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15.4Resonance and damping
We now examine the effect of resistance forces and other externally applied forces on an oscillating system.
Oscillations and damping
In ChapterÂ 12 we saw examples of systems performing oscillations. Those were free oscillations, i.e. oscillations without loss of energy and without externally applied forces. In this case the amplitude of oscillations stays constant. In this chapter we will examine the effects of the presence of such forces. If the forces resist the motion we will have damping; in other words, loss of energy in an oscillating system.
In light damping the amplitude of oscillation decreases slowly with time and eventually becomes zero; the oscillations stop, Figure 15.13. The amplitude decreases exponentially (dotted blue line).

Figure 15.13: In damped oscillations the amplitude is decreasing with time exponentially.

Driven oscillations and resonance
We will now examine qualitatively the effect of an externally applied force F on a system that is free to oscillate with frequency f0. This is called the natural frequency of the system. The force F will be assumed to vary periodically with time with frequency fD, called the driving frequency, for example F = F0 cos (2Ï fDt). The oscillations that take place in this case are called forced or driven oscillations.
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As an example consider a pendulum of length L that hangs vertically as in Figure 15.14. The point of support is made to oscillate with some frequency fD. How does the pendulum react to this force?

Figure 15.14: Forced oscillations: the point of support is acted upon by a periodic force that makes the point of support vibrate.

In general, after the external force is first applied, the system will eventually switch to oscillations with a frequency equal to the driving frequency fD. This is true both when the system is initially at rest and when it is oscillating at its own frequency before the force is applied. In the absence of damping, the amplitude of oscillations approaches infinity as the driving frequency approaches the natural frequency, Figure 15.15. This is impossible; it shows that we have to take damping into account.

Figure 15.15: Graph showing the variation with driving frequency of the amplitude of driven SHM when the system is driven by an external periodic force and no damping. The natural frequency is 5 Hz here.

The amplitude of the oscillations will depend on fD and the amount of damping. A detailed analysis produces the graph in 15.16 showing how the amplitude of oscillation of a system with natural frequency f0 varies as it is subjected to a periodic force of frequency fD. The degree of damping increases as we move from the top curve down.

Figure 15.16: Graph showing the variation with driving frequency of the amplitude of oscillations when the system is driven by an external periodic force. For very small damping, the peak occurs at a frequency very close to the natural frequency. As the damping increases, the peak occurs at a frequency less than f0, as indicated by the dark dashed curve.

The general features of the graph in Figure 15.16 are as follows:
âFor very small damping, the peak of the curve occurs very close to the natural frequency of the system, f0. In other words, when the driving frequency is equal to the natural frequency the amplitude of oscillations is a maximum.
This condition is called resonance. Resonance has been experienced by anyone trying to push a swing: even a small force exerted on the swing at the right time intervals results in large oscillations of the swing.
âAs the amount of damping increases, the peak shifts to lower frequencies and becomes wider and shorter. In other words, with large damping the amplitude is a maximum when the driving frequency is slightly less than the natural frequency.
âAt very low frequencies, the amplitude is essentially constant, and at high frequencies the amplitude goes to zero.
In all cases the system oscillates at the driven frequency.
We distinguish three cases in the amount of damping, Figure 15.17. In light damping the system returns to equilibrium after many oscillations of decreasing amplitude. In critical damping, the system returns to equilibrium in the shortest possible time without performing oscillations. Finally, in the heavy or overdamped case the system returns to equilibrium after a very long time without performing oscillations.
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Figure 15.17: Light, critical and heavy damping.

CHECK YOURSELF 6
The same system is exposed to the same driving force at different levels of damping. Which curve in Figure 15.18 corresponds to more damping?

Figure 15.18: For check yourself question 6.

Resonance can be disastrous: we do not want an aircraft wing to resonate, nor is it good for a building or a bridge to be set into resonance by an earthquake or the wind, Figure 15.19.

Figure 15.19: The Tacoma Narrows Bridge in Washington State in the United States oscillating before collapsing in 1940, a victim of resonance.

Resonance can be irritating if the car in which you drive is set into resonance by bumps on the road or by a poorly tuned engine. But resonance can also be a good thing: resonance is used by a microwave oven to warm food; and your radio uses resonance to tune into one specific station and not another.
Links
âThe fact that greenhouse gases in the atmosphere absorb infrared radiation emitted by the earthâs surface is an example of resonance: the frequency of the infrared photons matches the natural frequency of the greenhouse gases. The natural frequency in this case is the frequency of the photons the gases can emit.
âIn damped driven oscillations, energy is being dissipated and so the driving force must provide the energy needed to keep the amplitude constant. Application of the law of energy conservation gives the interesting result that the power supplied by the driving force is a maximum when the frequency of the driving force is exactly equal to the natural frequency of the system. So resonance can also be understood as the condition under which the power supplied is a maximum.
CHECK YOURSELF 7
In driven damped oscillations, the maximum amplitude occurs when the driving frequency:
ais equal to the natural frequency of the system
bis slightly greater than the natural frequency of the system
cis slightly smaller than the natural frequency of the system
dapproaches zero.
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NATURE OF SCIENCE
The phenomenon of resonance is ubiquitous in physics and engineering: microwave ovens emit electromagnetic radiation in the microwave region at wavelengths that match the vibration wavelengths of water molecules so they can be absorbed and warm food; in a technique known as MRI in medicine, resonance is used to force absorption of radio frequency radiation by protons in the body that is exposed to a magnetic field making possible superior images of body organs and cellular functions; detailed studies are necessary in any structure in order to avoid the structure coming into resonance with a variety of sources that might create catastrophic large amplitude oscillations.
TEST YOUR UNDERSTANDING
22aIn the context of oscillations, state what is meant by damping.
bDistinguish between free and driven oscillations.
23State what is meant by resonance.
24If you walk at one step a second holding a cup of water (diameter 8 cm) the water will spill out of the cup. Use this information to estimate the speed of the waves in water.
25Figure 15.20 shows the variation of the displacement of a system performing oscillations.

Figure 15.20: For question 25.

aDetermine the frequency of oscillations.
bDraw a sketch graph to show the variation with time of the energy of the system (no numbers required on the axes).
cThe amount of damping increases. State and explain the effect of this on the graph.
26aState the conditions that must be satisfied for simple harmonic oscillations to take place.
bDraw a graph of displacement versus time for a particle undergoing lightly damped simple harmonic oscillations.
cA system has natural frequency f0. A periodic force of variable frequency f acts on the system.
Copy the axes from Figure 15.21.
Draw graphs to show the variation with f of the amplitude of oscillation of the system for:
ilight damping
iiheavy damping.

Figure 15.21: For question 26c.

27aAt what frequency should a 2.0 m long pendulum be pushed in order to achieve large oscillations?
bAre there any other frequencies which will achieve large oscillations?
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28Figure 15.22 shows a set of weights of mass 0.50 kg hanging at the end of a spring of spring constant 220 N mâ1. The other end of the spring is attached to an oscillator. The oscillations are very slightly damped.

Figure 15.22: For question 28.

aAt what frequency must the oscillator be set in order to achieve large amplitude oscillations of the weights?
bWhat happens to the frequency in part a if the damping increases?
29Describe a situation where resonance is desirable and one where it is not.
30Distinguish between light, critical and heavy damping. Suggest how the oscillations in the shock absorbers of a car should be damped.