Coupled Oscillations
A single spring-mass system
2
2
xx
x Acos( t+ )
dm k
dt
One mass+ Two springs
kx
xkxkxm
2
)()(
m
k2
k3
k4
m
One mass+ Many springs
Coupled Oscillators
)(kkm
)(kkm
c
c
1222
2111
xxxx
xxxx
SHM term
Coupling term
m
k21 Let Natural freq. of each pendulum
122212
211211
xxk
xx
xxk
xx
m
m
c
c
21 2 1 1 2x x x x 0 Adding
Subtracting 21 2 1 1 2
2kx x x x 0c
m
221
121
xx
xx
q
q
21 1 1
22 1 2
0
20c
q q
kq q
m
Normal Co-ordinates
Normal modes
Normal Co-ordinates
Normal modes of vibration
Normal frequencies
Normal mode frequencies
m
k21
221
c21
22 2
2kkm
In-phase vibration
1 2
2
21 1 1
x x
0
0
q
q q
t
x2
x1
Out-of-phase vibration
1 2
1
2 C2 0 2
x x
0
2k0
q
q qm
t
x2
x1
1 1 2 10 1 1
2 1 2 20 2 2
x x cos
x x cos
q q t
q q t
Normal mode amplitudes : q10 and q20
Normal mode frequencies:
m
k21
mmck2k2
2
Initial conditions
oaqq 212010 & 2
0at 0 & 2 21 txax
Mass displacements
2
cos2
cos2
coscos2
1x
2112
21211
tta
ttaqq
2
sin2
sin 2
coscos2
1x
2112
21212
tta
ttaqq
Behavior with time for individual pendulum
x1(0) = 0 and x2(0) = 1
t
x1
x2
Tc
TB/2
Condition for complete energy exchange(Resonance)
12
4
nt x2=0
x1 0
12
)12(2
n
t x1=0
x2 0
2 21k
1
2
12
2/1
21
2
12
21
k
k
Stiff coupling
2 21k
2 1
Slow oscillation will be missing
11 1 0
00 0 1
( )
( )
xmx mg k x x
lx
mx mg k x xl
Equation of motion
SHM Couplingterm term
With assumption
Compare model Coupled Pendula and Experiment
Check two pendula are identical
Determine inphase and out of phase mode
Check whether frequency of inphase mode (w1) is less than thatof out of phase mode (w2)
•Coupled system
•Normal Co-ordinates
•Normal modes of vibration
•Normal frequencies
1. THE PHYSICS OF VIBRATIONS AND WAVES
AUTHOR: H.J. PAIN
IIT KGP Central Library
Class no. 530.124 PAI/P