Chapter 14 - Oscillations
• Harmonic Motion Circular Motion
• Simple Harmonic Oscillators– Linear - Horizontal/Vertical
Mass-Spring Systems– Angular - Simple Pendulum
• Energy of Simple Harmonic Motion
• Damped Oscillators• Driven Oscillators -
Resonance
Harmonic
Horizontal mass-spring
F ma
Hooke’s Law: sF kx
2
2
d xkx mdt
2
2
d x k x 0dt m
Solutions to differential equations
• Guess a solution• Plug the guess into the differential equation
– You will have to take a derivative or two• Check to see if your solution works. • Determine if there are any restrictions (required
conditions).• If the guess works, your guess is a solution, but it
might not be the only one.• Look at your constants and evaluate them using
initial conditions or boundary conditions.
Our guess
x A cos t
The restriction on the solution
2 km
1 kf2 2 m
2 mT 2k
Vertical Springs
The constant – phase angle x t 0 A v t 0 0 0
x t 0 0 0v t 0 v 2
x A cos t v A sin t
2a A cos t
Definitions• Amplitude - (A, m) Maximum value of the displacement
(radius of circular motion). Determined by initial displacement and velocity.
• Period - (T) Time for a particle/system to complete one cycle.
• Frequency - (f) The number of cycles or oscillations completed in a period of time
• Phase - t Time varying argument of the trigonometric function.
• Phase Constant - Initial value of the phase. Determined by initial displacement and velocity.
• Angular Frequency (Velocity) - Time rate of change of the phase.
Relation to circular motion
x A cos A cos t
Energy in the SHO
2 2 21 1 1E mv kx kA2 2 2
2 2kv A xm
Simple pendulum
I
2
22
dmg sin L mLdt
2
2
d g sin 0dt L
m cos t
The restriction on the solution
2 gL
1 gf2 2 L
2 LT 2g
“Dashpot”
dampingF bv
dxkx b madt
2
2
d x dxm b kx 0dt dt
b t
2mx Ae cos t
Equation of Motion
Solution
Damped Oscillations
b t
2mx Ae cos t
b bt t
2m 2mdx bv Ae sin t A e cos tdt 2m
2b b b b2 t t t t22m 2m 2m 2m
2
d x b b ba Ae cos t A e sin t A e sin t A e cos tdt 2m 2m 2m
2b t 22m b bAe sin t cos t
m 2m
b t
2m bAe sin t cos t2m
2
2
d x b dx k x 0dt m dt m
b b bt t t
2m 22
m 2m2 b kAe Ae Ab bcos t cos t cos tb sin t sin tm 2m 2m
e 0m m
2
22
b t2m
2b b k cos t2m 2m m
b b sin tm m
Ae 0
22k b 0
m 2m
2k bm 2m
“Dashpot”
dampingF bv
dxkx b madt
2
2
d x dxm b kx 0dt dt
tx Ae cos t
Equation of Motion
Solution
Damped Oscillations
tx Ae cos t
t tdxv Ae sin t A e cos tdt
2
t 2 t t 2 t2
d xa Ae cos t A e sin t A e sin t A e cos tdt
t 2 2Ae 2 sin t cos t
tAe sin t cos t
2
2
d x b dx k x 0dt m dt m
t t2 2 tb kAe Ae Ae 0m m
cos t cos t cos2 sin tt sin t
t 2 2
b2m b k cAe 0b2 s oin t
ms t
m m
22k b 0
m 2m
2k bm 2m
b2m
Damped frequency oscillation
2
2
k bm 4m
2b 4mk
B - Critical damping (=)C - Over damped (>)
b2m
Forced vibrations
ext 0F F cos t 0dxkx b F cos t madt
2
02
d x dxm b kx F cos tdt dt
0 0x A sin t
Resonance
0km
Natural frequency
0 0x A sin t
0
0 2 222 20 2
FAbmm
2 201
0
mtan
b
Quality (Q) value
• Q describes the sharpness of the resonance peak
• Low damping give a large Q• High damping gives a small Q• Q is inversely related to the
fraction width of the resonance peak at the half max amplitude point.
0mQb
0
1Q
Tacoma Narrows Bridge
Tacoma Narrows Bridge (short clip)
b t
2mx Ae cos t
b bt t
2m 2mdx bv Ae sin t A e cos tdt 2m
2b b b b2 t t t t22m 2m 2m 2m
2
d x b b ba Ae cos t A e sin t A e sin t A e cos tdt 2m 2m 2m
2b t 22m b bAe sin t cos t
m 2m
b t
2m bAe sin t cos t2m
2
2
d x b dx k x 0dt m dt m
b b bt t t
2m 22
m 2m2 b kAe Ae Ab bcos t cos t cos tb sin t sin tm 2m 2m
e 0m m
2
22
b t2m
2b b k cos t2m 2m m
b b sin tm m
Ae 0
22k b 0
m 2m
2k bm 2m
tx Ae cos t
t tdxv Ae sin t A e cos tdt
2
t 2 t t 2 t2
d xa Ae cos t A e sin t A e sin t A e cos tdt
t 2 2Ae 2 sin t cos t
tAe sin t cos t
2
2
d x b dx k x 0dt m dt m
t t2 2 tb kAe Ae Ae 0m m
cos t cos t cos2 sin tt sin t
t 2 2
b2m b k cAe 0b2 s oin t
ms t
m m
22k b 0
m 2m
2k bm 2m
b2m
