Bauchau, O. A. A solution of the eigenproblem for undamped gyroscopic systems with the Lanczos algorithm. (English) Zbl 0596.73076 Int. J. Numer. Methods Eng. 23, 1705-1713 (1986). Summary: This paper presents an efficient numerical solution of the quadratic eigenproblem arising in the analysis of gyroscopic systems. Such problems are known to reduce to a generalized linear eigenproblem defined by two real non-singular matrices, one symmetric and one skew-symmetric. For this class of problems, the general Lanczos algorithm for unsymmetric matrices is shown to simplify considerably and yields an efficient solution of the problem. Full advantage can be taken of the sparsity of the matrices and of the specific nature of gyroscopic systems. Numerical examples are presented, which demonstrate the efficiency and accuracy of the solution procedure. Cited in 6 Documents MSC: 74S30 Other numerical methods in solid mechanics (MSC2010) 74H45 Vibrations in dynamical problems in solid mechanics 65F15 Numerical computation of eigenvalues and eigenvectors of matrices Keywords:spinning structures; rotating frequencies; selective; orthogonalization; quadratic eigenproblem; gyroscopic systems; generalized linear eigenproblem; real non-singular matrices; symmetric; skew-symmetric; general Lanczos algorithm; unsymmetric matrices PDFBibTeX XMLCite \textit{O. A. Bauchau}, Int. J. Numer. Methods Eng. 23, 1705--1713 (1986; Zbl 0596.73076) Full Text: DOI References: [1] Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, 1982. [2] Bathe, Comp. Meth. Appl. Mech. Eng. 23 pp 313– (1980) [3] Computational Method in Structural Dynamics, Stijthoff & Noordhoff, 1980. [4] Clint, The Computer Journal 13 pp 76– (1970) [5] Parlett, Mathematics of Computation 23 pp 217– (1979) [6] Borri, Comp. Meth. Appl. Mech. Eng. 12 pp 19– (1977) [7] Likins, Int. J. Solids and Structures 8 pp 709– (1972) [8] Likins, AIAA Journal 11 pp 1251– (1973) [9] Hodges, AIAA J. 19 pp 1459– (1981) [10] Analytical Methods in Vibrations, The Macmillan Co., 1967. · Zbl 0166.43803 [11] Gupta, Int. j. numer. methods eng. 7 pp 509– (1973) [12] Gupta, Int. j. numer. methods eng. 17 pp 187– (1981) [13] L, AIAA J. 12 pp 1337– (1974) [14] Bathe, Int. j. numer. methods eng. 6 pp 213– (1973) [15] Wittrick, J. Sound and Vibration 82 pp 1– (1982) [16] Williams, ASCE J. Structural Engineering 109 pp 169– (1983) [17] Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. · Zbl 0258.65037 [18] The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, 1980. · Zbl 0431.65017 [19] Ojalvo, AIAA J. 8 pp 1234– (1970) [20] Bauchau, J. Amer. Helic. Soc. [21] Nour-Omid, Int. j. numer. methods eng. 19 pp 859– (1983) [22] Gupta, Int. j. numer. methods eng. 7 pp 17– (1973) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.