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Forced Harmonic Oscillator
DAMPEDUNDAMPED
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x
y=y0coswt
Undamped Forced Oscillator :
Effect of external force on the oscillator.
Force applied : End of the spring + Spring moves(y=y0coswt)
Change in the length of the spring = x - y(Equilibrium Position)
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)( yxkxm
Eqn. of motion without friction
Substituting for y
tm
Ft
m
kyxx coscos 002
0
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t
m
Ft
m
kyxx coscos 002
0
F0cost = Driving force
F0 = Amplitude = Driving frequency
Very special form of driving force !!!
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Examples of such forced oscillators in Nature :
1. Response of a bound electron to an EM field2. Tidal response of a lake to the periodic force of the
moon or sun.
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Guess
tAx cos
Why ?
RHS of eqn. has cos(wt)
LHS of eqn. must have cos(wt)
Therefore
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Substituting tAx cos
In LHS of eqn.
tm
Fxx cos
02
0
2
mk
FA o
22
1
o
o
m
FA
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tm
Fx
cos
122
0
0
The solution
Correct, Not complete
No arbitrary
constants
Must able to
specify x0and v0
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Complete solution
)cos(cos1
022
0
0
tBtm
Fx
Steady state solution General solution
0
2
0 xx Motion : Free undamped oscillator
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)cos(cos122
0
0
tBtmFx
For a damped system
B : Decreases exponentially
Steady state solution
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Amplitude of oscillation vs driving frequency
220
0 1
mFA
0 20 40 60 80 100
-0.010
-0.005
0.000
0.005
0.010
0
A
A finite at0
Resonance
0A
A0
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0 20 40 60 80 100
-0.010
-0.005
0.000
0.005
0.010
0
A
221
o
o
mFA
00;A 00;A
0;A
-ve A ?
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0
Displacement 180 degree out of phase with driving force
Mathematically
)180cos(cos
cos
ttntDisplaceme
tForce
NegativeAmplitude
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0
Displacement in phase with driving force
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Phenomenon of resonance :(+) ve and () ve aspects
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- ve aspects :Avoid motions of large amplitude in the springs of an automobile
To reduce response at resonance dissipative frictionforce is needed :
Analysis of the Forced Damped Harmonic Oscillator
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Forced damped harmonic oscillator
Motion of the oscillator = - bvRetarding force (Viscous)
FTot= Fspr+ Fvis+ Fdriving
= -kx - bv + F0cost
tcosFbv-kxxm 0
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tmFxxx oo cos
2
Willx =A cos t satisfy this differential equ.?No!
The velocityterm givessin
t
tcosFbv-kxxm 0
tcosmFx
mkx
mbx 0
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t
m
Fxxx oo cos2
How to find the solution?
Write the above equation in complex form
ti
em
F
zzz
02
0
Solution will be of the formz = z
oeit
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Real part of z = zo
eitgives the solution toForced damped harmonic oscillator
Substituting z = zo
eitin complex equation
im
Fz
em
Fiez titi
22
0
00
02
020
1
)(
tie
mFzzz
02
0
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i
1
m
Fz
22
0
00
2222
0
2200
)()(
i(
m
F
)
Put zo in cartesian form by multiplying the
numerator and denominator by the complexconjugate of the denominator
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In
polar
form
2
0
2
1
2
1
2222
0
0*
00
0
tan
)()(
1
Re
m
FzzR
z i
2222
0
22
000
)()(
i)(
m
F
z
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Real part
The complete solution isz = zo
eit
titii
ReeRez
)cos( tRx
1/22222
o
o
1
m
FRA
22
1tano
Phase difference between
the driving force
and the displacement
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1/22222
o
o
1
m
FRA
0d
dA
21
20m 2Q
11
At = max
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For light damping, Ais maximum for = oand the amplitude at resonanceis:
o
oo
m
FA )(
Behavior of A and as functions of ,
depends on the ratio / o
1/22222oo
1
m
FRA
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10
1
Q 01
1/22222oo
1
m
FA
2
1
20m 2Q
11
1F
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As increases, the maximum amplitude occurs at a
frequency less than the resonant frequency
1
0
2
1
20m 2Q
11
1/22222o
o
1
m
FA
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22
1tano
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0 20 40 60 80 100
-0.010
-0.005
0.000
0.005
0.010
0
A
Undamped FHO Damped FHO
1/22222
o
o
1
m
FA
22
1tan
o
22
1
o
o
m
FA
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