Analysis of modified structures by Rayleigh quotient

Abstract

In structural vibrations, the Rayleigh quotient may be used to calculate the natural frequency (or eigenvalue) of a structure given the corresponding mode shape (or eigenvector). Previous works have shown that the eigenvalue may be calculated relatively accurately using the Rayleigh quotient as long as an appropriate guess is made for the eigenvector. Typically, the Rayleigh quotient is used to predict the eigenvalues when the structure does not change. However, in this work the structure will be modified and the change in the eigenvalue will be predicted using the Rayleigh quotient. In order to use the Rayleigh quotient a guess for the eigenvector must be made. Here, the displacement vector of the nominal structure forced near a resonance will be used as the guess. Analysis and computations will be done for an undamped beam that is harmonically forced. To modify the structure, the stiffness elements will be scaled. Using this approach, it will be demonstrated that changes in the natural frequencies can be predicted relatively quickly for a modified structure. In addition, this method provides insight into which elements should be scaled to get the greatest frequency change. [Work supported by ONR under Grant N00014-19-1-2100.]