Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions

This paper investigates the thermo-electro-mechanical vibration of the rectangular piezoelectric nanoplate under various boundary conditions based on the nonlocal theory and the Mindlin plate theory. It is assumed that the piezoelectric nanoplate is subjected to a biaxial force, an external electric voltage and a uniform temperature rise. The Hamilton's principle is employed to derive the governing equations and boundary conditions, which are then discretized by using the differential quadrature (DQ) method to determine the natural frequencies and mode shapes. The detailed parametric study is conducted to examine the effect of the nonlocal parameter, thermo-electro-mechanical loadings, boundary conditions, aspect ratio and side-to-thickness ratio on the vibration behaviors.     

Nonlinear free vibration of a microscale beam based on modified couple stress theory

This paper presents a nonlinear free vibration analysis of the microbeams based on the modified couple stress Euler–Bernoulli beam theory and von Kármán geometrically nonlinear theory. The governing differential equations are established in variational form from Hamilton principle, with a material length scale parameter to interpret the size effect. These partial differential equations are reduced to corresponding ordinary ones by eliminating the time variable with the Kantorovich method following an assumed harmonic time mode. The resulting equations, which form a nonlinear two-point boundary value problem in spatial variable, are then solved numerically by shooting method, and the size-dependent characteristic relations of nonlinear vibration frequency vs. central amplitude of the microbeams are obtained successfully. The comparisons with available published results show that the current approach and algorithm are of good practicability. A parametric study is conducted involving the dependency of the frequency on the length scale parameter along with Poisson ratio, which shows that the nonlinear vibration frequency predicted by the current model is higher than that by the classical one.     

Nonlocal plate model for free vibrations of single-layered graphene sheets

Vibration analysis of single-layered graphene sheets (SLGSs) is investigated using nonlocal continuum plate model. To this end, Eringens's nonlocal elasticity equations are incorporated into the classical Mindlin plate theory for vibrations of rectangular nanoplates. In contrast to the classical model, the nonlocal model developed in this study has the capability to evaluate the natural frequencies of the graphene sheets with considering the size-effects on the vibrational characteristics of them. Solutions for frequencies of the free vibration of simply-supported and clamped SLGSs are computed using generalized differential quadrature (GDQ) method. Then, molecular dynamics (MD) simulations for the free vibration of various SLGSs with different values of side length and chirality are employed, the results of which are matched with the nonlocal model ones to derive the appropriate values of the nonlocal parameter relevant to each boundary condition. It is found that the value of the nonlocal parameter is independent of the magnitude of the geometrical variables of the system.     

The nano scale vibration of protein microtubules based on modified strain gradient theory

This paper investigates the free vibration of protein microtubules (MTs) embedded in the cytoplasm by using linear and nonlinear Euler–Bernoulli beam model based on modified strain gradient theory. The protein microtubule is modeled as a simply support or clamped–clamped beam. Beside, the elastic medium surrounding of MTs is modeled with Pasternak foundation. Vibration equations are obtained by using Hamilton principle and these equations are solved according to boundary conditions. Finally the dependency of vibration frequencies on environmental conditions, MTs size, changes of temperature and material length scale parameters (size effects) is studied. By comparing the findings, it could be said that the MTs' frequency is greatly increased in the presence of cytoplasm and it is very dependent to material length scale parameters.     

Eigenvalue analysis of Timoshenko beams and axisymmetric Mindlin plates by the pseudospectral method

A study of the free vibration of Timoshenko beams and axisymmetric Mindlin plates is presented. The analysis is based on the Chebyshev pseudospectral method, which has been widely used in the solution of fluid mechanics problems. Clamped, simply supported, free and sliding boundary conditions of Timoshenko beams are treated, and numerical results are presented for different thickness-to-length ratios. Eigenvalues of the axisymmetric vibration of Mindlin plates with clamped, simply supported and free boundary conditions are presented for various thickness-to-radius ratios.     

