LECTURE 4 DAMPED AND DRIVEN OSCILLATIONS

14.6 Damped oscillationsDifferent amounts of damping

14.7 Driven oscillations and resonanceResonance and hearing

LECTURE 4DAMPED AND DRIVEN OSCILLATIONS

Shock absorbers in suspension of a vehicle can damp oscillation quickly.

14.6 Damped oscillations

! A damped oscillation causes the maximum displacement !"#$ to exponentially decay with a time constant %&as a function of time from the initial amplitude of '.

!"#$ ( = '*+, -⁄

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14.6 Different amounts of damping / Demo

! Mathematically, the oscillation never ceases, however the amplitude will become undetectably small.

! For practical purposes, the time constant ! is the lifetime of the oscillation, the measure of how long it takes to decay.

! If ! ≫ #, the oscillation persists over many periods, and the amplitude decrease is small.

! If ! ≪ #, the oscillation will damp quickly.

! Applications:! Shock absorbers: piston in viscus oil! Building dampers

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Taipei 101 (509.2 m)

730-ton suspended sphere

Quiz: 14.6-1

! After an object on a horizontal spring starts oscillating, its maximum displacement from its equilibrium position gradually decreases with time because of friction on the surface. What happens to the mechanical energy of the object-spring system during this time?A. Mechanical energy increases. B. Mechanical energy does not change. C. Mechanical energy decreases.

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Quiz: 14.6-1 answer

! Mechanical energy decreases.

! Mechanical energy dissipates as thermal energy.! Friction causes the object and the surface to warm up.

! When there is no damping, the mechanical energy of the system is constant, ! = #

$%&$.

! But with friction, the constant amplitude is replaced with '()* which decreases exponentially as a function of time, so the mechanical energy of the system becomes

! = #$%'()*

$ = #$ % &+,- .⁄ $

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Quiz: 14.6-2

A 0.5-kg object oscillates on a spring. Its energy decays to 50% of its initial value in 30 s. What is the value of the time constant?

Quiz: 14.6-2 answer

! A 0.5-kg object oscillates on a spring. Its energy decays to 50% of its initial value in 30 s. What is the value of the time constant?

! ! = #

$% &'() *⁄ $

!,-./01/2,343.3-5

=678 9:; 01/2 <⁄ 7

678 9:; 1/2 <⁄ 7 = '($ =>/? *⁄ = 0.50

! C = ($ =>/?

DE >.F>= 90/s

14.7 Driven oscillations and resonance / Demos

! The natural frequency !" of an oscillator is the frequency of the system if it is displaced from equilibrium and released.

! Driven oscillation is the motion of an oscillator that is subjected to a periodic external force with a driving frequency !#$%.

! At resonance, when the driving frequency is equal to the natural frequency on an oscillator (resonance frequency), its amplitude becomes the maximum.

! Demos: ! Ball on a string! Driven oscillator! Resonant driven pendula

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Quiz: 14.7-1

! When you drive a pendulum at a frequency !", you observe an amplitude of 1 cm. When you increase the driving frequency gradually to !#, the amplitude continually increases to 2 cm. What is the natural frequency?A. Less than !"B. Between !" and !#C. Greater than or equal to !#

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Quiz: 14.7-1 answer

! Greater than or equal to !"! An oscillator’s response curve peaks at the resonance frequency.! As the amplitude is continually increasing without going through a maximum, it

indicates that it has not reached or has just reached the natural frequency. Thus it must be greater than or equal to !".

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Quiz: 14.7-2

! When the driving frequency is near natural frequency of an oscillator, if the damping is small, the oscillator absorbsA. less energy from the driving force than it does at other frequencies. B. more energy from the driving force than it does at other frequencies. C. the same amount of energy from the driving force than it does at other

frequencies.

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Quiz: 14.7-2 answer

! When the driving frequency is near resonance frequency of an oscillator, if the damping is small, the oscillator absorbs more energy from the driving force than it does at other frequencies.

! The amplitude of oscillation becomes large when driven near the natural frequency.

! The energy of an oscillator is greater for a greater !"#$, ~!"#$& .

! The response curve of an oscillator with a small damping is more sharply peaked.

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Quiz: 14.7-3 (Knight P14.43)

! Vision is blurred if the head is vibrated at 29 Hz because the vibrations are resonant with the natural frequency of the eyeball held by the musculature in its socket. If the mass of the eyeball is 7.5 g, a typical value, what is the effective spring constant in N/m of the musculature attached to the eyeball?

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Quiz: 14.7-3 (Knight P14.43) answer

! Vision is blurred if the head is vibrated at 29 Hz because the vibrations are resonant with the natural frequency of the eyeball held by the musculature in its socket. If the mass of the eyeball is 7.5 g, a typical value, what is the effective spring constant of the musculature attached to the eyeball?

! !" = $%&

'(

! ) = 2+!" %, = 2+ 29.Hz % 7.5.×.1078kg = 250. N m⁄

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,)

For fun: Noisy neighbors

! Why can you hear bass notes from music played by your neighbors more distinctively than higher pitched notes?! Sound is vibration of air molecules.! The bass tones have lower frequencies than higher pitched notes.! Large structures like walls and ceilings are more easily set into forced vibrations and

resonance by bass notes since their natural frequencies are closer to the frequencies of bass notes.

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Examples of resonance

! Reed tachometer! https://www.youtube.com/watch?v=Q3NQ4XxOjsU

! Glass and speaker at 748 Hz! http://www.youtube.com/watch?v=17tqXgvCN0E&NR=1

! London Millennium Bridge! http://www.youtube.com/watch?v=eAXVa__XWZ8! The problem was fixed by the retrofitting of 37 fluid-

viscous dampers (energy dissipating) to control horizontal movement and 52 tuned mass dampers (inertial) to control vertical movement.

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Fluid (top) and mass dampers (bottom)