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Oscillations Notes

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Oscillatory Systems 2014

Contents 1 Introduction 2 The

2.1

2.2

2.3

2.4

2.5

2.6

2

Simple Harmonic Oscillator What you already know... . . . . . . . . . . . . . . . The simple pendulum . . . . . . . . . . . . . . . . . Beats . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical Note: Complex Numbers . . . . . . . The complex representation . . . . . . . . . . . . . . Mathematical Note: Ordinary Differential Equations

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3 3 4 4 6 7 8

3 Damped Harmonic Motion 9

3.1 Energy decay and the Quality Factor . . . . . . . . . . . . . . . . . . . . 11 4 Forced Oscillations 12

4.1 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2 Power and Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1

1

1

INTRODUCTION

2

Introduction

In this chapter we will be exploring the mathematical and physical ideas at the basis of our description of oscillations and waves. We have already explored the dynamics of the simple harmonic oscillator and we shall be building on this example to explore oscillating systems in general. We will first consider the effect of damping, and then analyse the response of the oscillating system to an external driving force (in both time and frequency domain) leading to the idea of resonance1 . We shall also be making liberal use of complex numbers, that have an important role in greatly simplifying the mathematics of oscillations, and really add a new dimension to the solution by helping us to focus our attention of the concept of phase. The strength of the simple harmonic oscillator model lies in its simplicity and widespread applicability. In fact the same mathematics that we will be developing in this chapter is used to describe motion of masses on springs, pendula (2.1), atomic resonances, perturbation to orbits of satellites and electrical circuits with reactive elements (capacitors and inductors). Indeed any physical system if displaced from equilibrium by a small enough amount will perform simple harmonic motion (as we proved in Mechanics, Part I). The logical development of the idea of a simple harmonic oscillator is to introduce a space dependants to the solution: imagine putting a number of harmonic oscillators one after the other with a mutual phase relationship depending on position: you have built yourself a wave! Waves, like harmonic oscillators, are everywhere in Physics and describe some of the most fascinating phenomena known to the inquiring mind: waves in materials (sound waves) and the electromagnetic spectrum (including visible light!).

Figure 1: A schematic representation of the Electromagnetic Spectrum. 1 In this chapter we will only be dealing with periodic sinusoidal driving, but through the theory of Fourier Transform, the results in this chapter can easily be generalised to any driving force.

2

THE SIMPLE HARMONIC OSCILLATOR

3

We will not have the time to make an in-depth study of optics or acoustics, but we shall lay down the basis of the behaviour of waves in general. We will start from the wave equation and study its solutions. Then we will take the usual complementary approach and study the energetics of the system, and show how waves transmit energy and phase.

2

The Simple Harmonic Oscillator The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction. Sidney Coleman

2.1

What you already know...

Before starting our new adventure into new physical concepts, let us take a step back and pause to revise our previous knowledge about the simple harmonic oscillator. In fact we already know almost everything, namely:

1. The force law F = −kx

(1)

m¨ + kx = 0 x

(2)

x = A sin ωt + B cos ωt = x0 cos(ωt + φ)

(3)

and the equation of motion

2. The general solution

where A, B and x0 and φ respectively are arbitrary constants (we generally refer to x0 as the amplitude and φ as the phase), fixed by the boundary conditions, k and ω 2 = m (the angular frequency, or natural frequency of oscillation) is
fixed by the system properties.

3. There is a π phase difference between displacement and velocity and between 2 velocity and acceleration.

4. Total energy is conserved in simple harmonic motion, and continuously interchanged between kinetic and potential energy. E =T +V =

1 1 mx2 + kx2 = const
˙
2 2

(4)

5. The Potential Energy is quadratic in the displacement. Since any potential well, close enough to equilibrium, appears parabolic, then any system will exhibit SHM close enough to equilibrium.

2

THE SIMPLE HARMONIC OSCILLATOR

2.2

4

The simple pendulum

To sum up on all this knowledge we will look in some detail at one of the most famous physical systems displaying SHM, that I have in fact saved until now: the simple pendulum. We consider the pendulum to be a point mass m, attached to light string of length l, making and angle θ to the vertical. Resolving the weight of the bob in the direction perpendicular to the string and applying N2 we get m¨ = −mg sin θ
x

(5) x2 l .

in the limit of small angles we can approximate sin θ θ =
Therefore in this limit the pendulum follows simple harmonic motion described by the equation: g x+ x=0
¨
(6) l Hence the system will oscillate with angular frequency ω = g . l We can also approach the problem from the energy side. If we start from scratch and write the potential energy of the pendulum when it is displaced by an angle θ, from the geometry of figure 2 we see this corresponds to V = mgl(1 − cos θ)

(7)

taking the potential energy to be zero when the bob is at θ = 0. Obviously this is nothing more than the integral of the force law (which we have previously found to be F = −mg sin θ) with the relevant integration constant. Note that if we apply the small angle approximation cos θ = 1 − 1 θ2 we recover a potential energy quadratic in 2
θ (V = 1 mglθ2 ), as necessary for SHM. 2

2.3

Beats

Consider the case where a system is subject to two oscillations at different frequencies (and for mathematical simplicity take them to have the same amplitude). For example set
ω1 + ω2
ω1 − ω2 x = a(cos ω1 t + cos ω2 t) = 2a cos t + cos t (8) 2 2 The amplitude of the oscillation varies between 2a (when the phase difference between the two oscillations (ω2 −ω1 )t = 2nπ) and the oscillations are said to be in phase) and zero (when the phase difference between the two oscillations (ω2 − ω1 )t = (2n + 1)π) and the oscillations are said to be out of phase). Acoustically this growth and decay of the amplitude is registered as beats of strong reinforcement when two sounds of almost equal frequency are heard. However this calculation is important for a number of other reasons:

• It gives an example of how cumbersome it is to add oscillation together when we use trig functions. 2 Indeed, the small angle formula, which is nothing more than the Taylor expansion of sine close to zero, is at the basis of the simple harmonic behaviour of pendula!! If we avoid making this approximation the equation of motion is non-linear (since it contains a sine term) and no analytic solution exists.

2

THE SIMPLE HARMONIC OSCILLATOR

5

Figure 2: On the left, a sketch of a simple pendulum. In blue the trajectory with the relevant definition of the variable x. In red the force of gravity acting on the bob, and its components parallel and perpendicular to the string. On the right, the shape of the true potential for the simple pendulum is showed. Close to equilibrium the potential is well approximated by a parabolic well, and hence the system exhibits simple harmonic motion for small angles (figure on the right from Kibble and Berkshire).

• Adding oscillations, generally of the same frequency, but with different phases constitutes the basis of the phenomenon of diffraction that we will be discussing at length later. The calculation for this case, however, proceeds essentially equivalently.

Figure 3: A representation of the phenomenon of beats.

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