Abstract
This paper presents methods for computing a second-order sensitivity matrix and the Hessian matrix of eigenvalues and eigenvectors of multiple parameter structures. Second-order perturbations of eigenvalues and eigenvectors are transformed into multiple parameter forms, and the second-order perturbation sensitivity matrices of eigenvalues and eigenvectors are developed. With these formulations, the efficient methods based on the second-order Taylor expansion and second-order perturbation are obtained to estimate changes of eigenvalues and eigenvectors when the design parameters are changed. The presented method avoids direct differential operation, and thus reduces difficulty for computing the second-order sensitivity matrices of eigenpairs. A numerical example is given to demonstrate application and accuracy of the proposed method.
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References
Fox, R. L. and Kapoor, M. P. Rates of change of eigenvalues and eigenvectors. AIAA 6(12), 2426–2429 (1968)
Nelson, R. B. Simplified calculations of eigenvector derivative. AIAA 14, 1201–1205 (1976)
Juang, J. N., Ghaemmaghami, P., and Lim, K. B. Eigenvalue and eigenvector derivatives of a nondefective matrix. Journal of Guidance, Control Dynamics 12, 480–486 (1989)
Lee, I. W. and Jung, G. H. An efficient algebraic method for computation of natural frequency and mode shape sensitivities-part I. distinct natural frequencies. Computers and Structures 62(3), 429–435 (1997)
Lee, I. W. and Jung, G. H. An efficient algebraic method for computation of natural frequency and mode shape sensitivities-part II. multiple natural frequencies. Computers and Structures 62(3), 437–443 (1997)
Lim, K. B. and Junkins, J. L. Re-examination of eigenvector derivative. Journal of Guidance 10, 581–587 (1987)
Liu, Z. S., Chen, S. H., and Zhao, Y. Q. An accurate method for computing eigenvector derivatives for free-free structures. Computers and Structures 52, 1135–1143 (1994)
Moon, Y. J., Kim, B. W., Ko, M. G., and Lee, I. W. Modified modal methods for calculating eigenpair sensitivity of asymmetric damped system. International Journal for Numerical Methods in Engineering 60, 1847–1860 (2004)
Gong, Y. L. and Xu, L. Sensitivity analysis of steel moment frame accounting for geometric and material nonlinearity. Computers and Structures 84, 462–475 (2006)
Maddulapalli, A. K., Azarm, S., and Boyars, A. Sensitivity analysis for product design selection with an implicit value function. European Journal of Operation Research 80, 1245–1259 (2007)
Choi, K. M., Jo, H. K., Kim, W. H., and Lee, I. W. Sensitivity analysis of non-conservative eigensystems. Journal of Sound and Vibration 274, 997–1011 (2004)
Chen, S. H. Matrix Perturbation Theory in Structural Dynamic Design, Science Press, Beijing (2007)
Liu, X. L. Accurate modal perturbation in non-self-adjoint eigenvalue problem. Communications in Numerical Methods in Engineering 17, 715–725 (2001)
Godoy, A., Taroco, E. O., and Feijoo, R. A. Second-order sensitivity analysis in vibration and buckling problems. International Journal for Numerical Methods in Engineering 37(23), 3999–4014 (1994)
Mirzaeifar, R., Bahai, H., Aryana, F., and Yeilaghi, A. Optimization of the dynamic characteristics of composite plates using an inverse approach. Journal of Composite Materials 41(26), 3091–3108 (2007)
Mirzaeifar, R., Bahai, H., and Shahab, S. Active control of natural frequencies of FGM plates by piezoelectric sensor/actuator pairs. Smart Materials and Structures 17(4) (2008) DOI: 10.1088/0964-1726/17/4/045003
Mirzaeifar, R., Bahai, H., and Shahab, S. A new method for finding the first- and second-order eigenderivatives of asymmetric non-conservative systems with application to an FGM plate actively controlled by piezoelectric sensor/actuators. International Journal for Numerical Methods in Engineering 75(12), 1492–1510 (2008)
Aryana, F. and Bahai, H. Sensitivity analysis and modification of structural dynamic characteristics using second order approximation. Engineering Structures 25(10), 1279–1287 (2003)
Bahai, H., Farahani, K., and Djoudi, M. S. Eigenvalue inverse formulation for optimizing vibratory behavior of truss and continuous structures. Computers and Structures 80, 2397–2403 (2002)
Farahani, K. and Bahai, H. An inverse strategy for relocation of eigenfrequencies in structural design, part II: second order approximate solutions. Journal of Sound and Vibration 274, 507–528 (2004)
Choi, K. M., Cho, S. W., Ko, M. G., and Lee, I. W. Higher order eigensensitivity analysis of damped systems with repeated eigenvalues. Computers and Structures 82, 63–69 (2004)
Guedria, N., Chouchane, M., and Smaoui, H. Second-order eigensensitivity analysis of asymmetric damped systems using Nelson’s method. Journal of Sound and Virbation 300, 974–992 (2007)
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Communicated by Li-qun CHEN
Project supported by the 985-Engineering Innovation of Graduate Students of Jilin University and the Science and Technology Development Foundation of Jilin Province (20070541)
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Chen, Sh., Guo, R. & Meng, Gw. Second-order sensitivity of eigenpairs in multiple parameter structures. Appl. Math. Mech.-Engl. Ed. 30, 1475–1487 (2009). https://doi.org/10.1007/s10483-009-1201-z
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DOI: https://doi.org/10.1007/s10483-009-1201-z