Oscillations An oscillation is a repetitive to-and-fro movement. There are two types of vibration: free and forced.

A forced vibration is produced when a system is made to oscillate by another system at the other system’s frequency. The frequency of the oscillation is not the object’s natural frequency.

A free vibration is produced when a system is displaced from its equilibrium position and then released. It then oscillates at its natural frequency.

A pendulum is an example of a free vibration. Other examples include a swing at a playground, or the string of a guitar when plucked. A free vibration has no driving mechanism and zero friction. Once started, theoretically, it would continue to oscillate for ever. In practice, no oscillation is ever truly ‘free’.

A forced vibration is something like a loudspeaker cone; the cone is forced to vibrate at the frequency of the current that passes in the coil. The amplitude of the vibration depends on the size of the current.

For a pendulum swinging, the time period does not depend on the amplitude of the oscillations, for small angles of θ where sin θ approximately equals θ in radians.

Amplitude: maximum displacement from the mean equilibrium position Period: time to complete one full oscillation Frequency: number of oscillations per unit time Angular frequency: 2πf – one oscillation corresponds to one rotation, this is an angle of 2π. If there are f oscillations per unit time, there will be a corresponding angle of 2πf radians per unit time. Phase Difference : angle in radians between two oscillations.

Simple Harmonic Motion: A body executes simple harmonic motion if its acceleration is directly proportional to its displacement from its equilibrium position and the acceleration is in the opposite direction to the displacement, always towards the equilibrium position.

-Xmax XmaxEquilibrium

A body has maximum velocity and zero acceleration when at equilibrium (displacement is zero)

A body has maximum acceleration to the right when at –A, and therefore maximum resultant force is exerted. It’s velocity is zero.

A body has maximum acceleration to the left when at A, and therefore maximum resultant force is exerted. It’s velocity is zero.

Acceleration is greatest at maximum displacement; velocity is greatest at zero displacement; resultant force greatest at maximum displacement.

Plotting a displacement against time graph for a body freely vibrating, we would get something that looked like this;

We can use this graph to a plot velocity-time graph…

The gradient of this graph gives us rate of change of displacement – which is velocity. So, where the gradient is zero, the velocity is zero; where the gradient is steepest, velocity is at its maximum.

V

TWe can then use this graph to a plot an acceleration-time graph…

The gradient of this graph gives us rate of change of velocity – which is acceleration. So, where the gradient is zero, the acceleration is zero; where the gradient is steepest, acceleration is at its maximum.

From these 3 graphs, we can see that velocity is radians out of phase with displacement, and acceleration is π radians out of phase with displacement (it is in antiphase).

The conclusion we can draw is that acceleration is proportional to – displacement, as the acceleration and displacement are in opposite directions.

𝑎=− (2π 𝑓 )2 𝑥

The displacement of an object in simple harmonic motion can be calculated using either of these equations, depending on the nature of the wave.

𝑥=𝐴 sin 2𝜋 𝑓𝑡

𝑥=𝐴𝑐𝑜𝑠2𝜋 𝑓𝑡The maximum speed of an object performing simple harmonic motion is given by:

𝑉𝑚𝑎𝑥=(2𝜋 𝑓 )𝐴

Note: 2πf appears in all of these equations. It is a quantity known as the angular frequency, denoted by the letter ω omega. Its units are rads-1.

Energy Changes in Simple Harmonic Motion

The energy of the oscillating mass is transformed back and forth between kinetic and potential forms as it oscillates. The total amount of energy stays constant.

At the endpoints, there is maximum gravitational potential energy and zero kinetic energy.

At the midpoints, there is zero gravitational potential energy and maximum kinetic energy.

Total Energy = Kinetic Energy + Potential Energy

Damping

Damping is the effect of resistive forces removing energy from a vibrating object.

Light damping gradually reduces the energy and therefore the amplitude of the vibrating object. An example is a swing in playground; it will gradually come to rest.

Critical damping returns the object to the equilibrium position in the shortest time possible. An example is shock absorbers in a car; they increase the resistive force so that after being displaced when going over a bump, the vehicle returns to its original position without oscillating.

Heavy damping is the result of a very large resistive force. An example could be a mass on a spring in a thick, viscous liquid. Heavy damping results in a long time before the object comes to rest.

Resonance is a maximum amplitude oscillation of a system when it is forced to vibrate at its natural frequency.

A system can be made to oscillate by means of a periodic external driving force. Resonance occurs when the driving frequency most efficiently transfers its energy to the driven system. The amplitude therefore increases as more of the driving force's energy is taken into the system.

Maximum amplitude when the driving frequency is at the natural resonant frequency of the object.

As the degree of damping increases, the amplitude of the resonant vibrations decreases. The resonant peaks become broader.

The frequency at which resonance occurs becomes lower for a damped system.

Resonance

Problems with Resonance: Example

What Oscillates?

What drives the oscillation?

Buildings in an Earthquake

Mechanical Structure of the building

Energy from waves of Earthquake

Components in Engines Components of engine

Energy output from engine – at certain rates of rotation components can be caused to crack or disintegrate.

Uses of Resonance: Example

What Oscillates?

What drives the oscillation?

Magnetic Resonance Imaging

Nuclei of atoms within the body

Electromagnetic waves cause the nuclei to vibrate, and because each nucleus has a different frequency, the structure of the molecules can be determined.

Microwave Cooker Water molecules Microwaves