Oscillations

The six words that describe an oscillation

An oscillation is a repeated back-and-forth motion about an equilibrium position. A pendulum bob, a mass hanging on a spring, a guitar string, a quartz crystal in a watch and the ions in a crystal lattice all oscillate. Each of them has a position where it would rest if it were left alone, and the whole motion is a repeating departure from and return to that position.

Displacement x is the distance of the oscillating object from the equilibrium position, measured in a stated direction. Because direction matters, displacement is positive on one side of the equilibrium position and negative on the other, and it is measured in metres. Amplitude x0 is the maximum magnitude of that displacement, so it is always positive and is measured from the equilibrium position out to an extreme, never from one extreme to the other: the peak-to-peak distance is twice the amplitude.

Period T is the time taken for one complete oscillation, that is the time for the object to return to the same position moving in the same direction. Frequency f is the number of complete oscillations each second, so f = 1 / T, measured in hertz (Hz), where 1 Hz is one oscillation per second. A 2.5 Hz oscillation repeats 2.5 times each second and has a period of 1 / 2.5 = 0.40 s.

Angular frequency omega is the same idea in radians: it is the phase change per second, omega = 2 pi f = 2 pi / T, measured in radians per second (rad s-1). One complete cycle is 2 pi radians, so an oscillation of frequency f completes 2 pi f radians of phase every second. This gives the two forms of the period, T = 1 / f and T = 2 pi / omega, and both are quoted in the syllabus. Phase difference describes how far one oscillation leads or lags another at the same instant, as a fraction of a cycle: measured in degrees or radians, a quarter of a cycle is 90 degrees (half pi radians) and half a cycle is 180 degrees (pi radians).

What makes an oscillation simple harmonic

An oscillation is simple harmonic when two conditions are both true: the acceleration is proportional to the displacement from a fixed point, and the acceleration is always directed towards that fixed point, that is in the opposite direction to the displacement. Written as an equation, a = -omega^2 x, where x is the displacement in metres, a is the acceleration in m s-2 and omega is the angular frequency in rad s-1. The constant of proportionality is omega^2, and the minus sign carries the physics: it is what makes the acceleration always point back towards the equilibrium position.

The minus sign also explains the shape of the motion. When the object is at the equilibrium position, x is zero, so the acceleration is zero and the object is travelling at its fastest. As it moves away from the centre the acceleration grows and points back, slowing it down, until at maximum displacement the speed is momentarily zero and the acceleration has its largest value, omega^2 x0, directed towards the centre. The object then accelerates back through the centre and out to the other side, where the displacement is negative and the acceleration is positive again. The repetition of this exchange is what makes the motion periodic.

This is why a mass on a spring oscillates with simple harmonic motion for any amplitude, while a pendulum is only approximately simple harmonic. For a spring, the restoring force is F = -kx by Hooke's law, so a = F / m = -(k / m) x, which has exactly the required form with omega^2 = k / m, giving omega = sqrt(k / m) and so T = 2 pi sqrt(m / k). A pendulum's restoring force is proportional to sin theta rather than to theta, so a = -g sin(theta) / L is proportional to the displacement only while the angle stays small; that is why a pendulum is treated as simple harmonic only for small oscillations, and why the small-angle approximation matters in the laboratory.

• Simple harmonic motion: a = -omega^2 x, so the acceleration is proportional to the displacement and always directed towards the fixed point.
• At the equilibrium position: displacement zero, acceleration zero, speed a maximum.
• At maximum displacement: displacement x0, speed zero, acceleration omega^2 x0 towards the centre.
• Mass on a spring: omega = sqrt(k / m) and T = 2 pi sqrt(m / k), exact at all amplitudes.
• Simple pendulum: T = 2 pi sqrt(L / g), a good approximation only for small angles.

The equations of simple harmonic motion

The defining equation a = -omega^2 x is a relation between the acceleration and the displacement at the same instant. Because the acceleration is the rate of change of velocity and the velocity is the rate of change of displacement, the whole motion can be built out of it. The displacement that satisfies it, starting from the equilibrium position and moving in the positive direction, is x = x0 sin omega t. You can check the solution by differentiating: the velocity is v = dx/dt = omega x0 cos omega t, which is written as v = v0 cos omega t with the maximum speed v0 = omega x0, and differentiating once more gives the acceleration a = -omega^2 x0 sin omega t = -(omega^2) x, which is the equation you started from. The solution really does satisfy the definition.

