Damped and forced oscillations

Why does a swing stop?

Give a swing one push and leave it. Each swing is a little smaller than the one before. After a minute or two it hangs still. Where did its energy go?

Air pushes against the moving swing. The chains rub at the top. These forces always act against the motion. They take energy away, a little in every swing. We call this damping.

Damped oscillations

For many bodies the damping force is in proportion to the speed and opposite to it:

The constant $b$ is the damping constant, in kg/s. When $b$ is small, the body still oscillates at nearly its natural frequency. But its amplitude falls with time:

This kind of fall has one special property. In each equal time, the amplitude is multiplied by the same factor. So if it halves in $10\ \text{s}$, it halves again in the next $10\ \text{s}$.

The time to halve. We want $t$ when $A = \tfrac12 A_0$. So $e^{-bt/2m} = \tfrac12$, and

The energy. The energy of an oscillator is in proportion to the square of its amplitude. So the energy falls as $e^{-bt/m}$, twice as fast in the exponent. When the amplitude halves, the energy falls to one quarter.

A worked example. A bob of mass $0.20\ \text{kg}$ swings in air with $b = 0.010\ \text{kg/s}$.

1. Amplitude. $t_{1/2} = \dfrac{2 \times 0.20\ \text{kg} \times 0.693}{0.010\ \text{kg/s}} = 27.7\ \text{s}$.
2. Energy. It halves in $\dfrac{0.20\ \text{kg} \times 0.693}{0.010\ \text{kg/s}} = 13.9\ \text{s}$.

Shock absorbers. A car sits on springs. After a bump it would bounce for a long time. A shock absorber pushes oil through small holes, which gives a large $b$. The bounce dies away in about one swing. Here damping is useful.

Forced oscillations

Now keep pushing. A periodic force, $F_0\cos\omega_d t$, acts on the body all the time. The body starts to move in a mixed way. After a while the natural motion has died away. Then the body oscillates steadily, at the driving frequency $\omega_d$, not at its own.

How large is the steady amplitude? It depends on how close $\omega_d$ is to the natural angular frequency $\omega$:

Look at the bottom. Far from $\omega$, the first term is large, and $A$ is small. When $\omega_d = \omega$, the first term is zero. Then only the damping term is left, and $A$ is largest.

Resonance

The large amplitude when the driving frequency matches the natural frequency is called resonance.

With little damping, the peak is tall and narrow. The body answers strongly to the right frequency and hardly at all to others. With heavy damping, the peak is low and broad.

You meet resonance every day.

• A swing. Push once in each swing, at the right moment, and the swing goes higher and higher.
• Soldiers on a bridge. Soldiers marching in step push a bridge at one frequency. If it matches a natural frequency of the bridge, the bridge can swing dangerously. So soldiers break step on a bridge.
• A radio. Turning the knob changes the natural frequency of a circuit. When it matches the frequency of a station, that station is loud and the others are not heard.

Sweep the driving frequency slowly through $1.00\ \text{Hz}$. Watch the amplitude rise and fall. Then make the damping as small as you can and sweep again. Is the peak taller? Is it narrower?

Practise with new numbers

Try it in Python

The program follows a damped oscillator in tiny steps of time. Every ten swings it prints the amplitude and its ratio to the amplitude ten swings before.

Common mistakes

• Letting the amplitude fall in a straight line. It falls by the same factor in each equal time.
• Mixing up amplitude and energy. The energy falls twice as fast in the exponent: $e^{-bt/m}$.
• Saying a forced body moves at its natural frequency. In the steady state it moves at the driving frequency.
• Thinking more damping gives a taller peak. More damping gives a lower, broader peak.

Remember these

• Damping force $-bv$. Amplitude $A_0e^{-bt/2m}$. Energy $\propto e^{-bt/m}$.
• Time for the amplitude to halve: $t_{1/2} = \dfrac{2m\ln 2}{b}$.
• A forced body moves at the driving frequency.
• Resonance: largest amplitude when $\omega_d = \omega$. Small damping, sharp peak.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. The amplitude of a pendulum halves in $8\ \text{s}$. What fraction of it is left after $24\ \text{s}$?

2. The amplitude of an oscillator falls to half. What fraction of its energy is left?

3. At resonance, which two quantities are equal?

4. A car's shock absorbers are worn out. What do you notice after a bump?