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4.2 damped harmonic motion

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4.2 damped harmonic motion

1. 1. Topic 4 – Oscillations and Waves4.2 Damped and Forced Harmonic Motion
2. 2. Damped● In SHM there is only the one restoring forceacting in the line of the displacement.● In damped harmonic motion (DHM) anadditional damping force acts in the oppositedirection to the velocity of the object todissipate energy and stop the vibrations.
3. 3. Damping Forces● The damping force acts so as to cause the amplitude of thevibrations to decay naturally dissipating energy.● The general equation of this decay is A=A0e-ˠt● Here ˠ is a damping factor.● The system can be under-damped● This means the system can make more than one full oscillationbefore it comes to a stop.● The system can be over-damped● The system comes to a stop before it completes one oscillation● The system can be critically-damped● The system completes exactly one oscillation before stopping.
4. 4. Damping Forces
5. 5. Damping Forces (beyond Syllabus)● The general equations governing the motion ofa damped harmonic oscillation are:x=x0 e−γ tcos(ωt+ ϕ)v=−x0(γ e−γ tcos(ωt+ ϕ)+ ωe−γ tsin(ωt+ ϕ))a=−x0 (−γ2e−γ tcos(ωt+ ϕ)+ ω2e− γtcos(ωt+ ϕ))
6. 6. Natural Frequency● The frequency with which a system oscillates ifit is started and allowed to move freely is calledits natural frequency.● Simple harmonic motion occurs at the naturalfrequency.● Often, extra energy is imparted into the systemeach oscillation by another external periodicforce.● This is like a child pushing a swing to keep it going.● Such a system is said to be a forced harmonicoscillator.
7. 7. Forced Harmonic Motion● The equation for forced harmonic motion (withsome damping) would be:● Here the first part of the equation is the normalSHM equation with natural frequency ω0andamplitude x0● The second part of the equation is due to theforcing (driving) force of magnitude F anddriving frequency ωx=x0e− γtcos(ω0t)+Fmω2cos(ωt)
8. 8. Forced Harmonic Motion and Resonance● As the driving frequency of the systemapproaches the natural frequency of thesystem, the amplitude of the system increasesdramatically.● The force adds energy to each swing makingthe amplitude continue to increase andincrease.● When the two frequencies are identical, thenthe system is said to be at resonance.
9. 9. Resonance● The state in which the frequency of theexternally applied periodic force equals thenatural frequency of the system is calledresonance.● This causes oscillations with large amplitudes.● Damping causes the maximum amplitude to belimited.
10. 10. Resonance-5 01 9 5 03 9 5 05 9 5 07 9 5 09 9 5 01 1 9 5 01 3 9 5 00 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0MaximumAmplitudeD r iv in g F r e q u e n c yV e r y L ig h t d a m p in gL ig h t d a m p in gM e d iu m D a m p in gH e a v y D a m p in g
11. 11. Dangerous Resonance● Resonance can bedisastrous● If a bridge happens tohave a natural frequencythat is in the range of thefrequencies that can begenerated by the windthen the bridge canoscillate.● The bridge can thenvibrate it can collapse!● This is resonance at itsworst!!!
12. 12. Useful Resonance● Resonance can be useful.● A radio is tuned by causing a quartz crystal toresonate at a particular frequency.● Wind instruments rely on the resonance of avibrating air column to make an audible sound.– Because of the sharp spike on the frequency responsecurve, other frequencies are cancelled out and not heard.