Contents 
Free
and Forced Vibrations
An
oscillation is any toandfro movement. It
can arise from:
a
swinging pendulum;
a
mass bouncing on a spring;
a
vibrating system.
We
need to define some terms:
Cycle
– a complete toandfro movement;
Period
– time taken for a complete toandfro movement.
It is given the Physics code
T and measured in seconds (s);
Frequency – how many cycles there are in a second. Physics code f and measured in Hertz (Hz).
A
rope hanging from a tree swings with a period of 5 s.
What is its frequency?

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:
What is the phase relationship between these two waves? Which one is lagging? 
If the frequencies are different, the phase relationship changes all the time.
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.
What is the difference between natural and forced oscillations? 
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:
Barton’s
pendulums
A mass on a spring being forced up and down with a vibration generator.
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:
The pendulums with the shorter strings than the target pendulum swing in phase with the driver;
The target pendulum has a string the same length as the driver. It swings with the largest amplitude, as its natural frequency is the same as that of the driver. It is out of phase with the driver by p/2 rad;
The pendulums with the longer strings oscillate p rad out of phase.
Click HERE to see this demonstrated.
Therefore:
If the forcing frequency is less than the natural frequency, the oscillators and the driver move in phase.
If the forcing frequency is the same as the natural frequency, the driver leads the oscillator by 90 degrees (or p/2 rad)
If the forcing frequency is bigger than the natural frequency, the oscillators oscillate in antiphase.
In the exam, you may be asked to describe resonance. Remember to include discussion about phase.
At a certain engine speed, a car’s wing mirror starts to vibrate strongly. Why does this happen? 
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 f_{0}. When considering the resonant frequency of strings and columns of air, we often call this the fundamental frequency.
Resonance
has many uses:
In
order to sound heavy church bells (which may have masses of several tonnes),
bell ringers swing them at the resonant frequency of the bell in its
carriage. They cannot swing
them at any other rate.
Resonance
of strings at their fundamental frequency and multiples of them give us
musical sounds. Wind
instruments are sounded by making a column of air resonate by either blowing
a whistle or a raspberry (an embouchure) at one end.
Resonance
of electrons makes radio waves and allows them to be received.
Resonance
can also be a nuisance or even dangerous:
Panels
in a bus rattling.
Resonance
in a car suspension needs to be damped.
If the dampers (shock absorbers) are not working properly, the car
could go out of control.
A
suspension bridge started rocking in a wind at its resonant frequency.
Its oscillations got so large that the deck collapsed into the sea.
Marching troops are ordered to "break step" when crossing bridges.
Resonance in an aircraft can make the machine difficult to control, or even destroy it.
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) 
The amplitude of resonant oscillations can be
reduced by damping.
Light
damping reduces oscillations slowly.
Heavy
damping
reduces oscillations quickly.
Critical
damping stops the oscillation within one cycle.
The graph above shows light damping.
Sketch graphs to show heavy damping and critical damping. 
Overdamped 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
Explain why a car shock absorber needs to be a critically damped system rather than an overdamped system. 
The graph below shows the effect on amplitude of damping. The resonant frequency reduces slightly.
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.
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. 
Plastic Deformation and Oscillation
So for we have assumed that the oscillators have obey Hooke's Law and showed perfectly elastic deformation. But what happens to the the way the oscillators behave if the they exceed the elastic limit?
One example is what happens with a climbing rope. An ordinary rope would slow the climber down in a very short time, causing serious (if not lifethreatening) injury. The speedtime graph would be like this:
The distance travelled is very small, so the body undergoes a very high value of deceleration. The energy will be dissipated as heat. The climber will not bounce up and down.
Suppose you used a bungee rope instead. The motion of the climber would be:
The climber will not suffer any injury after the fall, but would bounce up and down, as shown in this graph:
There would be every possibility of the climber being injured as he (or she) bounces about.
The idea of a climbing rope is that it should be strong enough to hold the climber's weight during climbing, but when the climber has a fall, the rope plastically deforms to allow the climber to slow down to zero speed without suffering injury.
The energy is dissipated as heat, so there are are fewer oscillations of reduced amplitude. It is heavily damped: