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Journal of Engineering Mathematics

Erschienen in:

06.05.2015

Eigensensitivity of symmetric damped systems with repeated eigenvalues by generalized inverse

verfasst von: Pingxin Wang, Hua Dai

Erschienen in: Journal of Engineering Mathematics | Ausgabe 1/2016

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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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Nächster Artikel Erratum to: Natural convection heat transfer of a viscous fluid in a vertical porous channel

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

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

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

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

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

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

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

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

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

10.

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

11.

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

12.

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

13.

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

14.

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

15.

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

16.

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

17.

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

18.

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

19.

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

20.

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

21.

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

22.

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

23.

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

24.

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

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 20:78–100MathSciNetCrossRefMATH

26.

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

27.

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

28.

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

29.

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

30.

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

31.

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

32.

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

33.

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

34.

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

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 223:399–412ADSCrossRef

36.

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

37.

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

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 68:674–689CrossRefMATH

39.

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

40.

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

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 223:413–424ADSCrossRef

42.

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

43.

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

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 34:1089–1111MathSciNetCrossRefMATH

45.

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

Titel: Eigensensitivity of symmetric damped systems with repeated eigenvalues by generalized inverse
verfasst von: Pingxin Wang
Hua Dai
Publikationsdatum: 06.05.2015
Verlag: Springer Netherlands
Erschienen in: Journal of Engineering Mathematics / Ausgabe 1/2016
Print ISSN: 0022-0833
Elektronische ISSN: 1573-2703
DOI: https://doi.org/10.1007/s10665-015-9790-1

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