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Eigensensitivity of symmetric damped systems with repeated eigenvalues by generalized inverse

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Abstract

We consider computation of the derivatives of the semisimple eigenvalues and corresponding eigenvectors of a symmetric quadratic eigenvalue problem. Using the normalization condition, we can compute the derivatives of the differentiable eigenvalues of the quadratic eigenvalue problem. Using the constrained generalized inverse, we present an efficient algorithm to compute the particular solutions to the governing equation of the derivatives of eigenvectors. The proposed method is suitable for the computation of the eigenpair derivatives of a symmetric quadratic eigenvalue problem when the first-order derivatives of eigenvalues are distinct or repeated. A numerical example is included to illustrate the validity of the proposed method.

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References

  1. Haug EJ, Choi KK, Komkov V (1986) Design sensitivity analysis of structural systems. Academic Press, New York

    MATH  Google Scholar 

  2. Adhikari S (2013) Structural dynamics with generalized damping models: analysis. Wiley-ISTE, Hoboken

    Book  Google Scholar 

  3. Friswell MI, Mottershead JE (1995) Finite element model updating in structural dynamics. Kluwer, Norwell

    Book  MATH  Google Scholar 

  4. Messina A, Williams EJ, Contursi T (1998) Structural damage detection by a sensitivity and statistical-based method. J Sound Vib 216:791–808

    Article  ADS  Google Scholar 

  5. Adhikari S (2013) Structural dynamics with generalized damping models: identification. Wiley-ISTE, Hoboken

    Book  Google Scholar 

  6. Lancaster P (1964) On eigenvalues of matrices dependent on a parameter. Numer Math 6:377–387

    Article  MathSciNet  MATH  Google Scholar 

  7. Kato T (1966) Perturbation theory for linear operators. Spring, New York

    Book  MATH  Google Scholar 

  8. Rellich F (1969) Perturbation theory of eigenvalue problems. Gordon and Breach, New York

    MATH  Google Scholar 

  9. Meyer CD, Stewart GW (1988) Derivatives and perturbations of eigenvectors. SIAM J Numer Anal 25:679–691

    Article  MathSciNet  MATH  Google Scholar 

  10. Lancaster P, Markus AS, Zhou F (2003) Perturbation theory for analytic matrix functions: the semisimple case. SIAM J Matrix Anal Appl 25:606–626

    Article  MathSciNet  MATH  Google Scholar 

  11. Sun JG (1985) Eigenvalues and eigenvectors of a matrix dependent on several parameters. J Comput Math 3:351–364

    MathSciNet  MATH  Google Scholar 

  12. Sun JG (1989) A note on local behavior of multiple eigenvalues. SIAM J Matrix Anal Appl 10:533–541

    Article  MathSciNet  MATH  Google Scholar 

  13. Sun JG (1990) Multiple eigenvalue sensitivity analysis. Linear Algebra Appl 137(138):183–211

    Article  MathSciNet  MATH  Google Scholar 

  14. Chu K-WE (1990) On multiple eigenvalues of matrices depending on several parameters. SIAM J Numer Anal 27:1368–1385

    Article  MathSciNet  MATH  Google Scholar 

  15. Xie HQ, Dai H (2003) On the sensitivity of multiple eigenvalues of nonsymmetric matrix pencils. Linear Algebra Appl 374:143–158

    Article  MathSciNet  MATH  Google Scholar 

  16. Andrew AL, Chu K-WE, Lancaster P (1993) Derivatives of eigenvalues and eigenvectors of matrix functions. SIAM J Matrix Anal Appl 14:903–926

    Article  MathSciNet  MATH  Google Scholar 

  17. Fox RL, Kapoor MP (1968) Rates of change of eigenvalues and eigenvectors. AIAA J 6:2426–2429

    Article  ADS  MATH  Google Scholar 

  18. Rogers LC (1970) Derivatives of eigenvalues and eigenvectors. AIAA J 8:943–944

    Article  ADS  MathSciNet  Google Scholar 

  19. Plaut RH, Huseyin K (1973) Derivatives of eigenvalues and eigenvectors in non-self-adjoint systems. AIAA J 11:250–251

    Article  ADS  MathSciNet  Google Scholar 

  20. Garg S (1973) Derivatives of eigensolutions for a general matrix. AIAA J 11:1191–1194

    Article  ADS  MathSciNet  Google Scholar 

  21. Rudisill CS (1974) Derivatives of eigenvalues and eigenvectors for a general matrix. AIAA J 12:721–722

    Article  ADS  MATH  Google Scholar 

  22. Rudisill CS, Chu Y (1975) Numerical methods for evaluating the derivatives of eigenvalues and eigenvectors. AIAA J 13:834–837

    Article  ADS  MathSciNet  Google Scholar 

  23. Nelson RB (1976) Simplified calculation of eigenvector derivatives. AIAA J 14:1201–1205

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Mills-Curran WC (1988) Calculation of eigenvector derivatives for structures with repeated eigenvalues. AIAA J 26:867–871

