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In this paper, a novel four-variable refined theory of plate, called RPT, has been proposed for free vibration of composite laminated plates, using a hyperbolic sine function, for calculating out-of-plane shear strains. It is one of the properties of this theory that the boundary condition of zero shear stress is satisfied over upper layer and under lower layer of plate, with no reference to Timoshenko shape factor. In contrast to other higher-order shear deformation theories, in RPT theory, equations of motion are coupled dynamically only in inertial terms, while elastic energy terms are not coupled for the variables used. From this viewpoint, RPT theory is similar to classical plate theory (CLPT). Some of the objectives of this paper are the investigation of effect of influential parameters on fundamental frequency, such as modulus ratio, angle of plies, and plate length-to-thickness ratio. The results of this proposed version of RPT are compared and validated with those of first-order shear deformation theory (FSDT), higher-order shear deformation theory (HSDT), and the original version of RPT.

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In this paper the meshless local Petrov-Galerkin (MLPG) method is implemented to study the vibration of a Functionally Graded Material (FGM) cylindrical shell. Displacement field equations, based on Donnell and first order shear deformation theory, are taken into consideration. Material properties are assumed to be temperature-dependent and graded in the thickness direction according to different volume fraction functions. A FGM cylindrical shell made up of a mixture of ceramic and metal is considered herein. The set of governing equations of motion are numerically solved by the Meshless method in which a new variational trial-functional is constructed to derive the stiffness and mass matrices so the natural frequencies are obtained in various boundary conditions by using discretization procedure and solving the general eigenvalue problem. The influences of some commonly used boundary conditions, variations of volume fractions and effects of shell geometrical parameters are studied. The results show the convergence characteristics and accuracy of the mentioned method.

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In this paper, the free vibration of a two-dimensional functionally graded circular cylindrical shell is analyzed.To describe the material properties of the two-phased FGM material Mori–Tanaka micromechanical model is used. The spatial derivatives of the equations of motion and boundary conditions are discretized using the methods of generalized differential – Integral quadrature (GDIQ). To validate the results, comparisons are made with the solutions for FG cylindrical shells available in the literature. The results of this study show that the values of natural frequency of 2D FGMs are higher than those of 1D FGMs in parallel conditions. Furthermore, application of a confining elastic foundation increases the value of natural frequencies. The results of this study show that the values of natural frequency of 2D FGMs are higher than those of 1D FGMs in parallel conditions. Furthermore, application of a confining elastic foundation increases the value of natural frequencies. The results of this study show that the values of natural frequency of 2D FGMs are higher than those of 1D FGMs in parallel conditions. Furthermore, application of a confining elastic foundation increases the value of natural frequencies.

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In this paper a two dimensional elasticity for free vibrations and the effect of elastic foundantion on a two-direction functionally graded beams with integrated surface piezoelectric layers with combination of differential quadrature method and space-state method is presented here. Differential quadrature method in axial direction and space-state method in transverse direction is used. It’s considered that two parameters model or winkler-pasternak for elastic foundation which has been considered two kinds of boundary conditions include simply support and clamped-clamped. Also, It is assumed that beam properties in thickness and axial direction varying exponentially and poison factor is constant which has been considered the effects of materials properties gradient index and number waves on free vibrations beams. The obtained results show that this method has good accuracy and high speed of convergence.

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In this paper, for the free vibration analysis of bilayer graphenes with interlayer shear effect the sandwich beam model is introduced. Because of the similarity between the bilayer graphene and the sandwich structures, in which at the top and the bottom of the bilayer graphene there is a single layer graphene and between them there is Vander walls bindings, the bilayer graphene is modeled as a sandwich beam and its free vibration is investigated for free-clamp end condition. To obtain the governing equations, each graphene layer is modeled based on the Euler-Bernoulli theory and in-plane displacements are also considered in addition to the transverse displacement. It is also assumed that the graphene layers do not have relative displacement during vibration. The effect of the Vander walls bindings is introduced in the governing equations as the shear modulus. The results obtained by the sandwich beam model, presented in this paper for the first time, include the first five natural frequencies of the bilayer graphenes with 7 to 20 nanometer lengths. These results are validated by the molecular dynamic and the Multi-Beam-Shear model results.

