doPhysics · Oscillations
Superposition, Damped and Forced Oscillations
Two SHMs compared and added, along one line and at right angles. Real oscillators that lose energy to damping. Oscillators driven by a periodic force, and the resonance that makes a small push produce a large swing.
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Why this chapter
Why this chapter exists
The last two chapters followed one ideal oscillator, alone and for ever. Real oscillators are neither. A particle is often pushed by two oscillations at once, and its motion is their sum. Every real swing loses energy and dies away, and many are kept going by a periodic push from outside. This chapter compares and adds SHMs, follows an oscillation as damping drains it, and drives it with a periodic force. The last idea, resonance, explains a rattling window, a tuned radio and a bridge that fell to a marching step.
You will learn
By the end of this chapter you will
- Compare two SHMs that differ in amplitude, in period or in phase, and find a phase difference from two positions.
- Add two SHMs of one frequency along one line with phasors, and find the resultant amplitude and phase.
- Find the path of two perpendicular SHMs, and read a phase difference or a frequency ratio from a Lissajous figure.
- Write the equation of a damped oscillator, and use \(A = A_0e^{-bt/2m}\), the energy decay and the logarithmic decrement.
- Tell underdamped, critically damped and overdamped motion apart, and find the critical damping constant.
- Explain why a driven oscillator settles at the driving frequency, and use the amplitude formula of a forced oscillation.
- Describe resonance and how damping sets the height and sharpness of its peak, with its uses and dangers.
Before you begin
Before you begin
From the chapter Kinematics of SHM, you should know \(x = A\sin(\omega t + \varphi)\), the meaning of amplitude, period and phase, and the phasor, the rotating arrow whose shadow is an SHM. From the chapter Dynamics of SHM, you should know \(\omega = \sqrt{k/m}\) and the energy \(\tfrac12 kA^2\). You should be able to add vectors, and use the sine of a sum and \(e^x\) with its logarithm.
Lessons
1
Comparing Two SHMs
Two oscillations about the same mean position can differ in amplitude, in period, or in how far each one has gone through its cycle. The phase difference measures the last, and it stays fixed only when the periods are equal.
Same period different amplitudesSame amplitude different periodsPhase difference of two shms
2
Adding SHMs of One Frequency
A particle driven by two SHMs along one line moves with their sum. When the two have the same frequency, the sum is again an SHM of that frequency. The phasor diagram gives its amplitude and phase in one triangle.
Principle of superposition for shmPhasor addition of two shmsConstructive and destructive superposition
3
Perpendicular SHMs and Lissajous Figures
A body shaken along x and along y at once moves in a plane. With equal frequencies its path is a line, an ellipse or a circle, set by the phase difference. With unequal frequencies it traces closed curves called Lissajous figures.
Perpendicular shms of equal frequencyLissajous figures
4
Damped Oscillations
Real oscillators lose energy to friction and drag, so their swings shrink. With a drag proportional to velocity the amplitude falls exponentially, the energy falls twice as fast, and the period hardly changes. Enough damping stops the oscillation altogether.
Damping drains mechanical energyEquation of damped shmEnergy and logarithmic decrementUnderdamped critical and overdamped
5
Forced Oscillations and Resonance
A periodic push makes an oscillator settle into motion at the frequency of the push. Its amplitude is largest when the push matches the natural frequency, and damping decides how tall and how sharp that peak of resonance is.
Forced oscillations settle at the driving frequencyResonance and the amplitude curveDamping sets the sharpness of resonanceResonance in the world
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- constructive superposition
- Adding two SHMs in phase, so that their amplitudes add: \(A_R = A_1 + A_2\).
- critically damped
- Damping with \(b = 2\sqrt{mk}\). The body returns to the mean position in the least possible time without crossing it.
- damped oscillation
- An oscillation whose amplitude decreases with time because damping takes away its energy.
- damping
- The steady loss of the mechanical energy of an oscillator to forces that oppose its motion, such as friction and drag. It makes the amplitude decrease with time.
- damping constant
- The constant \(b\) in a drag force \(F = -bv\) proportional to the velocity. Its unit is kg/s; a larger \(b\) means stronger damping.
- destructive superposition
- Adding two SHMs in opposite phase, so that their amplitudes subtract: \(A_R = |A_1 - A_2|\). Equal amplitudes cancel completely.
- driving frequency
- The frequency of the periodic force that drives a forced oscillation. It is set from outside, not by the oscillator.
- forced oscillation
- The oscillation of a body driven by an outside periodic force. In the steady state it has the frequency of the force.
- free oscillation
- The oscillation of a body disturbed and then left to itself, at its natural frequency. With damping it dies away.
- in phase
- Two SHMs with a phase difference of zero, or a whole multiple of \(2\pi\). They pass the mean position and reach their extremes at the same instants.
- lissajous figures
- The closed paths traced by a body with two perpendicular SHMs whose frequencies are in a ratio of whole numbers. The ratio 1 : 2 gives a figure of eight.
- logarithmic decrement
- The natural logarithm of the ratio of two successive maxima of a damped oscillation, \(\delta = \ln(A_n/A_{n+1}) = bT'/2m\).
- natural frequency
- The frequency at which an oscillator oscillates when disturbed and left to itself. For a block on a spring it is \(\omega_0 = \sqrt{k/m}\), in rad/s.
- opposite phase
- Two SHMs with a phase difference of \(\pi\). One is at its positive extreme when the other is at its negative extreme, and they always move in opposite directions.
- overdamped
- Damping with \(b > 2\sqrt{mk}\). The body returns to the mean position without crossing it, but slowly.
- phase difference
- The difference of the phases of two SHMs of the same frequency, \(\Delta\varphi = \varphi_2 - \varphi_1\). It stays constant in time; the motion with the larger phase leads the other by a time \(\Delta\varphi/\omega\).
- phasor
- A rotating arrow that stands for an SHM. It has the length of the amplitude and turns at the angular frequency; the height of its tip is the displacement.
- principle of superposition
- When a particle is subject to two or more SHMs along one line, its displacement at every instant is the sum of their displacements.
- resonance
- The large response of an oscillator driven at a frequency close to its natural frequency. Its height is limited only by the damping.
- resultant amplitude
- The amplitude of the single SHM formed by adding two SHMs of one frequency, \(A_R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos\Delta\varphi}\).
- steady state
- The lasting motion of a driven oscillator after the free, damped part has died away: an oscillation at the driving frequency with a constant amplitude.
- underdamped
- Damping with \(b < 2\sqrt{mk}\). The body oscillates about the mean position with a shrinking amplitude.