4.3.1 State what is meant by damping.

4.3.2 Describe examples of damped oscillations.

4.3.3 State what is meant by the natural frequency of vibration and forced oscillations.

4.3.4 Describe graphically the variation with forced frequency of the amplitude of vibration of an object close to its natural frequency.

4.3.5 What is meant by resonance.

4.3.6 Describe examples of resonance where it is useful, and where it is not.

Topic 4: Oscillations and waves4.3 Forced oscillations and resonance

State what is meant by damping.

The underlying implication of the above is that there is no friction force and no drag force. Then ET is constant.

Note that the amplitude of the displacement x remains constant, shown in A.

If there is friction or drag, ET decreases over time as does the amplitude, shown in B and C.

We say that the oscillations have been damped.

Note that damping does not affect period T or frequency f.

Topic 4: Oscillations and waves4.3 Forced oscillations and resonance

EK + EP = ET = CONST relation between EK and EP

d/cm

A

d/cm

B

d/cm

C

Describe examples of damped oscillations.

We say that A is undamped.We cay that B is lightly damped.

We say that C is heavily damped.

If the oscillation is stopped immediately we say that the system is critically damped.

Topic 4: Oscillations and waves4.3 Forced oscillations and resonance

d/cm

A

d/cm

B

d/cm

C

d/cm

D

U N D

A M P

E D

L I G

H T

D A M

P I N

G

H E A

V Y

D A M

P I N

G

C R I

T I C

A L

D A M

P I N

G

FYI

Without the shocks the springs would oscillate in SHM.

EXAMPLE:

The following three scenarios describe three different damping effects for the same mass/spring system.

Describe examples of damped oscillations.

Topic 4: Oscillations and waves4.3 Forced oscillations and resonance

Undamped

(AIR)

heavily damped(WATER)

critically damped(OIL)

d/cm

C Water

d/cm Oil

d/cm Air

Describe examples of damped oscillations.

Topic 4: Oscillations and waves4.3 Forced oscillations and resonance

PRACTICE:

The displacement vs. time plot for a particle in SHM is shown above. Tell which type of damping is evidenced in the graph, and explain your reasoning.The particle is undergoing undamped SHM as evidenced by the unchanging amplitude of the displacement.

Describe examples of damped oscillations.

Topic 4: Oscillations and waves4.3 Forced oscillations and resonance

EXAMPLE:The grandfather clock has weights which relinquish potential energy as they slowly descend, overcoming the damped SHM of the pendulum.The pendulum would eventually stop if it weren’t for these weights.

Cogsworth, on the other hand, must be wound daily to replenish his energy losses and keep his clock ticking.

State what is meant by the natural frequency of vibration and forced oscillations.

Imagine being on a swing. In order to get as high as you can you kick out at precisely the right time or frequency.

If you don’t match the optimal frequency, called the natural frequency, your swinging motion will not grow. In fact, it might be impeded.

The natural frequency depends on your mass, the swing-seat’s mass, and the length and mass of the chain or rope holding up the swing.

The swing does not swing of its own accord. You must power it. We say that the swing is undergoing forced oscillations because you are applying a force as you kick.

Topic 4: Oscillations and waves4.3 Forced oscillations and resonance

State what is meant by the natural frequency of vibration and forced oscillations.

By the way, if you don’t kick at the right frequency you won’t swing very effectively.

But even if you are kicking at the right frequency, if you do not kick at the right time you will still fail to swing effectively.

Thus, if you want to increase the amplitude of SHM, the forced oscillations must be in phase with and at the natural frequency of the SHM you are driving.

Topic 4: Oscillations and waves4.3 Forced oscillations and resonance

PRACTICE:The mass/spring system shown has a period of 1.5 seconds. The rotating cog transfers energy to the mass through friction. What should the angular speed of the cog be so that it matches the natural frequency of the mass/spring system? For the mass/spring system: T = 1.5 s.For the cog: = 2/T = 2/1.5 = 4.2 rad s-1.

State what is meant by the natural frequency of vibration and forced oscillations.

Topic 4: Oscillations and waves4.3 Forced oscillations and resonance

FYI

A system such as this could be used to keep ET from diminishing so the oscillation continues.

State what is meant by the natural frequency of vibration and forced oscillations.PRACTICE:

The cam shown has an adjustable angular speed. A board is held tight against the cam with a spring and vibrates with an amplitude of 5.0 mm in response to the cam’s rotation. A mass is placed on the top of the vibrating board. The cam’s angular speed is increased until the mass starts to bounce. What is this minimum angular speed?We need the acceleration of the board to equal g (or be bigger). Why? Thus |aboard| = 2x0 = 10. Then 10 = 2(.005) so that board = 48 rad s-1.Note that each cam cycle produces two board cycles. Why? So cam = (1/2)board = 24 rad s-1.

Topic 4: Oscillations and waves4.3 Forced oscillations and resonance

CAM

SPRING

TENSIONER

What is meant by resonance.

If a system is forced to oscillate at its natural frequency f0 the amplitude of the motion will increase and we say that the system is in resonance.

The swing is undergoing a forced oscillation and is in resonance.

All objects have a natural frequency. For example a tapped wine glass will ring at its natural frequency f0. It can also be made to resonate if someone sings loudly enough at f0.

If its amplitude grows enough it may even shatter!

Topic 4: Oscillations and waves4.3 Forced oscillations and resonance

Describe graphically the variation with forced frequency of the amplitude of vibration of an object close to its natural frequency.

Observe the amplitude x0 as we adjust the red dri- ving force’s frequency f:

Topic 4: Oscillations and waves4.3 Forced oscillations and resonance

Driving force at freq. f

Oscillating system at freq. f0Driving force frequency f

f0

Oscillating system

amplitude x0

A

B

CA

B

C

f < f0

f = f0

f > f0

x0

x0

x0

Amplitude vs. Frequency plot near resonance

The frequency of the

system is always f0

Describe examples of resonance where it is useful, and where it is not.

Topic 4: Oscillations and waves4.3 Forced oscillations and resonance

EXAMPLE:The Sears Tower was designed to dampen the energy from the windy city’s wind so that the period of the top is in minutes, rather than seconds and its amplitude is small. Why?The Tacoma Narrows Bridge was not properly designed for dampening. On a windy day it began to resonate.Tacoma Narrows movie

Describe examples of resonance where it is useful, and where it is not.

Topic 4: Oscillations and waves4.3 Forced oscillations and resonance

EXAMPLE:When you tune in a radio receiver you are changing the resonant frequency of the circuitry so that one particular station comes in loud and clear.Which curve is best?

Tuner f

amplitude