Damped and Forced SHM

Physics 202Professor 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(t+) 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 FrequencyThe 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:’ = [(k/m) - (b2/4m2)]½

For small values of b: ’ ~

Exponential Damping

Damped SystemsAll real systems of SHM experience

dampingMost damping comes from 2 sources:Air resistance

Example: the slowing of a pendulumEnergy dissipation

Example: heat generated by a springLost energy usually goes into heat

Damping

Forced OscillationsIf this force is applied periodically then

you have If you apply an additional force to a SHM system you create forced oscillationsExample: pushing a swing2 frequencies for the system

= the natural frequency of the systemd = the frequency of the driving forceThe amplitude of the motion will

increase the fastest when =d

ResonanceThe condition where =d is called

resonance Resonance occurs when you apply

maximum driving force at the point where the system is experiencing maximum natural forceExample: pushing a swing when it is all the

way upAll 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 itselfA 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 travelExamples: waves on a string, ocean wavesSometimes called shear waves

Longitudinal waves are waves where the oscillations are parallel to the direction of travelExamples: slinky, sound wavesSometimes called pressure waves

Transverse Wave

Longitudinal Wave

Waves and MediumWaves travel through a medium (string, air

etc.)The wave has a net displacement but

the medium does notEach individual particle only moves up or down

or side to side with simple harmonic motionThis only holds true for mechanical waves

Photons, electrons and other particles can travel as a wave with no medium (see Chapter 33)

Wave PropertiesConsider a transverse wave traveling in the x

direction and oscillating in the y directionThe y position is a function of both time and

x position and can be represented as:y(x,t) = ym sin (kx-t)

Where:ym = amplitudek = angular wave number = angular frequency

Wavelength and NumberA wavelength () is the distance

along the x-axis for one complete cycle of the waveOne 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=

Period and FrequencyPeriod is the time for one wavelength to

pass a pointFrequency is the number of oscillations

(wavelengths) per second (f=1/T)We will again use the angular

frequency, =2/T

The quantity (kx-t) is called the phase of the wave

Speed of a WaveOur equation for the wave, tells us the “up-down”

position of some part of the medium y(x,t) = ym sin (kx-t)

But we want to know how fast the waveform moves along the x axis:

v=dx/dtWe 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-t = constante.g. the peak of the wave is when (kx-t) = /2

we want to know how fast the peak moves

Wave Speed

VelocityWe can take the derivative of this expression

w.r.t time (t): k(dx/dt) - = 0 (dx/dt) = /k = v

Since = 2f and k = v = /k = 2f/2

v = fThus, 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