On The Modal Effective Nonlinearity of A Class of Tapered-Cantilever Beams

The objective of this dissertation is to investigate the influence of i) higher-order nonlinearities, and ii) assumed mode discretization on the approximate modal effective nonlinearity associated with the flexural response of thin metallic cantilever beams of a constant thickness and a linearly-varying width. To this end, in the first part of this dissertation, a nonlinear model of an Euler-Bernoulli metallic cantilever beam with an arbitrarily variable cross-section is derived. Unlike the common literature wherein the Lagrangian is truncated using a fourth-order Taylor series; here the series is truncated at the eighth order to study the influence of higher-order nonlinearities on the modal effective nonlinearity of a given vibration mode. The influence of the cubic, quintic, and septic nonlinearities on the modal effective nonlinearity is then investigated. It is shown that, in general, quintic and septic nonlinearities have a negligible influence on the estimates of the effective nonlinearity and, therefore, can be neglected in any further analysis. In the second part, the dissertation investigates the validity and accuracy of using assumed modes methods to estimate the effective nonlinearities of the beam’s vibration modes. Since the linear eigenvalue problem associated with our problem is very hard to solve analytically for the exact mode shapes, an approximate set is assumed 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 prismatic cantilever beam, ii) a finite element approach wherein the mode shapes are obtained computationally in ANSYS, then fit 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 and the effective nonlinearities of the first three vibration modes are compared for eight beams with different tapering along the width. It is shown that, even when the modal

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