Oscillation_summary

oscillation_summary
Download Document
Showing pages : 1 of 1
This preview has blurred sections. Sign up to view the full version! View Full Document
This is the end of the preview. Sign up to access the rest of the 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