9702 · 17.3Damped and forced oscillations, resonance

Energy loss and damping

0:00Give a child on a swing one push and walk away. The swing goes back and forth, but each swing is a little smaller than the last, and eventually it stops. Now stand behind the swing and give it a small push every time it comes back to you. Even gentle pushes, timed just right, can build the swing up to a huge height. And yet if you push at the wrong moments, say every time it's rushing towards you, you'll barely get it moving at all.

0:21By the end of this video, you'll understand why real oscillations die away, the difference between light, critical and heavy damping and how to sketch each one, and resonance: why an oscillator responds most strongly when it's pushed at its own natural frequency. In the last video, we saw that the total energy of a simple harmonic oscillator is one-half m omega squared x-nought squared. So for a given oscillator, energy is proportional to amplitude squared.

0:46If it loses energy, its amplitude gets smaller. If it gains energy, its amplitude grows. This whole video is about energy leaving an oscillator, or being fed into it. In the real world, there's always something resisting the motion. A pendulum pushes through the air, so there's air resistance. A mass on a spring might rub against a surface, so there's friction. And an object moving through a liquid, like oil or water, feels viscous drag.

1:11All of these are resistive forces. A resistive force always acts opposite to the velocity, so the oscillator is doing work against it the whole time. That work transfers energy out of the oscillation, mostly into thermal energy of the surroundings. The energy isn't destroyed, but it's no longer part of the oscillation. So the amplitude decreases. This reduction in amplitude caused by a resistive force is called damping. A pendulum in air is only slightly damped.

1:37The same pendulum in thick oil is very heavily damped. That gives us three named types of damping: light, critical and heavy.

Light, critical and heavy damping

1:46Start with light damping, like a pendulum in air. The resistive force is small, so only a small fraction of the energy is lost on each cycle. The system still oscillates, but the amplitude gradually decreases, swing after swing, until it eventually dies away. Let's look carefully at that graph, because this is exactly what you'll need to sketch. The curve still looks like a sine wave, crossing the time axis again and again.

2:09But the peaks get smaller. If you join up the tops of the peaks, you get a smooth curve that falls away, steeply at first and then more gently, heading towards zero. And here's a detail that's easy to miss. The peaks are still evenly spaced in time. Light damping reduces the amplitude, but the period stays very nearly the same as it would be with no damping at all.

2:29So when you sketch it, keep the crossings of the time axis evenly spaced, and only let the height shrink. Now go to the other extreme. Put the pendulum in a jar of thick treacle, pull it to one side and let it go. It doesn't swing at all. It just creeps slowly back towards the middle. This is heavy damping. There's no oscillation, and the system returns to equilibrium slowly.

2:50On the displacement–time graph, heavy damping starts at the same initial displacement, then falls very gradually. It never crosses the time axis. It just gets closer and closer to zero, taking a long time to settle. Between those two extremes is critical damping. This is the amount of damping that brings the system back to its equilibrium position in the shortest possible time, without oscillating. Too little damping, and the system overshoots and wobbles before settling.

3:16Too much, and it crawls back slowly. Critical damping is just enough to stop the overshoot, and no more. On the graph, the critical damping curve drops quickly and smoothly down to zero displacement and stays there. It doesn't cross the time axis. And compared with the heavy damping curve, it's always lower, because it gets back to equilibrium sooner. Car suspension is the classic use. Without shock absorbers, a car would keep bouncing after every bump.

3:41With far too much damping, the suspension would be slow to recover. So shock absorbers are designed to give damping close to critical, and the car settles quickly. The needle on an analogue meter is another example: you want it to go straight to the reading and stop.

Free and forced oscillations

3:54So far, we've displaced an oscillator once and let it go. That's a free oscillation. A freely oscillating system oscillates at its own frequency, set by the system itself, like the length of a pendulum or the mass and stiffness of a spring. This is its natural frequency. Now suppose you hold the top of the spring and keep moving your hand up and down. That periodic push is a driving force, its frequency is the driving frequency, and the result is a forced oscillation.

4:20Here's an important point. Once things settle, a forced oscillator oscillates at the driving frequency, not its natural frequency. Move your hand twice a second, and the mass moves twice a second. What the natural frequency controls is how big those oscillations are. Let's do the experiment. Keep the size of the driving push the same, and slowly change the driving frequency, from very low to very high. Each time, wait for things to settle, then measure the amplitude of the forced oscillation.

4:47At a very low driving frequency, the mass just follows your hand slowly. The amplitude is small. At a very high driving frequency, your hand is jiggling far too quickly. The mass can't keep up, and it barely moves at all. The amplitude is tiny. But as the driving frequency gets close to the natural frequency, the amplitude grows dramatically. At the natural frequency, the amplitude is at its maximum.

5:09This is resonance. Resonance is when the amplitude of a forced oscillation is at a maximum, and it happens when the driving frequency is equal to the natural frequency of the system.

Energy transfer and resonance curves

5:21But why does this happen? Think about the swing again. When you push at exactly the swing's natural frequency, every push arrives at the same point in the swing's motion. You can push in the same direction the swing is already moving. So every single push does positive work on the swing, and every push adds energy. If you push at some other frequency, your pushes drift in and out of step with the swing.

5:43Sometimes you're pushing with its motion and adding energy, but at other times you're pushing against its motion and taking energy away. Overall, you get far less energy in. So at the natural frequency, energy is transferred from the driver to the oscillator most efficiently. And since more energy means a bigger amplitude, that's where the amplitude is largest. So why doesn't the amplitude at resonance just keep growing forever?

