111

Stochastic Structural Dynamics

Lecture-14

Dr C S ManoharDepartment of Civil Engineering

Professor of Structural EngineeringIndian Institute of ScienceBangalore 560 012 India

manohar@civil.iisc.ernet.in

Random vibration of MDOF systems - 2

2

Coupling and non-diagonal nature of structural matricesNatural coordinatesNormal modes & natural frequenciesOrthogonality of normal modesUncoupling of equations

Review of dynamics of mdof systems

of motionClassical damping modelsInput-output relations in frequency domain

3

MDOF system with -th dof driven by an unit harmonic forces

exp

0 0 1 0 0t

MX CX KX F i t

F

th entrys

response of the -th coordinate due to unit harmonic driving at -th coordinate.

rsX t rs

lim ?rstX t

4

0

0

20

20 0 0

20

20

exp

0 0 1 0 0

lim exp

exp

exp

exp exp exp exp

exp exp

t

t

MX CX KX F i t

F

X t X i t

X t X i i t

X t X i t

MX i t CX i i t KX i t F i t

M i C K X i t F i t

M i C K X F

5

20

0 0

20

20

20

20

exp exp

& is classical (Diagonal) with 2

t t

tnn n n

t t

t t t t

t

M i C K X F

X t X i t Z i t

M I KC C

M i C K Z F

M i C K Z F

M i C K Z F

I i Z F

Diagonal

6

20

1 10 2 2 2 2

0 2 2

0 01

2 21

2 2

Recall0 0 1 0 0 ( -th entry=1; rest=0)

2

lim exp exp

2

t

N Ntnk k kn k

k kn

n n n n n n

t

snn

n n n

N

r rn nt nN

rn sn

n n n n

I i Z F

F FZ

i i

F s

Zi

X t Z i t X t Z i t

i

2 21

exp

exp

2

rs rs

Nrn sn

rsn n n n

i t

X t H i t

Hi

7

2 21

2 21

122 2

1

exp2

2

is symmetric but not Hermitian

2

Nrn sn

rsn n n n

Nrn sn

rsn n n n

rs sr

rs sr

rs

Nrn sn

n n n n

X t i ti

Hi

X t X t

H H

H H

H

H M i C ki

Remarks

8

*

12

2 21

Conceptually simple Computationally difficult to implement

2

Computationally easier to implement Not al

N Nrn sn

n n n n

H M i C k

Hi

l modes need to be included

(Nor it is advisable to include all modes)

9

1x t 2x t 3x t

exp i t

1 11 11

2 12 12

3 13 13

exp

exp

exp

exp

rs rsX t H i t

x t X t H i t

x t X t H i t

x t X t H i t

rsH H

10

6 24.5 10 Nm for all columns=0.03 for all modes

EI

Example

11

1 1 1 1 2 1 2

2 2 2 2 1 3 2 3

3 3 3 3 2

1 16

2 2

3 3

6 2 6

6 6

( ) 0( ) ( ) 0( ) 0

5000 0 0 2 1 00 4000 0 4 10 1 2 1 00 0 3000 0 1 1

8 10 5000 4 10 04 10 8 10 400

m x k x k x xm x k x x k x xm x k x x

x xx xx x

2 6

6 6 2

6 4 7 2 9

0 4 10 00 4 10 4 10 3000

4933.3 5.867 10 1.066 10 014.86, 38.78, 56.64 rad/s

0.0058 0.0114 0.00610.0100 0.0014 0.01220.0120 0.0107 0.0087

12

4

1.1407 0.3244 0.06561 10 0.3244 0.9549 0.3419 Ns/m

0.0656 0.3419 0.5846C

13

0 10 20 30 40 50 60 70 8010

-8

10-7

10-6

10-5

frequency rad/s

abs(

H11

)

0 10 20 30 40 50 60 70 80-4

-3

-2

-1

0

frequency rad/s

angl

e(H

11)

14

0 10 20 30 40 50 60 70 8010

-10

10-5

frequency rad/s

abs(

H13

)

0 10 20 30 40 50 60 70 80-10

-5

0

frequency rad/s

angl

e(H

13)

15

0 10 20 30 40 50 60 70 8010

-8

10-7

10-6

10-5

frequency rad/s

abs(

H12

)

0 10 20 30 40 50 60 70 80-8

-6

-4

-2

0

frequency rad/s

angl

e(H

12)

