MULTI DEGREES OF FREEDOM SYSTEMS

Vibration of Structures Prof. Dr. Tuncer TOPRAK

MULTI DEGREES OF FREEDOM SYSTEMS

Vibration of Structures Prof. Dr. Tuncer TOPRAK

MULTI DEGREES OF FREEDOM SYSTEMS

Vibration of Structures Prof. Dr. Tuncer TOPRAK

MULTI DEGREES OF FREEDOM SYSTEMS

Newton’s Second Law

Vibration of Structures Prof. Dr. Tuncer TOPRAK

MULTI DEGREES OF FREEDOM SYSTEMS

)()()(

)()()(

212212222

12111111

tFkxxkkcxxccxm

tFkxxkkxccxm

2

1

2

1

2

1

2

1

2

1

2

1

2

1

)(

..)(

)(

..)(0

0 F

F

x

x

kk

k

k

kk

x

x

cc

c

c

cc

x

x

m

m

)(tFxKxCxM x : Deplesman vektörüx’ : Velocity Vectorx’’ : Acceleration VectorM : Mass MatrixC : Damping MatrixK : Rigidity Matrix

)()()(

)()()(

21222122222

12111211111

tFxxcxcxxkxkxm

tFxxcxcxxkxkxm

Vibration of Structures Prof. Dr. Tuncer TOPRAK

MULTI DEGREES OF FREEDOM SYSTEMS

Example : Write the differential equations of the system below and find the natural frequencies, plot the displacement of each mass in time domain.

0)(

0)(

12222

21111

kxxkkxm

kxxkkxm

st

st

eBx

eBx

22

11

.).sin(

.).sin(

22

11

tAx

tAx

0)(

0)(

222

21

2112

1

AmkkkA

kAAmkk

0)(21

21212

2

2

1

14

mm

kkkkkk

m

kk

m

kk

)sin()sin()(

)sin()sin()(

222211212

221211111

tAtAtx

tAtAtx

Vibration of Structures Prof. Dr. Tuncer TOPRAK

MULTI DEGREES OF FREEDOM SYSTEMS

kkk

mmm

21

21For a special case and x(0)=1 ve v(0)=0

snradm

k

snradm

k

/3

.

/.

2

1

2

11211 AA

2

12221 AA

0. 2.1

tm

kt

m

kx

tm

kt

m

kx

3cos2

1cos2

1

3cos2

1cos2

1

2

1