Oscillations - German University in...

Prof. Reinhard Sigel PHYS202 2013

Oscillations(Revision and more)


Mass Spring System: Harmonic Oscillation (1502)

Hooke’s law

kxFx

Force?

2

2

d

d

t

xmkx

maFx

differential

equation

Solution: )cos()( tAtx m

k

Newton


Simple Harmonic motion

• A periodic motion: displacement changes as a periodic

function in time,

)cos()( tAtx

Amplitude Angular

frequency

rad/s

Time sPhase

constant/ rad

The whole bracket is the phase of motion/rad


Simple Harmonic Motion

Characteristics:

• Amplitude A

• Frequency f and periodic time T=1/f

• Angular frequency ω=2πf

• Phase constant Φ

• Phase (ωt + Φ)

• Total energy is constant and leads to conversion of

kinetic energy into potential energy and back.


The Phase Constant Φ

)cos()( tAtx

0 o90

tx

t

xtv

d

d

2

2

d

d

t

xta

→ chose A and Φ to fulfill the initial conditions x(t=0s) and v(t=0s)


The Amplitude A (1507)

Which oscillates

faster?

)cos()( tAtx m

k

a) the upper

b) the lower

c) equal

higher A

→ higher xmax, vmax

ω independet of A!

(linear equation)


x

x xFU0

' 'd

2

0

0

'

2

1'd'

'd

kxxkx

xFU

x

x

x

Potential Energy

222 sin2

1 tAmK

Energy of the Harmonic Oscillator

'' kxFx

Kinetic Energy2

d

d

2

1

t

xmK

)cos()( tAtx

22

222

sin2

1

sin2

1

tkA

tAmK

m

k 22 cos

2

1 tkAU

2222

2

1sincos

2

1kAttkAKUE

)sin(d

d tA

t

x


Energy of Simple Harmonic Oscillations (1510)


Why do We Find Often Harmonic Oscillations?

Example: atoms in a crystal lattice

Conversion of kinetic and potential energy

For small deformations, the potential energy is well

approximated by a parabola → Hooke’s law

Other examples: houses, cars, engineering constructions


The Pendulum

Tangential force

)cos()( 0 txtx Solution:

2

2

d

dsin

t

smmgFt

arc length Ls

Small angle approx.

2for sin

L

g

t

2

2

d

ddifferential

equation

L

g

ω independent of m !!!


The Pendulum (1517)


Damped oscillations (1521)


Damped oscillations

amplitude of the oscillation decreases

dissipation of Total energy

In reality: friction causes damping of the oscillation

2

2

.dt

xdm

dt

dxbkxSo

dt

dxbkxFx

)cos()( 2

tAetxt

m

b 2

2

m

b

m

k

For weak damping („underdamped“):

Restoring forceDamping Force


Damped oscillations

• How strong is the friction?

underdamped oscillation

critically damped oscillation

overdamped oscillation

02

2

m

b

m

k0

2

2

2

m

b

m

k

02

2

2

m

b

m

k

02

2

2

m

b

m

k


For a smooth ride: spring-damping system

Critical damping would be preferred, but: mass of the car changes

change of the damping type. Better: Slightly underdamped (few cycles)


Forced oscillations - Resonance

Resonant excitation for 0: large oscillation amplitude

)cos( tAx Solution:

How to keep an oscillation running despite damping?

Periodic external force driving the system besides the restoring force and

the damping force

System has then the own frequency: 0,

not identical to the frequency of the driving force

)sin(0 tFFdrive

The system oscillates at the driving frequency.

Amplitude and phase will depend on -0

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