Recursive second order convergence method for natural frequencies and modes when using dynamic stiffness matrices

Abstract

When exact dynamic sti ness matrices are used to compute natural frequencies and vibration modes for skeletal and certain other structures, a challenging transcendental eigenvalue problem results. The present paper presents a newly developed, mathematically elegant and computationally e cient method for accurate and reliable computation of both natural frequencies and vibration modes. The method can also be applied to buckling problems. The transcendental eigenvalue problem is ÿrst reduced to a generalized linear eigenvalue problem by using Newton's method in the vicinity of an exact natural frequency identiÿed by the Wittrick-Williams algorithm. Then the generalized linear eigenvalue problem is e ectively solved by using a standard inverse iteration or subspace iteration method. The recursive use of the Newton method employing the Wittrick-Williams algorithm to guide and guard each Newton correction gives secure second order convergence on both natural frequencies and mode vectors. The second order mode accuracy is a major advantage over earlier transcendental eigenvalue solution methods, which typically give modes of much lower accuracy than that of the natural frequencies. The excellent performance of the method is demonstrated by numerical examples, including some demanding problems, e.g. with coincident natural frequencies, with rigid body motions and large-scale structures.

Adaptive finite element method for eigensolutions of regular second and fourth order Sturm-Liouville problems via the element energy projection technique

“…(10) and so can be safely and efficiently solved for by using inverse iteration (Yuan et al, 2003) which guarantees convergence on the eigenpair ) , ( D  for which the absolute eigenvalue is least. The inverse iteration is terminated when , 1974;Strang and Fix, 1973).…”

“…Four of the present authors developed a recursive second order convergence method (Yuan et al, 2003) for accurate solution of both eigenvalues (natural frequencies) and eigenfunctions (modes) by using the exact Dynamic Stiffness Method (DSM). This critical success led to further progress in a series of research projects using the DSM (Djoudi et al, 2005;Yuan et al, 2007c.…”

An exact finite strip for the calculation of relative post-buckling stiffness of I-section struts

“…It can be seen in Fig. 1 that the nodal line forces and moments can be expressed in terms of internal in-plane forces and normal displacements on the edges as follows: P xi ¼ ÀN xy j y¼0 ; P xj ¼ N xy j y¼b ; P yi ¼ ÀN y j y¼0 ; P yj ¼ N y j y¼b P zi ¼ D½w; yyy þ ð2 À vÞw; xxy y¼0 ; P zj ¼ ÀD½w; yyy þ ð2 À vÞw; xxy y¼b M yi ¼ ÀD½w; yy þ vw; xx y¼0 ; M yj ¼ D½w; yy þ vw; xx y¼b (10) It is noted that in the remaining of the paper, the subscripts 0 , 1 , 2 are used for pre-buckling, buckling and post-buckling stages, respectively.…”

“…(10). The outcome can be re-arranged to obtain the following set of linear simultaneous equations for the strip, which are designated as the strip stiffness equations:…”

“…The W-W algorithm [10] is a theoretically proven, reliable and efficient tool to obtain the number of eigenvalues of transcendental eigenvalue equations to any required accuracy. The algorithm does not directly compute the eigenvalues, but instead simply finds J, the total number of eigenvalues below an arbitrarily given trial value.…”