Transcript Document

Summary so far:
• Free, undamped, linear (harmonic) oscillator
• Free, undamped, non-linear oscillator
• Free, damped linear oscillator
Starting today:
• Driven, damped linear oscillator
• Laboratory to investigate LRC circuit as
example of driven, damped oscillator
• Time and frequency representations
• Fourier series
THE DRIVEN, DAMPED HARMONIC
OSCILLATOR
Reading:
Main 5.1, 6.1
Taylor 5.5, 5.6
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Natural motion of damped, driven harmonic oscillator
Force = m˙x˙
restoring + resistive + driving force = mx˙˙
w0 =
w1 = w 0
k
k
m
b2
1- 2
w0
k
F0coswt
m
viscous medium
m
x
-kx - bx + F0 cos (w t ) = mx
mx + w 02 x + 2b x+ = F0 cos (w t )
Note w and w0 are not
the same thing!
w is driving frequency
w0 is natural frequency
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Natural motion of damped, driven harmonic oscillator
Apply Kirchoff’s laws
L
I
Vocoswt
C
R
q
V0 cos (w t ) - Lq - - Rq = 0
C
V0 iw t
2
q + 2b q + w 0 q = e
L
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http://www.sciencejoywagon.com/physicszone/lesson/otherpub/wfendt/accircuit.htm
underdamped
b < w0
Damping time or "1/e" time is t = 1/b > 1/w0
(>> 1/w0 if b 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.
t
w0
Q=p =
T0 2 b
large if b is small compared to w0
LRC circuit
L
I
C
R
LCR circuit obeys precisely the
same equation as the damped
mass/spring.
Q factor:
Q
w0 L
R
Natural (resonance)
frequency determined by
the inductor and capacitor
1
LC
w0 =
Typical numbers:
L≈500µH; C≈100pF; R≈50W
Damping determined by
w0 ≈106s-1 (f0 ≈700 kHz)
resistor & inductor
t=1/b≈2µs;
R
(your lab has different parameters)
b=
2L
Menu off button
“push”=enter
Measure the frequency!
“ctrl-alt-del” for osc
save to usb drive
8
Put cursor in track mode, one
measure Vout across R Vin to func gen to track ch1, one for ch2
iw t
Vext = Re éëV0 e ùû
V0 real, constant, and known
q ( t ) = Re éë q0 e ùû
Let's assume this form for q(t)
iw t
But now q0 is complex:
q0 = q0 e
ifq
This solution makes sure q(t) is oscillatory (and at the
same frequency as Fext), but may not be in phase with
the driving force.
Task #1: Substitute this assumed form into the equation
of motion, and find the values of |q0| and fq in terms of
the known quantities. Note that these constants depend
on driving frequency w (but not on t – that's why they're
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"constants"). How does the shape vary with w?
iw t
Vext = Re éëV0 e ùû
q ( t ) = Re éë q0 e ùû
iw t
Assume V0 real, and constant
I (t ) = ?
Task #2: In the lab, you'll actually measure I (current)
or dq/dt. So let's look at that: Having found q(t), find
I(t) and think about how the shape of the amplitude
and phase of I change with frequency.
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iw t
Vext = Re éëV0 e ùû
Assume V0 real, and constant
q ( t ) = Re éë q0 e ùû
Task #1: Substitute this assumed form
into the equation of motion, and find the
values of |q0| and f in terms of the
known quantities. Note that these
constants depend on w (but not on t –
that's why they're “constants”). How
does the shape vary with w?
iw t
q0 =
V0 L
(
é w2 -w
ë 0
-2bw
tan f = 2
2
w0 - w
)
2 2
1/2
+ 4b w ù
û
2
2
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"Resonance"
Charge
Amplitude
|q0|
Driving Frequency------>
Charge
Phase fq
0
-π/2
-π
12
q ( t ) = Re éë q0 e ùû
iw t
iw t
Vext = Re éëV0 e ùû
i (fq + p /2 ) iw t ù
é
I(t) = q ( t ) = Re w q0 e
e
ë
û
I0 =
wV0 L
(
é w2 -w
ë 0
p
)
2 2
1/2
+ 4b w ù
û
2
-2bw
f I = + arctan 2
2
2
w0 - w
2
Task #2: In the lab, you’ll actually
measure I (current) or dq/dt. So
let's look at that: Having found q(t),
find I(t) and think about how the
shape of the amplitude and phase
of
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I change with frequency.
“Resonance”
Current
Amplitude
|I0|
Driving Frequency------>
Current
Phase
π/2
0
-π/2
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“Resonance”
Charge
Amplitude
|q0|
w0
Driving Frequency------>
Current
Amplitude
|I0|
w0
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Charge
Phase fq
0
-π/2
-π
w0
Current
Phase
Driving Frequency------>
π/2
0
-π/2
w0
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