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In this paper, exact closed-form solutions in explicit forms are presented to investigate small scale effects on the transverse vibration behavior of Lévy-type rectangular nanoplates based on the Reddy’s nonlocal third-order shear deformation plate theory. Two other edges may be restrained by different combinations of free, simply supported, or clamped boundary conditions. Hamilton’s principle is used to derive the nonlocal equations of motion and natural boundary conditions of the nanoplate. Two comparison studies with analytical and numerical techniques reported in literature are carried out to demonstrate the high accuracy of the present new formulation. Comprehensive benchmark results with considering the small scale effects on frequency ratios and non-dimensional fundamental natural frequencies of rectangular nanoplates with different combinations of boundary conditions are tabulated for various values of nonlocal parameters, aspect ratios and thickness to length ratios. Due to the inherent features of the present exact closed-form solution, the present findings will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future. Also, the present study may be useful for static and dynamic analysis of thicker nano scale plate-like structures, multi-layer graphene and graphite as composite or sandwich structures.

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In this study, static and free vibration behaviors of two type of sandwich plates based on the three dimensional theory of elasticity are investigated. The core layer of one type is functionally graded (FG) with the homogeneous face sheets where as in second type the core layer is isotropic with the face sheets FG material. Plate is under uniform pressure at the top surface and free from traction in the bottom surface. The effective material properties of FG layers are estimated to vary continuously through the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. State space differential equations are obtained from equilibrium equations and constitutive relations. The obtained governing differential equations are solved by using Fourier series expansion along the in plane directions and state space technique across the thickness direction. Accuracy and exactness of the present approach is validated by comparing the numerical results with the published results. Furthermore it is possible to validate the exactness of the conventional two dimensional theories. Finally the influences of volume fraction, width-to-thickness ratios and aspect ratio on the vibration and static behaviors of plate are investigated.

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In this paper, Free Vibration analysis of truncated conical shell Reinforced with single-walled carbon nanotubes for uniformly distribution (UD), resting on Pasternak elastic foundation, based on the first order shear deformation plate theory is investigated. The rule of mixture is used to effect of the properties of nanotubes in the mentioned structure. Based on the displacement field according to the first order shear deformation theory, after determining the strain components in the curvilinear coordinates and simplifying derived relation, we compute the strain components in conical coordinate. Then, the stress components are derived by the Hook’s law. In the next stage, by computing the total potential energy of system by regarding the effect of Pasternak elastic foundation and regarding the suitable functions for displacements, by applying the Ritz method the natural frequency of system have been derived. At the end, the effect of volume fraction of nanotubes, ratio of thickness to radius of cone, elastic constants and other parameters, on the natural frequency of structure have been investigated. Also, it can be observe close agreements between present results and other papers.

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In the present study the free vibration analysis of functionally graded rectangular nanoplates in thermal environment is investigated. The modified coupled stress theory based on the first order shear deformation theory has been used to obtain the natural frequencies of the nanoplate. Modified coupled stress theory is a non-classical theory. In this theory material length scale parameter is applied to capture the size effect of the microstructures which the earlier classical plate theories were not able to explain these effects. The functionally graded material properties are varied continuously and smoothly along the thickness. The Poisson’s ratio of the FGM plate is assumed to be constant in the whole plate. In order to validate the present method, the natural frequencies of the both functionally graded rectangular plate and rectangular nanoplates are compared with those are reported in the literature, separately. Finally, the effect of various parameters such as; the power law index, the thickness to length scale parameter ratio, aspect ratio, thickness ratio on the natural frequencies of plates in thermal environments with different temperatures are presented and discussed in detail.

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In this paper, free vibration analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical shells surrounded by elastic foundation and subjected to uniform temperature rise loading is investigated. The material properties of FG-CNTRC are assumed to be graded through the thickness direction. Two kinds of carbon nanotube reinforced composites including uniformly distributed (UD) and functionally graded (FG) are considered. The elastic foundation is modeled by two-parameter Pasternak model, which is obtained by adding a shear layer to the Winkler model. The effect of thermal loading is considered as a initial stress. Applying the Hamilton’s principle based on first-order shear deformation theory and considering Sanders and Donnell strain-displacement relation, the governing equations are obtained. Using the generalized differential quadrature method in axial direction and periodic differential matrix operators in circumferential direction, the equilibrium equations are discretized. The results are compared with those presented in literature. In addition, the effect of various parameters such as thermal loading, boundary conditions, elastic foundation and different geometrical conditions are studied. The results show that increase in the elastic foundation coefficients and initial thermal loading increase and decrease the non-dimensional fundamental frequency, respectively.

