Nonlinear Vibration of Nanobeam With Quadratic Rational Bezier Arc Curvature

Author(s):  
Hassan Askari ◽  
Zia Saadatnia ◽  
Ebrahim Esmailzadeh

Nonlinear vibration of nanobeam with the quadratic rational Bezier arc curvature is investigated. The governing equation of motion of the nanobeam based on the Euler-Bernoulli beam theory is developed. Accordingly, the non-uniform rational B-spline (NURBS) is implemented in order to write the implicit form of the governing equation of the structure. The simply-supported boundary conditions are assumed and the Galerkin procedure is utilized to find the nonlinear ordinary differential equation of the system. The nonlinear natural frequency of the system is found and the effects of different parameters, namely, the waviness amplitude, oscillation amplitude, aspect ratio, curvature shape and the Pasternak foundation coefficient are fully investigated. The hardening and softening responses of the natural frequency of structure are detected for variations of the shape and amplitude of the curvature waviness. It is revealed that the ratio of nonlinear to linear frequency increases with an increase in the oscillation amplitudes. It is found that by increasing the Pasternak foundation coefficient, the ratio of nonlinear to linear frequency decreases.

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Yan-Shin Shih ◽  
Chen-Yuan Chung

This paper investigates the dynamic response of the cracked and flexible connecting rod in a slider-crank mechanism. Using Euler–Bernoulli beam theory to model the connecting rod without a crack, the governing equation and boundary conditions of the rod's transverse vibration are derived through Hamilton's principle. The moving boundary constraint of the joint between the connecting rod and the slider is considered. After transforming variables and applying the Galerkin method, the governing equation without a crack is reduced to a time-dependent differential equation. After this, the stiffness without a crack is replaced by the stiffness with a crack in the equation. Then, the Runge–Kutta numerical method is applied to solve the transient amplitude of the cracked connecting rod. In addition, the breathing crack model is applied to discuss the behavior of vibration. The influence of cracks with different crack depths on natural frequencies and amplitudes is also discussed. The results of the proposed method agree with the experimental and numerical results available in the literature.


2012 ◽  
Vol 468-471 ◽  
pp. 1344-1348
Author(s):  
Ying Wu ◽  
Han Bin Jia ◽  
Dong Xu Zhang ◽  
Lei Tian ◽  
Yan Jun Lü

The nonlinear dynamic behaviour of sandwich beams with different cores under transverse cycling loads is investigated in this paper. Based on Euler-Bernoulli beam theory, the second-order nonlinear ordinary differential equation of the sandwich beams with different cores is established by applying Hamilton’s principle and Galerkin method. The effects of the cores of metal foam and lightweight wood and porous metal fiber as well as pyramid lattice structure on the dynamic behaviour are studied through numerical simulations. It is shown that the dynamic behaviour of sandwich beams is not solely determined by and


Author(s):  
Hassan Askari ◽  
Ebrahim Esmailzadeh

Nonlinear forced vibration of the carbon nanotubes based on the Euler-Bernoulli beam theory is studied. The Euler-Bernoulli beam theory is implemented to find the governing equation of the vibrations of the carbon nanotube. The Pasternak and Nonlinear Winkler foundation is assumed for the objective system. It is supposed that the system is supported by hinged-hinged boundary conditions. The Galerkin procedure is employed in order to find the nonlinear ordinary differential equation of the vibration of the objective system considering two modes of vibrations. The primary and secondary resonant cases are developed for the objective system employing the multiple scales method. Influence of different factors such as length, thickness, position of applied force, Pasternak and Winkler foundation are fully shown on the primary and secondary resonance of the system.


Author(s):  
Rameshchandra P Shimpi ◽  
Rajesh A Shetty ◽  
Anirban Guha

This paper proposes a simple single variable shear deformation theory for an isotropic beam of rectangular cross-section. The theory involves only one fourth-order governing differential equation. For beam bending problems, the governing equation and the expressions for the bending moment and shear force of the theory are strikingly similar to those of Euler–Bernoulli beam theory. For vibration and buckling problems, the Euler–Bernoulli beam theory governing equation comes out as a special case when terms pertaining to the effects of shear deformation are ignored from the governing equation of present theory. The chosen displacement functions of the theory give rise to a realistic parabolic distribution of transverse shear stress across the beam cross-section. The theory does not require a shear correction factor. Efficacy of the proposed theory is demonstrated through illustrative examples for bending, free vibrations and buckling of isotropic beams of rectangular cross-section. The numerical results obtained are compared with those of exact theory (two-dimensional theory of elasticity) and other first-order and higher-order shear deformation beam theory results. The results obtained are found to be accurate.


2018 ◽  
Vol 148 ◽  
pp. 13004 ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herisanu

By using the Optimal Auxiliary Functions Method (OAFM), nonlinear free thermomechanical vibration of functionally graded beam (FGB) on Winkler-Pasternak elastic foundation is studied. Based on von Karman geometric nonlinearity, on Euler-Bernoulli beam theory and also on Galerkin procedure we obtain a second-order nonlinear differential equation with quadratic and cubic nonlinear terms. The results obtained by means of OAFM are compared and shown to be in an excellent agreement with available solutions known in the literature.


Author(s):  
Jiahao Zheng ◽  
Hongyuan Qiu ◽  
Jianming Yang ◽  
Stephen Butt

Based on linear damage accumulation law, this paper investigates the fatigue problem of drill-strings in time domain. Rainflow algorithms are developed to count the stress cycles. The stress within the drill-string is calculated with finite element models which is developed using Euler-Bernoulli beam theory. Both deterministic and random excitations to the drill-string system are taken into account. With this model, the stress time history in random nature at any location of the drill-string can be obtained by solving the random dynamic model of the drill-string. Then the random time history is analyzed using rainflow counting method. The fatigue life of the drill-string under both deterministic and random excitations can therefore be predicted.


Author(s):  
Wei-Jiun Su ◽  
Hsuan-Chen Lu

In this study, a dual-beam piezoelectric energy harvester is proposed. This harvester consists of a main beam and an auxiliary beam with a pair of magnets attached to couple their motions. The potential energy of the system is modeled to understand the influence of the potential wells on the dynamics of the harvester. It is noted that the alignment of the magnets significantly influences the potential wells. A theoretical model of the harvester is developed based on the Euler-Bernoulli beam theory. Frequency sweeps are conducted experimentally and numerically to study the dynamics of the harvester. It is shown that the dual-beam harvester can exhibit hardening effect with different configurations of magnet alignments in frequency sweeps. The performance of the harvester can be improved with proper placement of the magnets.


2015 ◽  
Vol 82 (9) ◽  
Author(s):  
X. Chen ◽  
S. A. Meguid

In this paper, we investigate the asymmetric bifurcation behavior of an initially curved nanobeam accounting for Lorentz and electrostatic forces. The beam model was developed in the framework of Euler–Bernoulli beam theory, and the surface effects at the nanoscale were taken into account in the model by including the surface elasticity and the residual surface tension. Based on the Galerkin decomposition method, the model was simplified as two degrees of freedom reduced order model, from which the symmetry breaking criterion was derived. The results of our work reveal the significant surface effects on the symmetry breaking criterion for the considered nanobeam.


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