Nonlinear Vibrational Analysis of Nanobeams Embedded in an Elastic Medium including Surface Stress Effects (original) (raw)
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Nonlinear free vibration of piezoelectric nanobeams incorporating surface effects
2014
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 K´ arm´ an geometric nonlinearity. In order to satisfy the balance conditions between the nanobeam bulk and its surfaces, the component of the bulk stress, zz, 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 pie...
This study presents an examination of nonlinear free vibration of a nanobeam under electro-thermo-mechanical loading with elastic medium and various boundary conditions, especially the elastic boundary condition. The nanobeam is modeled as an Euler–Bernoulli beam. The von Kármán strain-displacement relationship together with Hamilton's principle and Eringen's theory are employed to derive equations of motion. The nonlinear free vibration frequency is obtained for simply supported (S-S) and elastic supported (E-E) boundary conditions. E-E boundary condition is a general and actual form of boundary conditions and it is chosen because of more realistic behavior. By applying the differential transform method (DTM), the nanobeam's natural frequencies can be easily obtained for the two different boundary conditions mentioned above. Performing a precise study led to investigation of the influences of nonlocal parameter, temperature change, spring constants (either for elastic medium or boundary condition) and imposed electric potential on the nonlinear free vibration characteristics of nanobeam. The results for S-S and E-E nanobeams are compared with each other. In order to validate the results, some comparisons are presented between DTM results and open literature to show the accuracy of this new approach. It has been discovered that DTM solves the equations with minimum calculation cost.
In this paper, surface effects including surface elasticity, surface stress and surface density, on the free vibration analysis of Euler-Bernoulli and Timoshenko nanobeams are considered using nonlocal elasticity theory. To this end, the balance conditions between nanobeam bulk and its surfaces are considered to be satisfied assuming a linear variation for the component of the normal stress through the nanobeam thickness. The governing equations are obtained and solved for Silicon and Aluminum nanobeams with three different boundary conditions, i.e. Simply-Simply, Clamped-Simply and Clamped-Clamped. The results show that the influence of the surface effects on the natural frequencies of the Aluminum nanobeams follows the order CC<CS<SS while this is not the case for Silicon nanobeams. On the other hand, the influence of the nonlocal parameter is opposite and follows the order SS<CS<CC. In addition, it is seen that considering rotary inertia and shear deformation has more...
Journal of Thermal Stresses, 2020
Nonlinear vibration of nanobeams embedded in the linear and nonlinear elastic materials under magnetic and temperature effects is investigated in this study. Von Karman's strain-displacement relation is applied to a nonlocal Euler-Bernoulli beam model. Equation of motion is derived using Hamilton's principle. Galerkin's method is applied to decompose the nonlinear partial differential equation into a nonlinear ordinary differential equation (NODE). The NODE is solved using He's method. The nanobeams are embedded in the Winkler, Pasternak, and nonlinear elastic media. The effects of low and high temperatures, nonlocal parameter, magnetic force, amplitude, and linear and nonlinear elastic materials are examined.
Vibration analysis of three-layered nanobeams based on nonlocal elasticity theory
Journal of Theoretical and Applied Mechanics, 2017
In this paper, the first investigation on free vibration analysis of three-layered nanobeams with the shear effect incorporated in the mid-layer based on the nonlocal theory and both Euler Bernoulli and Timoshenko beams theories is presented. Hamilton's formulation is applied to derive governing equations and edge conditions. In order to solve differential equations of motions and to determine natural frequencies of the proposed three-layered nanobeams with different boundary conditions, the generalized differential quadrature (GDQM) is used. The effect of the nanoscale parameter on the natural frequencies and deflection modes shapes of the three layered-nanobeams is discussed. It appears that the nonlocal effect is important for the natural frequencies of the nanobeams. The results can be pertinent to the design and application of MEMS and NEMS.
Vibration analysis of nonlocal nanobeams based on Euler-Bernoulli and Timoshenko beam theories is considered. Nonlocal nanobeams are important in the bending, buckling and vibration analyses of beam-like elements in microelectromechanical or nanoelectromechanical devices. Expressions for free vibration of Euler-Bernoulli and Timoshenko nanobeams are established within the framework of Eringen's nonlocal elasticity theory. The problem has been solved previously using finite element method, Chebyshev polynomials in Rayleigh-Ritz method and using other numerical methods. In this study, numerical results for free vibration of nanobeams have been presented using simple polynomials and orthonormal polynomials in the Rayleigh-Ritz method. The advantage of the method is that one can easily handle the specified boundary conditions at the edges. To validate the present analysis, a comparison study is carried out with the results of the existing literature. The proposed method is also validated by convergence studies. Frequency parameters are found for different scaling effect parameters and boundary conditions. The study highlights that small scale effects considerably influence the free vibration of nanobeams. Nonlocal frequency parameters of nanobeams are smaller when compared to the corresponding local ones. Deflection shapes of nonlocal clamped Euler-Bernoulli nanobeams are also incorporated for different scaling effect parameters, which are affected by the small scale effect. Obtained numerical solutions provide a better representation of the vibration behavior of short and stubby micro/nanobeams where the effects of small scale, transverse shear deformation and rotary inertia are significant.
