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