The two forms of the velocity equation are used in different situations. Use v = v0 cos omega t when you want the velocity as a function of time, and use the displacement form v = +/- omega sqrt(x0^2 - x^2) when you are told where the object is and want its speed there. The square root gives the magnitude of the speed, so you add the sign yourself from the direction of travel: positive on the way out, negative on the way back. Two limits are worth remembering and they follow straight from the equation: at x = x0 the square root is zero, so the speed is zero at the extremes, and at x = 0 the square root is x0, so the speed is omega x0 at the centre.

Which trig function describes the motion depends only on where the object is at t = 0. Starting at the equilibrium position, the displacement is x = x0 sin omega t. Starting at maximum displacement, the motion is a quarter of a cycle ahead, so the same motion is described by x = x0 cos omega t, with v = -v0 sin omega t. The choice of starting point never changes the amplitude, the period or the maximum speed, only the phase.

A worked example. A 0.40 kg mass on a spring oscillates with period 1.20 s and amplitude 4.0 cm. The angular frequency is omega = 2 pi / T = 2 x 3.14 / 1.20 = 5.23 rad s-1. The maximum speed is v0 = omega x0 = 5.23 x 0.040 = 0.21 m s-1, and the speed at a displacement of 2.0 cm is v = omega sqrt(x0^2 - x^2) = 5.23 x sqrt(0.040^2 - 0.020^2) = 5.23 x sqrt(0.0016 - 0.0004) = 5.23 x 0.0346 = 0.18 m s-1. The acceleration there is a = -omega^2 x = -27.4 x 0.020 = -0.55 m s-2, the minus sign meaning that it is directed back towards the equilibrium position, while at maximum displacement the magnitude would be 27.4 x 0.040 = 1.10 m s-2.

Displacement, velocity and acceleration graphs

The three graphs of an oscillator are all sinusoidal curves with the same period, and they differ only in amplitude and in phase. Because the velocity is the gradient of the displacement-time graph, the velocity curve is a quarter of a cycle ahead of the displacement curve: a crest of displacement happens where the displacement-time graph is momentarily flat, so the velocity is zero there, and the velocity is largest where the displacement-time graph is at its steepest, which is where it crosses the axis. The velocity curve is therefore 90 degrees (a quarter of a cycle) ahead of the displacement curve, and the speed is greatest at the equilibrium position.

The acceleration is the gradient of the velocity-time graph, and it is also given by a = -omega^2 x. Both ways of looking at it give the same answer: the acceleration-time graph is the displacement-time graph turned upside down and stretched by a factor of omega^2. So acceleration and displacement are 180 degrees out of phase (half a cycle apart), and the acceleration peaks are omega^2 x0 at the instants when the displacement peaks are x0. The three amplitudes are linked: if x = x0 sin omega t then a0 = omega^2 x0, so the acceleration amplitude is omega^2 times the displacement amplitude, and the velocity amplitude is omega times the displacement amplitude.

There is a fourth graph worth knowing, because it is the cleanest test of simple harmonic motion: acceleration plotted against displacement. Since a = -omega^2 x, this graph is a straight line through the origin with a negative gradient, and the gradient is -omega^2. A curve, or a line that does not pass through the origin, means the motion is not simple harmonic. This is exactly what a data-logging experiment with a mass on a spring and a motion sensor produces, and the period can be found from the gradient by omega = sqrt(-gradient), then T = 2 pi / omega.

The interchange of kinetic and potential energy

During an oscillation the restoring force is doing work on the object on the way in and the object is doing work against the restoring force on the way out, so energy is passed backwards and forwards between two stores. In a mass-spring system those stores are the kinetic energy of the moving mass and the elastic potential energy stored in the deformed spring; in a pendulum they are the kinetic energy of the bob and its gravitational potential energy. At the equilibrium position the whole of the energy is kinetic and the potential energy is zero, and at the extremes the whole of it is potential and the kinetic energy is zero because the object is momentarily at rest.

If the amplitude is steady, the forces doing work are conservative and the total energy stays constant through the cycle. The total energy is easiest to find at the equilibrium position, where all of it is kinetic: E = 1/2 m v0^2, and with v0 = omega x0 this becomes E = 1/2 m omega^2 x0^2, the syllabus expression for the total energy. Because the total energy is constant, the energy at any displacement is the sum of the two parts: kinetic energy = 1/2 m omega^2 (x0^2 - x^2) and potential energy = 1/2 m omega^2 x^2, adding to 1/2 m omega^2 x0^2, which is the same answer from every displacement.

