Abstract
This work deals with an improved plane frame formulation whose exact dynamic stiffness matrix (DSM) presents, uniquely, null determinant for the natural frequencies. In comparison with the classical DSM, the formulation herein presented has some major advantages: local mode shapes are preserved in the formulation so that, for any positive frequency, the DSM will never be ill-conditioned; in the absence of poles, it is possible to employ the secant method in order to have a more computationally efficient eigenvalue extraction procedure. Applying the procedure to the more general case of Timoshenko beams, we introduce a new technique, named "power deflation", that makes the secant method suitable for the transcendental nonlinear eigenvalue problems based on the improved DSM. In order to avoid overflow occurrences that can hinder the secant method iterations, limiting frequencies are formulated, with scaling also applied to the eigenvalue problem. Comparisons with results available in the literature demonstrate the strength of the proposed method. Computational efficiency is compared with solutions obtained both by FEM and by the Wittrick-Williams algorithm.
Highlights
Using an improved dynamic stiffness matrix (DSM), exact mode shape and natural frequency calculation of EulerBernoulli skeleton structures is discussed at length by Dias and Alves [4]
Dias / Power Secant Method applied to natural frequency extraction of Timoshenko beam structures concomitantly the bending and shear deflections as well as the mass and the rotatory inertia per unit length
Dias / Power Secant Method applied to natural frequency extraction of Timoshenko beam structures 309 section 6 the technique called “power deflation”: in the calculation of the nth-order eigenvalue, the determinant is deflated by a product of monomial powers; each one of these containing an eigenvalue whose order is lower than n
Summary
Using an improved DSM, exact mode shape and natural frequency calculation of EulerBernoulli skeleton structures is discussed at length by Dias and Alves [4]. For linear eigenvalue problems, whose similarity has been established by the finite elements method [1], the dynamic stiffness matrix determinant is a polynomial of equal or lesser order than the number of degrees of freedom In this case, polynomial deflation [1, 13] works perfectly as an auxiliary technique of the secant method, allowing for iterative calculation of a certain eigenvalue, in a recursive process, if the lower order eigenvalues are known. C.A.N. Dias / Power Secant Method applied to natural frequency extraction of Timoshenko beam structures 309 section 6 the technique called “power deflation”: in the calculation of the nth-order eigenvalue, the determinant is deflated by a product of monomial powers; each one of these containing an eigenvalue whose order is lower than n
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