## Date of Award

12-2016

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## Legacy Department

Mechanical Engineering

## Committee Member

Dr. Mohammed Daqaq, Committee Chair

## Committee Member

Dr. Gang Li

## Committee Member

Dr. Phanindra Tallapragada

## Committee Member

Dr. Lonny Thompson

## Abstract

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 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 beam's structural modes. Specifically, it is clearly shown that, 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, it is recommended that a stricter measure based on the convergence of the nonlinear coefficients be implemented for discretizing a nonlinear system using approximate mode shapes. Finally, in the third part, the dissertation presents a closed-form analytical solution of the linear eigenvalue problem associated with a beam of a constant thickness and a linearly-varying width in the form of the general Meijer-G functions. Using this approach, the exact linear modal frequencies and shapes are obtained and, for the first time, used in the discretization of the nonlinear partial differential equation describing the dynamics of the system. The discretized system of ordinary differential equations is then solved using the method of multiple scales to obtain an approximate analytical solution describing the modal effective nonlinearities of the beam. It is shown that the effective nonlinearity of the first mode is always of the hardening type regardless of the degree of tapering and that the magnitude of the effective nonlinearity increases for beams that are tapered inwards, e.g. wider at the clamped end. It is also demonstrated that, the second and third modes always exhibit a softening nonlinearity which increases in magnitude for beams that are tapered inwards. Results were also compared to a finite element (FE) solution of the linear eigenvalue problem wherein the modal shapes obtained using the FE method are fit into a set of orthogonal polynomial functions and used to discretize the nonlinear problem. It is shown that, while the modal frequencies obtained using the FE method approximate those obtained analytically with negligible error (less than 1\%), there is a substantial error in the resulting estimates of the modal effective nonlinearity. This indicates that, even negligible errors in the approximate solution of the linear problem can propagate to become significant when analyzing the nonlinear problem reinforcing the importance of the exact solution.

## Recommended Citation

da Silva, Clodoaldo Jose, "On The Modal Effective Nonlinearity of A Class of Tapered-Cantilever Beams" (2016). *All Dissertations*. 1848.

https://open.clemson.edu/all_dissertations/1848