The usefulness and simplicity of an existing stiffness matrix method is emphasized by presenting it without its relatively complex proofs, and with simple examples. The method ensures that no natural frequencies are missed in undamped free vibration problems, and that no critical load factors are missed in buckling problems. This is maintained even when using the ‘exact’ member equations obtained by solving the appropriate differential equations. The method applies to all structures for which ‘exact’ member equations exist. The published applications are surveyed. They include rigidly jointed plane frames, space frames, and prismatic assemblies of flat or curved plates which are isotropic or anisotropic, and subjected to in‐plane normal or shear stresses. They also cover efficient multi‐level substructuring, design instead of analysis, damping, forced vibrations, elastic‐plastic collapse and programs for large computers, small computers, and programmable pocket calculators.

Exact Buckling and Frequency Calculations Surveyed

Substructuring techniques—status and projections

On considere 2 classes de sous-structures a symetrie cyclique autour d'un axe longitudinal. L'une n'est reliee a sa structure-mere qu'aux extremites de son axe, tandis que l'autre est reliee par un nombre quelconque de nœuds a sa structure-mere qui partage alors sa periodicite rotationnelle

Exact eigenvalue calculations for structures with rotationally periodic substructures

On the free vibration analysis of spinning structures by using discrete or distributed mass models

When exact dynamic stiffness 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 efficient 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 first reduced to a generalized linear eigenvalue problem by using Newton's method in the vicinity of an exact natural frequency identified by the Wittrick–Williams algorithm. Then the generalized linear eigenvalue problem is effectively 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. Copyright © 2003 John Wiley & Sons, Ltd.

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

Recent publications have emphasized the advantages of the sparseness of the over-all stiffness matrix of a structure. An alternative to setting up the over-all stiffness matrix is to use sub-structures. The main purpose of this paper is to show that there is equivalence between these two approaches when rows are not interchanged. It follows that a sparse matrix method can never be more efficient than the sub-structure method, if a suitable choice of sub-structures is assumed. However when identical sub-structures are contained within a structure the repetition can be utilized by the sub-structure method. In such cases the sub-structure method will often be quicker than the best sparse matrix solution.

F. W. Williams

Papers

Comparison between sparse stiffness matrix and sub‐structure methods