yeong-jong moon*: graduate student, kaist, korea kang-min choi: graduate student, kaist, korea...
TRANSCRIPT
Yeong-Jong Moon*: Graduate Student, KAIST, Korea
Kang-Min Choi: Graduate Student, KAIST, Korea
Hyun-Woo Lim: Graduate Student, KAIST, KoreaJong-Heon Lee: Professor, Kyungil University, Korea
In-Won Lee: Professor, KAIST, Korea
Modified Modal Methods for Calculating Eigenpair Sensitivity of Asymmetric Damped
Systems
Modified Modal Methods for Calculating Eigenpair Sensitivity of Asymmetric Damped
Systems
EASEC-9, Bali, IndonesiaEASEC-9, Bali, Indonesia
16-18, December, 200316-18, December, 2003
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 2 2
Contents
Introduction
Previous Studies
Proposed Methods
Numerical Example
Conclusions
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 3 3
• Recently Adhikari and M. I. Friswell proposed a modal method for asymmetric damped systems.
• Many real systems have asymmetric mass, damping
and stiffness matrices.
- moving vehicles on roads
- ship motion in sea water
- offshore structures
Introduction
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 4 4
jjj yzs , ,
,K ,C ,M
K, C, M,
,,,
,,, , , jjj yzs
Given:
Find:
• Sensitivity Analysis
:
,,,
jj
jj
jj
yy
zz
sswhere
Design parameter
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 5 5
- Propose the modal method for sensitivity technique of
symmetric system
- The accuracy is dependent on the number of modes used
in calculation
• K. B. Lim and J. L. Junkins, “Re-examination of Eigenvector
Derivatives”, Journal of Guidance, 10, 581-587, 1987.
Previous Studies
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 6 6
- Modified modal method for symmetric system
- This method achieved highly accurate results using
only a few lower modes.
• Q. H. Zeng, “Highly Accurate Modal Method for Calculating
Eigenvector Derivative in Viscous Damping Systems”, AIAA
Journal, 33(4), 746-751, 1994.
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 7 7
- Propose the modal method for sensitivity technique of
asymmetric system
- The accuracy is dependent on the number of modes used
in calculation
- The truncation error may become significant
• S. Adhikari and M. I. Friswell, “Eigenderivative Analysis of
Asymmetric Non-Conservative Systems”, International Journal
for Numerical Methods in Engineering, 51, 709-733, 2001.
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 8 8
– Expand and as complex linear combinations of
and
N
kkjkj zaz
2
1,
N
kkjkj yby
2
1, (2
)
(1)
,jz ,jy
jz jy
• Modal Method for Asymmetric System
where
,jz
: the j-th right eigenvector: the j-th left eigenvector: the derivatives of j-th right eigenvector: the derivatives of j-th left eigenvector,jy
jz
jy
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 9 9
jjjjjjj
Tjj
N
jkk kjk
Tkk
kjk
Tkk
j
zafsss
yz
sss
yz
sss
yzz
)(2
)(
)(2
)(
)(2
**
*
,1**
*
,
jjjjjjj
Tjj
N
jkk kjk
Tkk
kjk
Tkk
j
ybgsss
zy
sss
zy
sss
zyy
)(2
)(
)(2
)(
)(2
**
*
,1**
*
,
(3)
(4)
- The derivatives of right eigenvectors
- The derivatives of left eigenvectors
• From this idea, the eigenvector derivatives can be obtained
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 10 10
• Objective
- Develop the effective sensitivity techniques for
asymmetric damped systems
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 11 11
Proposed Methods
1. Modal Acceleration Method
2. Multiple Modal Acceleration Method
3. Multiple modal Acceleration Method