The expression E = 1/2 m omega^2 x0^2 has two consequences you should be able to quote. First, the total energy depends on the square of the amplitude: doubling the amplitude multiplies the total energy by four, and halving the amplitude divides it by four. Second, because E = 1/2 m v0^2 as well, doubling the amplitude only doubles the maximum speed, since v0 = omega x0. A graph question usually asks for the shape of the energy curves: plotted against time, the kinetic and potential energies are squared sine curves that swap with each other, and each of them reaches its maximum twice per cycle, because the object passes through the equilibrium position twice and comes to rest at the extremes twice. That is why the energy curves repeat at twice the frequency of the displacement curve.

A worked example. A 0.20 kg mass oscillates with amplitude 5.0 cm and angular frequency 8.0 rad s-1. The total energy is E = 1/2 m omega^2 x0^2 = 0.5 x 0.20 x 8.0^2 x 0.050^2 = 0.016 J, and this is the kinetic energy at the centre. The maximum speed is v0 = omega x0 = 8.0 x 0.050 = 0.40 m s-1, and the kinetic energy at the centre calculated the other way is 0.5 x 0.20 x 0.40^2 = 0.016 J, which agrees. At a displacement of 3.0 cm the potential energy is 0.5 x 0.20 x 64 x 0.030^2 = 0.0058 J and the kinetic energy is therefore 0.016 - 0.0058 = 0.010 J.

Damping: what a resistive force does to an oscillation

A resistive force such as friction, air resistance or the viscous drag of a liquid removes energy from an oscillating system and transfers it to the thermal store of the surroundings. This is damping, and its visible effect is that the amplitude of the oscillation falls as time goes on. The restoring force is unchanged, so damping does not alter the period much at all, and the oscillation continues at almost the same frequency with a steadily shrinking amplitude.

With light damping, the amplitude decreases gradually over many oscillations. The extremes lie on a smooth decaying curve, usually an exponential envelope, and after each cycle the amplitude has multiplied by the same factor; since the energy is proportional to the amplitude squared, the energy of the oscillation falls away more quickly than the amplitude does. A pendulum swinging in air and a guitar string after it is plucked are both lightly damped: they lose only a fraction of their amplitude each swing, but they do eventually stop.

With critical damping, the system is displaced and released and returns to the equilibrium position in the shortest possible time without overshooting it. There is no oscillation at all, no period, and no amplitude to measure. Critical damping is what every measuring instrument with a moving pointer needs: the needle of a moving-coil galvanometer, a set of kitchen scales, the movement of an analogue ammeter and a well-designed door closer are all damped so that the reading settles quickly without flicking past the answer and swinging back.

Heavy damping also gives a return to equilibrium with no oscillation, but it is much slower than critical damping, because the resistive force is so large that it prevents the system from building up any speed. This is the part students most often get backwards: heavy damping is slow, not fast. A pendulum in thick oil, or a door closer with too stiff a hydraulic damper, creeps back to the rest position without ever crossing it. Between the two extremes there is a whole family of damped motions: light damping still oscillates, and both critical and heavy damping stop without crossing the equilibrium position.

Forced oscillations and resonance

A system that is displaced and then left alone oscillates at its natural frequency, which is fixed by its mass, its stiffness or its length. If instead a periodic driving force keeps acting on it, the system is forced to oscillate at the frequency of the driving force, whatever that frequency happens to be, and it soon settles into a steady oscillation at the driving frequency rather than at its natural frequency. Pluck a guitar string and it sounds its natural frequency; hold it against a loudspeaker and it is forced to move at the loudspeaker's frequency instead.

The amplitude of a forced oscillation depends on how the driving frequency compares with the natural frequency, and the dependence is the resonance curve. When the driving frequency is much lower than the natural frequency, the system simply follows the driving force and the amplitude stays modest; when the driving frequency is much higher, the system cannot keep up with the rapidly changing force and the amplitude is again small. The amplitude is a maximum when the driving frequency equals the natural frequency, and this is resonance: the driving force is always in step with the motion, so energy is transferred from the driving force to the system at the greatest possible rate. That energy is stored in the oscillation until the energy lost to damping each cycle equals the energy supplied, at which point the amplitude stops growing and stays at its maximum.

