Further Mechanics Tutorial 3 - Oscillations and Resonance

 

Free and Forced Vibrations

An oscillation is any to-and-fro movement.  It can arise from:

 

We need to define some terms:

 

Question 1

A rope hanging from a tree swings with a period of 5 s.  What is its frequency?  

Answer

 

Phase

Suppose we have two oscillators swinging at the same frequency.  If they are swinging in step, we say that they are in phase.  If they are swinging so that one reaches its maximum displacement to the left at the same time as the other reaches its maximum displacement to the right, we say that they are 180 o (or p rad) out of phase.

 

 

Of course we can have any number of degrees out of phase.  The oscillator that gets to a particular point first is leading.  The other one is lagging.

 

We can show the displacements of two oscillators that are out of phase, but with the same frequency:

 

 

Question 2

What is the phase relationship between these two waves?  Which one is lagging?

Answer

 

If the frequencies are different, the phase relationship changes all the time.

 

 

Free and Forced Oscillations

If we swing a pendulum at let it swing freely, it will swing at its natural frequency.  The same will apply to a mass bouncing up and down on a spring.

 

If we try to make the oscillator oscillate, we apply a forcing frequency.  An example of this is the push we give to a child on a swing.

 

Question 3

What is the difference between natural and forced oscillations?

Answer

 

Resonance

If the forced vibrations have the same frequency as the natural frequency, the amplitude of the oscillations will get very large.  We can show this with our child on the swing.  If we apply the push at the same point of the swing every time, the child swings higher and higher.  We call this situation resonance.

We can demonstrate resonance in the lab in several ways including:

 

 

If we alter the frequency we see that the mass bounces with varying amplitude.  However at the resonant frequency, the amplitude gets very large.  It is not unknown for the masses to fly off!  Typically the resonant frequency of this kind of system is about 1.5 Hz.

 

Another demonstration is to show Barton's Pendulums.  It was named after Edwin Henry Barton (1858 – 1925), Senior Lecturer in Physics at University College, Nottingham.  It consists of a number of pendulums of different lengths which are mounted from a string as shown:

 

 

The apparatus demonstrates the phase as well as the amplitude of the oscillations.  Phase difference describes how much oscillations are "out of step".  The driver pendulum is set swinging, applying a torque to the string.  This in turn sets the others swinging:

Click HERE to see this demonstrated.

 

Therefore:

In the exam, you may be asked to describe resonance.  Remember to include discussion about phase.

 

Question 4

At a certain engine speed, a car’s wing mirror starts to vibrate strongly.  Why does this happen? 

Answer

 

If we plot a graph of amplitude against frequency, we see a very large peak.  It occurs at the resonant frequency, which we give the code f0.  When considering the resonant frequency of strings and columns of air, we often call this the fundamental frequency.

Resonance has many uses:

 

Resonance can also be a nuisance or even dangerous:

Question 5

Explain why worn shock absorbers can make a car fail its annual MOT (an annual check in which about 30 safety items are checked.  It is illegal to drive a car without a current MOT)

Answer

 

The amplitude of resonant oscillations can be reduced by damping.

 

The graph above shows light damping. 

 

Question 6

Sketch graphs to show heavy damping and critical damping.

Answer

 

Over-damped systems do not oscillate.  They take a long time to return to the equilibrium position.  An example is the return spring on a door.  The graph looks like this

 

 

Question 7

Explain why a car shock absorber needs to be a critically damped system rather than an over-damped system.

Answer

 

The graph below shows the effect on amplitude of damping.  The resonant frequency reduces slightly.

      

Damping and Period

When a damping occurs, the period of the damped oscillation remains the same, as shown in this diagram:

 

 

When damping occurs, the amplitude decays as a constant fraction of the amplitude of the previous oscillation.  So if the first oscillation has an amplitude of 100 % and the fraction is 0.9, the next oscillation has 90 %, the next 81 %, and so on.  As with any system that decays by a constant ratio, natural logarithms are involved.  Therefore it is sometimes called the logarithmic decrement, and you will study the relationships at university:

Image from Wikimedia Commons

 

The symbol d (delta, a Greek lower case letter 'd') is the physics code for the decrement.  x(t) is the amplitude at time t, while x(t + nT) is the amplitude a whole number of periods later.  n is the whole number.  This is not on the A level syllabus.

 

Question 8

Two students are told that when damping occurs, the amplitude decays as a constant fraction of the amplitude of the previous oscillation. Describe how they could investigate this.

Answer