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Adaptive Beamforming and Localization
Published in David C. Swanson, ®, 2011
The eigenvalues for the matrix in Equation 14.21 are found in M + 1 pairs of real and imaginary components. For large matrices, there are a number of generalized eigenvalue algorithms with much greater efficiency and robustness. This is particularly of interest when the matrix is ill-conditioned. A good measure of effectiveness to see if the eigenvector/eigenvalue solution is robust is to simply compute the covariance matrix inverse and multiply it by the original covariance matrix to see how close the result comes to the identity matrix. The covariance matrix inverse is found trivially from the eigenvalue solution. () R¯−1=AΛ−1AHwhereΛ−1=diag{λ0−1λ1−1⋯λM−1}
Hybrid composite shaft of High-Speed Rotor-Bearing System - A rotor dynamics preview
Published in Mechanics Based Design of Structures and Machines, 2021
Thimothy Harold Gonsalves, Mohan Kumar Garje Channabasappa, Ramesh Motagondanahalli Rangarasaiah
φi is the displacement vector and is the complex value at the mode i, ai is a real part of eigenvalue representing the stability of the Eigen problem, bi is the imaginary part of eigenvalue representing the frequency and j is √-1. QR eigenvalue algorithm along with the inverse iteration is used to calculate the complex eigenvalues and eigenvectors. The ith eigenvalue will be stable if ai is negative and unstable if ai is positive. Damping in the freely vibrating system can be measured using the logarithmic decrement which is a function of modal damping ratio ( Logarithmic decrement δi represents the logarithm of the ratio of two consecutive peaks in the dynamic response, a positive value indicating the instability threshold. Alternatively, modal damping ratio can be used by ascertaining a positive value, the stability of the rotor-bearing system may be ensured.