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
k
I
C
q
m
x
m!!
x = !kx
1
q!! = !
q
LC
! !
r; r = L
θ
Common notation for all
T
m
mg
g
!!
!"# !
L
2
!!
! + " 0! = 0
k
friction
m
x
m!!
x = !kx ! bx!
1
!
LI + q + RI = 0
C
1
Lq!! + q + Rq! = 0
C
!
r = Lcm
θ
Common notation for all
T
m
mg
g
!!
! " # ! # b'!!
L
!!! + 2 "!! + # 02! = 0
Natural motion of damped harmonic oscillator
Force = mx˙˙
restoring force + resistive force = mx˙˙
!
k
k
!kx
m
m
x
Need a model for this.
Try restoring force
proportional to velocity
!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! + " 02 x = 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
x(t) = Ce pt
Try
˙x˙(t) = p 2 x(t)
x˙ (t) = px ( t ),
Substitute:
(
Now p is known (and
there are 2 p values)
x(t) = Ce
p+ t
C, p are unknown constants
)
p 2 + 2 ! p + " 02 x(t) = 0
p = ! " ± " 2 ! # 02
+ C'e
p" t
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
!1 = ! 0
#2
1" 2
!0
time --->
p = ! " ± " 2 ! # 02 = ! " ± i# 1
" #t+i$1t
* " #t"i$1t
x(t) = Ce
+C e
" #t
x(t) = Ae [cos($1t + % )]
Keep x(t) real
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/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.
" #0
Q=! =
T0 2 $
large if β is small compared to ω0
LRC circuit
L
I
dI
q
VL = L ;VR = IR;VC =
dt
C
C
R
!
L (inductance), C (capacitance),
cause oscillation, R (resistance)
causes damping
2
q˙˙ + 2"q˙ + # 0 q
=0
dI
q
!L ! IR ! = 0
dt
C
q
Lq!! + Rq! + = 0
C
R
1
q!! + q! +
q=0
L
LC
LRC circuit
L
I
C
R
LCR circuit obeys precisely the
same equation as the damped
mass/spring.
Q factor:
Natural (resonance)
1
frequency determined by
Q=
! 0 RC
the inductor and capacitor
1
LC
!0 =
Typical numbers: L≈500µH; C≈100pF; R≈50Ω
Damping determined by
ω0 ≈106s-1 (f0 ≈700 kHz)
resistor & inductor
τ=1/β≈2µs; Q≈45
R
(your lab has different parameters)
!=
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