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Free vibration characteristics of rectangular composite plate with constrained layer damping and magneto-rheological fluid (MR) core are presented.. Hamilton principal is used to obtain the equation of motion of the sandwich plate. Based on the Navier method, a closed-form solution is presented for free vibration analysis of MR sandwich plate under simply supported boundary conditions. The governing equation of motion is derived on the base of classical lamination theory for the faceplates. Only shear strain energy density of the core is considered. Using displacement continuity conditions at the interface of the layers and core, shear strain of the core is expressed in terms of displacement components of the base and constraint layers. The complex shear modulus of the MR material in the pre-yield region was described by complex modulus approach as a function of magnetic field intensity. The validity of the developed formulation is demonstrated by comparing the results in terms of natural frequencies with those in the available literature. The effects of magnetic field intensity, plate aspect ratio, thicknesses of the MR core, base layer and constrained layer for three different stacking sequences of composite faceplates on the fundamental frequency and loss factor of the first mode are discussed. The results indicate significant effect of physical and geometrical parameters on the natural frequency and loss factor associated with the first mode.

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The linearity is a simplifying assumption in most vibration problems of real mechanical systems which may, is not turn, lead to a considerable error in predicting the system dynamic response. Determining a suitable mathematical model for a nonlinear vibrating system is an important step in order to analyze the structural dynamics behavior efficiently. When the amplitude of vibration is large, the system is said to be geometrically nonlinear. In this paper, the nonlinear identification of a cantilever slender beam undergoing large amplitude free vibration has been investigated. Because of no excitation force in this situation and lack of information about its response, the existing identification methods are not efficient. In present research a new approach based on optimum correction factor of terms having uncertainty is used and identification has been done by using nonlinear free vibration decay. In order to solve the geometrical and inertial nonlinear terms, the method of modified differential transform according to Padé approximation was used and resonant frequency is determined. Also, the resonant frequency of nonlinear system is calculated by generalized variational iteration method and compared with the obtained frequency from the modified differential transform method. Comparison of the current results with those of 4th order Runge-Kutta technique shows good agreement of the two approaches. Finally, Obtained results compared with the experimental results showed good accuracy identifying models for nonlinear beam.

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In this paper, equations of motion for a horizontal axis wind turbine with movable base are extracted and natural frequencies and vibration of the system is studied. The wind turbine tower is assumed rigid while its blades are modeled as flexible beams. In-plane bending and twisting are considered as two degrees of freedom for blades motion.The shaft connected the tower to blades is assumed rigid and its rotational velocity is considered.In this paper, specifically, a 5-megawattfloating horizontal axis wind turbine, which it’s basehas three angular velocities in different directions,is studied.Due to the complex shape and variation of the properties along the length, the turbine blade properties such as mass and geometric parameters are extracted by curve fitting in MATLAB.The equations of motion and boundary conditions are derived by Hamilton's principle and then are transformed to ordinary differential equations by Galerkin method. By setting the governing equations to standard form (space state), eigenvalues and frequencies are calculated. The numerical results are compared with published results and good agreement is observed.Then the effect of various parameters on turbine blades frequencies and time responses are demonstrated. Results show that the tower base angular velocity and blades rotational speed have considerable effects on turbine blades time response and vibration frequencies.

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Rotating cylindrical shells are applied in different industrial applications, such as gas turbine engines, electric motors, rotary kilns and rotor systems. So, it is of great interest to conduct some researches to improve the understanding of vibrational characteristics of rotating cylindrical shells. Grid stiffened laminated composite cylindrical shells are used as components of aerospace, marine industries and civil engineering structures. In this research free vibration of rotating grid stiffened composite cylindrical shell with various boundary conditions using the Fourier series expansion method is presented. Smeared method is employed to superimpose the stiffness contribution of the stiffeners with those of shell in order to obtain the equivalent stiffness parameters of the whole structure. The stiffeners are considered as a beam and support shear loads and bending moments in addition to the axial loads. Strain displacement relations from Sanders's shell theory are employed in the analysis. Using the Fourier series expansion and Stokes’ transformation, frequency determinant of laminated cylindrical shells is derived. The effects of shell geometrical parameters and changes in the cross stiffeners angle and axial loading on the natural frequencies are investigated. Results given are novel and can be used as a benchmark for further studies.

