Simple harmonic motion
An object experiencing simple harmonic motion is one which experiences a restoring force,
which acts towards the centre of equilibrium. This force is directly proportional to the object’s
distance from the equilibrium position and can be described using the equation below:
F = -kx
Simple Harmonic example and yap
An example of a simple harmonic oscillator is the simple pendulum, as shown in the diagram
above. The pendulum oscillates around a central midpoint known as the equilibrium position.
Marked on the diagram by an .x is the measure of displacement, and by an A is the amplitude of
the oscillations, this is the maximum displacement. You could also measure the time period (T) of
the oscillations by measuring the time taken by the pendulum to move from the equilibrium
position, to the maximum displacement to the left, then to the maximum displacement to the right
and back to the equilibrium position.
In a simple pendulum, the restoring force is provided by the horizontal component of gravity
acting on the pendulum bob.
There are many other examples of simple harmonic motion (SHM), all of which obey the condition
for SHM, which is the existence of the restoring force as described above.
Simple harmonic motion calculations
The acceleration of an object experiencing simple harmonic motion is directly proportional to
displacement and is in the opposite direction.
Angular speed (w)
Angular speed (w) is the angle an object moves through per unit time
Resonance
13.186 - Resonance
Resonance is where the amplitude of oscillations of a system drastically increases due to gaining
an increased amount of energy from the driving force. Resonance has many applications
Instruments
An instrument such as a flute has a long tube in which air resonates,
causing a stationary sound wave to be formed.
Radio
These are tuned so that their electric circuit resonates at the same frequency as
the desired broadcast frequency.
Swing
If someone pushes you on a swing they are providing a driving frequency, which
can cause resonance if it’s equal to the resonant frequency and cause you to swing higher.
Resonance Law
Resonance occurs when the driving frequency, which is the frequency of the force driving the
system, is equal to the natural frequency of the system
Damping and energy conservation
Alongside all the positive consequences of resonance described above, resonance can also have
negative consequences, such as causing damage to a structure, for example a bridge when the
people crossing it are providing a driving frequency close to the natural frequency, will begin to
oscillate violently which could be very dangerous and damage the bridge. Therefore damping can
be used to decrease the effect of resonance - damping is where a force acts on an oscillating
system and energy is lost from the system to its environment, leading to reduced amplitude of
oscillations.
Damping and energy conservation 2
An oscillating system cannot gain or lose energy unless there are any external forces acting on it
- this is the principle of conservation of energy.
For any simple harmonic motion system, kinetic energy is transferred to potential energy and back
as the system oscillates, the type of potential energy depends on the system.
Damping and energy conservation 3
At the amplitude of its oscillations the system will have the maximum amount of potential
energy, as it moves towards the equilibrium position, this potential energy is converted to kinetic
energy so that at the centre of its oscillations the kinetic energy is at a maximum, then as the
system moves away from the equilibrium again, the kinetic energy is transferred to potential energy
until it is at a maximum again and this process repeats for one full oscillation.
Free vibrations
Free vibrations occur when no external force is continuously acting on the system, therefore the
system will oscillate at its natural frequency.
Forced vibrations
Forced vibrations are where a system experiences an external driving force which causes it to
oscillate, the frequency of this driving force, known as driving frequency, is significant.
The effect of damping on resonance
As mentioned previously, if the driving frequency is equal to the natural frequency of a system
(also known as the resonant frequency), then resonance occurs, meaning the amplitude of forced
oscillation increases drastically.
Light damping
This is also known as under-damping and this is where the amplitude
gradually decreases by a small amount each oscillation.
Critical damping
This reduces the amplitude to zero in the shortest possible time (without
oscillating).
Heavy damping
This is also known as over-damping, and this is where the amplitude
reduces slower than with critical damping, but also without any additional oscillations.
Damping uses
Damping can be used to decrease the effect of resonance, different types of damping will have
different effects, as the degree of damping increases, the resonant frequency decreases (shifts
to left on a graph), the maximum amplitude decreases and the peak of maximum amplitude
becomes wider, these effects are shown in the graph below, where ? is the damping ratio, ( = 1
represents critical damping.
Reducing the amplitude of oscillation
In the diagram above, you can see the effects of damping on the amplitude of oscillation of a
system: as the degree of damping increases, the amplitude decreases.
A ductile material is one which can undergo a large amount of plastic deformation before
fracturing, meaning it will be permanently deformed. The plastic deformation of a ductile material
can be used to reduce the amplitude of oscillations, this happens because energy is used to deform the material, decreasing the kinetic energy of the system and so the amplitude of
oscillations decreases.
Reducing the amplitude of oscillation 2 (example)
For example, a climbing rope is manufactured so that it will reduce the amplitude of oscillations if a
climber falls as quickly as possible (this is critical damping), meaning that they can stay safe,
while not having to bounce many times before stopping (as you would with a bungee cord). As the
climbing rope suffers plastic deformation in order to reduce the amplitude of oscillations, it cannot
be used a second time after a climber falls with it.
This is just a single example, but many ductile materials (such as metals) are used in this way, to
reduce the amplitude of oscillations.
