Damped and Coupled Oscillations - extension material

A damped oscillator is...

Losing energyrun the following s

preadsheet to see this

damping can be: underdamped overdamped critically damped

Damped Harmonic Oscillator4th order Runge-Kutta Solution

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Damped Harmonic Oscillator4th order Runge-Kutta Solution

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)Damped Harmonic Oscillator

4th order Runge-Kutta Solution

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Coupled Oscillations...

Sometimes an object vibrates in a number of different ways and at different frequencies. Sometimes these vibrations can interact - we call this a coupled oscillation

• coupled pendula• coupled springs• longitudinal & torsional coupling

Resonance and Resonant Coupling...

What happens if you “jiggle” a system (maybe a spring-mass) at the same rate that it naturally vibrates?

What if coupled oscillators have matching frequencies?

Resonance is the condition that occurs when we drive a system (by coupling or other means) at one of its natural frequencies.

Wave Motion in General

Wave motion is one of “nature’s” most common ways of transporting energy.

Mechanical waves propagate through a medium

Three basic flavours:• transverse• longitudinal• torsional

go to this applet to illustrate this...

Wave Equation

Wavelength (crest-crest or trough-trough distance), frequency and wave speed are related in a very simple way:

v f Wave speed frequency

“lambda” = wavelength

Wave speed...

The speed of a wave is NOT the same as the speed with which parts of the propagating medium move!

For mechanical waves we can show that

where T is the tension of the medium and is the density.

wave

Tv

Wave Pulses and Superposition

Waves and wave pulses add together! This is a very important property and the Principle of Superposition is vital to our understanding of wave phenomena.

Go to a Maple Worksheet illustrating thisGo to an Applet that illustrates this

Wave phase...

The idea of wave phase relates the position of the crest (or any other point) on one wave to the same point on another wave. If these points match then the waves are “in phase”, otherwise we talk about a phase difference between the waves.

go to this applet to see this

Wave reflection, phase and the influence of the medium...

When waves encounter a change in media there is always transmission and reflection at the interface. We can easily show this (and will) and you will see that: when a wave passes from a rare medium to a dense

medium the reflected wave travels back with its phase inverted (flipped by 180o)

when a wave travels from a dense medium to a rare medium NO PHASE inversion occurs