A numerical method for quadratic eigenvalue problems of gyroscopic systems

Abstract

We consider the quadratic eigenvalues problem (QEP) of gyroscopic systems (λ 2 M + λG + K)x = 0, with M = M being positive definite, G = −G , and K = K being negative semidefinite. In [1], it is shown that all eigenvalues of the QEP can be found by finding the maximal solution of a nonlinear matrix equation Z + A Z −1 A = Q under the assumption that the QEP has no eigenvalues on the imaginary axis. Although for some cases when the QEP has eigenvalues on the imaginary axis, the algorithm proposed in [1] also works, but the convergence is much slower. In this paper, we consider the general case when the QEP has eigenvalues on the imaginary axis. We propose an eigenvalue shifting technique to modify the original gyroscopic system to a new gyroscopic system, which changes the eigenvalues on the imaginary axis to eigenvalues with nonzero real parts, while keeps other eigenpairs unchanged. This ensures that the algorithm for the maximal solution of the nonlinear matrix equation converges quadratically. Numerical examples illustrate the efficiency of our method.

Efficient computation of enclosures for the exact solvents of a quadratic matrix equation

“…The non-singularity condition of the matrix A in not so restrictive, because the matrix A in the quadratic matrix equation (1.2) is often nonsingular, e.g., in applications in the overdamped quadratic eigenvalue problems [19], gyroscopic systems [12,27], noisy Wiener-Hopf problems for …”

“…Introduction. Many applications such as multivariate rational expectations models [3], noisy Wiener-Hopf problems for Markov chains [11], quasi-birth death process [14], and the quadratic eigenvalue problem Q(λ)ν = (λ 2 A + λB + C)ν = 0, λ ∈ C, ν ∈ C n , (1.1) which comes from the analysis of damped structural systems, vibration problems [14,15], and gyroscopic systems [12,27] require the solution of the quadratic matrix equation…”

Semi-definite programming techniques for structured quadratic inverse eigenvalue problems

“…Numerical techniques for forward gyroscopic eigensystems have been discussed in [10, 22, 36], but the attention mostly is on the damping free case, i.e., C = G is skew-symmetric. A hybrid optimization method employing genetic algorithms and simulated annealing to identify bearing parameters of rotating machinery from bearing forces [4] is somewhat close to an inverse problem, but we have found no discussion on the gyroscopic inverse eigenvalue problem.…”

Numerical analysis for the quadratic matrix equations from a modification of fixed‐point type

“…The quadratic matrix equations are intimately connected with quadratic eigenvalue problems Q ( λ ) x =0, where A , B , C are n × n real matrices. This equation comes from the analysis of damped structural systems, vibration problems, and gyroscope systems …”