An improved method for computing eigenpair derivatives of damped system. (English) Zbl 1427.70047
Summary: The calculation of eigenpair derivatives plays an important role in vibroengineering. This paper presents an improved algorithm for the eigenvector derivative of the damped systems by dividing it into a particular solution and general solution of the corresponding homogeneous equation. Compared with the existing methods, the proposed algorithm can significantly reduce the condition number of the equation for particular solution. Therefore, the relative errors of the calculated solutions are notably cut down. The results on two numerical examples show that such strategy is effective in reducing the condition numbers for both distinct and repeated eigenvalues.
MSC:
| 70J30 | Free motions in linear vibration theory |
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
| 70J50 | Systems arising from the discretization of structural vibration problems |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
References:
| [1] | Haug, E. J.; Choi, K. K.; Komkov, V., Design Sensitivity Analysis of Structural Systems (1986), Orlando, Fla, USA: Academic Press, Orlando, Fla, USA · Zbl 0618.73106 |
| [2] | Adhikari, S., Structural Dynamics with Generalized Damping Models: Analysis (2013), Wiley-ISTE |
| [3] | Yuan, Y. Y.; Liu, H., An iterative updating method for undamped structural systems, Meccanica, 47, 3, 699-706 (2012) · Zbl 1293.74419 · doi:10.1007/s11012-011-9483-4 |
| [4] | Messina, A.; Williams, E. J.; Contursi, T., Structural damage detection by a sensitivity and statistical-based method, Journal of Sound and Vibration, 216, 5, 791-808 (1998) · doi:10.1006/jsvi.1998.1728 |
| [5] | Adhikari, S., Structural Dynamics with Generalized Damping Models: Identification (2013), Wiley-ISTE |
| [6] | Fox, R. L.; Kapoor, M. P., Rates of change of eigenvalues and eigenvectors, AIAA Journal, 6, 12, 2426-2429 (1968) · Zbl 0181.53003 · doi:10.2514/3.5008 |
| [7] | Adhikari, S., Rates of change of eigenvalues and eigenvectors in damped dynamic system, AIAA Journal, 37, 11, 1452-1458 (1999) · doi:10.2514/2.622 |
| [8] | Adhikari, S., Calculation of derivative of complex modes using classical normal modes, Computers & Structures, 77, 6, 625-633 (2000) · doi:10.1016/S0045-7949(00)00016-X |
| [9] | Adhikari, S.; Friswell, M. I., Eigenderivative analysis of asymmetric non-conservative systems, International Journal for Numerical Methods in Engineering, 51, 6, 709-733 (2001) · Zbl 0992.70016 · doi:10.1002/nme.186.abs |
| [10] | Adhikari, S., Derivative of eigensolutions of nonviscously damped linear systems, AIAA Journal, 40, 10, 2061-2069 (2002) · doi:10.2514/2.1539 |
| [11] | Lee, I.-W.; Jung, G.-H., An efficient algebraic method for the computation of natural frequency and mode shape sensitivities. Part I. Distinct natural frequencies, Computers & Structures, 62, 3, 429-435 (1997) · Zbl 0912.73080 · doi:10.1016/s0045-7949(96)00206-4 |
| [12] | Lee, I.-W.; Kim, D.-O.; Jung, G.-H., Natural frequency and mode shape sensitivities of damped systems: Part I. Distinct natural frequencies, Journal of Sound and Vibration, 223, 3, 399-412 (1999) · doi:10.1006/jsvi.1998.2129 |
| [13] | Lee, I.-W.; Kim, D.-O.; Jung, G.-H., Natural frequency and mode shape sensitivities of damped systems: part II, multiple natural frequencies, Journal of Sound and Vibration, 223, 3, 413-424 (1999) · doi:10.1006/jsvi.1998.2130 |
| [14] | Zhao, Y. Q.; Liu, Z. S.; Chen, S. H.; Zhang, G. Y., An accurate modal truncation method for eigenvector derivatives, Computers & Structures, 73, 609-614 (1999) · Zbl 0955.74516 |
| [15] | Moon, Y.-J.; Kim, B.-W.; Ko, M.-G.; Lee, I.-W., Modified modal methods for calculating eigenpair sensitivity of asymmetric damped system, International Journal for Numerical Methods in Engineering, 60, 11, 1847-1860 (2004) · Zbl 1060.70511 · doi:10.1002/nme.1025 |
| [16] | Choi, K.-M.; Jo, H.-K.; Kim, W.-H.; Lee, I.-W., Sensitivity analysis of non-conservative eigensystems, Journal of Sound and Vibration, 274, 3-5, 997-1011 (2004) · doi:10.1016/S0022-460X(03)00660-6 |
