Oscillations: Damping & Resonance Big Picture

Big Picture

I Simple Harmonic Motion handles the steady state (constant amplitude) – but how does oscillation start and stop? I Stops through damping: friction or drag robs energy I also slows frequency: ω ω0 → I Starts in 2 ways: 1. system given initial speed and/or displacement, then released 2. continuous driving at system’s natural frequency ω I driving can be slight if continuous and at the right frequency

I Monday: Waves – oscillations that travel through a medium I Wednesday: quiz on Oscillations (topics 36–38) Language & Notation

I xmax0 : my notation for initial amplitude (book uses xm) I I use xmax for the (always decreasing) amplitude later

I ω0: angular frequency of damped oscillations

I damping constant b, damping force Fdamp = bv − I Diﬀerential equation: involves a variable and its derivatives I arises in Fnet = ma when Fnet depends on x or v d2x dx d2x I example: kx = m 2 or kx b = m 2 − dt − − dt dt I solution is any function x(t) that makes the equation true I resonance: when system is driven by an oscillating force at the system’s natural frequency, it responds with large-amplitude oscillations; other driving frequencies produce small oscillations

Main Results

I Fdamp = bv (opposite the velocity) − 1 2 I not the drag model we used before (FD = 2 CρAv ) I but makes the math doable I also same math as RLC circuit (SP212)

I Damped oscillations: amplitude decreases from xmax0 bt/2m I x(t) = xmax0 e− cos(ω0t + φ) I comes from including damping force in diﬀerential equation: kx bv = ma − − k b2 I note ω replaced by ω0: ω0 = 2 rm − 4m bt/2m I new time-dependent amplitude is xmax = xmax0 e− 1 2 bt/m I Mechanical energy dies out: E(t) = 2 kxmax = E0e− What you need to be able to do

I Calculate amplitude as fraction of initial amplitude in damped oscillator

I Calculate mechanical energy as fraction of initial Emech in damped oscillator I Identify which driving frequency will cause large oscillations in a system with a known natural frequeny