TY - JOUR

T1 - Numerical algorithms for undamped gyroscopic systems

AU - Ferng, W. R.

AU - Lin, Wen Wei

AU - Wang, Chern Shuh

N1 - Funding Information:
This work was supported in part by the National Science Council, Taiwan. Part of this research was performed while the authors were visiting the Boeing Computer Services at Bellevue, Washington, U.S.A. This work was inspired by J. Lewis while he was visiting Tsing Hua University, Taiwan. The authors would thank him for his valuable comments and discussions. The authors are also grateful to D. J. Pierce and his family and R. Grimes for their warm hospitality.

PY - 1999/1

Y1 - 1999/1

N2 - The solutions of a gyroscopic vibrating system oscillating about an equilibrium position, with no external applied forces and no damping forces, are completely determined by the quadratic eigenvalue problem (-λ2iM + λiG + K)xi = 0, for i = 1, . . . , 2n, where M, G, and K are real n x n matrices, and M is symmetric positive definite (denoted by M > 0), G is skew symmetric, and either K > O or -K > 0. Gyroscopic system in motion about a stable equilibrium position (with -K > 0) are well understood. Two Lanczos-type algorithms, the pseudo skew symmetric Lanczos algorithm and the J-Lanczos algorithm, are studied for computing some extreme eigenpairs for solving gyroscopic systems in motion about an unstable equilibrium position (with K > 0). Shift and invert strategies, error bounds, implementation issues, and numerical results for both algorithms are presented in details.

AB - The solutions of a gyroscopic vibrating system oscillating about an equilibrium position, with no external applied forces and no damping forces, are completely determined by the quadratic eigenvalue problem (-λ2iM + λiG + K)xi = 0, for i = 1, . . . , 2n, where M, G, and K are real n x n matrices, and M is symmetric positive definite (denoted by M > 0), G is skew symmetric, and either K > O or -K > 0. Gyroscopic system in motion about a stable equilibrium position (with -K > 0) are well understood. Two Lanczos-type algorithms, the pseudo skew symmetric Lanczos algorithm and the J-Lanczos algorithm, are studied for computing some extreme eigenpairs for solving gyroscopic systems in motion about an unstable equilibrium position (with K > 0). Shift and invert strategies, error bounds, implementation issues, and numerical results for both algorithms are presented in details.

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U2 - 10.1016/S0898-1221(98)00241-7

DO - 10.1016/S0898-1221(98)00241-7

M3 - Article

AN - SCOPUS:0032777001

SN - 0898-1221

VL - 37

SP - 49

EP - 66

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

IS - 1

ER -