1 / 19

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

zeroun
Download Presentation

Damped and Forced SHM

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related