On utilizing approximate modal shapes to estimate the effective nonlinearity of a structural mode

This effort investigates the accuracy of estimating the effective nonlinearity of a given vibration mode using approximate modal shapes. As an example, the problem of approximating the modal effective nonlinearities of a linearly-tapered cantilever beam (along the width) is considered. This example was intentionally selected because the linear eigenvalue problem cannot be solved analytically for the exact eigenfrequencies and actual linear mode shapes of the structure, which permits investigating the influence of approximating the mode shapes on the effective nonlinearity. The nonlinear partial differential equation governing the beamâTMs motion is first discretized into an infinite set of nonlinear ordinary differential equations. The method of multiple scales is then utilized to obtain an approximate analytical expression for the effective nonlinearity which depends on the assumed mode shapes used in the series discretization. To approximate the mode shapes, three methods were utilized: i) a crude approach which directly utilizes the linear modes of a regular (untapered) cantilever beam to estimate the effective nonlinearity, ii) a finite element approach wherein the structural modes are obtained in ANSYS, then fitted into orthonormal polynomial curves while minimizing the least square error in calculating the eigenfrequencies, and iii) a Rayleigh-Ritz approach which utilizes a set of orthonormal trial basis functions to construct the structural mode shapes as a linear combination of the trial functions used. A comparison among the three methods for eight different taperings reveals that, while the modal frequencies are well-approximated yielding less than 2% deviation among the three methods, there is a huge discrepancy in approximating the nonlinear coefficients including the effective nonlinearity. This leads to the conclusion that convergence of the eigenfrequencies is not sufficient for accurate estimation of the nonlinear parameters.