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**Unformatted text preview: **Oscillations Summary Free Oscillations: • Diﬀerential 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
• Classiﬁcation: 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 diﬀerence between driver and oscillation: • ωresonant = ω0
b 1 − 2( 2mω0 )2
tan(ξ ) =
−bω 2 m(ω0 −ω 2 )
...

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