Damped and Forced SHM

1
Damped and Forced SHM

• Physics 202
• Professor Vogel
• (Professor Carkners 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(wtf) 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
• xm 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 wwd

8
Resonance

• The condition where wwd 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,
• k2p/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 (f1/T)
• We will again use the angular frequency w,
• w2p/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
• vdx/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