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