Geometrically nonlinear free and forced vibrations of Euler-Bernoulli multi-span beams

Hatim Fakhreddine,Rhali Benamar,Ahmed Adri,Saïd Rifai,N Maia,Z Dimitrovová

doi:10.1051/matecconf/201821102001

Hatim Fakhreddine, Rhali Benamar + Show 4 more

Open Access

https://doi.org/10.1051/matecconf/201821102001

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Abstract

The objective of this paper is to establish the formulation of the problem of nonlinear transverse forced vibrations of uniform multi-span beams, with several intermediate simple supports and general end conditions, including use of translational and rotational springs at the ends. The beam bending vibration equation is first written at each span and then the continuity requirements at each simple support are stated, in addition to the beam end conditions. This leads to a homogeneous linear system whose determinant must vanish in order to allow nontrivial solutions to be obtained. The formulation is based on the application of Hamilton’s principle and spectral analysis to the problem of nonlinear forced vibrations occurring at large displacement amplitudes, leading to the solution of a nonlinear algebraic system using numerical or analytical methods. The nonlinear algebraic system has been solved here in the case of a four span beam in the free regime using an approximate method developed previously (second formulation) leading to the amplitude dependent fundamental nonlinear mode of the multi-span beam and to the corresponding backbone curves. Considering the nonlinear regime, under a uniformly distributed excitation harmonic force, the calculation of the corresponding generalised forces has led to the conclusion that the nonlinear response involves predominately the fourth mode. Consequently, an analysis has been performed in the neighbourhood of this mode, based on the single mode approach, to obtain the multi-span beam nonlinear frequency response functions for various excitation levels.

Highlights

The analysis of structural vibration is of a crucial importance in many fields, such as aerospace, aeronautics, mechanical and civil engineering
The main aim of this study is to examine the effect of geometrical nonlinearity, occurring at large vibration amplitudes, as well as to extend the Benamar’s method to the case of multi-span beams with intermediates simple supports
The non-linear frequency ratio versus the non-dimensional vibration amplitude curves of a uniform beam with three intermediate simple supports is shown in figure Fig. 2. for the first non-linear mode shape

Summary

Introduction

The analysis of structural vibration is of a crucial importance in many fields, such as aerospace, aeronautics, mechanical and civil engineering. Due to the improvement of the structure characteristics and the increasing use of new materials, the nonlinear vibration analysis is increasingly becoming a necessary step in the design and verification processes in many situations. The purpose of this work is to establish the formulation of the nonlinear transverse forced vibration of uniform multi-span beams with intermediate simple supports under various end conditions. Different beam end conditions may be examined by the present formulation involving at the beam ends two types of springs i.e. Rotational and translational springs. The main aim of this study is to examine the effect of geometrical nonlinearity, occurring at large vibration amplitudes, as well as to extend the Benamar’s method to the case of multi-span beams with intermediates simple supports. By the appropriate use of the single mode approach in the neighbourhood of the predominant mode excited by a uniformly distributed applied load, the nonlinear frequency response functions have been determined and plotted for various excitation levels

Linear mode shapes of an Euler-Bernoulli multi-span beam

Nonlinear formulation

Numerical results and discussion

Conclusion