with Shifted Poles
1. Modal Acceleration Method
2. Multiple Modal Acceleration Method
3. Multiple modal Acceleration Method
with Shifted Poles
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 12 12
0)( zBsA
fzBsAAszBsA ,,,,)(
• Differentiate the Eq. (5) with a design parameter
0)( BsAyT
(5)
(6)
(7)
1. Modal Acceleration Method (MA)
• The general equation of motion for asymmetric systems
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 13 13
00, ds ddz
fBd s1
0
fYs
s
sssZ
fBfBsAd
T
kkk
d
)(2
1
)( 110
where
(8)
(9)
(10)
• Separate the response into and,z 0sd 0dd
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 14 14
jjjjjjj
Tjj
j
j
N
jkk kjk
Tkk
k
j
kjk
Tkk
k
jTj
ybgsss
zy
s
s
sss
zy
s
s
sss
zy
s
sBy
)(2
)(
)(2
)(
)(2)(
**
*
*
,1**
*
*1
,
jjjjjjj
Tjj
j
j
N
jkk kjk
Tkk
k
j
kjk
Tkk
k
jj
zafsss
yz
s
s
sss
yz
s
s
sss
yz
s
sBz
)(2
)(
)(2
)(
)(2
**
*
*
,1**
*
*1
,
• Substituting the Eq. (9) and (10) into the Eq. (8)
• By the similar procedure, the left eigenvector derivatives can be obtained
(11)
(12)
k
j
s
s
*k
j
s
s
k
j
s
s
*k
j
s
s
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 15 15
2. Multiple Modal Acceleration Method (MMA)
11, ds ddz
fsABIBd s ][ 111
fYs
s
sssZ
dfBsAdzd
T
kkk
ssd
2
11
1,1
)(2
1
)(
where
(13)
(14)
(15)
• Separate the response into and,z1sd 1dd
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 16 16
• Therefore the right eigenvector derivatives are given as
jjjjjjj
Tjj
j
j
kjk
Tkk
k
j
N
jkk kjk
Tkk
k
jjj
zafsss
yz
s
s
sss
yz
s
s
sss
yz
s
sABsIBz
)(2
)(
)(2
)(
)(2)(
**
*2
***
*2
*
,1
2
11,
• By the similar procedure,
jjjjjjj
Tjj
j
j
kjk
Tkk
k
j
N
jkk kjk
Tkk
k
jTTj
Tj
ybgsss
zy
s
s
sss
zy
s
s
sss
zy
s
sBAsIBy
)(2
)(
)(2
)(
)(2)(
**
*2
***
*2
*
,1
2
,
(16)
(17)
2
k
j
s
s
2
*
k
j
s
s
2
k
j
s
s
2
*
k
j
s
s
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 17 17
jjjjjjj
Tjj
M
j
j
kjk
Tkk
M
k
j
N
jkk kjk
Tkk
M
k
jM
m
mjj
zafsss
yz
s
s
sss
yz
s
s
sss
yz
s
sABsBz
)(2
)(
)(2
)(
)(2)(
**
*
***
*
*
,1
1
0
11,
• Based on the similar procedure, we can obtain the higher order equations
jjjjjjj
Tjj
M
j
j
kjk
Tkk
M
k
j
N
jkk kjk
Tkk
M
k
jM
m
mTTj
Tj
ybgsss
zy
s
s
sss
zy
s
s
sss
zy
s
sBAsBy
)(2
)(
)(2
)(
)(2)(
**
*
***
*
*
,1
1
0,
(18)
(19)
M
k
j
s
s
M
k
j
s
s
*
M
k
j
s
s
M
k
j
s
s
*
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 18 18
1
0
11
111
11
])()([)(B
]))(([)(B
)])(([()(
M
m
mj
j
jj
ABAsA
AABsIA
AsABBAs
3. Multiple Modal Acceleration with Shifted-Poles (MMAS)
• For more high convergence rate, the term is expanded in Taylor’s series at the position
(20)
1)( BAs j
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 19 19
• Using the Eq. (20), we can obtain the following equation
(21)
jjjjjjj
Tjj
M
j
j
kjk
Tkk
M
k
j
N
jkk kjk
Tkk
M
k
j
M
m
mjj
zafsss
yz
s
s
sss
yz
s
s
sss
yz
s
s
ABAsABz
)(2
)(
)(2
)(
)(2
])()([)(
**
*
*
**
*
*
,1
1
0
11,
M
k
j
s
s
M
k
j
s
s
*
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 20 20
jjjjjjj
Tjj
M
j
j
kjk
Tkk
M
k
j
N
jkk kjk
Tkk
M
k
j
M
m
mTTj
Tj
ybgsss
zy
s
s
sss
zy
s
s
sss
zy
s
s
ABAsABy
)(2
)(
)(2
)(
)(2
])()([)(
**
*
*
**
*
*
,1
1
0,
(22)
• By the similar procedure
M
k
j
s
s
M
k
j
s
s
*
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 21 21
M
L
Y
X
Z, z
x
y
t
Figure 1. The whirling beam
L. Meirovitch and G. Ryland, “A Perturbation Technique for Gyroscopic Systems with Small Internal and External Damping,” Journal of Sound and Vibration, 100(3), 393-408, 1985.