16

MDOF system with -th dof driven by an unit impulse forces

0 0; 0 0

0 0 1 0 0t

MX CX KX F t

X X

F

th entrys

response of the -th coordinate due to unit impulse driving at -th coordinate.

rsX t rs

17

0 0 1 0 0

& is classical (Diagonal) with 2

t

t t

tnn n n

t t t t

t

MX CX KX F t

F

X t Z t

M I KC C

M Z t C Z t K Z t F t

M Z t C Z t K Z t F t

IZ Z Z F t

18

2

1

1

1

2

0 0; 0 0

exp sin

1 exp sin

t

N

n n n n n n jn j nsj

n n

snn ns n n n dn

dn

N

r rn nnN

rs rn sn n n dnn dn

IZ Z Z F t

z z z F t t

z z

z t h t t t

X Z

X t z t

h t t t

19

1

1 exp sin

Remarks

Matrix of impulse response functions

Not all modes need to be included in the summationIf an arbitrary load

N

r rs rn sn n n dnn dn

rs sr

rs

t

s

X t h t t t

h t h t

h t h t

h t h t

f

0

10

is applied at the -th dof (instead ofunit impulse excitation)

1 exp sin

t

rs rs s

t N

s rn sn n n dnn dn

s

X t h t f d

f t t d

20

1x t 2x t 3x t

t

1 11 11

2 12 12

3 13 13

( )

( )

( )

x t X t h t

x t X t h t

x t X t h t

rsh t h t

21

0 1 2 3 4 5 6 7 8 9 10-4

-3

-2

-1

0

1

2

3

4

5x 10

-6

time s

h11(

t)All modes participate in vibration

22

0 1 2 3 4 5 6 7 8 9 10-4

-3

-2

-1

0

1

2

3

4

5x 10

-6

time s

h12(

t)

23

0 1 2 3 4 5 6 7 8 9 10-6

-4

-2

0

2

4

6x 10

-6

time s

h13(

t)

24

Recall

exp

1 exp2

F f t i t d

f t F i t d

exp

1 exp2

ij ij

ij ij

H h t i t d

h t H i t d

ExerciseShow that

25

MDOF system under stationary vector random excitationResponse analysis in frequency domain

*

* *

( ); (0) 0; 0 0

1

( ) 0 ;

is a matrixWe are interested in characterizing the response in the steady state.

1lim

1lim

tFF

FF

T T

tXX T TT

t tT TT

MX CX KX F t X X

X t N

F t F t F t R

R N N

X H F

S X XT

H F F HT

H

* *

*

1lim t tT TT

tFF

F F HT

H S H

26

MDOF system under stationary vector random excitationResponse analysis in time domain

0

0

( ); (0) 0; 0 0

1

( ) 0 ;

is a matrix

0

tFF

FF

t

t

MX CX KX F t X X

X t N

F t F t F t R

R N N

X t h t F d

X t h t F d

27

1

2

2

1 2

1 2

1 1 1 1 10

2 2 2 2 10

2 2 2 2 20

1 2 1 1 2 2 2 2 1 20 0

1 2 1 1 1 2 2 2 1 20 0

1 2 1 1

, ,

,

t

t

ttt t

t ttt t

t tt

XX FF

XX FF

X t h t F d

X t h t F d

X t F h t d

X t X t h t F F h t d d

R t t h t R h t d d

R t t h t R

1 2

2 1 2 2 1 20 0

t tt

h t d d

28

12

1 1

10

1 1 1 10

System starts from rest.

2

N

k kn nn

n n n n n n n

N Nt

n ns s sn ss st N

n n sn ss

tN N N N

k kn n s kn sn nn s n s

MX CX KX F t

X t Z t X t Z t

Z Z Z p t

p t F t F t

Z t h t F d

X t h t F d h t

0

Based on this expression, we can evaluatemean, covariance and other moments of .

