probabilistic analysis of dynamic systems with …...49th aiaa sdm conference, schaumburg, il, april...
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49th AIAA SDM Conference, Schaumburg, IL, April 2008
PROBABILISTIC ANALYSIS OF DYNAMIC SYSTEMS WITH COMPLEX-VALUED
EIGENSOLUTIONS
Sharif RahmanThe Uni ersit of Iowa
EIGENSOLUTIONS
The University of IowaIowa City, IA 52245
Work supported by U.S. National Science Foundation (CMMI-0653279)
OUTLINE
Introduction Introduction Dimensional Decomposition Method Examplesp Conclusions & Future Work
INTRODUCTION
A General Random Eigenvalue Problem (XN)
random eigenvector L or L: ( , ) ( , )N N X
1( ); ( ), , ( ) ( )Kf X A X A X X Φ 0
random eigenvector L or L
random matrices LLrandom eigenvalue or
Example Random Eigenvalue Problem Problem/Application1 Linear; undamped or 1 Linear; undamped or
proportionally damped systems
2 Quadratic; non-proportionally damped systems; singularity problems
( ) ( ) ( ) ( ) X M X K X XΦ 0
2 ( ) ( ) ( ) ( ) ( ) ( ) X M X X C X K X XΦ 0p
3 Palindromic; acoustic emissions in high speed trains (M0 = M1
T)
4 Polynomial; optimal control problems
1 0 1( ) ( ) ( ) ( ) ( ) ( )TM M X M X X X X XΦ 0
( ) ( ) ( )kkA
X X XΦ 0
5 Rational; plate vibration (m = 1) & fluid-solid structures (m = 2); vibration of viscoelastic materials
k ( ) ( )( ) ( ) ( ) ( )
( )
mk
k ka
X C XX M X K X XX
Φ 0
INTRODUCTION
Random Matrix Theory Pioneering works by Wishart
(1928) Wigner Mehta and Dyson
Approximate Methods Dominated by perturbation
methods(1928), Wigner, Mehta, and Dyson Analytical solutions for classical
ensembles (GOE, GUE, GSE) and others
methods Other methods
Iteration method (Boyce) Crossing theory (Grigoriu)
Asymptotic result yields statistical solutions dependent only on the global symmetry property of
d t i
C oss g t eo y (G go u) Reduced basis (Nair & Keane) Asymptotic method (Adhikari) Polynomial chaos (Ghanem)
random matrices Impossible or highly non-trivial to
apply for non-asymptotic or general random matrices
Dim. Decomposition (Rahman) Mostly used for real eigensolutions.
Complex-valued eigensolutions not studied extensivelygeneral random matrices studied extensively
Obj ti D l di i l d iti th d Objective: Develop dimensional decomposition method for solving complex-valued random eigenvalue problems
DIMENSIONAL DECOMPOSITION
D iti f Q d ti Ei l P bl Decomposition for Quadratic Eigenvalue Problem
( )m x2 ( ) ( ) ( ) ( ) ( ) ( ) x M x x C x K x x 0
NONLINEARSYSTEMInput Nx ( ) ( ) 1 ( )
Output ( ) ( ) 1 ( )
R I
R I
x x x
x x x
( ) ( ) ( ) ( ) ( ) ( ) x M x x C x K x x 0
( ) ( ) ( )R I
Univariate 1 2 1 2
1 2
1
1
1,1
,,0 ,, , 1, 1
( ) ()) , ,( ,( )S
S s
N
m i i i ii ii i
N
m i i i i
N
m i ii
mi ii i
m x x xx x
x
Univariate (individual
effects)
1 2
,2
,
,1
1ˆ ( )
ˆ ( )
ˆ ( )m
S
m
m
S
i i i i
x
x
x
Bivariate (2D cooperative effects)
S variate (SD cooperative effects)
Conjecture: Component functions arising in proposed
S-variate (SD cooperative effects)
decomposition will exhibit insignificant S-variate effects cooperatively when S N.
