© John Parkinson 1 VIBRATIONS & RESONANCE © John Parkinson 2 Natural Frequency / Free Vibrations the frequency at which an elastic system naturally tends.

©John Parkinson

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VIBRATIONSVIBRATIONS

&&

RESONANCERESONANCE

©John Parkinson

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Natural Frequency / Free Vibrationsthe frequency at which an elastic system naturally tends to vibrate, if it is displaced and then released

The natural frequency of a body depends on its elasticity and its shape.

At this frequency, a minimum energy is required to produce a forced vibration.

Free vibration is the vibration of an object that has been set in motion and then left.

©John Parkinson

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Forced vibrations

are the result of a vibration caused by the continuous application of a repetitive

forceUnless the forcing frequency is equal to the

natural frequency, the amplitude of oscillation will be small.

e.g. a swing pushed at “the wrong frequency”

©John Parkinson

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the result of forced vibrations in a body when the applied frequency matches

the natural frequency of the body

The resulting vibration has a high amplitude --

and can destroy the body that is vibrating.Resonance allows energy to be transferred

efficiently

©John Parkinson

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ON NOVEMBER, 7 1940 THE TACOMA NARROWS BRIDGE IN WASHINGTON STATE WAS BUFFETED BY 40 MPH WINDS

AT APPROXIMATELY 11:00 AM, IT COLLAPSED DUE TO WIND-INDUCED VIBRATIONS

http://www.civeng.carleton.ca/Exhibits/Tacoma_Narrows/

http://www.glendale-h.schools.nsw.edu.au/faculty_pages/ind_arts_web/bridgeweb/commentary.htm

WATCH A VIDEO AT

OR AT

©John Parkinson

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Other Resonance Examples

RUMBLE STRIPS Wheels hit the strips at regular time intervals as the car travels at a steady speed and this makes the suspension resonate so the car vibrates with a larger and larger amplitude and makes the driver slow down.

Bus windows

At low engine revs the windows natural frequency can be the same as that of the engine.

©John Parkinson

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Tuning Circuit

The circuit contain the coil and the capacitor resonates to a certain frequency of AC that is picked up in the aerial.

The variable capacitor enables different frequencies to be received

A wine glass can be broken by a singer finding its resonant frequency

©John Parkinson

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DRIVER FREQUENCY IN PURPLE

DRIVEN FREQUENCY IN ORANGE

©John Parkinson

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applied frequency

amp

litu

de

Resonant frequency f0

RELATIONSHIP BETWEEN AMPLITUDE AND DRIVER

FREQUENCY

LIGHT DAMPING

HEAVY DAMPING

©John Parkinson

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Phase lag in degrees

f0

0

180

90

Applied frequency

Phase lag of the driven system behind the driver frequency

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Damping

Damping is the term used to describe the loss of energy of an oscillating system(due to friction/air resistance/ elastic hysteresis etc.)

slight damping

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time

DAMPING

DISPLACEMENT

INITIAL AMPLITUDE

THE AMPLITUDE DECAYS EXPONENTIALLY WITH TIME

©John Parkinson

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With Critically Damped motion the body will return to the equilibrium in the shortest time - about T/4.

Heavy damping or overdamping

©John Parkinson

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Longitudinal Waves

• Each point or particle is moving parallel or antiparallel to the direction of propagation of the wave.

• Common examples:- Sound, slinky springs sesmic p waves

• Longitudinal waves cannot be polarised

Direction of travel

VIBRATION

©John Parkinson

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A longitudinal sound wave in air produced by a tuning fork

Observe the compressions and rarefactions

©John Parkinson

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transverse wave

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Transverse

• Each point or particle is moving perpendicular to the direction of propagation of the wave.

• Common examples:- Water, electromagnetic, ropes, seismic s waves

• You can prove that you have a transverse wave if you can polarise the wave (especially important with light (electromagnetic) as you cannot “see” the wave!!)

Direction of travel

vibration

©John Parkinson

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Formation of a STANDING WAVE

Two counter-propagating travelling waves of same frequency and amplitude superpose to form a

standing wave, characterised by nodes (positions of zero disturbance) and antinodes (positions of maximum

disturbance

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NODESANTINODES

Node to Node = ½ λBETWEEN ANY PAIR OF ADJACENT NODES, ALL PARTICLES

ARE MOVING IN PHASE

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©John Parkinson

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STANDING WAVES ON A STRING

Fundamental length = λ/2

length

length

length

First overtone length = λ

Second overtone length = 3λ/2

©John Parkinson

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LONGITUDINAL STANDING WAVES

OPEN ENDED PIPE

FUNDAMENTAL l = λ/21st harmonic

actual air vibration

1st overtone l = λ2nd harmonic

2nd overtone l = 3λ/23rd harmonic

©John Parkinson

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CLOSED PIPE

FUNDAMENTAL l = λ/41st harmonic

1st overtone l = 3λ/43rd harmonic

2nd overtone l = 5λ/45th harmonic