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PROBABILISTIC ANALYSIS OF DYNAMIC SYSTEMS WITH COMPLEX-VALUED - - PowerPoint PPT Presentation
PROBABILISTIC ANALYSIS OF DYNAMIC SYSTEMS WITH COMPLEX-VALUED - - PowerPoint PPT Presentation
49 th AIAA SDM Conference, Schaumburg, IL, April 2008 PROBABILISTIC ANALYSIS OF DYNAMIC SYSTEMS WITH COMPLEX-VALUED EIGENSOLUTIONS EIGENSOLUTIONS Sharif Rahman The Uni ersit of Iowa The University of Iowa Iowa City, IA 52245 Work supported
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SLIDE 3
INTRODUCTION
A General Random Eigenvalue Problem (XN)
random eigenvector L or L
: ( , ) ( , )
N N
X
1
( ); ( ), , ( ) ( )
K
f X A X A X X Φ
random eigenvector L or L random matrices LL random eigenvalue or
Example Random Eigenvalue Problem Problem/Application 1 Linear; undamped or 1 Linear; undamped or proportionally damped systems 2 Quadratic; non-proportionally damped systems; singularity problems
( ) ( ) ( ) ( ) X M X K X X Φ
2(
) ( ) ( ) ( ) ( ) ( ) X M X X C X K X X Φ p 3 Palindromic; acoustic emissions in high speed trains (M0 = M1
T)
4 Polynomial; optimal control problems
1 1
( ) ( ) ( ) ( ) ( ) ( )
T
M M X M X X X X X Φ ( ) ( ) ( )
k k
A
X X X Φ 5 Rational; plate vibration (m = 1) & fluid-solid structures (m = 2); vibration of viscoelastic materials
k
( ) ( ) ( ) ( ) ( ) ( ) ( )
m k k k
a
X C X X M X K X X X Φ
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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
- thers
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 extensively general 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
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DIMENSIONAL DECOMPOSITION
D iti f Q d ti Ei l P bl Decomposition for Quadratic Eigenvalue Problem
( )
m
x
2( )
( ) ( ) ( ) ( ) ( ) x M x x C x K x x
NONLINEAR SYSTEM
Input
N
x ( ) ( ) 1 ( ) Output ( ) ( ) 1 ( )
R I R I
x x x x x x ( ) ( ) ( ) ( ) ( ) ( ) x M x x C x K x x ( ) ( ) ( )
R I
Univariate
1 2 1 2 1 2 1 1 1
, 1 , ,0 , , , 1 , 1
( ) ( ) ) , , ( , ( )
S S s
N m i i i i i i i i N m i i i i N m i i i m i i i i m
x x x x x
x
Univariate (individual effects)
1 2 ,2 , , 1 1
ˆ ( ) 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.
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DIMENSIONAL DECOMPOSITION
L V i t A i ti Lower-Variate Approximations
N
Univariate Approximation
reference point
, ,0
,1 ,1 1 1 1 1 1 ( )
ˆ ˆ ( ) ( , , ) ( , , , , , , ) ( 1) ( )
m i i m
N m m N m i i i N m i 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 i
x x N m m N m i i i i i i N i i i i
x x c c x c c x c c
x
Bivariate Approximation
1 2 ,
1 1 1 1 ( )
( 1)( 2) ( 2) ( , , , , , , ) ( ) 2
m i i
i i N m i i i N m i x
N N N c c x c c
c
,0 m
( ) , 1 1 1 1
( ) ( ) ( , , , , , , )
n j m i i j i m i i i N j
x x c c x c c
,0 m
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 n j j m i i i i j i j i m i i i i i i N j j
x x x x c c x c c x c c
shape functions
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DIMENSIONAL DECOMPOSITION
E li it F Explicit Forms
N
Univariate Approximation
( 1 1 ,1 ) 1 1 1
( , , , ˆ ( ) ( ) ( , , , ) ( ) 1)
N n m j i i j j m i i i N m
c c x c N c X
X c
Bivariate Approximation
1 2 1 1 1 1 2 1 2 2 1 2 2 2 2 2 1 1
( ) ( ) 1 1 1 1 1 ,2 , 1 1 1
( , , , , , , , , , ˆ ( ) , ) ( ) ( )
N n n m j i j i i i j j i i j j m i i i i i i N
c c x c c x c c X X
X
Bivariate Approximation
( ) 1 1 1 1 1
( , ( 1)( 2) ( 2) ( , , , ) , ( 2 , ) )
N n j i i j m i i i N j m
c c x c c N N N X
c
S variate Approximation
1 1
, 1 1 1
1 ˆ ( ) ( 1) ( ) ( )
S i S i
S N n n i m S j k j k i k k j j
N S X X i i
X
S-variate Approximation
1 1 1 1 1 1 1
( ) ( ) 1 1 1 1 1 , , 1 1 1
( , , , , , , , , , ) ,
S i S i S S i S i i i S i S
j j m i k k j j k k k k k k k k N
c c x c c x c i c
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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 1
2( )
( ) det ( ) ( ) ( ) c c c M c C K c
2 ( ) 1 1 1 ( ) 1 1 1
( , , , , , , ) det ( , , , , , , )
j i i i N j i i i N
c 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, , )
j i i i N j j i 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
( ) ( ) 2
d ( )
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
