chapter 5 french
TRANSCRIPT
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Chapter V
Coupled Oscillations
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Two, or more, oscillators coupled together
Examples
1. Two pendulums, coupled by a spring
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2. A molecule
Molecule2CO
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3. A crystalline solid
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~
4. Coupled electrical oscillators
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5. An elastic medium, like a vibrating string
A wave in a medium is a continuum of
coupled oscillators
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Two Simple Pendulums Coupled by a Spring
1x 2x
m mk
Equations of Motion :
)xx(k x
gmdt
xdm 12
1
2
1
2
)xx(k x
gmdt
xdm 12
2
2
2
2
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0)xx(
m
k x
g
dt
xd1212
1
2
0)xx(m
k x
g
dt
xd1222
2
2
The above two differential equations are
coupled.
Adding and subtracting the above equations :Decoupling
0qdt
qd
1
2
02
1
2
0qdt
qd
; 2
2
2
2
2
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122211 xxq;xxq
Where,
m
k 2;
g 2
0
22
0
The solutions for are :21 q&q
)t(cosAq 1011
)tcos(Aq 222
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21 x&x can then be obtained as :
)qq(2
1x 211 )qq(
2
1x; 212
21 q&q are known as the normal
coordinates and are known as normal
frequencies.
&0
Initial Conditions on are obtained
from those of 21 x&x21 q&q
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Normal Coordinates : New coordinates, that
are linear combinations of the original, inwhich the equations of motion are
decoupled.
Normal Frequencies : Frequencies of
oscillation of the normal coordinates.
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Let the initial conditions be such that
remains zero forever.2
q
0)0(q)0(q 22
Or, )0(x)0(x;)0(x)0(x 2121
Normal Mode Vibrations
Thus, 0)t(q&)tcos(A)t(q 21011
For this to happen, the initial conditions
must be such that
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0)t(q&tcosA)t(q 201
tcos2
A)t(x)t(x 021 And,
Taking :
0)0(x)0(x;2A)0(x)0(x 2121
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This mode of vibration is called a Normal
Mode Vibration
Normal Mode Vibration : A mode of vibration inwhich only one normal coordinate is excited,
the other normal coordinates remaining zero.
First Mode for the Coupled Pendulums
0t 4Tt 0
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2Tt 0 4T3t 0
0Tt
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Second normal mode vibration :
0)t(q1
)0(x)0(x&)0(x)0(x 2121
Required initial conditions :
0)0(q;0)0(q 11
Or,
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Complete solution :
tcosA)t(q2
1)t(x
21
0)0(x)0(x;A)0(x)0(x 2112
0)0(q;A2)0(q 22
tcosA2)t(q2
tcosA)t(q2
1)t(x 22
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0t 4Tt
2Tt 4T3t
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Tt
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General Motion (Both Modes Excited)
A
Initial Conditions :
A)0(x;0)0(x 21
0)0(x)0(x 21
A)0(q)0(q 21 0)0(q)0(q 21
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tcosA)t(q;tcosA)t(q 201
Solutions for 21 q&q
Solutions for 21 x&x
]tcost[cos2
A)t(x 01
t2sint2sinA00
t
2
cost
2
cosA)t(x 002
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)t(x2
t
)t(x1
t
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Summary
1. There exist normal coordinates, which
are such that, the equations of motion in
them, are decoupled
2. Each normal coordinate behaves likea simple harmonic oscillation with its own
frequency, the normal frequency
3. With appropriate initial conditions, onecan excite only one normal coordinate, the
other remaining dormant. Such vibrational
modes are called normal mode vibrations.
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4. In a normal mode vibration, each mass in
the coupled system, executes a SHO with
the same frequency, the corresponding
normal frequency. The amplitudes of motion
of the different masses, and their phases
are in general different.
5. In the most general motion, which results
from arbitrary initial conditions, the motion of each mass is rather complicated. There is no
definite frequency of vibration. However, the
motion is a superposition of SHMs
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No Class on Saturday
Combined Class for Both Sections
On
Sunday (25/11)
At
10.00 AM
InRoom No. 5102
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General Approach for Finding Normal Modes
Back to Coupled Pendulums :
0)xx(x
dt
xd12
2
s1
2
02
1
2
0)xx(xdt
xd12
2
s2
2
02
2
2
Since in a normal mode vibration, all
masses execute SHM of a common
frequency, put :
mk ;g 2
s20
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tcosAx;tcosAx 2211
0AA)( 2
2
s1
22
s
2
0
0A)(A 2
22
s
2
01
2
s
In the matrix form :
0A
A
2
1
22
s
2
0
2
s
2s
22s
20
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For non-trivial solutions for
( ), the determinant of
the matrix must be zero :
21 A&A
zeronotA&ABoth 21
04
s
222
s
2
0
Or,2
s
2
00 2or
First Normal Mode
1A
A
2
101
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tcosAx;tcosAx 1211
Second Normal Mode
1A
A2
2
12
s
2
02
tcosAx;tcosAx 2221
Normal Coordinates
0xx 21
First Mode : 0xx 21
Second Mode :
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Since, in the second normal mode vibration,
the first normal coordinate is identically zero
211 xxq
Since, in the first normal mode vibration, thesecond normal coordinate is identically zero
122 xxq
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Prob. 5.10
m
m
k
k
Consider the vertical motion
of the system.