Exact analytical solution for free vibration of functionally graded micro/nanoplates via three-dimensional nonlocal elasticity

Using three-dimensional (3-D) nonlocal elasticity theory of Eringen, this paper presents closed-form solutions for in-plane and out-of-plane free vibration of simply supported functionally graded (FG) rectangular micro/nanoplates. Elasticity modulus and mass density of FG material are assumed to vary exponentially through the thickness of micro/nanoplate, whereas Poisson's ratio is considered to be constant. By employing appropriate displacement fields for the in-plane and out-of-plane modes that satisfy boundary conditions of the plate, ordinary differential equations of free vibration are obtained. Boundary conditions on the lateral surfaces are imposed on the analytical solutions of the equations to yield the natural frequencies of FG micro/nanoplate. The natural frequencies of FG micro/nanoplate are obtained for different values of nonlocal parameter and gradient index of material properties. The results of this investigation can be used as a benchmark for the future numerical, semi-analytical and analytical studies on the free vibration of FG micro/nanoplates.     

Transverse Vibration of Tapered Single-Walled Carbon Nanotubes Embedded in Viscoelastic Medium

Based on the nonlocal theory, Euler-Bernoulli beam theory and Kelvin viscoelastic foundation model, free transverse vibration is studied for a tapered viscoelastic single-walled carbon nanotube (visco-SWCNT) embedded in a viscoelastic medium. Firstly, the governing equations for vibration analysis are established. And then, we derive the natural frequencies in closed form for SWCNTs with arbitrary boundary conditions by applying transfer function method and perturbation method. Numerical results are also presented to discuss the effects of nonlocal parameter, relaxation time and taper parameter of SWCNTs, and material property parameters of the medium. This study demonstrates that the proposed model is available for vibration analysis of the tapered SWCNTs-viscoelastic medium coupling system.     

Nonlinear size-dependent longitudinal vibration of carbon nanotubes embedded in an elastic medium

In this paper, we study the longitudinal linear and nonlinear free vibration response of a single walled carbon nanotube (CNT) embedded in an elastic medium subjected to different boundary conditions. This formulation is based on a large deformation analysis in which the linear and nonlinear von Kármán strains and their gradient are included in the expression of the strain energy and the velocity and its gradient are taken into account in the expression of the kinetic energy. Therefore, static and kinetic length scales associated with both energies are introduced to model size effects. The governing motion equation along with the boundary conditions are derived using Hamilton's principle. Closed-form solutions for the linear free vibration problem of the embedded CNT rod are first obtained. Then, the nonlinear free vibration response is investigated for various values of length scales using the method of multiple scales.     

On the crossing points of the Lamb modes and the maxima and minima of displacements observed at the surface

This article elaborates on the crossing points of the frequency–wavenumber branches for the symmetric and anti-symmetric Lamb modes in a homogeneous plate. It is shown both theoretically as well as experimentally that at these crossing points either the normal or the longitudinal components of modal displacement attain an extreme value, i.e. a maximum or it vanishes. This behavior is assessed herein using a method due to Mindlin, who showed that the dispersion curves for a plate with mixed boundary conditions – which are associated with uncoupled shear and dilatational modes – provide bounds to the spectral lines of the free plate. Therefore, a subset of the crossing points of the symmetric and antisymmetric Lamb modes for a free plate coincide with the crossing points for a plate with mixed boundary conditions.     

Free vibration of super elliptical plates with constant and variable thickness by Ritz method

In this study free vibration of simply supported and clamped super elliptical plates is investigated. This class of plates includes a wide range of external boundaries varying from an ellipse to a rectangle. Although studies on the upper and lower bounds of these plate geometries, namely circle and rectangle, are quite extensive, contributions on the mid-shapes, especially for simply supported boundary edges are very limited. The Kirchhoff plate model with isotropic and homogeneous material is studied. The super elliptical powers are chosen from 1 to 10. The Ritz method is employed for the solution of the energy equations of the plates. The effects of Poisson's ratio, which should not be neglected for simply supported plates with curved boundaries, and the aspect ratio of the plate are both examined in detail. The effect of thickness variation is also considered in this study. In order to avoid long computational run times, physically pertinent trial functions are utilized. The frequency parameters obtained are presented and compared with published results for plate shapes that match the current cases.     

Thermomechanical bending and free vibration of single-layered graphene sheets embedded in an elastic medium

In the present paper, the sinusoidal shear deformation plate theory (SDPT) is reformulated using the nonlocal differential constitutive relations of Eringen to analyze the bending and vibration of the nanoplates, such as single-layered graphene sheets, resting on two-parameter elastic foundations. The present SDPT is compared with other plate theories. The nanoplates are assumed to be subjected to mechanical and thermal loads. The equations of motion of the nonlocal model are derived including the plate foundation interaction and thermal effects. The governing equations are solved analytically for various boundary conditions. Nonlocal theory is employed to bring out the effect of the nonlocal parameter on the bending and natural frequencies of the nanoplates. The influences of nonlocal parameter, side-to-thickness ratio and elastic foundation moduli on the displacements and vibration frequencies are investigated.     

