THIS LECTURE rom single single to oupled oscillators oupled oscillators
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THIS LECTURE
From singlesingle to coupled oscillatorscoupled oscillators
Coupled pendulaCoupled pendula
Coupled pendulaCoupled pendula
Propagation of energy in space
Two thin rods connected by a pivot “Chaotic” motion
Coupled pendulaCoupled pendula
Coupled oscillators
• Normal modesModes of vibration in which each oscillator vibrates with same frequency
• Coupled motion implies energy exchange
Two coupled oscillators
Newton equations
k’k k
x1 x2
)(
)(
12'
22
21'
11
xxkkxxm
xxkkxxm
)cos(2
)cos(2
)cos(2
)cos(2
222
111
2
222
111
1
tA
tA
x
tA
tA
x
mkk
mk
/)'2(
/
2
1
A1, A2, 1 and 2 are constants that depend on initial conditions
Two coupled oscillators
Newton equations
k’k k
x1 x2
)(
)(
12'
22
21'
11
xxkkxxm
xxkkxxm
)cos(2
)cos(2
)cos(2
)cos(2
222
111
2
222
111
1
tA
tA
x
tA
tA
x
mkk
mk
/)'2(
/
2
1
A1, A2, 1 and 2 are constants that depend on initial conditions
Two coupled oscillators
Newton equations
k’k k
x1 x2
)(
)(
12'
22
21'
11
xxkkxxm
xxkkxxm
)cos(2
)cos(2
)cos(2
)cos(2
222
111
2
222
111
1
tA
tA
x
tA
tA
x
mkk
mk
/)'2(
/
2
1
A1, A2, 1 and 2 are constants that depend on initial conditions
NORMAL MODESNORMAL MODESEach mass vibrates with the same frequency
m
k1
NORMAL MODE 1
Masses are in phase and vibratewith frequency
m
kk '22
Masses are in antiphase and vibratewith frequency
NORMAL MODE 2
)cos(2 22
221 t
Axx)cos(
2 111
21 tA
xx
ProblemAt t =0, one mass is moved a distance 2a, but the second mass is held fixed. Find the time-dependence of x1 and x2.
x1
x2
t
t
2
2
121T
2
2
122T
T2
ttax2
cos2
cos2 12121
ttax2
sin2
sin2 12122
T1
Two coupled oscillators: general case
Newton equations
k2k1
k3
x1 x2
Mm
2121111 xaxax
2221212 xaxax
To find the normal modes, we assume that x1 and x2 are harmonic functions with same frequency .
)cos( 111 tAx )cos( 222 tAx
12
1 xx 22
2 xx
Normal modes
Two coupled oscillators: general case
Newton equations
k2k1
k3
x1 x2
Mm
To find the normal modes, we assume that x1 and x2 are harmonic functions with same frequency .
)cos( 111 tAx )cos( 222 tAx
12
1 xx 22
2 xx
Normal modes
Newton equations 12
2121111 xxaxax
22
2221212 xxaxax
Two coupled oscillators: general case
Newton equations
k2k1
k3
x1 x2
Mm
Newton equations 12
2121111 xxaxax
22
2221212 xxaxax
0))(( 21122
222
11222
12
21
211
aaaaa
a
a
a
This system of equations has solution if the determinant of its coefficients vanishes
This is a quadratic equation in the variable . It has two solutions, 1 and 2, corresponding to two normal modes.
1 and 2 corresponding to two normal modes
)cos( 11 tAx
)cos( 12 tBx
)cos( 21 tCx
)cos( 22 tDx
)cos()cos( 211 tCtAx
)cos()cos( 212 tDtBx
NORMAL MODE 1 NORMAL MODE 2
The most general solution is the superposition of the two modes.
Two coupled oscillators: general case
k2k1
k3
x1 x2
Mm
Crystals
Coupled oscillators
N coupled oscillators
z
y
x
Method to describe the motionMethod to describe the motion• Consider separate motion along x, y, z
• Write down Newton equations for the displacement of each mass along a given direction, for example x. Displacements are called x1, x2, x3…xN.
Normal modesDetermine the normal modes for each variable x1, x2…xN to obtain equations that describe harmonic motion NNNNNNN
NN
NN
xaxaxaxx
xaxaxaxx
xaxaxaxx
...
............................................................
...
...
22112
222212122
2
121211112
1
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