a) Find the normal modefrequencies and the ratio of the
amplitude of the two masses in
each mode
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1x
2x
m
m
Equations of Motion :
)xx(k xk td
xdm 1212
12
)xx(k td
xdm 1222
2
Or,
0)xx2(td
xd21
2
02
1
2
0)xx(td
xd12
2
02
2
2
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Substituting : tcosAx;tcosAx 2211
0AA)2( 2
2
01
22
0
0A)(A 2
22
01
2
0
Writing the above as a matrix equation and
setting the determinant to zero, we get :
02
22
0
2
0
2
0
22
0
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Or,2
0
2
2
53
We have : 22
0
2
0
2
1
2A
A
First Mode :15
2
A
A
2
1
Second Mode :15
2
A
A
2
1
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2 cm
1.23 cm
2 cm
3.23 cm
First Mode
Second Mode
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b*) Find the normal coordinates
First Mode : 0x2x15 21
Second Mode :
211 x2x15q
212 x2x15q
0x2x1521
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c*) If the lower mass is pulled down a
distance A, while the upper mass is held
fixed, and the system released, describe thesubsequent motion.
0)0(q)0(q;A2)0(q;A2)0(q 2121
)t(cosA)t(q 1111
We have :
)t(cosA)t(q2222
Initial Conditions :
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)tcos(B2)tcos(A2)t(x 22111
)tcos(B)15()tcos(A)15()t(x 22111
One can directly write down the complete
solutions for , without requiring to
obtain the normal coordinates .
21 x&x
21 q&q
When B is zero, the system oscillates in
the first mode with the requisite ratio of
amplitudes, and, so is the other way
around.
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tcosA2)t(q 11
tcosA2)t(q 22
Now,
)qq(52
1x 211
212 q)15(q)15(54
1x
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Prob. 5.9 The carbon
dioxide molecule can be
likened to a central
mass connected to two
other identical masses,
by identical springs of spring constant k.
a) Set up the equations of motion, find thenormal frequencies and ratios of the
amplitudes in the normal modes
2m
1m1m
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Equations of Motion :
)xx(k dt
xdm 122
1
2
1
)xx(k )xx(k dt
xdm 231222
2
2
)xx(k
dt
xdm 232
3
2
1
1x2x 3x
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Or,0)xx(
m
k
dt
xd21
1
2
1
2
0)xx2x(
m
k
dt
xd321
2
2
2
2
0)xx(m
k
dt
xd23
1
2
3
2
Substituting :
tcosAx;tcosAx;tcosAx 332211
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0Am
k A
m
k 2
1
1
2
1
0Am
k A
m
k 2A
m
k 3
2
2
2
2
1
2
0Am
k A
m
k 3
2
1
2
1
Setting the determinant of the coefficient
matrix to zero, the normal frequencies are
obtained as :
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21
3
1
21m
2
m
1k ;
m
k ;0
From the equations for the coefficients :
1
2
11
321m
k :
m
k :
m
k A:A:A
First mode
1:1:1A:A:A 321
No oscillations, rigid shift of the molecule
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Second Mode
1:0:1A:A:A 321
AAA
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Third Mode
1:m
m2:1A:A:A
2
1321
1:7.2:1
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Coupled and Driven Oscillators
tcos
m
F)xx(x
dt
dx
dt
xd 012
2
s2
2
02
2
2
2
0)xx(xdtdx
dtxd 12
2s1
20
121
2
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Adding and subtracting the equations
tcosm
Fq
dt
dq
dt
qd 01
2
01
2
1
2
tcosmFq
dtdq
dtqd 0
222
22
2
Steady-state solutions of the above
equations are :
)t(cos)(A)t(q 111 )t(cos)(A)t(q 222
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)t(cos)(A)tcos()(A2
1)t(x 22111
)t(cos)(A)tcos()(A2
1)t(x 22112
)(A1 )(A2
0
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When the driving frequency matches one of
the two frequencies , the
displacements of the two pendulums
become large.