Numerical Example
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 22 22
)()()()()()( tttt FuHKuGCuM
matrix ycirculator :H
matrix gyroscopic :G
0H
H0H,
K0
0KK
,0G
G0G,
C0
0CC,
M0
0MM
12
12
22
11
12
12
22
11
22
11
• Equation of motion
where
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 23 23
5 ,rad6.21
,5/9EI ,5/4EI ,4/1
,20/KK ,5 ,10M ,/10
1
2232231
2210
ps
NmLNmLNsmhc
NmLmLkgmkgm
yx
Design parameter : L
• Material Property
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 24 24
Mode Number Eigenvalues Derivatives
1-8.4987e-03
+2.3563e+00i1.3251e-03
+1.5799e+00i
2-2.7151e-03
+6.3523e+01i2.2533e-03
+8.5934e-01i
31.6771e-02
+1.0548e+01i3.3394e-03
+3.4034e-01i
8-5.8579e-02
+1.8650e+01i -3.7909e-03 -3.3918e-01i
9-4.7285e-02
+2.2774e+01i -2.2533e-03 -8.2215e-01i
10-3.6890e-02
+2.6214e+01i -1.2833e-03
-1.0644e+00i
• Eigenvalues and their derivatives of system
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 25 25
DOF Number Eigenvector Derivative
16.0874e-03
-6.2442e-06i 6.3118e-04
+6.3342e-06i
20.0000e+00
+0.0000e+00i 0.0000e+00
+0.0000e+00i
3-7.4415e-03
+6.7358e-06i -7.6005e-04 -7.1917e-06i
8+1.4785e-05 -1.4677e-02i
-1.2799e-05 +5.9162e-03i
90.0000e+00
+0.0000e+00i 0.0000e+00
+0.00005e+00i
108.3733e-05
-5.7187e-02i -3.7941e-05
+1.6957e-02i
• First right eigenvector and its derivative
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 26 26
DOF Number
Error (%)
MA MMA MMAS
1 0.831 0.202 0.072
2 0.000 0.000 0.000
3 38.258 1.506 0.478
4 0.000 0.000 0.000
5 4.631 0.121 0.035
6 0.080 0.053 0.012
7 0.000 0.000 0.000
8 1.679 0.588 0.118
9 0.000 0.000 0.000
10 0.520 0.157 0.030
• Errors of modified modal methods using six modes (%)
• MA : Modal Acceleration Method
• MMA : Multiple Modal
Acceleration Method (M=2)
• MMAS : Multiple Modal Accelerations
with Shifted Poles
(M=2, =eigenvalue –1)
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 27 27
DOF Number
Error (%)
6 modes 4 modes 2 modes
1 0.072 3.406 2.090
2 0.000 0.000 0.000
3 0.478 0.454 3.140
4 0.000 0.000 0.000
5 0.035 0.035 0.052
6 0.012 0.626 0.383
7 0.000 0.000 0.000
8 0.118 0.114 0.542
9 0.000 0.000 0.000
10 0.030 0.030 0.038
• Errors of MMAS method using 2, 4 and 6 lower modes (%)
(M=2, =eigenvalue –1)
Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea 28 28
• The modified modal methods for the eigenpair derivatives of asymmetric damped systems is derived
• In the proposed methods, the eigenvector derivatives of
asymmetric systems can be calculated by using only a few
lower modes
• Multiple modal acceleration method with shifted poles
is the most efficient technique of proposed methods
Conclusions