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Numerical method for sensitivity analysis of eigensystems with non-repeated and repeated eigenvalues. (English) Zbl 1235.74363
Summary: A numerical method is presented for computation of eigenvector derivatives used an iterative procedure with guaranteed convergence. An approach for treating the singularity in calculating the eigenvector derivatives is presented, in which a shift in each eigenvalue is introduced to avoid the singularity. If the shift is selected properly, the proposed method can give very satisfactory results after only one iteration. A criterion for choosing an adequate shift, dependent on computer hardware is suggested; it is directly dependent on the eigenvalue magnitudes and the number of bits per numeral of the computer. Another merit of this method is that eigenvector derivatives with repeated eigenvalues can be easily obtained if the new eigenvectors are calculated. These new eigenvectors lie “adjacent” to them (number of repeated eigenvalues) distinct eigenvectors, which appear when the design parameter varies. As an example to demonstrate the efficiency of the proposed method in the case of distinct eigenvalues, a cantilever plate is considered. The results are compared with those of Nelson’s method which can find the exact eigenvector derivatives. For the case of repeated eigenvalues, a cantilever beam is considered. The results are compared with those of Dailey’s method, which also can find the exact eigenvector derivatives, The design parameter of the cantilever plate is its thickness, and that of the cantilever beam its height.
MSC:
74S30Other numerical methods in solid mechanics
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
74K20Plates (solid mechanics)
74H45Vibrations (dynamical problems in solid mechanics)