A Method for Determining Mode Shapes and Frequencies Above the Fundamental by Matrix Iteration

Recently, an increasing use has been made of the matrix-iteration method for determining mode shapes and frequencies, particularly with regard to dynamic problems in aircraft design. Its particular advantage is the relative ease with which it handles complex discontinuous structures whose elastic properties can be defined adequately only in terms of influence coefficients. The disadvantage of tedious calculations has been alleviated greatly by an “acceleration method” for convergence which has been described by Isakson. The predominant disadvantage to matrix iteration, however, has been the difficulty in obtaining mode shapes and frequencies higher than the fundamental. The purpose of this paper is to establish a technique for accomplishing this in a manner that is practical for use in industry, as proved by its successful application to many problems of this type in the Aero-Elastic and Structures Research Laboratory at the Massachusetts Institute of Technology. This is accomplished by applying a device worked out by L. A. Pipes, and extending it to the general case, at the same time organizing the computations in tabular form. Only a basic knowledge of matrix notation and dynamic systems is necessary to understand this development, and this can be obtained easily by a review of von Kármán and Biot’s work on this subject.