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Damped and Forced SHM

Damped and Forced SHM. Physics 202 Professor Vogel (Professor Carkner’s notes, ed) Lecture 4. Damped SHM. Consider a system of SHM where friction is present The mass will slow down over time The damping force is usually proportional to the velocity

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Damped and Forced SHM

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  1. Damped and ForcedSHM Physics 202 Professor Vogel (Professor Carkner’s notes, ed) Lecture 4

  2. Damped SHM • Consider a system of SHM where friction is present • The mass will slow down over time • The damping force is usually proportional to the velocity • The faster it is moving, the more energy it loses • If the damping force is represented by • Fd = -bv • Where b is the damping constant • Then, • x = xmcos(wt+f) e(-bt/2m) • e(-bt/2m) is called the damping factor and tells you by what factor the amplitude has dropped for a given time or: x’m = xm e(-bt/2m)

  3. Energy and Frequency • The energy of the system is: • E = ½kxm2 e(-bt/m) • The energy and amplitude will decay with time exponentially • The period will change as well: • w’ = [(k/m) - (b2/4m2)]½ • For small values of b: w’ ~ w

  4. Exponential Damping

  5. Damped Systems • All real systems of SHM experience damping • Most damping comes from 2 sources: • Air resistance • Example: the slowing of a pendulum • Energy dissipation • Example: heat generated by a spring • Lost energy usually goes into heat

  6. Damping

  7. Forced Oscillations • If this force is applied periodically then you have If you apply an additional force to a SHM system you create forced oscillations • Example: pushing a swing • 2 frequencies for the system • w = the natural frequency of the system • wd = the frequency of the driving force • The amplitude of the motion will increase the fastest when w=wd

  8. Resonance • The condition where w=wd is called resonance • Resonance occurs when you apply maximum driving force at the point where the system is experiencing maximum natural force • Example: pushing a swing when it is all the way up • All structures have natural frequencies • When the structures are driven at these natural frequencies large amplitude vibrations can occur

  9. What is a Wave? • If you wish to move something (energy, information etc.) from one place to another you can use a particle or a wave • Example: transmitting energy, • A bullet will move energy from one place to another by physically moving itself • A sound wave can also transmit energy but the original packet of air undergoes no net displacement

  10. Transverse and Longitudinal • Transverse waves are waves where the oscillations are perpendicular to the direction of travel • Examples: waves on a string, ocean waves • Sometimes called shear waves • Longitudinal waves are waves where the oscillations are parallel to the direction of travel • Examples: slinky, sound waves • Sometimes called pressure waves

  11. Transverse Wave

  12. Longitudinal Wave

  13. Waves and Medium • Waves travel through a medium (string, air etc.) • The wave has a net displacement but the medium does not • Each individual particle only moves up or down or side to side with simple harmonic motion • This only holds true for mechanical waves • Photons, electrons and other particles can travel as a wave with no medium (see Chapter 33)

  14. Wave Properties • Consider a transverse wave traveling in the x direction and oscillating in the y direction • The y position is a function of both time and x position and can be represented as: • y(x,t) = ym sin (kx-wt) • Where: • ym = amplitude • k = angular wave number • w = angular frequency

  15. Wavelength and Number • A wavelength (l) is the distance along the x-axis for one complete cycle of the wave • One wavelength must include a maximum and a minimum and cross the x-axis twice • We will often refer to the angular wave number k, k=2p/l

  16. Period and Frequency • Period is the time for one wavelength to pass a point • Frequency is the number of oscillations (wavelengths) per second (f=1/T) • We will again use the angular frequency w, • w=2p/T • The quantity (kx-wt) is called the phase of the wave

  17. Speed of a Wave • Our equation for the wave, tells us the “up-down” position of some part of the medium • y(x,t) = ym sin (kx-wt) • But we want to know how fast the waveform moves along the x axis: • v=dx/dt • We need an expression for x in terms of t • If we wish to discuss the wave form (not the medium) then y = constant and: • kx-wt = constant • e.g. the peak of the wave is when (kx-wt) = p/2 • we want to know how fast the peak moves

  18. Wave Speed

  19. Velocity • We can take the derivative of this expression w.r.t time (t): • k(dx/dt) - w = 0 • (dx/dt) = w/k = v • Since w = 2pf and k = 2p/l • v = w/k = 2pfl/2p • v = lf • Thus, the speed of the wave is the number of wavelengths per second times the length of each • i.e. v is the velocity of the wave form

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