Oscillations
Prof. Yury Kolomensky
Apr 2-6, 2007
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
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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 an
infinite sum:
"
& 2$ n
)
f (t) = # an cos(
t + %n +
' T
*
n= 0
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
x(t ) = xm cos (!t + " )
xm: amplitude of oscillations
(max displacement)
ω= 2πf = 2π/T : angular frequency
φ: initial phase (determines velocity
and position at t=0)
(see examples)
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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
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YGK, Physics 8A
Energy of SHO
1 2 1 2
kx = kxm cos 2 (!t + " )
2
2
1
1
1 k
Kinetic energy K = mv 2 = m! 2 xm2 sin 2 (!t + " ) = m xm2 sin 2 (!t + " )
2
2
2 m
1
1
Mechanical energy E = U + K = kxm2 #%cos 2 (!t + " ) + sin 2 (!t + " )$& = kxm2
2
2
In the figure we plot the potential energy U (green line), the kinetic energy K
Potential energy U =
(red line) and the mechanical energy E (black line) versus time t. While U and
K vary with time, the energy E is a constant. The energy of the oscillating object
transfers back and forth between potential and kinetic energy, while the sum of
the two remains constant
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YGK, Physics 8A
Example: Physical Pendulum
• Worked out on the board
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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 velocitydependent friction




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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)
YGK, Physics 8A
Damped Oscillations
Newton's second law for the damped harmonic oscillator:
d 2x
dx
m 2 + b + kx = 0
dt
dt
The solution has the form:
x(t ) = xm e # bt / 2 m cos (! $t + " )
where
" '=
k
b2
#
-- slightly smaller than natural frequency
2
m 4m
ω2 - natural (undamped) frequency
Equivalently, can say that
!A(t) = x e"bt / 2m
m
kA 2 (t) kx m2 "bt / m
E(t) =
=
e
2
2
τ=m/b -- lifetime
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YGK, Physics 8A
Driven Oscillations
Moving
support
• Free oscillations

Move system out of equilibrium and
let oscillate freely
 Oscillate with natural frequency ω (or
smaller ω’ with damping)
 E.g. ω2=k/m for a spring
• Driven oscillations: apply periodic
force
 F(t)=Fmcos(ωdt)
 x(t)=xmcos(ωdt+φ)
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YGK, Physics 8A
Resonance
• Amplitude of driven oscillations xm
depends strongly on driving frequency
xm =

Fm /m
(" d 2 # " 2 ) 2 + b 2" d 2 /m 2
Highest if ωd=ω
 Max amplitude (and sharpness of the
frequency dependence) inversely proportional
to damping b
!

Each solid body has a set of “resonance”
frequencies ω (typically, the larger the
object, the smaller ω is)
 Resonance is an important phenomenon, as
resonant excitations can be quite destructive
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YGK, Physics 8A