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The Damped Harmonic Oscillator

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The Damped Harmonic Oscillator THE DAMPED HARMONIC OSCILLATOR Reading: Main 3.1, 3.2, 3.3 Taylor 5.4 Giancoli 14.7, 14.8 Free, undamped oscillators – other examples k m L No friction I C k m q 1 x m!x! = !kx q!! = ! q LC ! ! r; r L = θ Common notation for all g !! 2 T ! " # ! !!! + " ! = 0 m L 0 mg k friction m 1 LI! + q + RI = 0 x C 1 Lq!!+ q + Rq! = 0 C m!x! = !kx ! bx! ! r L = cm θ Common notation for all g !! ! 2 T ! " # ! # b'! !!! + 2"!! +# ! = 0 m L 0 mg Natural motion of damped harmonic oscillator Force = mx˙˙ restoring force + resistive force = mx˙˙ ! !kx ! k Need a model for this. m Try restoring force proportional to velocity k m x !bx! How do we choose a model? Physically reasonable, mathematically tractable … Validation comes IF it describes the experimental system accurately Natural motion of damped harmonic oscillator Force = mx˙˙ restoring force + resistive force = mx˙˙ !kx ! bx! = m!x! ! Divide by coefficient of d2x/dt2 ! and rearrange: x 2 x 2 x 0 !!+ ! ! + " 0 = inverse time β and ω0 (rate or frequency) are generic to any oscillating system This is the notation of TM; Main uses γ = 2β. Natural motion of damped harmonic oscillator 2 x˙˙ + 2"x˙ +#0 x = 0 Try x(t) = Ce pt C, p are unknown constants ! x˙ (t) = px(t), x˙˙ (t) = p2 x(t) p2 2 p 2 x(t) 0 Substitute: ( + ! + " 0 ) = ! 2 2 Now p is known (and p = !" ± " ! # 0 there are 2 p values) p t p t x(t) = Ce + + C'e " Must be sure to make x real! ! Natural motion of damped HO Can identify 3 cases " < #0 underdamped ! " > #0 overdamped ! " = #0 critically damped time ---> ! underdamped " < #0 # 2 !1 = ! 0 1" 2 ! 0 ! time ---> 2 2 p = !" ± " ! # 0 = !" ± i#1 x(t) = Ce"#t+i$1t +C*e"#t"i$1t Keep x(t) real "#t x(t) = Ae [cos($1t +%)] complex <-> amp/phase System oscillates at "frequency" ω1 (very close to ω0) ! - but in fact there is not only one single frequency associated with the motion as we will see. ! underdamped " < #0 ! Damping time or "1/e" time is τ = 1/β > 1/ω0 % (>> 1/"0 if ! is very small) How many T0 periods elapse in the damping time? This number (times π) is the Quality factor or Q of the system. " # Q = ! = 0 large if β is small compared to ω0 T0 2$ LRC circuit dI q V = L ;V = IR;V = L L dt R C C I C dI q R !L ! IR ! = 0 ! dt C q L (inductance), C (capacitance), Lq!!+ Rq! + = 0 cause oscillation, R (resistance) C causes damping 2 R 1 q˙˙ + 2"q˙ +#0 q = 0 q!!+ q! + q = 0 L LC ! LRC circuit LCR circuit obeys precisely the L same equation as the damped mass/spring. I C Q factor: 1 Natural (resonance) R Q = frequency determined by ! 0 RC the inductor and capacitor 1 Typical numbers: ! 0 = LC L≈500µH; C≈100pF; R≈50Ω ω ≈106s-1 (f ≈700 kHz) Damping determined by 0 0 resistor & inductor τ=1/β≈2µs; Q≈45 (your lab has different parameters) R ! = 2L Does the model fit? Does the model fit? Summary so far: • Free, undamped, linear (harmonic) oscillator • Free, undamped, non-linear oscillator • Free, damped linear oscillator Next • Driven, damped linear oscillator • Laboratory to investigate LRC circuit as example of driven, damped oscillator • Time and frequency representations • Fourier series .
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