( ) ( ) 2 1 1 1 1 1 ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 1
det ( , , , , , , , , , , ) ( , , , , , , , , , , ) ( , , , , , , , , , , )
j j j i i i i i i N j j i i j i 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 c C
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 j i i i i i i N j j i 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)
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EXAMPLES
E l 1 3DOF S i M D Example 1: 3DOF Spring-Mass-Damper
y1 y2 y3
1
( )
M
M X X
M K C K M M K
2 3
( ) ( ) 1 kg
C K M
C X K X X X
K K K K
0.3 N-s/m 1 N/m
C K
( ) ( ) ( ) ( ) M M M X M X X X ( ) ( ) C C X X 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) 2 ( ) K K K K K K K X X K X X X X X X ( ) M X ( ) 2 ( ) K K X X
( ) ( ) ( )
{ ( )} { ( ) 1 ( )}; 1,2,3
i i i R I
i X X X 6 eigenvalues
3 1 2 3 2
{ , , } is independent lognormal vector ( , ; 0.25 or 0.5)
T
X X X v v
X X
X X I 1
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EXAMPLES
Variances of Eigenvalues
( ) ( ) ( )
{ ( )} { ( ) 1 ( )}; 1,2,3
i i i R I
i X X X
v = 0.25 v = 0.5 Univariate Bivariate Monte Carlo Univariate Bivariate Monte Carlo
(1) 2
R
(1), s-2
0.00078 0.00087 0.00087 0.00382 0.00596 0.00606 R
(2), s-2
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)
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EXAMPLES
Example 2: Flexural Vibration of Beam Example 2: Flexural Vibration of Beam
y7 M l
Random Input
Point masses ( ) M
l y6 y5 k6 k M
Point masses ( ) Damping coefficient at bottom ( ) Rotational stiffness at bottom ( ) M C K S i ll i iff ( )di i d b k
l l y4 k5 k4 M M
Spatially varying stiffness ( )discretized by 6 rotational stiffnesses ( ); 1, ,6
i i
k x k k x i
l l y3 y2 k3 M
9 1 9
{ , , } is independent and lognormally distributed
T
X X X
l x y1 k2 k1 M
7 7 ( ) ( ) ( )
( ), ( ), ( ) { ( )} { ( ) 1 ( )}; 1, ,7
i i i R I
i
M X C X K X X X X
l K C M
14 eigenvalues
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EXAMPLES
Marginal PDFs of Eigenvalues (Real) Marginal PDFs of Eigenvalues (Real)
5 6 7 8 9
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
0.0 1 2 3 4 5
f1(R
- 14
- 12
- 10
- 8
- 6
- 4
- 2
0.0 0.1 0.2
f2(R
- 70
- 60
- 50
- 40
- 30
- 20
- 10
0.00 0.02 0.04
f3(R)
R R R 0.02 0.03
(R)
0.015 0.020 0.025
(R)
0.015 0.020 0.025
(R)
- 200
- 150
- 100
- 50
R 0.00 0.01
f4
- 200
- 150
- 100
- 50
R 0.000 0.005 0.010
f5
- 250
- 200
- 150
- 100
- 50
R 0.000 0.005 0.010
f6(
R R R
0.012 0.018
f7(R)
Univariate Monte Carlo
Univariate: 37 analyses Bivariate: 613 analyses
- 300
- 250
- 200
- 150
- 100
- 50
R 0.000 0.006
Bivariate
y Monte Carlo: 105 analyses
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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 14 0.0 0.1 0.2
f1(I)
30 40 50 60 70 80 90 100 0.00 0.02 0.04
f2(I)
80 120 160 200 240 280 0.000 0.005 0.010
f3(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
f4(
400 550 700 850 1000 I, rad/s 0.000 0.002
f5(
800 1100 1400 1700 2000 I, rad/s 0.000 0.001
f6(
Univariate Monte Carlo
I, I, I,
0.0005 0.0010
f7(I)
Univariate: 37 analyses Bivariate: 613 analyses
Bivariate
2500 3400 4300 5200 6100 7000 I, rad/s 0.0000
f
y Monte Carlo: 105 analyses
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EXAMPLES
E l 3 B k S l A l i Example 3: Brake Squeal Analysis
Random Input
1 1111 2222 2 1122 2 1133 2233 2
125 GPa 5.94 GPa 0.76 GPa 0.98 GPa
c
E X D D X D X D D X
1133 2233 2 3333 2 1212 2 1313 2323 2
2.27 GPa 2.59 GPa 1.18 GPa 207 GP D X D X D D X E X
3 6 3 4 5
207 GPa 7.2 10 kg/mm
s c r
E X X f X
5
{ } LN
T
X X X
1 5
{ , , } LN X X X
Two-Step Analysis
- Apply pressure to develop
No damping DOF 881 460
pp y p p contact betw. rotor & pads
- Apply rot. vel. of 5 rad/s to
create steady-state motion
DOF = 881,460 Unsymmetric K(X)
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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 (frequency),
r
2000 4000
Imaginary pa
- 200
- 100
100 200
Real part
merged modes (fr = 0.5)
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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 ( )
u
N i i u u
U
X X X
1 i
10 0
[f ] 0 5
10 -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
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