Buckling analysis and smart control of SLGS using elastically coupled PVDF nanoplate based on the nonlocal Mindlin plate theory

This study presents an analytical approach for buckling analysis and smart control of a single layer graphene sheet (SLGS) using a coupled polyvinylidene fluoride (PVDF) nanoplate. The SLGS and PVDF nanoplate are considered to be coupled by an enclosing elastic medium which is simulated by the Pasternak foundation. The PVDF nanoplate is subjected to an applied voltage in the thickness direction which operates in control of critical load of the SLGS. In order to satisfy the Maxwell equation, electric potential distribution is assumed as a combination of a half-cosine and linear variation. The exact analysis is performed for the case when all four ends are simply supported and free electrical boundary condition. Adopting the nonlocal Mindlin plate theory, the governing equations are derived based on the energy method and Hamilton's principle. A detailed parametric study is conducted to elucidate the influences of the small scale coefficient, stiffness of the internal elastic medium, graphene length, mode number and external electric voltage on the buckling smart control of the SLGS. The results depict that the imposed external voltage is an effective controlling parameter for buckling of the SLGS. This study might be useful for the design and smart control of nano-devices.     

Nonlocal vibration of coupled DLGS systems embedded on Visco-Pasternak foundation

In this paper, vibration analysis of the coupled system of double-layered graphene sheets (CS-DLGSs) embedded in a Visco-Pasternak foundation is carried out using the nonlocal elasticity theory of orthotropic plate. The two DLGSs are coupled by an enclosing viscoelastic medium which is simulated as a Visco-Pasternak foundation. Considering the Von Kármán nonlinear strain-displacement-relations, the motion equations are derived using the Hamilton's principle. Differential quadrature method (DQM) is applied to obtain the frequency ratio for various boundary conditions. The detailed parametric study is conducted, focusing on the combined effects of the nonlocal parameter, aspect ratio, graphene sheet's size, boundary conditions and the elastic and viscoelastic medium coefficients on the frequency ratio of CS-DLGSs. In this coupled system, two case of DLGSs vibration are investigated and compared with each other: (1) In-phase vibration (2) Out-of-phase vibration. The results indicate that the frequency ratio of the CS-DLGSs is more than the single-layered graphene sheet (SLGS). The results are in good agreement with the previous researches.     

The determination of electrical parameters of quartz crystal resonators with the consideration of dissipation

Recently, as the dissipation of quartz crystal through material viscosity is being considered in vibrations of piezoelectric plates, we have the opportunity to obtain electrical parameters from vibration solutions of a crystal plate representing an ideal resonator. Since the solutions are readily available with complex elastic constants from Mindlin plate equations for thickness-shear vibrations, the calculation of resistance and other parameters related to both mechanical deformation and electrical potential is straightforward. We start with the first-order Mindlin plate equations of a piezoelectric plate for the thickness-shear vibration analysis of a simple resonator model. The electrical parameters are derived with emphasis on the resistance that is related to the imaginary part of complex elastic constants, or the viscosity. All the electrical parameters are frequency dependent, enabling the study of the frequency behavior of crystal resonators with a direct formulation. Through the full consideration of complications like partial electrodes and supporting structures, we should be able obtain electrical parameters for practical applications in resonator design.     

Nonlocal three-dimensional theory of elasticity with application to free vibration of functionally graded nanoplates on elastic foundations

In the present work, a three-dimensional (3D) elastic plate model capturing the small scale effects is developed for the free vibration of functionally graded (FG) nanoplates resting on elastic foundations. The theoretical model is formulated employing the nonlocal differential constitutive relations of Eringen in conjunction with the 3D equations of motion of elasticity.The material properties are assumed to vary continuously along the thickness of the nanoplate in accordance with the power law formulation. Through extending the generalized differential quadrature (GDQ) method to the three-dimensional case, the governing equations are simultaneously discretized in every three coordinate directions and are then recast to the standard form of an eigen value problem. Solving the acquired problem, the natural frequencies of the nanoplates with different boundary conditions are calculated. The convergence behavior of the numerical results is checked out and comparison studies are conducted to make sure of the accuracy and reliability of the present model. Finally, the dependence of the vibration behavior of the nanoplate on edge conditions, elastic coefficients of the foundation, scale coefficient, mode number, material and geometric parameters are discussed.     

