Vibrations

• Simple harmonic motion - s.h.m.

• Examples include….

​

a mass bouncing on a spring

…or bungee jumping

…and a swinging pendulum

Simple harmonic motion is a special type of repetitive motion…..

• The time period of the oscillation stays the same even if the amplitude varies.
• The time taken to get from a to b and back to a in all three cases below is the same.

The usual equations and terms apply

• T (s) = Time Period = time for one oscillation
• f (hz) = frequency = no. of oscillations/sec
• A = amplitude = maximum displacement from equilibrium position

t

T

A

Also…

• … the acceleration of the body is directly proportional to its displacement from a fixed point
• and is always directed towards that point.

Let’s consider a pendulum…�(taking positive to be to the right)

• Displacement, x = max = amplitude, A
• Acceleration, a = max = -a_max (left)
• Velocity = zero
• So…
• x = x_max = A
• a = -a_max
• v = 0

​

As it swings through the centre…

• Displacement, x = 0
• Acceleration, a = 0
• Velocity = max = -v_max (left)
• So…
• x = 0
• a = 0
• v = -v_max

It stops and then…

• Displacement, x = max = amplitude, -A
• Acceleration, a = max = a_max (right)
• Velocity = zero
• So…
• x = x_max = -A
• a = a_max
• v = 0

As it swings through the centre again…

• Displacement, x = 0
• Acceleration, a = 0
• Velocity = max = v_max (right)
• So…
• x = 0
• a = 0
• v = v_max

So the acceleration is always doing what the displacement is doing…they are directly proportional

• x = x_max
• a = -a_max
• v = 0

• x = x_max
• a = a_max
• v = 0

• x = 0
• a = 0
• v = -v_max

• x = 0
• a = 0
• v = v_max

Similarly with a mass on a spring…

So the defining equation for shm is…

• The minus sign means the acceleration and displacement are oppositely directed.

..and the definition in words is…

• If the acceleration of a body is directly proportional to its distance from a fixed point and is always directed towards that point, the motion is simple harmonic.

Simple harmonic motion can be characterized by a sine function.

Plotting the pendulum’s displacement

displacement

time

The bigger the amplitude of the oscillation the higher the peak of the sine wave

Consider the pendulum again…

• Starting with the pendulum pulled up to the right…
• Displacement is a maximum (equal to the amplitude) …and velocity is zero…

• Then displacement decreases as…
• …velocity increases, but to the left, so in the negative direction.

Putting all three together…

• Displacement and velocity we’ve talked about…
• …and acceleration and displacement do the same as each other but in opposite directions i.e. both go from max to min but in opposite directions.

Timing your pendulum…(as you do)

• You can start timing from when x = A
• Or from when x = 0
• These give different displacement-time graphs

t=0

x=0

t=0

x=-A

t=0

x=A

t=0

x=0

Cosine curve

Sine curve

x = Acos(ωt + ε)

x = Asin(ωt + ε)

All the equations!

• Period, T (s) = time for one oscillation
• Frequency, f (Hz) = number of oscillations per second
• Angular frequency, ω (rad/s) = 2πf