| [17] | Guedria, N.; Smaoui, H.; Chouchane, M., A direct algebraic method for eigensolution sensitivity computation of damped asymmetric systems, International Journal for Numerical Methods in Engineering, 68, 6, 674-689 (2006) · Zbl 1128.74015 · doi:10.1002/nme.1732 |
| [18] | Chouchane, M.; Guedria, N.; Smaoui, H., Eigensensitivity computation of asymmetric damped systems using an algebraic approach, Mechanical Systems and Signal Processing, 21, 7, 2761-2776 (2007) · doi:10.1016/j.ymssp.2007.01.007 |
| [19] | Xu, Z. H.; Zhong, H. X.; Zhu, X. W.; Wu, B. S., An efficient algebraic method for computing eigensolution sensitivity of asymmetric damped systems, Journal of Sound and Vibration, 327, 3-5, 584-592 (2009) · doi:10.1016/j.jsv.2009.07.013 |
| [20] | Li, L.; Hu, Y. J.; Wang, X. L.; Ling, L., Computation of eigensolution derivatives for nonviscously damped systems using the algebraic method, AIAA Journal, 50, 10, 2282-2284 (2012) · doi:10.2514/1.J051664 |
| [21] | Li, L.; Hu, Y. J.; Wang, X. L.; Ling, L., Eigensensitivity analysis for asymmetric nonviscous systems, AIAA Journal, 51, 3, 738-741 (2013) · doi:10.2514/1.J051931 |
| [22] | Li, L.; Hu, Y. J.; Wang, X. L., A study on design sensitivity analysis for general nonlinear eigenproblems, Mechanical Systems & Signal Processing, 88-105 (2013), 34 |
| [23] | Li, L.; Hu, Y. J.; Wang, X. L., Numerical methods for evaluating the sensitivity of element modal strain energy, Finite Elements in Analysis and Design, 64, 13-23 (2013) · Zbl 1282.65089 · doi:10.1016/j.finel.2012.09.006 |
| [24] | Qian, J.; Andrew, A. L.; Chu, D. L.; Tan, R. C. E., Computing derivatives of repeated eigenvalues and corresponding eigenvectors of quadratic eigenvalue problems, SIAM Journal on Matrix Analysis and Applications, 34, 3, 1089-1111 (2013) · Zbl 1286.65049 · doi:10.1137/120879841 |
| [25] | Sutter, T. R.; Camarda, C. J.; Walsh, J. L.; Adelman, H. M., Comparison of several methods for calculating vibration mode shape derivatives, AIAA Journal, 26, 12, 1506-1511 (1988) · doi:10.2514/3.10070 |
| [26] | Andrew, A. L.; Tan, R. C. E., Computation of derivatives of repeated eigenvalues and corresponding eigenvectors by simultaneous iteration, AIAA Journal, 34, 10, 2214-2216 (1996) · Zbl 0899.65015 · doi:10.2514/3.13382 |
| [27] | Xie, H. Q.; Dai, H., Simultaneous iterative method for eigenpair derivatives of damped systems, AIAA Journal, 43, 1, 1403-1405 (2013) · doi:10.2514/1.J051759 |
| [28] | Xie, H. Q., Simultaneous iterative method for eigenpair derivatives of damped systems, AIAA Journal, 51, 1, 236-239 (2013) · doi:10.2514/1.J051759 |
| [29] | Nelson, R. B., Simplified calculation of eigenvector derivatives, American Institute of Aeronautics and Astronautics. Journal, 14, 9, 1201-1205 (1976) · Zbl 0342.65021 · doi:10.2514/3.7211 |
| [30] | Andrew, A. L.; Tan, R. C., Computation of derivatives of repeated eigenvalues and the corresponding eigenvectors of symmetric matrix pencils, SIAM Journal on Matrix Analysis and Applications, 20, 1, 78-100 (1999) · Zbl 0919.65026 · doi:10.1137/S0895479896304332 |
| [31] | Friswell, M. I.; Adhikari, S., Derivatives of complex eigenvectors using Nelson’s method, AIAA Journal, 38, 12, 2355-2357 (2000) · doi:10.2514/2.907 |
| [32] | Guedria, N.; Chouchane, M.; Smaoui, H., Second-order eigensensitivity analysis of asymmetric damped systems using Nelson’s method, Journal of Sound and Vibration, 300, 5, 974-992 (2007) · doi:10.1016/j.jsv.2006.09.003 |
| [33] | Adhikari, S.; Friswell, M. I., Calculation of eigensolution derivatives for nonviscously damped systems using Nelson’s method, AIAA Journal, 44, 8, 1799-1806 (2006) · doi:10.2514/1.20049 |
| [34] | Wu, B. S.; Xu, Z. H.; Li, Z. G., A note on computing eigenvector derivatives with distinct and repeated eigenvalues, Communications in Numerical Methods in Engineering, 23, 3, 241-251 (2007) · Zbl 1111.65034 · doi:10.1002/cnm.895 |