DIMENSIONAL DECOMPOSITION
L V i t A i ti Lower-Variate Approximations
N
Univariate Approximation reference point
, ,0
,1 ,1 1 1 1 11 ( )
ˆ ˆ( ) ( , , ) ( , , , , , , ) ( 1) ( )m i i m
N
m m N m i i i N mi x
x x c c x c c N
x c
Bivariate Approximation, 1 2 1 2
1 1 1 2 2 2
1 2
( , )
,2 ,2 1 1 1 1 1 1, 1
ˆ ˆ( ) ( , , ) ( , , , , , , , , , , )m i i i ix x
N
m m N m i i i i i i Ni ii i
x x c c x c c x c c
x
Bivariate Approximation
1 2
,
1 1 11 ( )
( 1)( 2)( 2) ( , , , , , , ) ( )2
m i i
i i
N
m i i i N mi x
N NN c c x c c
c
,0m
( ), 1 1 1
1
( ) ( ) ( , , , , , , )n
jm i i j i m i i i N
j
x x c c x c c
,0m
Lagrange h
1 2
1 2 1 2 1 1 2 2 1 1 1 2 2 2
2 1
1
( ) ( ), 1 1 1 1 1
1 1
( , ) ( ) ( ) ( , , , , , , , , , , )
j
n nj j
m i i i i j i j i m i i i i i i Nj j
x x x x c c x c c x c c
shape
functions
DIMENSIONAL DECOMPOSITION
E li it F Explicit Forms
N
Univariate Approximation
(1 1,1
)
11
1
( , , ,ˆ ( ) ( ) (, , , ) ( )1)N n
m j ii j
jm i i i N mc c x c NcX
X c
Bivariate Approximation
1 2
1 1 1 1 21 2 2
1 22
2
2
2
11
( ) ( )1 1 1 1 1,2
, 1 1 1( , , , , , , , , ,ˆ ( ) , )( ) ( )
N n n
m j i j ii i j ji i
j jm i i i i i i Nc c x c c x c cX X
X
Bivariate Approximation
( )1 1
1 11( , ( 1)( 2) ( 2) ( , , ,) , (
2, ) )
N n
j ii
jm i i i N
jmc c x c c N NN X
c
S variate Approximation
1 1,0 1 1 1
1ˆ ( ) ( 1) ( ) ( )S i S i
S N n ni
m S j k j ki k k j j
N SX X
ii
X
S-variate Approximation
1
1
1
1 1
11
( )( )1 1 1 1 1
0 , , 1 1 1
( , , , , , , , , , ) , S i
S i S
S i S i
i i
S i
S
jjm
i k k j jk
k
k
k k k k k Nc c x c c x c
i
c
DIMENSIONAL DECOMPOSITION
C t ti l Eff t (C l l ti C ffi i t ) Computational Effort (Calculating Coefficients)2 ( ) ( ) ( ) ( ) ( ) ( ) X M X X C X K X X 0
12 ( ) ( )det ( ) ( ) ( ) 0 c ccM c C K c
2 ( )1 1 1
( )1 1 1( , , , , , , )det ( , , , , , , )j
i i i Nj
i i i Nc c x c c c c x c c M Char.Eq.
