On estimating the effective nonlinearity of structural modes using approximate modal shapes

We investigate the validity and accuracy of using assumed modes methods to estimate the effective nonlinearities of vibration modes. For this purpose, a problem concerning the nonlinear response of a linearly tapered cantilever beam is considered. Since, in the selected example, the linear eigenvalue problem cannot be solved analytically for the exact mode shapes, an approximate set is required to discretize the partial differential equation governing the beam's motion. To approximate the mode shapes, three methods are utilized: (i) a crude approach, which directly utilizes the linear mode shapes of a regular (untapered) cantilever beam; (ii) a finite-element approach wherein the mode shapes are obtained in ANSYS, then fitted into orthonormal polynomial curves while minimizing the least square error in the modal frequencies; and (iii) a Rayleigh-Ritz approach which utilizes a set of orthonormal trial basis functions to construct the mode shapes as a linear combination of the trial functions used. Upon discretization, the modal frequencies, the geometric and inertia nonlinearity coefficients, as well as the effective nonlinearities of the first three vibration modes are compared for eight beams with different tapering. It is shown that, even when the modal frequencies are well-approximated using the three methods, a large discrepancy is observed among the estimates of the inertia, geometric and, thereby, effective nonlinearities of the structural modes. In fact, when using the modal frequencies as a convergence measure for assumed modes methods, inaccurate, and sometimes, erroneous predictions of the effective nonlinearities can be obtained. As a result, this paper recommends that a stricter measure based on the convergence of the nonlinear coefficients be implemented for discretizing a nonlinear system using approximate mode shapes.