Abstract
The scope of this study is to present a contribution to the geometrically nonlinear free and forced vibration of multiple-stepped beams, based on the theories of Euler–Bernoulli and von Karman, in order to calculate their corresponding amplitude-dependent modes and frequencies. Discrete expressions of the strain energy and kinetic energies are derived, and Hamilton’s principle is applied to reduce the problem to a solution of a nonlinear algebraic system and then solved by an approximate method. The forced vibration is then studied based on a multimode approach. The effect of nonlinearity on the dynamic behaviour of multistepped beams in the free and forced vibration is demonstrated and discussed. The effect of varying some geometrical parameters of the stepped beams in the free and forced cases is investigated and illustrated, among which is the variation in the level of excitation.
Highlights
In different engineering fields, such as the automotive, aeronautical, civil, or mechanical engineering, many structural components that permit the reinforcement of the whole system, a good resistance to the working loads, or the transmission of motions, can be modeled as stepped beams
A literature survey on the transverse linear vibration of stepped beams goes backwards to Taleb and Suppiger [1], who presented some of the results of the integral equation theory, applied the Cauchy function method to achieve an approximate estimation of the fundamental frequency and modal configuration of the transverse vibration of beams with one step, and compared it with the exact solution. e study of the performance of a four-degree-of-freedom-per-node element model for the vibration analysis of uniform and multistepped beams was carried out by Balasubramanian and Subramanian [2]
E main objective of this work was to present a contribution to the geometrically nonlinear free and forced vibration of beams with multiple steps based on the Euler–Bernoulli and von Karman theories
Summary
In different engineering fields, such as the automotive, aeronautical, civil, or mechanical engineering, many structural components that permit the reinforcement of the whole system, a good resistance to the working loads, or the transmission of motions, can be modeled as stepped beams. Most work on stepped beams has focused on linear vibration analysis, few papers have included the effect of geometric nonlinearity in their studies. E main objective of this work was to present a contribution to the geometrically nonlinear free and forced vibration of beams with multiple steps based on the Euler–Bernoulli and von Karman theories. E presented method can be extended for other types of beams including nano-/microbeams, previously investigated in [27] using the modified strain gradient and hyperbolic shear deformation beam theories and in [28] using the Euler–Bernoulli beam theory via the enhanced Eringen differential model It can be extended for the case of free vibration nanotubes, earlier studied in [29], using the discrete singular convolution technique
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