nN( )1 1 1
( )1 1 1
( )1 1 1
( , , , , , , ( , , , , , , )
( , , , , , , ) 0; 1, , ; 1, ,
) ji i i N
j
ji i i
i i N
N
i
c c x c c x c c
c c x c c i N j
c
n
c
C
K
( ) ( )2d ( )j j
q(FEA)
N(N-1)n2/2
1 2
1 1 1 2 2 2
1 2
1 1 1
1 2
1
2
2 2
2
1
2
1 2
( ) ( )21 1 1 1 1
( ) ( )
( ) ( )1
1
1 1 1 1
11 1 1
det
( , , , , , , , , , , )
( , , , , , , , , , , )
( , , , , , , , , , , )
j
j ji i i i i i N
j ji i
ji i i i i i N
i i i i N
c c x c c x c c
c c x c
c c x c c
c x
x c
c c
c
M
1 ( , ,c cC 1 2
1 1 1 2 2 2
1 2
1 1 1 2 2 2
( ) ( )1 1 1 1
( ) ( )1 1 1 1 1 1 2 1 2
, , , , , , , , )
( , , , , , , , , , , ) 0; , 1, , ; , 1, ,
j ji i i i i i N
j ji i i i i i N
x c c x c c
c c x c c x c c i i N j j n
K
Univariate: nN + 1 (linear)Bivariate: N(N-1)n2/2 + nN + 1 (quadratic)
EXAMPLES
E l 1 3DOF S i M D Example 1: 3DOF Spring-Mass-Damper
y1 y2 y3 1( ) MM X X
MK
C
KM M
K
2
3
( )( )
1 kg
C
K
M
C XK X
XX
KK
KK0.3 N-s/m1 N/m
C
K
( ) 0 0( ) 0 ( ) 0
0 0 ( )
MM
M
XM X X
X
0 0 0( ) 0 ( ) 0
0 0 0C
C X X2 ( ) ( ) 0
( ) ( ) 2 ( ) ( )0 ( ) 2 ( )
K KK K K
K K
X XK X X X X
X X0 0 ( )M X 0 0 0 0 ( ) 2 ( )K K X X
( ) ( ) ( ){ ( )} { ( ) 1 ( )}; 1,2,3i i iR I i X X X 6 eigenvalues
31 2 3
2
{ , , } is independent lognormal vector
( , ; 0.25 or 0.5)
TX X X
v v
X X
X
X I
1
EXAMPLES
Variances of Eigenvalues
( ) ( ) ( ){ ( )} { ( ) 1 ( )}; 1,2,3i i iR I i X X X
v = 0.25 v = 0.5
Univariate
BivariateMonte Carlo
Univariate
Bivariate
Monte Carlo
(1) 2R(1), s-2 0.00078 0.00087 0.00087 0.00382 0.00596 0.00606
R(2), s-2 0 0 0 0 0 0
R(3), s-2 0.0007 0.00076 0.00076 0.00317 0.004 0.00422
I(1), s-2 0.01813 0.01859 0.01859 0.07097 0.07746 0.07795
I(2), s-2 0.06201 0.06361 0.06361 0.24293 0.26576 0.26655
I(3), s-2 0.10479 0.10741 0.10742 0.40935 0.44593 0.44669
10,000 samples; n = 5
Max Error by Univariate: 10% (v = 0.25); 37% (v = 0.5)
p
Max Error by Bivariate: 0% (v = 0.25); 5% (v = 0.5)
EXAMPLES Example 2: Flexural Vibration of Beam Example 2: Flexural Vibration of Beam
y7
MlRandom InputPoint masses ( )M
l
y6
y5
k6
k
M
Point masses ( )Damping coefficient at bottom ( )Rotational stiffness at bottom ( )
MC
K
S i ll i iff ( )di i d bkl
l
y4
k5
k4
M
M
Spatially varying stiffness ( )discretized by 6 rotational stiffnesses ( ); 1, ,6i i
k xk k x i
l
l
y3
y2
k3M
91 9{ , , } is independent
and lognormally distributed
TX X X
l
x
y1
k2
k1
M7 7
( ) ( ) ( )
( ), ( ), ( )
{ ( )} { ( ) 1 ( )}; 1, ,7i i iR I i
M X C X K X
X X X
l KC
M
14 eigenvalues
EXAMPLES Marginal PDFs of Eigenvalues (Real) Marginal PDFs of Eigenvalues (Real)
56789
R) 0. 3
0. 4
0. 5
R)
0. 06
0. 08
)
-0. 7 -0. 6 -0. 5 -0. 4 -0. 3 -0. 2 -0. 1 0. 0012345
f 1(R
-14 -12 -10 -8 -6 -4 -2 00. 0
0. 1
0. 2f 2(R
-70 -60 -50 -40 -30 -20 -10 00. 00
0. 02
0. 04
f 3(R)
R RR
0. 02
0. 03
(R) 0 . 015
0. 020
0. 025(
R) 0 . 015
0. 020
0. 025
(R)
-200 -150 -100 -50 0