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The beam theory is used in the analysis and design of a wide range of structures, from buildings to bridges to the load-bearing bones of the human body. Beams resting on elastic foundation have wide application in many branches of engineering problems namely geotechnics, road, railroad and marine engineering and bio-mechanics. The foundation is very often a rather complex medium; e.g., a rubberlike fuel binder, snow, or granular soil. The key issue in the analysis is modelling the contact between the structural elements and the elastic bed. Since of interest here is the response of the foundation at the contact area and not the stresses or displacements inside the foundation material, In most cases the contact is presented by replacing the elastic foundation with simple models, usually spring elements. The most frequently used foundation model in the analysis of beam on elastic foundation problems is the Winkler foundation model. In the Winkler model, the elastic bed is modeled as uniformly distributed, mutually independent, and linear elastic vertical springs which produce distributed reactions in the direction of the deflection of the beam. However since the model does not take into account either continuity or cohesion of the bed, it may be considered as a rather crude representation of the elastic foundation. In order to find a physically close and mathematically simple foundation model, Pasternak proposed a so-called two-parameter foundation model with shear interactions. The first foundation parameter is the same as the Winkler foundation model and the second one is the stiffness of the shearing layer in the Pasternak foundation model.
Dynamic analysis is an important part of structural investigation and the results of free vibration analysis are useful in this context. Vibration problems of beams on elastic foundation occupy an important place in many fields of structural and foundation engineering.With the increase of thickness, existence of simplifying hypotheses in beam theories such as the ignorance of rotational inertial and transverse shear deformation in classic theory, application of determination coefficient in first-order shear theory and expression of one or few unknown functions based on other functions in higher-order shear theories is accompanied by reduction in accuracy of these theories. This represents the necessity of precise and analytical solutions for beam problems with the least number of simplifying hypotheses and for different thicknesses.
In the present study, the analytical solution for the problem of free vibration of homogeneous prismatic simply supported beam with rectangular solid sections and desired thickness resting on Pasternak elastic foundation is provided for completely isotropic behaviors under two-dimensional theory of elasticity and functions of displacement potentials. Characteristic equations of natural vibration are defined by solving one partial differential equations of fourth order through separation of variables and application of boundary conditions. The major characteristics of present study are lack of limitation of thickness and its validity for beams of low, medium and large thickness. To verify, the results of present study were compared with those of other studies. The results show that increases of foundation parameters is associated with an increased natural frequency, The intensity by increasing the ratio of thickness to length and in values larger than 0.2 and in the higher modes of vibration is reduced considerably.

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In this study, the hydrostatic vibration analysis of an isotropic rectangular microplate in partial contact with a bounded fluid is studied. Modified couple stress theory based on the Kirchhoff plate assumptions are used to mathematically model the problem. The extended Hamilton’s principle is employed to drive the governing differential equation of motion and the corresponding boundary conditions. The transverse displacement of the microplate is approximated by a set of admissible functions which must satisfy the geometric boundary conditions. The fluid is assumed to be incompressible, inviscid and irrotational and the fluid velocity potential is obtained using the boundary and compatibility conditions. Natural frequencies of the microplate are calculated using the Rayleigh-Ritz method. To validate the present results, the natural frequencies of an isotropic macroplate in contact with fluid are compared with the available data in the literature and very good agreements are observed. Finally using the numerical data, the effect of different parameters such as thickness to length scale parameter, aspect ratio, length to thickness ratio and boundary conditions on the natural frequencies of the microplate are discussed in detail. We have observed that the difference between the natural frequencies predicted using the classical theory and the one evaluated by the modified couple stress theory is significant when thickness of the microplate is small, but diminishes as thickness increases.

