This lecture

Driven oscillators

• Dependence of x on time, t

• Power

Driven oscillatorsNewton equation of motion

)cos(2 020 t

m

Fxxx

Transientmotion

Steady-statemotion

Solution

t

x)cos(0 tAx

)cos(0 tFF

Driving force

COMPLEX NOTATION

)cos(0 tAx

)cos(0 tFF

i

mFeAA i

2

/22

0

00

)()(0

titi AeeAx

)(0

tieFF

tiem

Fxxx 02

02

DIFFERENTIATION AND SUBSTITUION gives:

Newton equation in complex form

Method of differentiation and substitution with complex notation complex notation to find to find AA00 and and

*AAA 0

ARe

AImtan

220 2 res

2220

22

0

0

4 m

F

A

A0

0

Magnitude of displacement x: A0

A0 depends on

A0 has a maximum if 220 2

The frequency corresponding to the maximum of A0 is

220 2res

Increasing damping,

0 res

2220

22

0

0

4

m

F

A

A0

0

A0has always a maximum

The frequency corresponding to the maximum of A0 is

0 res

Increasing damping,

Magnitude of velocity x: A0

The frequency at which a relevant parameter becomes maximum is known as the resonance frequency.

Resonance frequency

The sharpness of the resonance curves depends on the strength of damping (). In the limit , both displacement and velocity tend to infinity.

Two resonance frequencies

for the displacement

for the velocity

The two values are equal if

220 2 res

0 res

0

Phase of displacement,

Displacement and force are out of phase

)cos(0 tAx)cos(0 tFF

depends on

Phase of displacement,

20

2

2

tan

])[()(

/

imF

A 24

22022222

0

0

sincos 000 iAAeAA i

The displacement always lags the force

A

A

Re

Imtan

Phase of velocity, *

At resonance the velocity is in phase

with the force

)cos( tAx 0

)cos()sin(200

tAtAx

2

**)cos( tAx 0

0

)cos(

)cos(

tFF

tAx

0

0

)cos( tFF 0

“Sloshing” of waterFind the resonance frequency in the

absence of damping

ExamplesLakesTea-cup Bathtub…

“Sloshing” of water• In the absence of damping the resonance frequency res

is equal to the natural frequency 0

• Natural frequency 0

2R

R

Axis of rotation

• 0 = natural frequency of a physical pendulum

I

mgd0

d = distance of the centre of mass from axis of rotation

m = mass of water I = momentum of inertia

d

“Sloshing” of water Natural frequency 0

2R

d

R

g

R

g

I

mgd~

3

80

3

4Rd

2

2

1mRI

“Sloshing” of water

L

h

Consider general geometry

Lock Ness monster or a sloshing mode?

R

g~0

Power Power The oscillator absorbs more energy

when the frequency of the driving force, , matches the natural frequency,

0

Po

we

r, P

Driving frequency,

increasingdamping

Light,

Atom,

Power Power The oscillator absorbs more energy

when the frequency of the driving force, , matches the natural frequency,

Light,

Atom,

Power Power The oscillator absorbs more energy

when the frequency of the driving force, , matches the natural frequency,

0

Po

we

r, P

Driving frequency,

increasingdamping