In this paper, buckling analysis of functionally graded Rectangular Nano-Plates considering the surface effect is investigated. Also, to define the material properties the Mori-Tanaka scheme is used which according to this scheme the material properties change continuously along the thickness direction. Displacement field is obtained using modified shear deformation theories. In these theories, against classical plate theory, the effects of rotary inertia and transverse shear deformations are considered and various functions such as exponential, trigonometric, hyperbolic and parabolic functions are used to considering these effects along the thickness direction. To considering nonlocal and surface effects the nonlocal elasticity theory and surface elasticity theory are employed respectively. The governing equations of motion are obtained by Hamilton’s principle and the Galerkin method is used to solve these equations. To show the accuracy of the present formulations, the presented results in this thesis are compared with those reported in the literature. Finally, the effects of various parameters which is related to the surface parameters such as residual surface stress, surface elasticity constant and also other parameters such as thickness ratio, aspect ratio, material index and nonlocal parameter of functionally graded Nano-Plate are investigated.

The aim of this paper is to study the free vibration of composite Rectangular piezoelectric nanoplate subjected to an electro-mechanical loading includes a biaxial force and an external voltage based on exponential shear deformation theory and trigonometric shear deformation theory in conjunction with the nonlocal elasticity theory under the simply supported boundary condition. The nonlocal theory states that stress at a point is a function of strains at all points in the continuum. The nonlocal elasticity theory becomes significant for small length scale in micro and nanostructures. In exponential shear deformation theory andtrigonometric shear deformation theory, exponential and trigonometric functions are used in terms of thickness coordinate to include the effect of transverse shear deformation and rotary inertia.  Nonlocal elasticity theory is employed to investigate effect of small scale on natural frequency of composite Rectangular piezoelectric nanoplate. It is assumed that the composite Rectangular piezoelectric nanoplate made of PZT 4 composite piezoelectric material includes crystal compounds of Pb, Zr and Ti to achieve metal-ceramic and piezoelectric properties. The governingdifferential equations of the vibration of the composite Rectangular piezoelectric nanoplate are derived by using the Hamilton’s principle, which are then solved by using the Navier method to obtain the natural frequencies of the composite Rectangular piezoelectric nanoplate. The detailed parametric study is conducted to discuss the influences of the nonlocal parameter, biaxial force external electric voltage and geometrical ratios on the first six nondimensional frequencies of the composite Rectangular piezoelectric nanoplate.

In this paper, a Mindlin Rectangular nanoplate model is developed for the bending and vibration analysis of a graphene nanoplate based on a modified couple stress theory. In order to consider the small scale effects, the modified couple stress theory, with one length scale parameter, is used. In modified couple stress theory, strain energy density is a function of strain tensor, curvature tensor, stress tensor and symmetric part of couple stress tensor. After obtaining the strain and kinetic energy, external work and substituting them in the Hamilton’ s principle, the main and auxiliary equations of the nanoplate are obtained. Then, by manipulating the boundary conditions the governing equations are solved using Navier approach for bending and vibration of the nanoplate. The bending rates and dimensionless bending values under uniform surface traction and sinusoidal load and different mode frequencies are all obtained for various plate's dimensional ratios and material length scale to thickness ratios. The effect of material length scale, length, width and thickness of the nanoplate on the bending and vibration ratios are investigated and the results are presented and discussed in details.

In the present study the free vibration analysis of functionally graded composite Rectangular nanoplates in thermal environment is investigated. The nonlocal elasticity theory based on the exponential shear deformation theory has been used to obtain the natural frequencies of the nanoplate. In exponential shear deformation theory an exponential functions are used in terms of thickness coordinate to include the effect of transverse shear deformation and rotary inertia. Nonlocal elasticity theory is employed to investigate effect of small scale on natural frequency of the functionally graded Rectangular nanoplate. The temperature is assumed to be constant in the plane of the plate and to vary in the thickness direction only. Material properties are assumed to be temperature dependent, and vary continuously through the thickness according to a power law distribution in terms of the volume fraction of the constituents. The govering equations are derived by implementing Hamilton’s principle. To show the accuracy of the formulations, present result’s inspecific cases are compared with available results in literature and good agreement are seen. Finally, the effect of various parameters such as nonlocal parameter, power law indexes, width to length ratio, the thickness to length ratio, and temperature fields on the natural frequencies of Rectangular FG nanoplates are presented and discussed in detail.

In the present work, the free vibration behavior of Rectangular graphene sheet under shear in-plane load is studied. Nonlocal elasticity theory has been implemented to study the vibration analysis of orthotropic single-layered graphene sheets (SLGSs) subjected to shear in-plane load. The SLGSs is embedded on a viscoelastic medium which is simulated as a Visco-Pasternak foundation. Using the principle of virtual work, the governing equations are derived for the Rectangular nanoplates. Differential quadrature method (DQM) is employed and numerical solutions for the vibration frequency are obtained. The influence of surrounding elastic medium, material property, aspect ratio, nonlocal parameter, length of nanoplate and effect of boundary conditions on the vibration analysis of orthotropic single-layered graphene sheets (SLGSs) is studied. Six boundary conditions are investigated. Numerical results show that the vibration frequencies of SLGSs are strongly dependent on the small scale coefficient and shear in-plane load. The present analysis results can be used for the design of the next generation of nanodevices that make use of the vibration properties of the graphene.