6:06Damping. As the amplitude grows, the oscillator moves faster, so the resistive force gets bigger and more energy is lost on every cycle. Eventually, the energy lost per cycle to damping equals the energy supplied per cycle by the driver. At that point, the amplitude stops growing and stays steady. That last idea tells us exactly how damping affects resonance. With very light damping, very little energy is lost per cycle, so the amplitude can grow very large before the losses catch up with the energy supplied.

6:34The resonance peak is tall and sharp. With heavier damping, energy is lost much faster, so the amplitude levels off sooner. The peak is lower. Damping also makes the peak broader, or flatter. The system responds noticeably over a wider range of driving frequencies, instead of just a narrow band right around the natural frequency. So the rule is: more damping gives a lower, broader resonance peak. Notice that far away from the natural frequency, at very low or very high driving frequencies, the curves are almost the same.

7:04Damping matters most near resonance, because that's where the oscillation is biggest and fastest, so that's where the most energy is being lost. One small extra detail, which is useful context but not something the syllabus asks you to explain: as damping increases, the peak also shifts slightly to a lower frequency, just below the natural frequency. For light damping that shift is tiny, so we still say that resonance happens at the natural frequency.

Uses and dangers of resonance

7:27Sometimes resonance is exactly what we want. Pushing a swing is the simplest example. The body of a guitar resonates with the string, making the sound much louder. A radio's tuning circuit has a natural frequency you adjust, so only the station at that frequency gives a large response. And MRI scanners send in radio waves at the natural frequency of hydrogen nuclei in a magnetic field, so the nuclei absorb energy by resonance.

7:51But resonance can also be a problem. In machinery, spinning motors and engines drive other parts at a frequency set by how fast they spin. If that matches a part's natural frequency, it can vibrate violently and even crack. A washing machine shaking hard at one particular spin speed is passing through resonance. When the Millennium Bridge in London opened in 2000, it swayed from side to side as crowds walked across.

8:13As it moved, people adjusted their steps to keep balance, which pushed the bridge in time with its sway and made it worse. The fix was to add dampers. That's the main fix whenever resonance is a problem: add damping to lower the resonance peak, or change the natural frequency so it's well away from likely driving frequencies.

Damping sketches and exam errors

8:32This sub-topic doesn't have a formula to plug numbers into, so exam questions test sketching and interpretation. Let's work through one. "A mass on a spring is pulled down 4.0 centimetres from its equilibrium position and released at time zero. With light damping, it oscillates with a period of 0.80 seconds. On the same axes, sketch displacement–time graphs for light, critical and heavy damping." Step one: set up the axes.

8:57Displacement up the vertical axis, time along the horizontal. Mark 4.0 centimetres on the displacement axis, and mark times of 0.8, 1.6 and 2.4 seconds along the time axis. Step two: every curve starts at 4.0 centimetres at time zero, and starts flat, because the mass is momentarily at rest when it's released. Step three: the light damping curve. It crosses the time axis and comes back to a peak after 0.8 seconds, but the peak is a bit lower than 4.0.

9:25The next peak at 1.6 seconds is lower again, and so on. Keep the peaks at 0.8, 1.6 and 2.4 seconds, because the period barely changes. Step four: the critical damping curve. It drops smoothly to zero quickly, within about one period, and then runs along the time axis without crossing it. Step five: the heavy damping curve. It starts at 4.0 as well, but falls slowly, staying above the critical curve everywhere, and it still hasn't quite reached zero by the end of the sketch.

9:53Now a follow-up that links back to energy. "With light damping, after a few oscillations the amplitude has fallen from 4.0 centimetres to 3.0 centimetres. What fraction of the original energy of the oscillation remains?" The energy is proportional to amplitude squared. So the fraction remaining is 3.0 divided by 4.0, all squared. Three quarters squared is nine sixteenths, which is 0.5625, or about 56 percent. So although the amplitude has only dropped by a quarter, about 44 percent of the energy has been transferred to the surroundings.

10:24Let's go through the mistakes that cost marks most often. Mistake one: drawing light damping with unevenly spaced peaks. Light damping reduces the amplitude, not the period. Mistake two: drawing the critical or heavy curve crossing the time axis. Crossing means overshooting, which is oscillation. Mistake three: defining critical damping as just "returns to equilibrium without oscillating". Heavy damping does that too. The key words are "in the shortest time".

10:55Mistake four: thinking a forced oscillator oscillates at its natural frequency. It oscillates at the driving frequency. Mistake five: saying damping removes resonance. It makes the peak lower and broader, still at or very close to the natural frequency. Mistake six: saying damping destroys energy. It transfers energy to the surroundings, mainly as thermal energy. Let's bring it all together. A resistive force acting on an oscillating system removes energy from it, so the amplitude falls.

11:25That's damping. With light damping, the system keeps oscillating with a gradually decreasing amplitude, and an almost unchanged period. With critical damping, the system returns to equilibrium in the shortest possible time without oscillating. With heavy damping, the system doesn't oscillate, and it returns to equilibrium slowly. A free oscillation happens at the system's natural frequency. A forced oscillation happens at the driving frequency. Resonance is maximum amplitude of a forced oscillation, when the driving frequency equals the natural frequency, because that's when energy is transferred most efficiently.

11:57And more damping gives a lower, broader resonance peak.