Matrix Equations for the Determination of Beam Natural Frequencies

Abstract

1. Introduction. A general matrix equation is by which it is possible to determine the frequencies and modes of a beam vibrating in coupled beam bending, beam shear and torsion, with the effects of mass, torsional moment of inertia and rotary moment of inertia included. Information provided by this equation permits the determination of beam deflections, slopes and torsional rotations at points where this information is desired. To illustrate the method of imposing boundary conditions when using the general matrix equation, four special cases are developed the cantilever, free-free, fixed-fixed and hinged-hinged beams. The primary purpose of this paper, however, is to present sufficient information to enable the reader to develop solutions for whatever boundary conditions are required by him. Formulas are shown for the calculation of the influence coefficients. These formulas make it possible to vary the stiffnesses continuously throughout the span and also to account for abrupt changes in stiffnesses. Other excellent methods of determining ,influence coefficients are available and may be used in lieu of the one here. A second purpose of this paper is to demonstrate the setting forth of all information required (including equations) to transform by a continuous operation the basic data of a problem (inertia data, dimensions and section moduli) into the final characteristic matrix whose eigenvalues and eigenvectors will yield the frequencies and modes of the beam. This, of course, is essential when the entire transformation from basic data to eigenvalues and eigenvectors is to be carried out in a single continuous operation by computing machine. It should be noted that to achieve this end the nomenclature used in the many equations involved must be carefully coordinated so that no symbol is used with more than one meaning. Again, it is important to realize that the work presented here is to be regarded as a foundation and as an example to be used in constructing a system, adapted to the user’s needs. For example, Convair-Astronautics has extended the system beyond the production of eigenvalues and eigenvectors so that the end product of the program includes generalized masses, beam bending moments and shear, and beam deflections and slopes. Cases using elastic foundations have lso been developed.