A Comparison of Higher-Order Difference Methods in the Solution of Beam-Vibration Problems

Several higher-order difference methods are investigated and compared for the problem of finding the natural frequencies of the lateral vibration of a beam. All of the methods considered are applicable to either digital or analog computers, although particular reference is made to the analog computer. The methods considered in most detail use the same basic central difference approximation, the variations occurring in the method of representing boundary conditions. Three higher-order approaches to the problem of boundary conditions are presented. They are 1) the use of one-sided differences of fourth order, 2) the use of symmetry assumptions, and 3) the passive-circuit approach. Each method is shown to have its advantages, the final choice depending upon the particular requirements of the problem. Results are presented in the form of curves of percentage mode-frequency error vs number of cells for the various approximation methods.