oscillation_summary - Oscillations Summary Free...

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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 ) ...
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