Rayleigh-Ritz vibration analysis of Mindlin plates

The Rayleigh-Ritz method is applied to the prediction of the natural frequencies of flexural vibration of square plates having general boundary conditions. The analysis is based on the use of Mindlin plate theory so that the effects of shear deformation and rotary inertia are included. The spatial variations of the plate deflection and the two rotations over the plate middle surface are assumed to be series of products of appropriate Timoshenko beam functions. Results are presented for a number of types of plate and these demonstrate the manner of convergence of the method as the number of terms in the assumed series increases.     

Analytical modeling and vibration analysis of internally cracked rectangular plates

This study proposes an analytical model for nonlinear vibrations in a cracked rectangular isotropic plate containing a single and two perpendicular internal cracks located at the center of the plate. The two cracks are in the form of continuous line with each parallel to one of the edges of the plate. The equation of motion for isotropic cracked plate, based on classical plate theory is modified to accommodate the effect of internal cracks using the Line Spring Model. Berger?s formulation for in-plane forces makes the model nonlinear. Galerkin?s method used with three different boundary conditions transforms the equation into time dependent modal functions. The natural frequencies of the cracked plate are calculated for various crack lengths in case of a single crack and for various crack length ratio for the two cracks. The effect of the location of the part through crack(s) along the thickness of the plate on natural frequencies is studied considering appropriate crack compliance coefficients. It is thus deduced that the natural frequencies are maximally affected when the crack(s) are internal crack(s) symmetric about the mid-plane of the plate and are minimally affected when the crack(s) are surface crack(s), for all the three boundary conditions considered. It is also shown that crack parallel to the longer side of the plate affect the vibration characteristics more as compared to crack parallel to the shorter side. Further the application of method of multiple scales gives the nonlinear amplitudes for different aspect ratios of the cracked plate. The analytical results obtained for surface crack(s) are also assessed with FEM results. The FEM formulation is carried out in ANSYS.     

Free vibration of non-uniform nanobeams using Rayleigh–Ritz method

In this article, boundary characteristic orthogonal polynomials have been implemented in the Rayleigh–Ritz method to investigate free vibration of non-uniform Euler–Bernoulli nanobeams based on nonlocal elasticity theory. Non-uniform cross section of nanobeams has been considered by taking linear as well as quadratic variations of Young's modulus and density along the space coordinate. Detailed analysis has been reported for all the possible cases of such variations. The objective of the present study is to analyze the effects of nonlocal parameter, boundary condition, length-to-diameter ratio and non-uniform parameter on the frequency parameters. It is found that clamped nanobeams are having highest frequency parameters than other types of boundary conditions for a particular set of parameters. It is also observed that frequency parameters decrease with increase in scaling effect parameter. First four deflection shapes of non-uniform nanobeams have also been incorporated. In this analysis, some of the new results in terms of boundary conditions have also been included.     

Effect of Longitudinal Magnetic Field on Vibration Characteristics of Single-Walled Carbon Nanotubes in a Viscoelastic Medium

The effect of longitudinal magnetic field on vibration response of a sing-walled carbon nanotube (SWCNT) embedded in viscoelastic medium is investigated. Based on nonlocal Euler-Bernoulli beam theory, Maxwell’s relations, and Kelvin viscoelastic foundation model, the governing equations of motion for vibration analysis are established. The complex natural frequencies and corresponding mode shapes in closed form for the embedded SWCNT with arbitrary boundary conditions are obtained using transfer function method (TFM). The new analytical expressions for the complex natural frequencies are also derived for certain typical boundary conditions and Kelvin-Voigt model. Numerical results from the model are presented to show the effects of nonlocal parameter, viscoelastic parameter, boundary conditions, aspect ratio, and strength of the magnetic field on vibration characteristics for the embedded SWCNT in longitudinal magnetic field. The results demonstrate the efficiency of the proposed methods for vibration analysis of embedded SWCNTs under magnetic field.