    Article  ADS  MATH  Google Scholar 

  25. Andrew AL, Tan RCE (1998) Computation of derivatives of repeated eigenvalues and the corresponding eigenvectors of symmetric matrix pencils. SIAM J Matrix Anal Appl 20:78–100

    Article  MathSciNet  MATH  Google Scholar 

  26. Adelman HM, Haftka RT (1986) Sensitivity analysis of discrete structural system. AIAA J 24:823–832

    Article  ADS  Google Scholar 

  27. Murthy DV, Haftka RT (1988) Derivatives of eigenvalues and eigenvectors of a general complex matrix. Int J Numer Methods Eng 26:293–311

    Article  MathSciNet  MATH  Google Scholar 

  28. Haftka RT, Adelman HM (1989) Recent development in structural sensitivity analysis. Struct Optim 1:137–151

    Article  Google Scholar 

  29. Lancaster P (1966) Lambda-matrices and vibrating systems. Pergamon, Oxford

    MATH  Google Scholar 

  30. Gohberg I, Lancaster P, Rodman L (1982) Matrix polynomials. Academic Press, New York

    MATH  Google Scholar 

  31. Tisseur F, Meerbergen K (2001) The quadratic eigenvalue problem. SIAM Rev 43:235–286

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Zeng QH (1994) Highly accurate modal method for calculating eigenvector derivative in viscous damping systems. AIAA J 33:746–751

    Article  ADS  MATH  Google Scholar 

  33. Liu XB (2013) A new method for calculating derivatives of eigenvalues and eigenvectors for discrete structural systems. J Sound Vib 332:1859–1867

    Article  ADS  Google Scholar 

  34. Adhikari S (1999) Rates of change of eigenvalues and eigenvectors in damped dynamic systems. AIAA J 37:1452–1458

    Article  ADS  Google Scholar 

  35. Lee IW, Kim DO, Jung GH (1999) Natural frequency and mode shape sensitivities of damped systems: Part I, distinct natural frequencies. J Sound Vib 223:399–412

    Article  ADS  Google Scholar 

  36. Adhikari S, Friswell MI (2001) Eigenderivative analysis of asymmetric non-conservative systems. Int J Numer Methods Eng 51:709–733

    Article  MATH  Google Scholar 

  37. Choi KM, Jo HK, Kim WH, Lee IW (2004) Sensitivity analysis of non-conservative eigensystems. J Sound Vib 274:997–1011

    Article  ADS  Google Scholar 

  38. Guedria N, Smaoui H, Chouchane M (2006) A direct algebraic method for eigensolution sensitivity computation of damped asymmetric systems. Int J Numer Methods Eng 68:674–689

    Article  MATH  Google Scholar 

  39. Friswell MI, Adhikari S (2000) Derivatives of complex eigenvectors using Nelson’s method. AIAA J 38:2355–2357

    Article  ADS  Google Scholar 

  40. Tang J, Ni WM, Wang WL (1996) Eigensolutions sensitivity for quadratic eigenproblems. J Sound Vib 196:179–188

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Lee IW, Kim DO, Jung GH (1999) Natural frequency and mode shape sensitivities of damped systems: Part II, multiple natural frequencies. J Sound Vib 223:413–424

    Article  ADS  Google Scholar 

  42. Xie HQ, Dai H (2007) Derivatives of repeated eigenvalues and corresponding eigenvectors of damped systems. Appl Math Mech (English Ed) 28:837–845

    Article  MathSciNet  MATH  Google Scholar 

  43. Xie HQ, Dai H (2008) Calculation of derivatives of multiple eigenpairs of unsymmetrical quadratic eigenvalue problems. Int J Comput Math 85:1815–1831

    Article  MathSciNet  MATH  Google Scholar 

  44. Qian J, Andrew AL, Chu D, Tan RCE (2013) Computing derivatives of repeated eigenvalues and corresponding eigenvectors of quadratic eigenvalue problems. SIAM J Matrix Anal Appl 34:1089–1111

    Article  MathSciNet  MATH  Google Scholar 

  45. Ben-Israel A, Greville TNE (1974) Generalized inverses: theory and applications. Wiley, New York

    MATH  Google Scholar 

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (11071118 and 61100116) and the Fundamental Research Funds of Jiangsu University of Science and Technology. The authors are grateful to the referees for their valuable comments and suggestions, which helped to improve the presentation of this paper.

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Authors and Affiliations

  1. School of Mathematics and Physics, Jiangsu University of Science and Technology, Zhenjiang, 212003, People’s Republic of China

    Pingxin Wang

  2. College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, People’s Republic of China

    Hua Dai

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  1. Pingxin Wang
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  2. Hua Dai
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Correspondence to Pingxin Wang.

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Wang, P., Dai, H. Eigensensitivity of symmetric damped systems with repeated eigenvalues by generalized inverse. J Eng Math 96, 201–210 (2016). https://doi.org/10.1007/s10665-015-9790-1

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  • DOI: https://doi.org/10.1007/s10665-015-9790-1

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