&0
There are, thus, two resonant frequencies,corresponding to the two normal frequencies
at which the system can oscillate
At the lower resonance frequency, the
two pendulums are in phase. At the
higher one, they are totally out of phase.
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Prob. 5.12 Two identical masses are
connected to three identical springs on a
frictionless surface as shown
The free end is driven with a displacement :
tcosXX 0
Find and draw the graphs of the displacements
of the two masses.
X 1x 2x
mm kkk
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X 1x 2x
mm kkk
Equations of Motion :
)xx(k )Xx(k dt
xdm
1212
1
2
2122
2
2
xk )xx(k dt
xdm
Or, tcosXxx2dt
xd0
2
02
2
01
2
02
1
2
0x2xdt
xd
2
2
01
2
02
2
2
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Addition and subtraction of the two equations
gives us :
tcosm
Fq
dt
qd 01
2
02
1
2
tcosm
Fq3
dt
qd 02
2
02
2
2
Where, 00212211 Xk F,xxq,xxq
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Steady-state solution :
tcos)](A)(A[2
1)t(x 211
tcos)](A)(A[2
1)t(x 212
where,
22
0
0
2
0222
0
0
2
01
3
X)(A;
X)(A
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tcos)3()(
2X)t(x
22
0
22
0
22
00
2
01
tcos
)3()(
1X)t(x
22
0
22
0
0
4
02
At a driving frequency , the mass
nearer the end being driven, becomesstationary
02
Indefinite Number of Coupled Oscillators
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Indefinite Number of Coupled Oscillators
N Light Beads Connected by Massless Rods
Equilibrium
General State of Vibration
m
T
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Equations of Motion
p P+1P-1
py1py
1py
1
2
)sinsin(Tdt
ydm 122
p
2
)]yy()yy[(T
1ppp1p
Or,0)yy2y(
dt
yd
1pp1p
2
02
p
2
TT
m
T2
0
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0)yy2y(dt
yd1pp1p
2
02
p
2
N.........,,2,1p
It is assumed that there are two mass points,
at and , which are
permanently fixed.
Normal Modes
Substitute : tcosAy pp 0AA)2(A 1p
2
0p
22
01p
2
0
0p 1Np
Boundary Conditions : 0AA 1N0
0)t(y)t(y 1N0
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2
0
22
0
p
1p1p 2
A
AA
The amplitudes must depend upon the
discrete index in such a manner that the
ratio on the left above is independent of p
and the boundary condition is satisfied.
pA
p
A clever guess :
0AA 1N0
)tstancons,&C(psinCAp
cos2A
AA
p
1p1p
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For the boundary conditions to be satisfied,
we must have :
0)1N(sin N........,,2,1n)1N(
n
Each of the above possible values for
leads to a normal mode frequency :
2sin2 0
)1N(2
nsin2 0n N........,,2,1n
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The amplitudes of the mass point in
the normal mode is :
1N
npsinCApn
thpthn
N,........,1n;N.......,,1p
px)1N(
xnsinC
pxL
xnsinC
L
nxsinC)x(A,where),p(AA nnpn
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The First Mode
)1N(2
sin2 01
L
xsinC)x(A1
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The Second Mode
Lx2sinC)x(A2
)1N(sin2 02
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Three Particles
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General Solution
N
1n
nnnp )t(cos1N
npsinC)t(y N.......,,1p
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Prob. 5.16 (Resonance in N coupled
Oscillators)
Consider a system of N coupled oscillators
driven at a frequency . Forcing is done at
the extreme end such that
tcosh)t(y 1N
Find the resulting amplitudes of the particles.
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In the steady state, each particle will oscillate
with the driving frequency.
tcos)(Ay pp
Here, is the driving frequency, andhence, given
Substituting the above into the equations of
motion, we get :
2
0
22
0
p
1p1p 2
A
AA
Boundary Conditions : hA;0A 1N0
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Solution : psinCAp
Second boundary condition requires :
h)1N(sinC
)1N(sin
hC
2
0
22
0
p
1p1p 2cos2
A
AA
)1N(sin
)psin(h)(Ap 2
0
22
0
2
2cos
, Where,
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The amplitudes blow up whenever
0)1N(sin
Or,1N
n
This happens when the driving frequency is
such that :
2
0
22
0
2
2
1N
ncos
Or,
)1N(2
nsin2 0
A li d f ill i b l h
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Amplitudes of oscillations become large when
the driving frequency is one of the normal
mode (natural) frequencies
N
1n
nnnp )t(cos1N
npsinC)t(y
Most general solution of the driven oscillators :
tcos)1N(sin
)psin(h
Transient Steady State