Temperature Effects on Nonlinear Vibration Behaviors of Euler-Bernoulli Beams with Different Boundary Conditions

Yaobing Zhao,Chaohui Huang

doi:10.1155/2018/9834629

Abstract

This paper is concerned with temperature effects on the modeling and vibration characteristics of Euler-Bernoulli beams with symmetric and nonsymmetric boundary conditions. It is assumed that in the considered model the temperature increases/decreases instantly, and the temperature variation is uniformly distributed along the length and the cross-section. By using the extended Hamilton’s principle, the mathematical model which takes into account thermal and mechanical loadings, represented by partial differential equations (PDEs), is established. The PDEs of the planar motion are discretized to a set of second-order ordinary differential equations by using the Galerkin method. As to three different boundary conditions, eigenvalue analyses are performed to obtain the close-form eigenvalue solutions. First four natural frequencies with thermal effects are investigated. By using the Lindstedt-Poincaré method and multiple scales method, the approximate solutions of the nonlinear free and forced vibrations (primary, super, and subharmonic resonances) are obtained. The influences of temperature variations on response amplitudes, the localisation of the resonance zones, and the stability of the steady-state solutions are investigated, through examining frequency response curves and excitation response curves. Numerical results show that response amplitudes, the number and the stability of nontrivial solutions, and the hardening-spring characteristics are all closely related to temperature changes. As to temperature effects on vibration behaviors of structures, different boundary conditions should be paid more attention.

Highlights

Due to the importance in many applications in many fields, such as the industrial, civil, mechanical, automotive, aerospace, and other structural systems, a flexible beam with nonlinear characteristics has attracted more attention in the past few years
The linear and nonlinear vibration characteristics have been investigated for many years and were reviewed, e.g., by Nayfeh and Mook [1], Nayfeh and Balachandran [2], Nayfeh and Pai [3], Luongo and Zulli [4], and Lacarbonara et al [5]
The nonlinear vibration analysis of a curved beam subjected to the uniform base harmonic excitation with both quadratic and cubic nonlinearities was investigated by Huang et al [18]

Summary

Introduction

Due to the importance in many applications in many fields, such as the industrial, civil, mechanical, automotive, aerospace, and other structural systems, a flexible beam with nonlinear characteristics has attracted more attention in the past few years. In many applications, these structures are often subjected to vibration under thermal and dynamic loadings [19]. Large amplitude vibrations and regular and chaotic oscillations of a Timoshenko beam under the influence of temperature were analysed by Warminska et al [29, 30], and mechanical and thermal loadings have been discussed. The nonlinear free and forced vibrations of a beam-mass system under five different boundary conditions were investigated by Ozkaya et al [36]. Large amplitude vibrations of rectangular plates subjected to the radical harmonic excitation were investigated by Amabili [37], and three boundary conditions were considered. To the best of our knowledge, no specific study has addressed temperature effects on the nonlinear free and forced oscillations of the beam with different boundary conditions. At the end of the paper (Section 5), some conclusions are drawn

Mathematics for Nonlinear Modeling

Perturbation Analysis

Numerical Examples and Discussions

Summary and Conclusions