**preview**has

**blurred**sections. Sign up to view the full version! View Full Document

**Unformatted text preview: **Oscillations Summary Free Oscillations: Dierential Equation: Solution:
d2 x dt2
+ ω2x = 0
x(t) = A cos(ωt + δ )
Common Oscillators: Mass on Spring: Simple Pendulum: Physical Pendulum: Uncommon Oscillators: Damped Oscillations: Solution: x(t) = A0 e 2m cos(ωt + δ ) A(t) = A0 e 2m ω=
2 ω0 ( 2b )2 m
bt bt
ω0 = ω0 =
k m g L mgd I
ω0 =
Use Forces or Work/Energy to obtain ω0
The amplitude is time-dependent !
Frequency Shift ! System oscillates at
Classication: if ω is real, the system is underdamped. If it is imaginary, the system is overdamped. If it is zero, the system is critically damped. Energy (Lightly Damped): E = 1 kA(t)2 2 Driven Oscillations: Solution: x(t) =
F0 2 m2 (ω0 ω 2 )2 +(bω )2
(or similar)
cos(ω t + ξ )
ω is the angular frequency of the driver. Phase dierence between driver and oscillation: ωresonant = ω0
b 1 2( 2mω0 )2
tan(ξ ) =
bω 2 m(ω0 ω 2 )
...

View Full Document