Changing mass or stiffness changes the system's natural frequency, but either change is not automatically an improvement. Judge whether it moves the system towards or away from the actual drive frequency.

Investigate the response with controlled conditions

Change the driver frequency in steps, let transient motion settle, and record the steady amplitude from the equilibrium position. Use the same calibrated displacement method and measure enough of the motion to distinguish the steady amplitude from a brief transient peak.

Keep drive strength, mass, restoring arrangement and damping fixed when obtaining one frequency-response curve. To compare damping, change damping deliberately while maintaining the other conditions, then repeat the frequency scan.

Record what the apparatus controls. A driver that maintains a support's displacement amplitude does not automatically maintain driving-force amplitude, so its curve should not be claimed to reproduce this fixed-force comparison without checking that condition.