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In this paper, the behavior of free vibrations and buckling of the thick cylindrical sandwich panel with a flexible core and simply supported boundary conditions using a new improved ‎high-order sandwich panel theory were investigated. An axial compressive load is applied on the edges of the top and bottom face sheets simultaneously. The formulation used the third-order polynomial description for the displacement fields of thick composite face sheets and for the displacement fields in the core layer based on the displacement field of Frostig's second model. In this model, there are twenty seven degree of freedom. The transverse normal stress in the face sheets and the in-plane stresses in the core were considered .For calculated exact solution, according to thick face sheets, all of the stress components were engaged. The equations of motion and boundary conditions were derived via the Hamilton principle. Moreover, the effect of some important parameters such as those of thickness ratio of the core to panel, the length to radius ratio of the core, cumferential wave number and composite lay-up sequences on free vibration response and buckling of the panel were investigated. In order to validate the results, the obtained results were compared with those obtained using finite element ABAQUS software. The advantage of this paper is simplicity, considering face sheets as thick, exact solution and the considering of important terms such as (1+z_c/R_c ) in equations.

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Due to unique properties, lattice composite structure are used extensively in aviation, marine and automotive industry. In this research, experimental and numerical investigation of the free vibration of composite sandwich plates with triangular grid has been studied. For the fabrication of this plates, silicone mold, filament winding, and hand lay-up method were used. Stiffened plates and simple plates are fabricated, separately. Then, composite sandwich plates with triangular grid were created by attaching the two parts together. The modal test is done on the plates and natural frequencies have been extracted.The comparison of numerical and experimental results showed that there is a good agreement between them. By using Taguchi method, a parametric study was performed on the vibrational behavior of sandwich plates with triangular cores via three parameters that such as stiffeners’ number, stiffener thickness and skin thickness. The results show that the natural frequency of sandwich plates with triangular grid has a most sensitive to the stiffener thickness, and least sensitive to stiffeners’ number. The sensitivity of natural frequency is almost identical to stiffener thickness and skin thickness.To evaluate the efficiency of sandwich plates with triangular grid, the natural frequency of sandwich plates are compared with simple plates in the different boundary condition. The results show that the natural frequency of sandwich plates with the triangular grid is 133% and 138% higher than an equivalent simple shell at free and clamp boundary condition, respectively.

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In this study, free vibrations and resonances analysis of a nanocomposite beam with internal damping is investigated. For this purpose the various distributions of carbon nanotubes with arbitrary average volume fractions are considered. System includes the geometry and inertia nonlinearities. With the aid of Hamilton principle the equations of motion are derived and using the Galerkin method are reduced to ordinary ones. To analyze the system the multiple scales method is utilized. In free analysis the analytical expressions for amplitude, phase and nonlinear natural frequency are obtained. Also, the effect of system parameters such as damping coefficients, kind of the carbon nanotube distribution, average volume fraction of nanotubes in them are probed. In free analysis, it is observed that by increasing the external damping the amplitude is decreased. Also, by increasing the average volume fraction, the nonlinear natural frequency is increased. In resonance analysis, by depicting the frequency response curves, it is observed that by increasing internal damping coefficient the amplitude is decreased and the loci of the bifurcations is changed. Also carbon nanotube distribution and average volume fractions of them on the solution and bifurcations have an important effect. Also, it is seen that by decreasing the external force, the amplitude of the system is decreased and bifurcations occur in higher internal damping coefficients. An isotropic beam in the highest and a nano-composite beam in the lowest values of internal damping coefficients become completely stable.

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In this study a novel solution method for dynamic analysis of clamped-free shape memory alloy beams is presented. It is assumed that the beam is entirely made of shape memory alloy. Based on Euler-Bernoulli beam theory the governing equations of motion and corresponding boundary conditions are derived by using extended Hamilton principle. In the derived PDEs the transformation strain is behaved as external force that changes with time and position. The Galrkin approach is employed to convert PDEs to ODE system equations of motion. The derived equations of motion are solved by using Newmark integration method. The shape memory alloy constitutive model that presented by Souza is applied for specifying the phase of material all over beam. The transformation strain as internal variable that is coupled with states of equations of motion is identified in every time and every position of beam by using return map algorithm. A parametric study on the control variables has been adopted and the results of parametric study are discussed. The results show that the hysteresis damping is increased by increasing the operating temperature. Moreover the damping of system is faster by increasing the initial displacement in free vibration.