Contributed by:

• Simple Harmonic Motion revision

• Displacement, velocity, and acceleration in SHM

• Energy in SHM

• Damped harmonic motion

• Forced Oscillations

• Resonance

• Displacement, velocity, and acceleration in SHM

• Energy in SHM

• Damped harmonic motion

• Forced Oscillations

• Resonance

1.
PHY 102: Waves & Quanta

Topic 1

Oscillations

John Cockburn (j.cockburn@... Room E15)

Topic 1

Oscillations

John Cockburn (j.cockburn@... Room E15)

2.
•Simple Harmonic Motion revision

•Displacement, velocity and acceleration in SHM

•Energy in SHM

•Damped harmonic motion

•Forced Oscillations

•Displacement, velocity and acceleration in SHM

•Energy in SHM

•Damped harmonic motion

•Forced Oscillations

3.
Simple Harmonic Motion

Object (eg mass on a spring, pendulum bob) experiences

restoring force F, directed towards the equilibrium position,

proportional in magnitude to the displacement from equilibrium:

F kx Solution:

d 2x x(t) =

m 2 kx

dt

2

d x 2

2

x

dt

k

m

Object (eg mass on a spring, pendulum bob) experiences

restoring force F, directed towards the equilibrium position,

proportional in magnitude to the displacement from equilibrium:

F kx Solution:

d 2x x(t) =

m 2 kx

dt

2

d x 2

2

x

dt

k

m

4.
Example: Simple Pendulum (small amplitude)

(NOT SHM for large amplitude)

(NOT SHM for large amplitude)

5.
SHM and Circular motion

Light

Displacement of oscillating object = projection on

x-axis of object undergoing circular motion

y(t) = Acos

For rotational motion with angular frequency ,

displacement at time t:

y(t) = Acos(t + )

= angular displacement at t=0 (phase constant)

A = amplitude of oscillation (= radius of circle)

Light

Displacement of oscillating object = projection on

x-axis of object undergoing circular motion

y(t) = Acos

For rotational motion with angular frequency ,

displacement at time t:

y(t) = Acos(t + )

= angular displacement at t=0 (phase constant)

A = amplitude of oscillation (= radius of circle)

6.
Velocity and acceleration in SHM

Displacement : x(t ) A cos(t )

dx

Velocity : v(t )

dt

d 2x

Acceleration : a (t ) 2

dt

Displacement : x(t ) A cos(t )

dx

Velocity : v(t )

dt

d 2x

Acceleration : a (t ) 2

dt

7.
Velocity and acceleration in SHM

Displacement : x(t ) A cos(t )

dx

Velocity : v(t ) A sin(t )

dt

d 2x

Acceleration : a(t ) 2 2 A cos(t )

dt

Displacement : x(t ) A cos(t )

dx

Velocity : v(t ) A sin(t )

dt

d 2x

Acceleration : a(t ) 2 2 A cos(t )

dt

8.
Energy in Simple Harmonic Motion

Potential energy:

Work done to stretch spring by an amount dx = Fdx = -kxdx

Total work done to stretch spring to displacement x:

W stored potential energy

So, at any time t, the potential energy of the oscillator is given by:

PE

Potential energy:

Work done to stretch spring by an amount dx = Fdx = -kxdx

Total work done to stretch spring to displacement x:

W stored potential energy

So, at any time t, the potential energy of the oscillator is given by:

PE

9.
Energy in Simple Harmonic Motion

Kinetic Energy:

At any time t, kinetic energy given by:

2

1 2 1 dx

KE mv m

2 2 dt

Total Energy at time t = KE + PE:

E

Kinetic Energy:

At any time t, kinetic energy given by:

2

1 2 1 dx

KE mv m

2 2 dt

Total Energy at time t = KE + PE:

E

10.
Energy in Simple Harmonic Motion

Total energy in SHM is constant

1 2 2 1 2

E m A kA

2 2

Throughout oscillation, KE continually being transformed into PE and

vice versa, but TOTAL ENERGY remains constant

Total energy in SHM is constant

1 2 2 1 2

E m A kA

2 2

Throughout oscillation, KE continually being transformed into PE and

vice versa, but TOTAL ENERGY remains constant

11.
Energy in Simple Harmonic Motion

12.
Damped Oscillations

In most “real life” situations, oscillations are always damped

(air, fluid resistance etc)

In this case, amplitude of oscillation is not constant, but

decays with

In most “real life” situations, oscillations are always damped

(air, fluid resistance etc)

In this case, amplitude of oscillation is not constant, but

decays with

13.
Damped Oscillations

For damped oscillations, simplest case

is when the damping force is

proportional to the velocity of the

oscillating object

In this case, amplitude decays

exponentially:

x(t ) Ae ( b 2 m ) t cos( ' t )

d 2x dx

Equation of motion: m 2 kx b

dt dt

For damped oscillations, simplest case

is when the damping force is

proportional to the velocity of the

oscillating object

In this case, amplitude decays

exponentially:

x(t ) Ae ( b 2 m ) t cos( ' t )

d 2x dx

Equation of motion: m 2 kx b

dt dt

14.
Damped Oscillations

NB: in addition to time dependent amplitude, the damped oscillator also has

modified frequency:

k b2 2 b2

' 0

2

2

m 4m 4m

Light Damping Heavy Damping Critical Damping

(small b/m) (large b/m)

k b2

2

m 4m

NB: in addition to time dependent amplitude, the damped oscillator also has

modified frequency:

k b2 2 b2

' 0

2

2

m 4m 4m

Light Damping Heavy Damping Critical Damping

(small b/m) (large b/m)

k b2

2

m 4m

15.
Forced Oscillations & Resonance

If we apply a periodically varying driving force to an oscillator (rather than just

leaving it to vibrate on its own) we have a FORCED OSCILLATION

Free Oscillation with damping:

Amplitude A0 e (b 2 m )t

d 2x dx

m 2 b kx 0 k b2 2 b2

dt dt Frequency ' 0

2

2

m 4m 4m

Forced Oscillation with damping:

d 2x dx F0

m 2 b kx F0 cos D t Amplitude

dt dt k - m b

D

2 2 2

D

2

MAXIMUM AMPLITUDE WHEN DENOMINATOR MINIMISED:

k = mD2 ie when driving frequency = natural frequency of the UNDAMPED

k

D 0

m

If we apply a periodically varying driving force to an oscillator (rather than just

leaving it to vibrate on its own) we have a FORCED OSCILLATION

Free Oscillation with damping:

Amplitude A0 e (b 2 m )t

d 2x dx

m 2 b kx 0 k b2 2 b2

dt dt Frequency ' 0

2

2

m 4m 4m

Forced Oscillation with damping:

d 2x dx F0

m 2 b kx F0 cos D t Amplitude

dt dt k - m b

D

2 2 2

D

2

MAXIMUM AMPLITUDE WHEN DENOMINATOR MINIMISED:

k = mD2 ie when driving frequency = natural frequency of the UNDAMPED

k

D 0

m

16.
Forced Oscillations & Resonance

When driving frequency =

natural frequency of

oscillator, amplitude is

maximum.

We say the system is in

RESONANCE

“Sharpness” of resonance

peak described by quality

factor (Q)

High Q = sharp resonance

Damping reduces Q

When driving frequency =

natural frequency of

oscillator, amplitude is

maximum.

We say the system is in

RESONANCE

“Sharpness” of resonance

peak described by quality

factor (Q)

High Q = sharp resonance

Damping reduces Q

17.
The Millennium Bridge