Transcript courses:lecture:oslec:damped_oscillator.ppt

THE DAMPED HARMONIC OSCILLATOR
Reading:
Main 3.1, 3.2, 3.3
Taylor 5.4
Giancoli 14.7, 14.8
Free, undamped oscillators - summary
k
m
L
No friction
k
m
x
kx  m&
x&
I
C
q
1
q
q
LC
r
r; r  L
q
Common notation for all
T
m
mg
g
q  q
L
   02  0
Natural motion of damped harmonic oscillator
Force  mÝ
xÝ
Ý
restoring force  resistive force  mxÝ

k
k
kx
Need a model for this.
Try restoring force
proportional to velocity
m
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  mÝ
xÝ
Ý
restoring force  resistive force  mxÝ

kx  bx& m&
x&
Divide by coefficient of d2x/dt2
and rearrange:
x  2 x&  02 x  0
inverse time
 and 0 (rate or frequency) are generic to any oscillating system
This is the notation of TM; Main uses g= 2.
Natural motion of damped harmonic oscillator
2
xÝ
Ý 2xÝ  0 x
0
x(t)  Ce pt
Try
xÝ(t)  pxt ,
2
Ý
Ý
x(t)  p x(t)


p 2  2  p   02 x(t)  0
Substitute:
Now p is known
(and there are 2)
x(t)  Ce
C, p are unknown constants
p t
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” 1 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 = 1/1/0
(>> 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.

0
Q 
T0 2 
large if  is small compared to 0
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
qÝ
Ý 2qÝ  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:
1
Q
 0 RC
Natural (resonance)
frequency determined by
the inductor and capacitor
1
LC
0 
Typical numbers:
L≈500µH; C≈100pF; R≈50W
Damping determined by
0 ≈106s-1 (f0 ≈700 kHz)
resistor & inductor
=1/≈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