| [35] | Omenzetter, P., Sensitivity analysis of the eigenvalue problem for general dynamic systems with application to bridge deck flutter, Journal of Engineering Mechanics, 138, 6, 675-682 (2012) · doi:10.1061/(ASCE)EM.1943-7889.0000377 |
| [36] | Wang, P. X.; Dai, H., Eigensensitivity of symmetric damped systems with repeated eigenvalues by generalized inverse, Journal of Engineering Mathematics, 96, 201-210 (2016) · Zbl 1360.65157 · doi:10.1007/s10665-015-9790-1 |
| [37] | Wang, P. X.; Dai, H., Calculation of eigenpair derivatives for asymmetric damped systems with distinct and repeated eigenvalues, International Journal for Numerical Methods in Engineering, 103, 7, 501-515 (2015) · Zbl 1352.65122 · doi:10.1002/nme.4901 |
| [38] | Wang, P. X.; Yang, X.; Mi, J., Computing eigenpair derivatives of asymmetric damped system by generalized inverse, Journal of Vibroengineering, 19, 7, 5290-5300 (2017) · doi:10.21595/jve.2017.18232 |
| [39] | Wang, P. X.; Dai, H., Calculation of eigenpair derivatives for symmetric quadratic eigenvalue problem with repeated eigenvalues, Computational & Applied Mathematics, 35, 1, 17-28 (2016) · Zbl 1339.65098 · doi:10.1007/s40314-014-0169-0 |
| [40] | Mehrmann, V.; Voss, H., Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods, GAMM-Mitteilungen, 27, 2, 121-152 (2005) · Zbl 1071.65074 |
| [41] | Voss, H., Projection methods for nonlinear sparse eigenvalue problems, In Annals of the European Academy of Sciences, 152-183 (2005) |
| [42] | Li, R. C., Compute multiply nonlinear eigenvalues, Journal of Computational Mathematics, 10, 1, 1-20 (1992) · Zbl 0752.65042 |
| [43] | Dai, H.; Bai, Z.-Z., On smooth LU decompositions with applications to solutions of nonlinear eigenvalue problems, Journal of Computational Mathematics, 28, 6, 745-766 (2010) · Zbl 1240.65122 · doi:10.4208/jcm.1004-m0009 |
| [44] | Guo, J.-S.; Lin, W. W.; Wang, C. S., Nonequivalence deflation for the solution of matrix latent value problems, Linear Algebra and its Applications, 231, 15-45 (1995) · Zbl 0841.65024 · doi:10.1016/0024-3795(95)90004-7 |
| [45] | Neumaier, A., Residual inverse iteration for the nonlinear eigenvalue problem, SIAM Journal on Numerical Analysis, 22, 5, 914-923 (1985) · Zbl 0594.65026 · doi:10.1137/0722055 |
| [46] | Ruhe, A., Algorithms for the nonlinear eigenvalue problem, SIAM Journal on Numerical Analysis, 10, 674-689 (1973) · Zbl 0261.65032 · doi:10.1137/0710059 |
| [47] | Voss, H., An Arnoldi method for nonlinear eigenvalue problems, BIT Numerical Mathematics, 44, 2, 387-401 (2004) · Zbl 1066.65059 · doi:10.1023/B:BITN.0000039424.56697.8b |
| [48] | Voss, H., A Jacobi-Davidson method for nonlinear and nonsymmetric eigenproblems, Computers & Structures, 85, 17-18, 1284-1292 (2007) · doi:10.1016/j.compstruc.2006.08.088 |
| [49] | Van Beeumen, R.; Jarlebring, E.; Michiels, W., A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems, Numerical Linear Algebra with Applications, 23, 4, 607-628 (2016) · Zbl 1413.65105 · doi:10.1002/nla.2043 |
| [50] | Asakura, J.; Sakurai, T.; Tadano, H.; Ikegami, T.; Kimura, K., A numerical method for nonlinear eigenvalue problems using contour integrals, JSIAM Letters, 1, 52-55 (2009) · Zbl 1278.65072 · doi:10.14495/jsiaml.1.52 |
| [51] | Beyn, W.-J., An integral method for solving nonlinear eigenvalue problems, Linear Algebra and its Applications, 436, 10, 3839-3863 (2012) · Zbl 1237.65035 · doi:10.1016/j.laa.2011.03.030 |
| [52] | Chen, X.-P.; Dai, H., Stability analysis of time-delay systems using a contour integral method, Applied Mathematics and Computation, 273, 390-397 (2016) · Zbl 1410.34210 · doi:10.1016/j.amc.2015.09.094 |
| [53] | Golub, G. H.; van Loan, C. F., Matrix Computations (1996), London, UK: Johns Hopkins University Press, London, UK · Zbl 0865.65009 |
| [54] | Sun, J. G., Multiple eigenvalue sensitivity analysis, Linear Algebra and its Applications, 137/138, 183-211 (1990) · Zbl 0709.65028 · doi:10.1016/0024-3795(90)90129-Z |
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