Strongly damped quadratic matrix polynomials. (English) Zbl 1320.15007
Summary: We study the eigenvalues and eigenspaces of the quadratic matrix polynomial \(M\lambda^2+sD\lambda+K\) as \(s\to\infty\), where \(M\) and \(K\) are symmetric positive definite and \(D\) is symmetric positive semidefinite. This work is motivated by its application to modal analysis of finite element models with strong linear damping. Our results yield a mathematical explanation of why too strong damping may lead to practically undamped modes such that all nodes in the model vibrate essentially in phase.
MSC:
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15A22 | Matrix pencils |
| 15A54 | Matrices over function rings in one or more variables |
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
| 70J10 | Modal analysis in linear vibration theory |
| 70J30 | Free motions in linear vibration theory |
| 70J50 | Systems arising from the discretization of structural vibration problems |
Keywords:
quadratic eigenvalue problem; principal angles; canonical angles; matrix polynomial; viscous damping; discrete damper; vibrating systemSoftware:
NLEVPReferences:
| [2] | T. Betcke, N. J. Higham, V. Mehrmann, C. Schröder, and F. Tisseur, {\it NLEVP: A collection of nonlinear eigenvalue problems}, ACM Trans. Math. Software, 39 (2013), 7. · Zbl 1295.65140 |
| [3] | \AA. Björck and G. H. Golub, {\it Numerical methods for computing angles between linear subspaces}, Math. Comp., 27 (1973), pp. 579-594. · Zbl 0282.65031 |
| [4] | R. A. Horn and C. R. Johnson, {\it Matrix Analysis}, Cambridge University Press, Cambridge, UK, 1985. · Zbl 0576.15001 |
| [5] | T. Kato, {\it Pertubation Theory for Linear Operators}, Springer-Verlag, Berlin, 1995, reprint of the 1980 edition. |
| [6] | K. Knopp, {\it Theory of Functions}, II. {\it Applications and Continuation of the General Theory}, Dover, Mineola, NY, 1947. |
| [7] | P. Lancaster, {\it On eigenvalues of matrices dependent on a parameter}, Numer. Math., 6 (1964), pp. 377-387. · Zbl 0133.26201 |
| [8] | P. Lancaster, {\it Lambda-Matrices and Vibrating Systems}, Pergamon Press, New York, 1966. Reprinted by Dover, Mineola, NY, 2002. · Zbl 0146.32003 |
| [9] | H. Langer, B. Najman, and K. Veselić, {\it Perturbation of the eigenvalues of quadratic matrix polynomials}, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 474-489. · Zbl 0752.15009 |
| [10] | D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, {\it Jordan structures of alternating matrix polynomials}, Linear Algebra Appl., 432 (2010), pp. 867-891. · Zbl 1188.15010 |
| [11] | D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, {\it Möbius transformations of matrix polynomials}, Linear Algebra Appl., 470 (2015), pp. 120-184. · Zbl 1309.65042 |
| [12] | D. Meyer and K. Veselić, {\it On some new inclusion theorems for the eigenvalues of partitioned matrices}, Numer. Math., 34 (1980), pp. 431-437. · Zbl 0433.65020 |
| [13] | G. W. Stewart, {\it Matrix Algorithms, Volume II: Eigensystems}, SIAM, Philadelphia, 2001. · Zbl 0984.65031 |
| [14] | D. B. Szyld, {\it The many proofs of an identity on the norm of oblique projections}, Numer. Algorithms, 42 (2006), pp. 309-323. · Zbl 1102.47002 |
| [15] | L. Taslaman, {\it Algorithms and Theory for Polynomial Eigenproblems}. Ph.D. thesis, The University of Manchester, Manchester, UK, 2014; available as MIMS EPrint 2015.4 from http://eprints.ma.man.ac.uk/2237. |
| [16] | L. Taslaman, {\it The Principal Angles and the Gap}, MIMS EPrint 2014.9, Manchester Institute for Mathematical Sciences, The University of Manchester, Manchester, UK, 2014. Available online at http://eprints.ma.man.ac.uk/2105. |
| [17] | L. Taslaman, {\it An algorithm for quadratic eigenproblems with low rank damping}, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 251-272. · Zbl 1315.65039 |
| [18] | F. Tisseur, {\it Backward error and condition of polynomial eigenvalue problems}, Linear Algebra Appl., 309 (2000), pp. 339-361. · Zbl 0955.65027 |
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