1. Simplified Algebraic Method
Fifth European Conference of Structural Dynamics
EURODYN 2002
Munich, Germany
Sept , 2002
Simplified Algebraic Method
for Computing Eigenpair Sensitivities of Damped System
with Repeated Eigenvalues
Kang-Min Choi, Graduate Student, KAIST, Korea
Woon-Hak Kim, Professor, Hankyong National University, Korea
In-Won Lee, Professor, KAIST, Korea

2. CONTENTS  INTRODUCTION  PREVIOUS METHODS  PROPOSED METHOD
 NUMERICAL EXAMPLES
 CONCLUSIONS
Structural Dynamics & Vibration Control Lab., KAIST, Korea

3. INTRODUCTION Applications of sensitivity analysis are
determination of the sensitivity of dynamic response
optimization of natural frequencies and mode shapes
optimization of structures subject to natural frequencies
Typical structures have many repeated or nearly equal eigenvalues, due to structural symmetry.
The second- and higher order derivatives of eigenpairs are important to predict the eigenpairs, which relies on the matrix Taylor series expansion.
Structural Dynamics & Vibration Control Lab., KAIST, Korea

4. Problem Definition Eigenvalue problem of damped system (1)
Structural Dynamics & Vibration Control Lab., KAIST, Korea

5. Objective of this study:
Given:
Find:
* represents the derivative of with respect
design parameter α (length, area, moment of inertia, etc.)
Structural Dynamics & Vibration Control Lab., KAIST, Korea

6. PREVIOUS STUDIES Damped system with distinct eigenvalues
H. K. Jo, J. H. Lee and I. W. Lee, “Sensitivity Analysis of Non-conservative Eigensystems: Part I, Symmetric Systems,” Journal of Sound and Vibration, (submitted)
(2)
The coefficient matrix is symmetric and non-singular.
Eigenpair derivatives are obtained simultaneously.
The algorithm is simple and guarantees stability.
Structural Dynamics & Vibration Control Lab., KAIST, Korea

7. Undamped system with repeated eigenvalues
R. L. Dailey, “Eigenvector Derivatives with Repeated Eigenvalues,” AIAA Journal, Vol. 27, pp , 1989.
Introduction of Adjacent eigenvector
Calculation derivatives of eigenvectors by the sum of homogenous solutions and particular solutions using Nelson’s algorithm
Complicated algorithm and high time consumption
Structural Dynamics & Vibration Control Lab., KAIST, Korea

8. Second order derivatives of Undamped system with repeated eigenvalues
M. I. Friswell, “Calculation of Second and Higher Eigenvector Derivatives”, Journal of Guidance, Control and Dynamics, Vol. 18, pp , 1995.
(3)
(4)
where
- Second eigenvector derivatives extended by Nelson’s algorithm
Structural Dynamics & Vibration Control Lab., KAIST, Korea

9. PROPOSED METHOD
First-order eigenpair derivatives of damped system with repeated eigenvalues
Second-order eigenpair derivatives of damped system with repeated eigenvalues
Numerical stability of the proposed method
Structural Dynamics & Vibration Control Lab., KAIST, Korea

10. First-order eigenpair derivatives of damped system with repeated eigenvalues
Basic Equations
Eigenvalue problem
(5)
Orthonormalization condition
(6)
Structural Dynamics & Vibration Control Lab., KAIST, Korea

11. Adjacent eigenvectors
(7)
where T is an orthogonal transformation matrix
and its order m
(8)
Structural Dynamics & Vibration Control Lab., KAIST, Korea

12. Rearranging eq.(5) and eq.(6) using adjacent eigenvectors
(9)
(10)
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13. Differentiating eq.(9) w.r.t. design parameter α
(11)
Pre-multiplying at each side of eq.(11) by and substituting
(12)
where
Structural Dynamics & Vibration Control Lab., KAIST, Korea

14. Differentiating eq.(9) w.r.t. design parameter α
(13)
Differentiating eq.(10) w.r.t. design parameter α
(14)
Structural Dynamics & Vibration Control Lab., KAIST, Korea

15. Combining eq.(13) and eq.(14) into a single equation
(15)
- It maintains N-space without use of state space equation.
- Eigenpair derivatives are obtained simultaneously.
- It requires only corresponding eigenpair information.
- Numerical stability is guaranteed.
Structural Dynamics & Vibration Control Lab., KAIST, Korea

16. Second-order eigenpair derivatives of damped system with repeated eigenvalues
Differentiating eq.(13) w.r.t. another design parameter β
(16)
Differentiating eq.(14) w.r.t. another design parameter β
(17)
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17. Combining eq.(16) and eq.(17) into a single equation
(18)
where
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18. Numerical stability of the proposed method
Determinant property
(19)
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19. Then,
(20)
(21)
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20. Using the determinant property of partitioned matrix
Arranging eq.(20)
(22)
Using the determinant property of partitioned matrix
(23)
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21. Numerical Stability is Guaranteed.
Therefore
(24)
Numerical Stability is Guaranteed.
Structural Dynamics & Vibration Control Lab., KAIST, Korea

