Nonlinear Free Vibration Of Nanobeams Research Articles

Nonlinear vibration analysis of two-phase local/nonlocal nanobeams with size-dependent nonlinearity by using Galerkin method

It has been proved that using pure nonlocal elasticity, especially in differential form, leads to inconsistent and unreliable results. Therefore, to obviate these weaknesses, Eringen’s two-phase local/nonlocal elasticity has been recently used by researchers to consider the nonlocal size dependency of nanostructures. Given this, for the first time, the size-dependent nonlinear free vibration of nanobeams is investigated in this article within the framework of two-phase elasticity by using the Galerkin method. Contrary to differential nonlocal elasticity, the size dependency of the axial tension force, due to von Karman nonlinearity, is considered, and its effect on the nonlinear vibration is examined. The correct procedure of using the Galerkin method for studying the nonlinear vibration of two-phase nanobeams is introduced. It is shown that, although it is possible to extract the linear mode shapes of two-phase nanobeams using its equal differential equation, integral form of two-phase elasticity should be considered in the Galerkin method. Furthermore, it is observed that in two-phase elasticity, applying classic mode shapes in the Galerkin method leads to significant errors, especially in higher nonlocal parameters and lower values of local phase fraction and amplitude ratios.

Nonlinear free vibration of nanobeams based on nonlocal strain gradient theory with the consideration of thickness-dependent size effect

Nonlinear free vibration of nanobeams based on nonlocal strain gradient theory with the consideration of thickness-dependent size effect

Exact solution for large amplitude flexural vibration of nanobeams using nonlocal Euler-Bernoulli theory

In this paper, nonlinear free vibration of nanobeams with various end conditions is studied using the nonlocal elasticity within the frame work of Euler-Bernoulli theory with von K´arm´an nonlinearity. The equation of motion is obtained and the exact solution is established using elliptic integrals. Two comparison studies are carried out to demonstrate accuracy and applicability of the elliptic integrals method for nonlocal nonlinear free vibration analysis of nanobeams. It is observed that the phase plane diagrams of nanobeams in the presence of the small scale effect are symmetric ellipses, and consideration the small scale effect decreases the area of the diagram.

Nonlocal nonlinear free vibration of nanobeams with surface effects

Abstract In this paper, nonlinear free vibration analysis of simply-supported nanoscale beams incorporating surface effects, i.e. surface elasticity, surface tension and surface density, is studied using the nonlocal elasticity within the frame work of Euler–Bernoulli beam theory with von karman type nonlinearity. A linear variation for the component of the bulk stress, σzz, through the nanobeam thickness is used to satisfy the balance conditions between the nanobeam bulk and its surfaces. An exact analytical solution to the governing equation of motion is presented for natural frequencies of nanobeams using elliptic integrals. The effect of the nanobeam length, thickness to length ratio, mode number, amplitude of deflection to radius of gyration ratio and nonlocal parameter on the normalized natural frequencies of aluminum and silicon nanobeams with positive and negative surface elasticity, respectively, is investigated. It is observed that the surface effects increase natural frequencies of the aluminum nanobeam for all values of the amplitude ratio and the silicon nanobeam at low amplitude ratios while at higher amplitude ratios the surface effects decrease the natural frequencies of the silicon nanobeam. Also, for all values of amplitude ratios, the normalized fundamental natural frequencies of silicon and aluminum nanobeams vary linearly with respect to the nonlocal parameter while this is not the case at higher mode numbers.

Nonlinear free vibration of piezoelectric nanobeams incorporating surface effects

In this study, the nonlinear free vibration of piezoelectric nanobeams incorporating surface effects (surface elasticity, surface tension, and surface density) is studied. The governing equation of the piezoelectric nanobeam is derived within the framework of Euler–Bernoulli beam theory with the von Kármán geometric nonlinearity. In order to satisfy the balance conditions between the nanobeam bulk and its surfaces, the component of the bulk stress, σzz, is assumed to vary linearly through the nanobeam thickness. An exact solution is obtained for the natural frequencies of a simply supported piezoelectric nanobeam in terms of the Jacobi elliptic functions using the free vibration mode shape of the corresponding linear problem. Then, the influences of the surface effects and the piezoelectric field on the nonlinear free vibration of nanobeams made of aluminum and silicon with positive and negative surface elasticity, respectively, have been studied for various properties of the piezoelectric field, various nanobeam sizes and amplitude ratios. It is observed that if the Young’s modulus of a nanobeam is lower, the effect of the piezoelectric field on the frequency ratios (FRs) of the nanobeam will be greater. In addition, it is seen that by increasing the nanobeam length so that the nanobeam cross section is set to be constant, the surface effects and the piezoelectric field with negative voltage values increases the FRs, whereas it is the other way around when the nanobeam cross section is assumed to be dependent on the length of the nanobeam.

Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory

Abstract In this article, the nonlinear free vibration behavior of Timoshenko nanobeams subject to different types of end conditions is investigated. The Gurtin–Murdoch continuum elasticity is incorporated into the Timoshenko beam theory in order to capture surface stress effects. The nonlinear governing equations and corresponding boundary conditions are derived using Hamilton's principle. A numerical approach is used to solve the problem in which the generalized differential quadrature method is applied to discretize the governing equations and boundary conditions. Then, a Galerkin-based method is numerically employed with the aim of reducing the set of partial differential governing equations into a set of time-dependent ordinary differential equations. Discretization on time domain is also done via periodic time differential operators that are defined on the basis of the derivatives of a periodic base function. The resulting nonlinear algebraic parameterized equations are finally solved by means of the pseudo arc-length continuation algorithm through treating the time period as a parameter. Numerical results are given to study the geometrical and surface properties on the nonlinear free vibration of nanobeams.

An analytical study on the nonlinear free vibration of nanoscale beams incorporating surface density effects

The nonlinear free vibration of nanobeams with considering surface effects (surface elasticity, tension and density) is studied using Euler–Bernoulli beam theory including the von kármán geometric nonlinearity. The component of plane stress, σzz, is assumed to vary linearly through the beam thickness and satisfy the balance conditions between nanobeam bulk and its surfaces. Accordingly, surface density is introduced into the governing equation of the nonlinear free vibration of nanobeams. It is seen that the effect of surface density is independent of amplitude ratio. In addition, it is observed that in lower modes, surface density has insignificant effects on the variation of the natural frequency versus mode number, whereas this is not the case in higher modes where the surface density causes the normalized natural frequencies of the nanobeams to increase drastically. Moreover, it is shown that the effect of the surface density on the variation of the natural frequency of the nanobeam versus the thickness ratio decreases consistently with the increase of the mode number.

Surface effects on nonlinear free vibration of nanobeams

The nonlinear flexural vibrations of micro- and nanobeams in presence of surface effects are studied within the framework of Euler–Bernoulli beam theory including the von Kármán geometric nonlinearity. Exact solution is obtained for the natural frequencies of a simply-supported nanobeam in terms of the Jacobi elliptic functions by using the free vibration modes of the corresponding linear problem. Numerical results include the normalized natural frequencies of vibration as functions of mode number, vibration amplitude, and nanobeam length and thickness. Also, the influence of surface effects on the system phase trajectory is considered.