t

sF d

X t

29

MDOF systems under random support motionsCase of uniform support motions

30

u t u t

1m

2m

3m

1k

2k

3k

1z t

2z t

3z t

A building frame under random support motion

0

uu

u t

u t u t R

31

1 1 1 1 2 1 2 1 1 2 1 2

2 2 2 2 1 3 2 3 2 2 1 3 2 3

3 3 3 3 2 3 3 2

0

0

0

m z c z u c z z k z u k z z

m z c z z c z z k z z k z z

m z c z z k z z

1 1

2 2

3 3

1 1 1 1 2 1 2 1 1 2 1 2 1

2 2 2 2 1 3 2 3 2 2 1 3 2 3 2

3 3 3 3 2 3 3 2 3

x z ux z ux z u

m x c x c x x k x k x x m u

m x c x x c x x k x x k x x m u

m x c x x k x x m u

32

1 1 1 2 2 1 1 2 2 1

2 2 2 2 3 3 2 2 2 3 3 2

3 3 3 3 3 3 3 3

0 0 0 00 00 0 0 0

m x c c c x k k k xm x c c c c x k k k k x

m x c c x k k x

1

2

3

0 0 1 0 0 1

0 0 1

mm u

m

33

*

*

*

12

T T

t tFF T T uu

tXX FF

t tXX uu

MX CX KX F t

F t M u t

F M U

S M U U M M S M

S H S H

S H M S MH

H M i C K

1

1

1 1 1 1

1 1

34

1 2

1 2 1 2

1 2

0 0

0

1 2 1 1 1 2 1 1 1 20 0

0

,

0

, ,

tFF

t

t t

t

t ttt

XX uu

MX CX KX F t

F t M u t

F t M u t

R t t F t F t N N

M u t u t M

X t h t F d h t M u d

X t h t M u d

R t t h t M R M h t d d

1

1

1 1

1

1

1 1

35

12

2

2

[Modal participation factor]

2

N

k kn nn

n n n n n n n

tn

t

n n n n n n n

x t y t

x t y t

y y y p t

p t M u t u t

M

y y y u t

1

1

36

12

0

1 0

1 0

2

0

N

k kn nn

n n n n n n n

t

n n n

tN

k kn n nn

tN

k kn n nn

x t y t

x t y t

y y y u t

y t h t u d

x t h t u d

x t h t u d

37

1 2

1 2

1 0

1 2 1 21 1

1 1 2 2 1 2 1 21 1 0 0

1 1 2 2 1 2 1 21 1 0 0

1 1 2

,

tN

k kn n nn

N N

k k kn km n mn m

t tN N

kn km n m n mn m

t tN N

kn km n m n m uun m

kn km n m

x t h t u d

x t x t y t y t

h t h t u u d d

h t h t R d d

h t h t

1 2

2 1 2 1 21 1 0 0

t tN N

n m uun m

R d d

21 2 1 2 1 2

1 1 0 0k

t tN N

x kn km n m n m uun m

t h t h t R d d

38

1

12

2 2

2

2

N

k kn nn

N

kT kn nTn

n n n n n n n

n TnT

n n n

x t y t

x t y t

X Y

y y y u t

UY

i

39

2 21

2 2 2 21 1

2 2

2 22 21

2 2 2 21

2

2 2

2

2 2

k k

Nn T

kT knn n n n

N Nkn km n m

x x UUn m n n n m m m

Nkn n UU

nn n n

kn km n m

m n n n m m mn

UX

i

S Si i

S

i i

1

2

0

Contribution ignoring modal interactions + correctiondue to modal interactions

k k k

N N

UUn

m

x x x

S

S d

40

u t u t

1m

2m

3m

1k

2k

3k

1z t

2z t

3z t

A building frame under random support motion

4 2 2 2

22 2 2 2 2

4

4

g g gUU

g g g

S I

41

0 10 20 30 40 50 60 70 800

0.5

1

1.5

2

2.5

frequency rad/s

psd

of a

ccln

2

Input PSD(m/s/s) /(rad/s)

=1; =4 ; 0.53g gI

42

0 10 20 30 40 50 60 70 8010

-10

10-8

10-6

10-4

10-2

100

frequency rad/s

auto

-psd

x1

x2x3

2

Displacement auto-PSD(m / rad/s)

43

0 10 20 30 40 50 60 70 8010

-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

frequency rad/s

cros

s-ps

d (1

2)Amplitude of Cross PSD (12)

44

0 10 20 30 40 50 60 70 800

0.5

1

1.5

2

2.5

3

frequency rad/s

phas

e ra

d (1

2)

Phase of Cross PSD (12)

45

Structures under differential support motions

46

0

~ 1, ~ 1;

, , ~, , ~

, , ~

Pseudo-dynamic response

T T Tg g g

t t tg gg g gg g gg gg g g

T

g g g T g

g g g g

gg gg gg g g

g

M M C C K Ku u uM M C C K K p tu u u

u Nu p t N N N N

M C K N NM C K N N

M C K N N

K K

1

1

0

0

p

t pg gg gg

p pg g g g g

g

p t pg g gg g

uK K p tu

Ku K u u K K u t u t

K K

p t K u K u