22. NUMERICAL EXAMPLES Cantilever beam - Proportionally damped system
5-DOF mechanical system Non-proportionally damped system
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23. Cantilever beam (proportionally damped system)
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24. First derivatives of eigenvalues Second derivatives of eigenvalues
Results of analysis (eigenvalues)
Mode
number
Eigenvalues
First derivatives of eigenvalues
Second derivatives of eigenvalues
1, 2
e-03
±j5.2496e+00
e-10
j3.5347e-10
4.3916e-09
±j1.0285e-08
3, 4
e-02
±j5.2494e+01
e-01
j6.1102e-02
5, 6
e-02
±j3.2895e+01
e-10
±j2.3445e-10
1.0084e-08
j2.4918e-09
7, 8
e+00
±j3.2886e+02
e-08
j2.6913e+00
9, 10
e-01
±j9.2090e+01
6.9247e-10
j6.9600e-10
±j1.1514e-08
11, 12
e+00
±j9.2029e+02
e+01
j1.8358e+01
Structural Dynamics & Vibration Control Lab., KAIST, Korea

25. First derivatives of eigenvectors Second derivatives of eigenvectors
Results of analysis (first eigenvectors)
DOF
number
Eigenvectors
First derivatives of eigenvectors
Second derivatives of eigenvectors 1 2 3
e+05
-j6.6892e+05
3.3446e-04
+j3.3446e-04
e+03
-j5.0169e+03 4
e+04
-j2.6442e+04
1.3221e-03
+j1.3221e-03
e+02
-j1.9596e+02 77 78 79
e+02
-j1.5577e+02
7.7887e-02
+j7.7887e-02
e+00
-j1.1683e+00 80
e+03
-j2.1442e+03
1.0721e-02
+j1.0721e-02
e+01
-j1.6082e+01 … … … …
Structural Dynamics & Vibration Control Lab., KAIST, Korea

26. Results of analysis (errors of approximations)
Mode number
Actual
Eigenvalues
Approximated eigenvalues
Variations of eigenpairs
Errors of approximations
Eigenvectors
1, 2
e-03
±j5.2496e+00
2.2281e-11
4.9628e-03
2.2283e-11
3.7376e-05
3, 4
e-03
±j5.3021e+00
e-03
1.0000e-02
9.9010e-03
2.6622e-08
1.0000e-04
5, 6
e-02
±j3.2895e+01
3.7084e-12
3.6899e-12
7, 8
e-02
±j3.3224e+01
e-02
9.9997e-04
9.9023e-03
1.6763e-07
1.0001e-04
9, 10
e-01
±j9.2090e+01
9.1400e-12
9.1432e-12
11, 12
e-01
±j9.3010e+01
e-01
9.9936e-03
9.9041e-03
4.6508e-07
1.0002e-04
Structural Dynamics & Vibration Control Lab., KAIST, Korea

27. 5-DOF mechanical system (Non-proportionally damped system)
Structural Dynamics & Vibration Control Lab., KAIST, Korea

28. First derivatives of eigenvalues Second derivatives of eigenvalues
Results of analysis (eigenvalues)
Mode
number
Eigenvalues
First derivatives of eigenvalues
Second derivatives of eigenvalues
1, 2
e-02
±j1.5023e+00
9.6943e-07
±j1.7995e-04
e-08
j2.4680e-07
3, 4
e-01
±j3.4558e+00
0.0000e+00
±j0.0000e+00
5, 6
±j8.6811e-04
1.6409e-08
j1.4913e-07
7, 8
e-02
±j6.1354e+00
e-07
±j2.9526e-05
e-09
±j1.4301e-08
9, 10
e-02
±j9.7000e+00
e-07
±j5.0001e-06
e-11
±j8.4803e-10
Structural Dynamics & Vibration Control Lab., KAIST, Korea

29. First derivatives of eigenvectors Second derivatives of eigenvectors
Results of analysis (first eigenvectors)
DOF
number
Eigenvectors
First derivatives of eigenvectors
Second derivatives of eigenvectors 1
1.0851e-02
-j1.0743e-02
e-07
+j4.2149e-07
3.9185e-10
-j2.7842e-10 2
1.0334e-02
-j1.0284e-02
e-07
+j5.1709e-07
5.2496e-10
-j4.2633e-10 3
9.4601e-03
-j9.5112e-03
e-07
+j6.7328e-07
7.4423e-10
-j6.7172e-10 4
0.0000e+00
+j0.0000e+00
5.4601e-06
-j6.2004e-06
e-09
+j8.1097e-09 5
e-06
+j6.2004e-06
6.5680e-09
-j8.1097e-09
Structural Dynamics & Vibration Control Lab., KAIST, Korea

30. Results of analysis (errors of approximations)
Mode number
Actual
Eigenvalues
Approximated eigenvalues
Variations of eigenpairs
Errors of approximations
Eigenvectors
1, 2
e-02
±j1.5040e+00
e-02
±j1.5041e+00
1.1893e-03
4.6721e-03
8.1631e-07
2.9463e-05
3, 4
e-01
±j3.4558e+00
0.0000e+00
5, 6
±j3.4645e+00
2.5039e-03
1.5461e-03
2.1632e-06
5.2014e-06
7, 8
e-02
±j6.1357e+00
e-02 ±j6.1357e+00
4.8257e-05
1.0987e-03
1.1763e-07
2.5394e-06
9, 10
e-02
±j9.7000e+00
5.1624e-06
1.9422e-04
4.3893e-09
1.6332e-07
Structural Dynamics & Vibration Control Lab., KAIST, Korea

31. CONCLUSIONS Proposed Method
is an efficient eigensensitivity method for the damped system with repeated eigenvalues
guarantees numerical stability
gives exact solutions of eigenpair derivatives
can be extended to obtain second- and higher order derivatives of eigenpairs
Structural Dynamics & Vibration Control Lab., KAIST, Korea

32. Thank you for your attention!
An efficient eigensensitivity method for the
damped system with repeated eigenvalues
Thank you for your attention!
Structural Dynamics & Vibration Control Lab., KAIST, Korea