oscillations_03 - 1 oscillations_03 Damped Oscillations...

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1 oscillations_03 Damped Oscillations Forced Oscillations and Resonance SHM shm_v.avi
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2 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 max ma 2 2 2 2 max 2 2 2 2 2 2 max x a max 2 m x cos sin cos 1 1 cos 2 2 1 1 1 sin sin 2 2 2 x t v t a t a x PE k x k x t KE mv m x t x x x k x t ϖ = = - = - = - = = = = = 2 2 max max 1 1 = constant 2 2 total E KE PE k x mv = + = = 2 2 2 2 2 max max 1 1 1 2 2 2 mv k x k x v x x + = = ± - 2 2 2 f T k m π = = = CP 445 Review: SHM mass/spring system
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3 0 10 20 30 40 50 60 70 80 90 100 -10 0 10 S HM position x 0 10 20 30 40 50 60 70 80 90 100 -5 0 5 velocity v 0 10 20 30 40 50 60 70 80 90 100 -1 0 1 acceleration a tim e t CP445
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0 2 4 6 8 10 12 -10 0 10 extension y (m) SHM (zero damping) 0 2 4 6 8 10 12 -20 0 20 velocity v (m/s) 0 2 4 6 8 10 12 0 2000 4000 6000 energy K U e E (J) time t (s) bungee3.avi
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5 0 2 4 6 8 0 0.02 0.04 0.06 0.08 0.1 0.12 b = 0 energy K U E (J) time t (s) KE PE E CP 455
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6 oscillations_03: MINDMAP SUMMARY Reference frame, restoring force, damping force, driving force (harmonic driving force), net (resultant) force, Newton’s Second Law, natural frequency of vibration, free oscillations, damping (underdamped, critical damping, overdamped), exponential decay, graphical interpretation of damping, forced oscillations, resonance, driving frequency, resonance curve, self-excited oscillations, examples of resonant phenomena, strategy for answering examination questions, ISEE 1 2 k f m π =
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7 Mathematical modelling for harmonic motion Newton’s Second Law can be applied to the oscillating system Σ F = restoring force + damping force + driving force Σ F ( t ) = - k x ( t ) - b v ( t ) + F d ( t ) For a harmonic driving force at a single frequency F d ( t ) = F max cos( ϖ t + ε ). This differential equation can be solved to give x ( t ), v ( t ) and a ( t ). CP 463
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8 Damped Oscillations Oscillations in real systems die away (the amplitude steadily decreases) over time - the oscillations are said to be damped For example: The amplitude of a pendulum will
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This note was uploaded on 09/01/2011 for the course YEAR 1 taught by Professor Various during the Three '11 term at University of Sydney.

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oscillations_03 - 1 oscillations_03 Damped Oscillations...

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