Abstract
This article outlines many existing and forthcoming methods that can be used alone, or in various combinations, to accelerate the solutions of the transcendental stiffness matrix eigenproblems that arise when the stiffness matrix is assembled from exact member stiffnesses, which are obtained by solving the member differential equations exactly. Thus distributed member mass and/or the flexural effect of axial loading are incorporated exactly, and the solutions are the natural frequencies for vibration problems or the critical load factors for buckling problems.
Highlights
When member stiffness matrices are obtained by solving the differential equations, which include the distributed member mass and/or the destabilizing effect of axial force, their elements are transcendental functions of frequency and/or load factor
The overall stiffness matrix K of a structure assembled from such members is a transcendental function of the eigenparameter, p, which is the frequency in vibration problems or the load factor in buckling problems
The use of the Wittrick-Williams algorithm gives substantial time savings even when used with simple bisection routines, because it is no longer necessary to calculate IKI at fine intervals of the eigenparameter. Such solution times can be further halved by curve following methods, reduced by perhaps 70% more by substructuring, reduced much further still if the structure is linearly or rotationally periodic, and can still be further reduced by introducing linear or quadratic matrix pencil approximations to the eigenproblem at the earliest permissible stage of convergence
Summary
When member stiffness matrices are obtained by solving the differential equations, which include the distributed member mass and/or the destabilizing effect of axial force, their elements are transcendental functions of frequency and/or load factor. It is possible to analyze rotationally periodic structures by analyzing a single repeating portion for each possible circumferential wavelength in turn This too accelerates convergence on the exact eigenvalues very considerably, if complex arithmetic is used, with the special procedures of Williams (1986a) used to permit nodes and members to lie along the axis of rotational periodicity. Williams and Kennedy (1988b) accelerated convergence by replacing bisection, when the curve is suitably shaped, by carefully designed routines for following the curve of IKI versus p Such routines switch to bisection when the curve departs in major ways from being parabolic, has poles in the range considered, etc., while ensuring convergence by using the Wittrick-Williams algorithm at each iteration. Convergence can be accelerated by using methods that improve on the speed of the Gauss elimination used to reduce K to K'\ e.g., the Gauss-Doolittle method of Williams and Kennedy (1988a), which typically saves up to 25% of the elimination time
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