Applied Mathematical Modelling, 2014
In this paper, to consider all surface effects including surface elasticity, surface stress, and surface density, on the nonlinear free vibration analysis of simply-supported functionally graded Euler-Bernoulli nanobeams using nonlocal elasticity theory, the balance conditions between FG nanobeam bulk and its surfaces are considered to be satisfied assuming a cubic variation for the component of the normal stress through the FG nanobeam thickness. The nonlinear governing equation includes the von Kármán geometric nonlinearity and the material properties change continuously through the thickness of the FG nanobeam according to a power-law distribution of the volume fraction of the constituents. The multiple scale method is employed as an analytical solution for the nonlinear governing equation to obtain the nonlinear natural frequencies of FG nanobeams. The effect of the gradient index, the nanobeam length, thickness to length ratio, mode number, amplitude of deflection to radius of gyration ratio and nonlocal parameter on the frequency ratios of FG nanobeams is investigated.
Acta Mechanica, 2020
Based on the modified couple stress theory and Gurtin-Murdoch surface elasticity theory, a sizedependent Timoshenko beam model is developed for investigating the nonlinear vibration response of functionally graded (FG) micro-/nanobeams. The model is capable of capturing the simultaneous effects of microstructure couple stress, surface energy, and von Kármán's geometric nonlinearity. Sigmoid function and power law homogenization schemes are used to model the material gradation of the beam. Hamilton's principle is exploited to establish the nonclassical nonlinear governing equations and corresponding higher-order boundary conditions. To account for the nonhomogeneity in boundary conditions, the solution of the problem is split into two parts: the nonlinear static response with the nonhomogeneous boundary conditions and the nonlinear dynamic response. The resulting boundary conditions for the dynamic response are homogeneous, and so Galerkin's approach is applied to reduce the set of PDEs to a nonlinear system of ODEs. The generalized differential quadrature method in terms of spatial variables is applied to obtain the static response and linear vibration mode. Considering the nonlinear system of ODEs in terms of time-related variables, both pseudo-arclength continuation and Runge-Kutta methods are used to obtain the nonlinear free vibration behavior of FG Timoshenko micro-/nanobeams with simply supported and clamped ends. Verification of the proposed model and solution procedure is performed by comparing the obtained results with those available in the open literature. The effects of the nonhomogeneous boundary conditions, surface elasticity modulus, surface residual stress, material length scale parameter, gradient index, and thickness on the characteristics of linear and nonlinear free vibrations of sigmoid function and power law FG micro-/nanobeams are discussed in detail.
Analytical Solutions for Vibration of Simply Supported Nonlocal Nanobeams with an Axial Force
International Journal of Structural Stability and Dynamics, 2011
This paper presents exact, analytical solutions for the transverse vibration of simply supported nanobeams subjected to an initial axial force based on nonlocal elasticity theory. Classical continuum theory is inherently size independent while nonlocal elasticity exhibits size dependence. The latter has significant effects on bending moment, which results in a conceptually different definition of a new effective nonlocal bending moment with respect to the corresponding classical bending moment. A sixth-order partial differential governing equation is subsequently obtained. The effects of nonlocal nanoscale on the vibration frequencies and mode shapes are considered and analytical solutions are solved. Effects of the nonlocal nanoscale and dimensionless axial force including axial tension and axial compression on the first three mode frequencies are presented and discussed. It is found that the nonlocal nanoscale induces higher natural frequencies and stiffness of the nano structures.
Vibration of layered nanobeams with periodic nanostructures
Mechanics Based Design of Structures and Machines, 2020
Experimental studies show that softening or hardening behaviors of micro/ nanostructures depend on the microstructure of the considered material. Some scale dependent theories like nonlocal elasticity, strain gradient models, and modified couple stress theory predict only softening or hardening behavior. Unlike these theories, doublet mechanics predicts both softening and hardening behaviors in some nanomaterial structures. In the present study, vibration of layered nanobeams with periodic nanostructures is investigated using doublet mechanics theory. After deriving the governing equations, Navier-type solution is applied in the vibration analysis of layered nanobeams. Parametric results are presented for different material and geometrical properties.