Oscillations

Prof. Yury KolomenskyApr 2-6, 2007

04/02/2007 YGK, Physics 8A

Periodic Motion• Most common type of motion

Pendulum swings Also swings, clocks, arms and legs, etc.

Springs Also anything elastic: strings, bouncing balls, etc.

Atoms and molecules E.g. atoms in crystal lattice

Humans: heartbeat, brain waves, etc.• Most general: constrained motion around

equilibrium

04/02/2007 YGK, Physics 8A

Math Def: Periodic Functions• f(t) = f(t+T) = f(t+2T) = …

Function that repeats itself at regular intervals T is called period Also define frequency of oscillations f=1/T

T

04/02/2007 YGK, Physics 8A

More Math: Fourier Theorem• There is a theorem in math that states that

Any periodic function with period T can be written as aninfinite sum:

!

f (t) = ann= 0

"

# cos2$ n

Tt + %n

&

' (

)

* +

Example: f(t)=sin(t)+0.5sin(2t)+0.25cos(4t) a0 = 0 a1 = 1, φ0=−π/2 a2 = 0.5, φ2=−π/2 a3 = 0 a4 = 0.25, φ4=0 Each component is called harmonic

04/02/2007 YGK, Physics 8A

Simple Harmonic Motion (SHM)Fundamental (simplest) oscillations: single harmonic

( ) ( ) cosm

x t x t! "= +

xm: amplitude of oscillations (max displacement)ω= 2πf = 2π/T : angular frequencyφ: initial phase (determines velocity and position at t=0) (see examples)

04/02/2007 YGK, Physics 8A

Examples• Mass on a spring

Horizontal Vertical

• Block of wood in water Behaves like a mass on a spring

• Pendulum Mass on a string Physical pendulum

04/02/2007 YGK, Physics 8A

Energy of SHO( )

( ) ( )

( ) ( )

2 2 2

2 2 2 2 2 2

2 2 2 2

1 1Potential energy cos

2 2

1 1 1Kinetic energy sin sin

2 2 2

1 1Mechanical energy cos sin

2 2

In the figure we plot the potential energy

m

m m

m m

U kx kx t

kK mv m x t m x t

m

E U K kx t t kx

U

! "

! ! " ! "

! " ! "

= = +

= = + = +

# $= + = + + + =% &

(green line), the kinetic energy

(red line) and the mechanical energy (black line) versus time . While and

vary with time, the energy is a constant. The energy of the oscillating object

t

K

E t U

K E

ransfers back and forth between potential and kinetic energy, while the sum of

the two remains constant

04/02/2007 YGK, Physics 8A

Example: Physical Pendulum• Worked out on the board

04/02/2007 YGK, Physics 8A

Damped Oscillations• Small friction forces

Energy lost in each period Expect energy and amplitude of oscillations to decrease over time This is called “damping”

• Simplest (but common) case: small velocity-dependent friction

Fd = -bv = -b dx/dt Example: (viscous) friction in air or liquid at small velocities Losses due to heating of springs or strings Warning: dry kinetic friction between surfaces does not work this

way (does not depend on velocity)

04/02/2007 YGK, Physics 8A

Damped Oscillations

2

2

Newton's second law for the damped harmonic oscillator:

0 The solution has the form:d x dx

m b kxdt dt

+ + =

( )/ 2 ( ) co sbt m

mx t x e t! "# $= +

!

"'=k

m#

b2

4m2

where -- slightly smaller than natural frequency

ω2 - natural (undamped) frequency

Equivalently, can say that

!

A(t) = xme"bt / 2m

!

E(t) =kA

2(t)

2=kx

m

2

2e"bt /m

τ=m/b -- lifetime

04/02/2007 YGK, Physics 8A

Movingsupport

Driven Oscillations• Free oscillations

Move system out of equilibrium andlet oscillate freely Oscillate with natural frequency ω (or

smaller ω’ with damping) E.g. ω2=k/m for a spring

• Driven oscillations: apply periodicforce F(t)=Fmcos(ωdt) x(t)=xmcos(ωdt+φ)

04/02/2007 YGK, Physics 8A

Resonance• Amplitude of driven oscillations xm

depends strongly on driving frequency

Highest if ωd=ω Max amplitude (and sharpness of the

frequency dependence) inversely proportionalto damping b

Each solid body has a set of “resonance”frequencies ω (typically, the larger theobject, the smaller ω is) Resonance is an important phenomenon, as

resonant excitations can be quite destructive

!

xm

=Fm/m

("d

2#" 2

)2

+ b2"

d

2/m

2