Oscillations

Free and Forced Vibrations

An oscillation is any to-and-fro 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 to-and-fro movement;

• Period – time taken for a complete to-and-fro 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).  f = 1/T

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.

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, 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

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:

• 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.

  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. 

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

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