In this paper, axisymmetric free vibration analysis of a circular Nano-Plate having variable thickness was studied. The variation in thickness of plate was considered as a linearly in radial direction. Nonlocal elasticity theory was utilized to take into account size-dependent effects. Ritz functions was utilized to obtain the frequency equations for simply supported and clamped boundary. To verify accuracy of Ritz method, differential transform method (DTM) also used to drive the size dependent natural frequencies of circular Nano-Plates. The validity of solutions was performed by comparing present results with those of the literature for both classical plate and nano plate. Effect of nonlocal parameter, mode number and taper parameter on the natural frequency are investigated. Results showed that taper parameter has significant effect on the non-dimensional frequency and its effects on the clamped boundary condition is more than simply support.

In this research, in order to investigate the effect of the piezoelectric patch which is used as a sensor or actuator in rotating flexible structures such as a helicopter blade, the free vibrations of the rotating Rectangular sheet with and without the piezoelectric patch have been presented. First-order shear deformation theory is considered for plate displacement and piezoelectric field. Considering the effect of Coriolis acceleration, centrifugal acceleration and centrifugal in-plane forces, the equations of motion are derived from Hamilton's principle and the electromechanical couple equation is obtained from Maxwell's equation. For piezoelectric, two electrical conditions, open circuit and closed circuit, which are used in sensors and actuators, respectively, have been considered. The equations are discretized with the help of the numerical method of generalized differential squares and the matrices of inertia mass, eccentricity, Coriolis and stiffness matrix are obtained. Natural frequency values for beam and rotating plate have been compared in Abaqus software. Also, the values obtained from the numerical solution in MATLAB have been verified with articles and ABAQUS, which have high accuracy. The effect of parameters such as hub radius, rotation speed, sheet thickness, aspect ratio, piezoelectric patch thickness and applied voltage on the natural frequency of the system has also been investigated.

The nanostructures under rotation have high promising future to be used in nano-machines, nano-motors and nano-turbines. They are also one of the topics of interests and it is new in designing of rotating nano-systems. In this paper, the scale-dependent vibration analysis of a nanoplate with consideration of the axial force due to the rotation has been investigated. The governing equation and boundary conditions are derived using the Hamilton’s principle based on nonlocal elasticity theory. The boundary conditions of the nanoplate are considered as free-free in y direction and two clamped-free (cantilever plate) and clamped-simply (propped cantilever) in x direction. The equations have been solved using differential quadrature method to determine natural frequencies of the rotating nanoplate. For validation, in special cases, it has been shown that the obtained results coincide with literatures. The effects of the nonlocal parameter, aspect ratio, hub radius, angular velocity and different boundary conditions on the first three frequencies have been investigated. Results show that vibration behavior of the rotating nanoplate with cantilever boundary condition is different from other boundary conditions.

In this paper, the Ritz method has been employed to analyze the free inplane vibration of heterogeneous (non-uniform) Rectangular nanoplates corresponding to Eringen’ s nonlocal elasticity theory. The non-uniformity is taken into account using combinations of linear and quadratic forms in the thickness, material density and Young’ s modulus. Two-dimensional boundary characteristic orthogonal polynomials are applied in the Ritz method in order to examine the nonlocal effect, aspect ratio, length of nanoplate and non-uniformity parameters on the vibrational behaviors of the nanoplate. Results are verified with the available published data and good agreements are observed. The outcomes confirm apparent dependency of inplane frequency of nanoplate on the small scale effect, non-uniformity, aspect ratio and boundary conditions. For instance, frequency parameter decreases by increasing the nonlocal parameter in all vibration modes; the frequency parameters increase with length and aspect ratio of nanoplates. Furthermore, the effect of nonlocal parameters on the frequency parameter is more prominent at the higher aspect ratios. Finally, the effect of nonlocal parameter on the in-plane modes is also presented in this analysis.

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After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After curvature tensor, stress and symmetric part of the couple tensor. After obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation obtaining the strain energy, external work, and kinetic energy equation inserting them in inserting them in inserting them in inserting them in inserting them in inserting them in inserting them in inserting them in inserting them in inserting them in inserting them in inserting them in inserting them in the Hamilton principle, main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nano the Hamilton principle, main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of nanothe Hamilton principle, the main and auxiliary equations of Nano-Plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in plate are obtained. Then, by applying the boundary and force conditions in governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nano governing equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof nanogoverning equations, the vibrations of Rectangular Kirschhof Nano-Plate with plate with plate with plate with plate with plate with plate with the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated the thickness are investigated with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method with simple support around. The solution method used in this study is the Navier methodused in this study is the Navier method used in this study is the Navier methodused in this study is the Navier methodused in this study is the Navier method used in this study is the Navier methodused in this study is the Navier method used in this study is the Navier method used in this study is the Navier method used in this study is the Navier methodused in this study is the Navier methodused in this study is the Navier methodused in this study is the Navier methodused in this study is the Navier method used in this study is the Navier method used in this study is the Navier methodused in this study is the Navier methodused in this study is the Navier method used in this study is the Navier methodused in this study is the Navier method used in this study is the Navier methodused in this study is the Navier methodused in this study is the Navier method used in this study is the Navier methodused in this study is the Navier method andandand the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, the effects of material length scale, length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated length and thickness of the nanoplate on vibration are investigated results are presented and discussed in details results are presented and discussed in details results are presented and discussed in details results are presented and discussed in details results are presented and discussed in details results are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in details results are presented and discussed in details results are presented and discussed in detailsresults are presented and discussed in details results are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in details results are presented and discussed in detailsresults are presented and discussed in details results are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in details results are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in detailsresults are presented and discussed in details.