Damping controls how big the resonance peak is. A very lightly damped system has a sharp, narrow peak with a large maximum amplitude, because very little energy is lost each cycle and the amplitude can grow a great deal; a more heavily damped system has a lower and broader peak, and if the damping is heavy enough the peak disappears altogether. This is why resonance needs both halves of the sentence: the driving frequency must equal the natural frequency, and the size of the resulting amplitude depends on the damping. Buildings are fitted with dampers, and tuned mass dampers are fitted inside tall towers and footbridges, precisely to keep the amplitude at resonance within safe limits.

The phase of a forced oscillation also changes as the driving frequency changes, and the pattern is worth remembering. At a driving frequency well below resonance the displacement is in step with the driving force, so the system is in phase with it; at resonance the displacement lags the force by a quarter of a cycle, 90 degrees; and at driving frequencies well above resonance the displacement is half a cycle behind the force, that is 180 degrees, or in anti-phase. This phase change is used in practice: in a magnetic resonance imaging machine the radio-frequency pulses are switched on at the resonance of the hydrogen nuclei so that energy is absorbed strongly by the tissue being imaged.

Oscillators and resonance in real systems

The simple pendulum is the standard laboratory oscillator, and its famous property is that the period depends only on the length: T = 2 pi sqrt(L / g), with no mass in the expression. A pendulum 1.00 m long has a period of 2 pi sqrt(1.00 / 9.81) = 2.0 s, which is why pendulum clocks used a rod of about a metre. Rearranged as g = 4 pi^2 L / T^2, the same equation turns a pendulum into a way of measuring the gravitational field strength, which is one of the standard practicals for this course. Pendulum clocks, the first of which was designed by Christiaan Huygens in 1656, keep time because the period is so nearly independent of the amplitude, and their accuracy improved enormously when the pendulum was made to swing independently of the escapement.

The quartz crystal oscillator in a wristwatch is a different kind of oscillator and a much more reliable one. A thin slice of quartz is a piezoelectric material: apply a voltage and it deforms, and deform it and it produces a voltage, so it can both drive and sense its own vibration. Cut and mounted in the right way it has an extremely stable natural frequency, and the common watch crystal is designed to vibrate at 32768 Hz, which is 2 to the power 15. Divide that frequency by two fifteen times with electronic counters and you have exactly one pulse per second to drive the hands. The natural frequency hardly depends on temperature, on the drive level or on the age of the watch, which is why a quartz watch is far better timekeeper than a mechanical one.

Resonance is used deliberately in many instruments and devices. Musical instruments work by it: a guitar or violin body has resonances that radiate the string's vibration into the air, a wind instrument's air column resonates at the frequencies that make its notes, and a tuning fork held over a tube of adjustable length will drive the air column into a loud resonance when the length is right. A tuned circuit in a radio receiver is a resonance: the inductance and capacitance are chosen so that the circuit resonates at the frequency of the one station wanted, giving the largest current at that frequency and rejecting the rest. A microwave oven heats food at 2.45 GHz, and a seismometer uses a mass on a spring whose natural frequency is far below that of the earthquakes it detects, so that it responds to ground motion rather than resonating with it.

Resonance can also be destructive, and engineers spend a great deal of effort preventing it. The Millennium Bridge across the Thames in London was opened in June 2000 and closed two days later, because pedestrians walking at about two steps per second pushed sideways on the deck at a frequency that matched the bridge's natural sideways oscillation of about 0.5 Hz, and the deck began to wobble so much that people had to hold the handrails; the problem was cured by fitting viscous dampers and tuned mass dampers before it reopened in 2002. In a similar way, a washing machine passes through a violent vibration as its spin speed rises through the speed that matches the natural frequency of the drum suspension, and every internal combustion engine is mounted on flexible, damped mountings so that the engine's vibration is not passed into the body of the car at its resonant speed.

• Simple pendulum: T = 2 pi sqrt(L / g); the 1 m pendulum has a period of 2.0 s and is used to measure g.
• Mass on a spring: T = 2 pi sqrt(m / k); used in seismometers, where the natural frequency is kept low.
• Quartz watch: a crystal at 32768 Hz divided down to one pulse per second.
• Millennium Bridge, London (2000): pedestrian-driven resonance in the sideways mode, cured by dampers.
• Washing machine: violent vibration when the spin frequency matches the natural frequency of the drum suspension.