Class 16: Damped and forced oscillation Class 16: Damped and forced oscillation
Damped SHM Equation of motion (homogeneous) + + = & & & m x b x k x 0 Causes decay in oscillation Causes decay in oscillation (hence transient) Transient (homogeneous) solution: β Ω Ω β = + = Ω − φ - t i t -i t - t x(t) Ae (e e ) Ae cos ( t ) b b β = ω 0 Ω 2m β
Damping cases p g β β < ω 2 2 2 2 Under damped 0 β ω = 2 2 Critically damped 0 β > ω 2 2 2 Over damped 0
Forced vibration Equation of motion + + = ω & & & m x b x k x F cos t General solution = Transient (homogeneous) solution + Steady state (particular) solution solution + Steady state (particular) solution
Mechanical oscillation and AC circuit conversion dictionary conversion dictionary Q & & & + + = ω + + = ω & & & L Q R Q V cos t m x b x k x F cos t C C Mechanical oscillation RLC series circuits P Position x(t) i i ( ) Ch Charge Q(t) Q( ) & Velocity v(t) = & Current I(t) = x (t) Q (t) Force F Voltage V Mass m Inductance L Damping coefficient b Resistance R Spring constant k Reciprocal of Capacitance 1/C ~ = ~ ~ ~ ⇐ = F v Z Ohm' s Law : V I Z
Impedance model for steady state solution + + = ω & & & m x b x k x F cos t ~ ~ F F = ⇒ = & Ohm' s Law : x (t) x (t) ω p p Z i Z Z is the total impedance of Z m , Z b , and Z k in series: Im = ω Z i m Z m m = Z b b b Re Z b k = Z k ω i Z k Resonance occurs when Z m = Z k
+ + = ω & & & m x b x k x F cos t ( ( ) ) ⎛ ⎛ ⎞ ⎞ k k m m k k m m = ω + + = + ω − = + ω − ω ⎜ ⎟ 2 2 2 Z i m b b i b i ω ω ω 0 ⎝ ⎠ i m ~ ~ F F ~ = = = = ( ( ) ) x x (t) (t) ω p ω − ω − ω 2 2 i Z i b m 0 ~ F F ~ = = x (t) ( ( ) ) ω ω p i i Z Z 2 2 ω + ω − ω 2 2 2 2 2 2 2 2 2 2 b m 0 F ∴ = ω φ x (t) cos ( t - ) ( ( ) ) p 2 ω ω + + ω ω − ω ω 2 2 2 2 2 b b m m 0 ⎛ ⎞ ω There is a phase between b ⎜ ⎟ φ = ( ( ) ⎟ ) ⎠ - 1 tan - ⎜ ω − ω displacement and applied force 2 2 ⎝ ⎝ ⎠ m 0 0
Resonance and Q(uality) factor + + = ω & & & m x b x k x F cos t F ( ) 2 ω + ω − ω 2 2 2 2 2 b m 0 ω = 0 Q Δ ω Energy stored = π × Q Q 2 Energy dissipated per cycle mk 1 L Q = = Q (or (or for for RLC RLC series series circuit) circuit) b R C
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