Simple Harmonic Motion (SHM)

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Introduction

In addition to linear motion, circular motion there is another kind of motion that is common in physics. This is the to and fro motion of oscilations or vibrations.

When something oscillates, it moves back and forth with time. It is helpful to trace out the oscillation of an oscillating particle with time so we can define some terminology.

Flash 1. Simple Harmonic Motion, with time produces a Sinusoidal wave

The time taken for the particle to complete one oscilation, that is, the time taken for the particle to move from its starting position and return to its original position is known as the period. The frequency ν is related to the period, it is defined as how many oscillations occur in one second. Since the period is the time taken for one oscillation, the frequency is given by

$\nu=1/T$

The frequency is measured in [s^-1]. This unit is known as the Hertz (Hz) in honour of the physicist Heinrich Hertz. The maximum displacement of the particle from its resting position is known as its amplitude.

Simple Harmonic Motion is closely related to circular motion as can be seen if we project the image of an object moving in a circular path on to a screen. The vertical component of the motion is given by,

$y=A_{0}\sin(\omega{t})$

Flash 2. Simple Harmonic Motion, is also a component of circular motion.

We can also see that the period of the motion is equal to the time it takes for one rotation of the circle. Therefore we can say the period is,

$T=2\pi\omega$

The particle can also vibrate at different speeds, this is conected with the period. The frequency is the number of oscilations per second.

Flash 3. Displacement, velocity and acceleration vectors of a particle undergoing simple harmonic motion

Consider the particle undergoing simple harmonic motion in the figure. The displacement has been shown to take the form of a sinusoid. The velocity of the particle can be calculated by differentiating the displacement. The result is also a wave but the maximum amplitude is delayed, so that when the displacement is at a maximum the velocity is at a minimum and when the displacement is zero the velocity has its greatest value.

Differentiating the velocity with respect to time we obtain the acceleration and from this we can see that this to is also a wave. The maximum acceleration occurs at the extreme displacement but is directed toward the centre

Simple Harmonic Motion is characterised by the acceleration a being oppositely proportional to the displacement, x.

$\frac{d^{2}y}{dt^{2}}=-ky$

We can see that the sine-wave displacement is a solution by differentiating this equation with respect to time, the initial equation is

$A_0\sin(\omega{t})$

To obtain the velocity we differentiate, with respect to time obtaining,

$A_0\omega\cos(\omega{t})$

Finally, the acceleration is the derivative of the velocity with respect to time.

$-A_0\omega^{2}\sin(\omega{t})$

Equating both (1) and (3) we see that it does indeed satisfy the condition for simple harmonic motion with the constant being equal to ω²

Examples of SHM

Spring Mass System

Flash 4. Spring Mass system undergoing SHM.

The spring mass system consists of a spring with a spring constant of k attached to a mass, m.The mass is displaced a distance x from its equilibrium position. When the system is set to oscillate, the spring stores potential energy. The amount of energy stored is 1/2kx². If the mass is released the potential energy is transferred into kinetic energy by moving the mass. For a particle undergoing simple harmonic motion, the displacement x is given by equation 1. The potential energy is gradually transferred to kinetic energy. Kinetic energy is given by 1/2 mv². The velocity is given by equation 3. Summing the kinetic energy and potential energy we obtain,

$\frac{1}{2} k A_{0}^{2}\sin^{2}(\omega{t})=\frac{1}{2}m\omega^{2}A_{0}^{2}\cos^{2}(\omega{t})$

$k=m\omega^{2}$

Since cos²(ωt)+sin²(ωt)=1.

By T=2π/ω,

The figure above shows the change in energy betweeen kinetic energy and potential energy. The blue line shows the potential energy. It is highest at the positions of maximum displacement. The red line shows the kinetic energy. It has a maximum when the velocity is greatest, ie. as it passes the equilibrium position. The green line shows the total mechanical energy of the system, ie. the sum of the potential energy and kinetic energy. The total energy remains constant because there are no losses to friction, heat or air resistance.

Flash. KE (red) and PE (blue) curves. Total energy (green) remains constant

Damped Oscillations

In real life, the oscillations lose energy which reduces the total mechanical energy of the system. damped oscillation

Figure. Damped Oscillations

Driven Oscillations

All objects can oscillate if the conditions are right. It will have a frequency at which it will vibrate if energy is given to it. This is called the nautral frequency. It is important in many engineering disiplines to ensure that the natural frequency of vibration is not driven by external factors. Bridges are a good example of this. The Tacoma Narrows bridge in Washington oscillated wildly because its natural frequency was very close to that caused by the wind. Eventually, this lead to the bridge falling into the river. The millenium bridge also suffered from vibration caused by the natural frequency of the people walking across it. Such oscillations were fixed by adding damping.

To keep the system oscillating, energy must be supplied. Otherwise the system will oscillate and the amplitude of the oscillations will decrease as energy is lost. The system can be given energy by supplying energy periodically to the system much in the same way that a parent would keeps a child on a swing swinging by pushing the swing. As the swing comes back and reaches its maximum displacement, a push emparts energy to keep it going.The frequency with which we supply energy is known as the driving frequency.

We can investigate the effect of the driving frequency on the oscillation of a spring mass system. A motor supplies a driving force to the spring which causes the mass to oscillate on the spring. If the driving frequency is much less than the driving frequency the amplitude of the oscillations of the spring mass system are small.

Similarly, if the driving frequency is much greater than the natural frequency, the inertia of the mass prevents the mass from reacting quickly enough and the oscillations have a small amplitude.

If the driving frequency is close to the natural frequency then energy is transferred to the system with very little resistance. This causes the amplitude of the resulting oscillations to become very exagerated and much larger than the amplitude of the driving force. This phenomena is known as resonance

Figure. Resonance Curves with increasing levels of damping

With increased damping, the peak of resonance decreases

Figure. Phase difference between driving and driven masses with increasing levels of damping

Looking at the phase difference between the driving oscillation and the mass, we can see that resonance occurs when the phases is π/2. This is what resonance really is, it is the efficient transfer of energy from the driving system to the pasive system.

Examples of Resonance

Many types of object resonate as a result of being driven by the wind. Furthermore, the oscillations can occur in many different ways, called oscillating modes.These oscillations are potentially dangerous and to limit their effect, helical strakes are placed on the top of chimneys.

Resonance is an important part in the design of musical instruments. Pianos acoustic guitars, violins use resonance to amplify and shape the sound produced. An interesting article on the Violin acoustics: an introduction

Simple Harmonic Motion (SHM)

Cut to the Chase

Introduction

Examples of SHM

Spring Mass System

Damped Oscillations

Driven Oscillations

Examples of Resonance

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