Reading: Main 3.1, 3.2, 3.3 Taylor 5.4 Giancoli 14.7, 14.8

THE DAMPED HARMONIC OSCILLATOR

x

m

m k

k

Free, undamped oscillators – other examples

m!!x = !kx

No friction

θ

mg

m T

!r; !r = L

!!! " #

gL!

L

C I q

!!q = ! 1

LCq

!!! +"02! = 0

Common notation for all

m!!x = !kx ! b !x

θ

mg

m T

!r = Lcm

!!! " # g

L! # b ' !!

L!I + 1Cq + RI = 0

L!!q + 1Cq + R !q = 0

!!! + 2" !! +# 02! = 0

Common notation for all

x

m k

friction

Natural motion of damped harmonic oscillator

Need a model for this. Try restoring force proportional to velocity!b !x

!

Force = m˙ ̇ x

!

restoring force + resistive force = m˙ ̇ x

!kx

How do we choose a model? Physically reasonable, mathematically tractable … Validation comes IF it describes the experimental system accurately

x

m

m k

k

Natural motion of damped harmonic oscillator

!kx ! b !x = m!!x

!!x + 2! !x +" 0

2x = 0

!

Force = m˙ ̇ x

!

restoring force + resistive force = m˙ ̇ x

β and ω0 (rate or frequency) are generic to any oscillating system This is the notation of TM; Main uses γ = 2β.

inverse time

Divide by coefficient of d2x/dt2

and rearrange:

Natural motion of damped harmonic oscillator

!

˙ x (t) = px t( ), ˙ ̇ x (t) = p2x(t)

Substitute: p2 + 2! p +" 0

2( )x(t) = 0

!

˙ ̇ x + 2" ˙ x +#02x = 0

!

x(t) = Cep+ t + C'ep" t

p = !" ± " 2 !#02Now p is known (and

there are 2 p values)

C, p are unknown constants x(t) = CeptTry

Must be sure to make x real!

Natural motion of damped HO

!

" <#0

underdamped

!

" >#0

overdamped

!

" =#0

critically damped

Can identify 3 cases

time --->

!

" <#0

underdamped

p = !" ± " 2 !#02 = !" ± i#1

!

x(t) =Ce"#t+i$1t +C*e"#t"i$1t Keep x(t) real

!

x(t) = Ae"#t cos $1t +%( )[ ] complex <-> amp/phase

!1 =!0 1" # 2

!02

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.

time --->

!

" <#0

underdamped

Q = !"T0

=#0

2$ large if β is small compared to ω0

Damping time or "1/e" time is τ = 1/β > 1/ω0 ���(>> 1/ω0 if β 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.

L (inductance), C (capacitance), cause oscillation, R (resistance) causes damping

LRC circuit

!

VL = L dIdt;VR = IR;VC =

qC

!L dIdt! IR ! q

C= 0

L!!q + R !q + q

C= 0

!!q + R

L!q + 1

LCq = 0

!

˙ ̇ q + 2" ˙ q +#02q = 0

L

R

C I

LCR circuit obeys precisely the same equation as the damped mass/spring.

LRC circuit

L

R

C I

Natural (resonance) frequency determined by the inductor and capacitor

!0 =1LC

! =R2L

Damping determined by resistor & inductor

Typical numbers: L≈500µH; C≈100pF; R≈50Ω ω0 ≈106s-1 (f0 ≈700 kHz) τ=1/β≈2µs; Q≈45 (your lab has different parameters)

Q factor:

Q =1

!0RC

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