R
0. 00
0. 01
f 4
-200 -150 -100 -50 0
R
0. 000
0. 005
0. 010f 5
-250 -200 -150 -100 -50 0
R
0. 000
0. 005
0. 010f 6(
R R R
0. 012
0. 018
f 7(R)
UnivariateMonte Carlo Univariate: 37 analyses
Bivariate: 613 analyses
-300 -250 -200 -150 -100 -50 0
R
0. 000
0. 006 Bivariatey
Monte Carlo: 105 analyses
EXAMPLES Marginal PDFs of Eigenvalues (Imaginary) Marginal PDFs of Eigenvalues (Imaginary)
0. 3
0. 4
0. 5
)
0. 06
0. 08
) 0. 015
0. 020
0. 025
I)
4 6 8 10 12 140. 0
0. 1
0. 2f 1(I)
30 40 50 60 70 80 90 1000. 00
0. 02
0. 04
f 2(I)
80 120 160 200 240 2800. 000
0. 005
0. 010
f 3(I
I, rad/s I, rad/s I, rad/s
0. 006
0. 009
0. 012
(I)
0 . 004
0. 006
0. 008
(I) 0 . 002
0. 003
0. 004
(I)
200 300 400 500 600
I, rad/s
0. 000
0. 003
f 4(
400 550 700 850 1000
I, rad/s
0. 000
0. 002
f 5(
800 1100 1400 1700 2000
I, rad/s
0. 000
0. 001
f 6(UnivariateMonte Carlo
I, I, I,
0. 0005
0. 0010
f 7(I)
Univariate: 37 analysesBivariate: 613 analyses
Bivariate
2500 3400 4300 5200 6100 7000
I, rad/s
0. 0000
f yMonte Carlo: 105 analyses
EXAMPLES
E l 3 B k S l A l i Example 3: Brake Squeal Analysis
Random Input1
1111 2222 2
1122 2
1133 2233 2
125 GPa5.94 GPa
0.76 GPa0.98 GPa
cE XD D XD XD D X
1133 2233 2
3333 2
1212 2
1313 2323 2
2.27 GPa2.59 GPa
1.18 GPa207 GP
D XD XD D XE X
3
6 34
5
207 GPa
7.2 10 kg/mms
c
r
E X
Xf X
5{ } LNTX X X 1 5{ , , } LNX X X
Two-Step Analysis- Apply pressure to develop No damping
DOF 881 460pp y p p
contact betw. rotor & pads- Apply rot. vel. of 5 rad/s to
create steady-state motion
DOF = 881,460Unsymmetric K(X)
EXAMPLES Effects of Friction Effects of Friction
8000
10000
, Hz
mean input(fr = 0.5)
Results of First 55 Eigenvalues
closely spaced modes (fr = 0)
6000
8000
art (
freq
uenc
y),
r
2000
4000
Imag
inar
y pa
-200 -100 0 100 200Real part
0
merged modes (fr = 0.5)
EXAMPLES
CDF f I t bilit I d b U i i t (21 FEA) CDF of Instability Index by Univariate (21 FEA)
( ) ( )( ) 2Re ( ) Im ( )uN
i iu uU X X X
1i
10 0
[f ] 0 510 -1
< u]
[fr] = 0.5
10 -2
P[U
(X) < [fr] = 0.3
10 -4
10 -3[fr] = 0.1
-0.10 -0.08 -0.06 -0.04 -0.02 0.00
Instability index (u)
10 4
CONCLUSIONS & FUTURE WORK
A novel decomposition method was developed for solving complex-valued random eigenvalue problems Yi ld t ffi i t & t l ti Yields accurate, efficient, & convergent solutions Univariate & bivariate methods entail linear &
quadratic cost scalings w r t no of random variablesquadratic cost scalings w.r.t. no. of random variables
Neither the derivatives of eigensolutions nor the assumption of small input variability neededp p y
Non-intrusiveness permits easy coupling with external codes for solving industrial-scale problems
Future works involve developing/solving Computationally efficient bivariate method Computationally efficient bivariate method Discontinuous/non-smooth problems