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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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