Transcript damped and driven oscillations, waves

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
 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)
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
Exponential Damping
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
Damping
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
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
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
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
Transverse Wave
Longitudinal Wave
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)
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
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
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
Speed of a Wave
 Our equation for the wave, tells us the “updown” 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
Wave Speed
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