Remarkable nature of the effect of boundary conditions on closed-form solutions for vibrating inhomogeneous Bernoulli-Euler beams

Author Type

Outside Researcher

Co-Author Type 1

Outside Researcher

Co-Author Type 2

Outside Researcher

Co-Author Type 3

Outside Researcher

College

Engineering and Computer Science

Department

Ocean and Mechanical Engineering

Document Type

Article

Publication/Event/Conference Title

Chaos Solitons and Fractals

Publication Status

Version of Record

Abstract

In this paper three sets of boundary conditions are considered for reconstructing the stiffness of the inhomogeneous Bernoulli-Euler beams. The essence of the paper consists in postulating the mode shape of the vibrating beam as a static deflection of associated uniform, homogeneous beam. This unconventional way of problem formulation turns out to lead to series of new closed-form solutions. For each combination of the boundary conditions several cases of the inertial coefficients are considered. All exact solutions for natural frequencies are represented as rational expressions of the involved coefficients. Solutions are written in terms of two positive integers: `m' representing the degree of the polynomial in the inertial term and `n' indicating power in the postulated mode shape. A remarkable conclusion is reached: For specified `m' and `n', the natural frequencies of the inhomogeneous beams with different boundary conditions coalesce. This remarkable nature does not imply that these beams share the same frequencies. In fact, these are different beams for each set of boundary conditions the expression for the stiffness is different. The paper should be considered as a first step towards analysis of uncertainty, inherently present in structures.

First Page

659

Last Page

704

DOI

10.1016/S0960-0779(00)00009-6

Publication Date

1-3-2001

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