random vibration of mdof systems-2nptel.ac.in/courses/105108080/module4/lecture14.pdf · 3 mdof...
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![Page 1: Random vibration of MDOF systems-2nptel.ac.in/courses/105108080/module4/Lecture14.pdf · 3 MDOF system with -th dof driven by an unit harmonic forces exp t 00 1 0 0 MXCX KX F it F](https://reader030.vdocuments.net/reader030/viewer/2022020215/5b8801ed7f8b9a46538ce95f/html5/thumbnails/1.jpg)
111
Stochastic Structural Dynamics
Lecture-14
Dr C S ManoharDepartment of Civil Engineering
Professor of Structural EngineeringIndian Institute of ScienceBangalore 560 012 India
Random vibration of MDOF systems - 2
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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
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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
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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
![Page 5: Random vibration of MDOF systems-2nptel.ac.in/courses/105108080/module4/Lecture14.pdf · 3 MDOF system with -th dof driven by an unit harmonic forces exp t 00 1 0 0 MXCX KX F it F](https://reader030.vdocuments.net/reader030/viewer/2022020215/5b8801ed7f8b9a46538ce95f/html5/thumbnails/5.jpg)
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
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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
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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
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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)
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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
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10
6 24.5 10 Nm for all columns=0.03 for all modes
EI
Example
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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
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12
4
1.1407 0.3244 0.06561 10 0.3244 0.9549 0.3419 Ns/m
0.0656 0.3419 0.5846C
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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)
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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)
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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)
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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
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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
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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
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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
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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
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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
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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)
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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)
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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
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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
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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
![Page 27: Random vibration of MDOF systems-2nptel.ac.in/courses/105108080/module4/Lecture14.pdf · 3 MDOF system with -th dof driven by an unit harmonic forces exp t 00 1 0 0 MXCX KX F it F](https://reader030.vdocuments.net/reader030/viewer/2022020215/5b8801ed7f8b9a46538ce95f/html5/thumbnails/27.jpg)
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
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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
![Page 29: Random vibration of MDOF systems-2nptel.ac.in/courses/105108080/module4/Lecture14.pdf · 3 MDOF system with -th dof driven by an unit harmonic forces exp t 00 1 0 0 MXCX KX F it F](https://reader030.vdocuments.net/reader030/viewer/2022020215/5b8801ed7f8b9a46538ce95f/html5/thumbnails/29.jpg)
29
MDOF systems under random support motionsCase of uniform support motions
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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
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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
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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
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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
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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
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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
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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
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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
![Page 38: Random vibration of MDOF systems-2nptel.ac.in/courses/105108080/module4/Lecture14.pdf · 3 MDOF system with -th dof driven by an unit harmonic forces exp t 00 1 0 0 MXCX KX F it F](https://reader030.vdocuments.net/reader030/viewer/2022020215/5b8801ed7f8b9a46538ce95f/html5/thumbnails/38.jpg)
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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
![Page 39: Random vibration of MDOF systems-2nptel.ac.in/courses/105108080/module4/Lecture14.pdf · 3 MDOF system with -th dof driven by an unit harmonic forces exp t 00 1 0 0 MXCX KX F it F](https://reader030.vdocuments.net/reader030/viewer/2022020215/5b8801ed7f8b9a46538ce95f/html5/thumbnails/39.jpg)
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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
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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
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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
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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)
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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)
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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)
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Structures under differential support motions
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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