Abstract
The Lindstedt-Poincaré method is applied to a nonuniform Euler-Bernoulli beam model for the free transverse vibrations of the system. The nonuniformities in the system include spatially varying and piecewise continuous bending stiffness and mass per unit length. The expression for the natural frequencies is obtained up to second-order and the expression for the mode shapes is obtained up to first-order. The explicit dependence of the natural frequencies and mode shapes on reference values for the bending stiffness and the mass per unit length of the system is determined. Multiple methods for choosing these reference values are presented and are compared using numerical examples.
Highlights
Nonuniformities in a mechanical system can be caused by sudden or gradual changes in geometry or material properties or a combination of multiple components into the system
A simple way to account for the dynamic effects of these additional components is to use ad hoc models that account for the additional mass and stiffness of the cables on the host structure
Since these additional cables are treated as mechanical elements with stiffness and mass, their local dynamic effects may be approximated by variable mass per unit length and stiffness specific to the point of attachment to the host structure
Summary
Nonuniformities in a mechanical system can be caused by sudden or gradual changes in geometry or material properties or a combination of multiple components into the system. Differential equations with spatially variable coefficients will be developed in this work for which a perturbation theory or other approximation modelling techniques should be used in order to obtain a solution. It is not always possible to determine an exact analytical solution or express the solution in terms of these special functions with exception of very specific types of nonuniformities some of which are discussed in [7,8,9,10,11,12,13,14] This is a major difficulty with obtaining a dynamic solution for spatially varying beams. With respect to a nonuniform EulerBernoulli beam, the recently developed method of varying amplitudes, a perturbation theory approach, was applied first in [19] to a system with continuous, and periodically varying, mass, and stiffness properties. Future work includes applying the developed approximation technique to more complex geometries, such as cable-harnessed structures discussed in [1,2,3,4,5,6]
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