Free vibration of functionally graded spatial curved beams

A formulation for the free vibration analysis of functionally graded (FG) spatial curved beams is presented by taking into account the effects of thickness-curvature. The governing equation is based on the first-order shear deformation theory (FSDT) and Ritz method is employed to obtain the natural frequencies. The curved beams presented are in the form of the cylindrical helical spring. The material distribution is in the direction of the curvature of the curved beam. The results for isotropic planar curved beams are validated with the known data in the literature. The effects of helix pitch angle, number of turns and boundary conditions on frequency parameters of spatial curved beams are investigated.     

Nonlinear forced vibration analysis of clamped functionally graded beams

Multiple time scale solutions are presented to study the nonlinear forced vibration of a beam made of symmetric functionally graded (FG) materials based on Euler?CBernoulli beam theory and von Kármán geometric nonlinearity. It is assumed that material properties follow either exponential or power law distributions through the thickness direction. A Galerkin procedure is used to obtain a second-order nonlinear ordinary equation with cubic nonlinear term. The natural frequencies are obtained for the nonlinear problem. The effects of material property distribution and end supports on the nonlinear dynamic behavior of FG beams are discussed. Also, forced vibrations of the system in primary and secondary resonances have been studied, and the effects of different parameters on the frequency-response have been investigated.     

Hierarchical theories for the free vibration analysis of functionally graded beams

This work addresses a free vibration analysis of functionally graded beams via several axiomatic refined theories. The material properties of the beam are assumed to vary continuously on the cross-section according to a power law distribution in terms of the volume fraction of the material constituents. Young’s modulus, Poisson’s ratio and density can vary along one or two dimensions all together or independently. The three-dimensional kinematic field is derived in a compact form as a generic N-order polynomial approximation. The governing differential equations and the boundary conditions are derived by variationally imposing the equilibrium via the Principle of Virtual Displacements. They are written in terms of a fundamental nucleo that does not depend upon the approximation order. A Navier-type, closed form solution is adopted. Higher-order displacements-based theories that account for non-classical effects are formulated. Classical beam models, such as Euler–Bernoulli’s and Timoshenko’s, are obtained as particular cases. Bending, torsion and axial modes are investigated. Slender as well as short beams are considered. Numerical results highlight the effect of different material distributions on natural frequencies and mode shapes and the accuracy of the proposed models.     

Hierarchical theories for the free vibration analysis of functionally graded beams

This work addresses a free vibration analysis of functionally graded beams via several axiomatic refined theories. The material properties of the beam are assumed to vary continuously on the cross-section according to a power law distribution in terms of the volume fraction of the material constituents. Young’s modulus, Poisson’s ratio and density can vary along one or two dimensions all together or independently. The three-dimensional kinematic field is derived in a compact form as a generic N-order polynomial approximation. The governing differential equations and the boundary conditions are derived by variationally imposing the equilibrium via the Principle of Virtual Displacements. They are written in terms of a fundamental nucleo that does not depend upon the approximation order. A Navier-type, closed form solution is adopted. Higher-order displacements-based theories that account for non-classical effects are formulated. Classical beam models, such as Euler–Bernoulli’s and Timoshenko’s, are obtained as particular cases. Bending, torsion and axial modes are investigated. Slender as well as short beams are considered. Numerical results highlight the effect of different material distributions on natural frequencies and mode shapes and the accuracy of the proposed models.     

Free vibration analysis of functionally graded conical shell panels by a meshless method

A free vibration analysis of metal and ceramic functionally graded conical shell panels is presented using the element-free kp-Ritz method. The first-order shear deformation shell theory is used to account for the transverse shear strains and rotary inertia, and mesh-free kernel particle functions are employed to approximate the two-dimensional displacement fields. The material properties of the conical shell panels are assumed to vary continuously through their thickness in accordance with a power-law distribution of the volume fractions of their constituents. Convergence studies are performed in terms of the number of nodes, and comparisons of the current solutions and those reported in literature are provided to verify the accuracy of the proposed method. Two types of functionally graded conical shell panels, including Al/ZrO₂ and Ti–6Al–4V/aluminum oxide, are chosen in the study, and the effects of the volume fraction, boundary condition, semi-vertex angle, and length-to-thickness ratio on their frequency characteristics are discussed in detail.     

An analytical method for temperature-dependent free vibration analysis of functionally graded beams with general boundary conditions

Exact solutions are presented to study the free vibration of a beam made of symmetric functionally graded materials. The formulation used is based on a unified higher order shear deformation theory. Material properties are taken to be temperature-dependent, and vary continuously through the thickness according to a power law distribution (P-FGM), or an exponential law distribution (E-FGM) or a sigmoid law distribution (S-FGM). The beam is assumed to be initially stressed by a temperature rise through the thickness. Temperature field is considered constant in xy plane of the beam. Hamilton’s principle is used to derive the governing equations of motion. Free vibration frequencies are obtained by solving analytically a system of ordinary differential equations, for different boundary conditions.     

复合材料封闭变截面薄壁梁自由振动分析

摘要：基于拉格朗日方程建立了复合材料封闭变截面薄壁梁的自由振动微分方程,给出了两种刚度配置下的变矩形截面悬臂直梁的自由振动方程简化形式及其相应的迦辽金法求解的固有频率。基于大型通用有限元软件ANSYS,计算了薄壁变截面悬臂梁的固有频率,并且与迦辽金法的求解结果进行了对比。分析了复合材料的弹性耦合,铺层角度和截面变化对薄壁梁的自由振动的影响。     

Free vibration analysis of functionally graded annular plates by state-space based differential quadrature method and comparative modeling by ANN

This paper deals with three-dimensional analysis of functionally graded annular plates through using state-space based differential quadrature method (SSDQM) and comparative behavior modeling by artificial neural network (ANN) for different boundary conditions. The material properties are assumed to have an exponent-law variation along the thickness. A semi-analytical approach which makes use of state-space method in thickness direction and one-dimensional differential quadrature method in radial direction is used to obtain the vibration frequencies. The state variables include a combination of three displacement parameters and three stress parameters. Numerical results are given to demonstrate the convergency and accuracy of the present method. Once the semi-analytical method is validated, an optimal ANN is selected, trained and tested by the obtained numerical results. In addition to the quantitative input parameters, support type is also considered as a qualitative input in NN modeling. Eventually the results of SSDQM and ANN are compared and the influence of thickness of the annular plate, material property graded index and circumferential wave number on the non-dimensional natural frequency of annular functionally graded material (FGM) plates with different boundary conditions are investigated. The results show that ANN can acceptably model the behavior of FG annular plates with different boundary conditions.     

Size-dependent free flexural vibrational behavior of functionally graded nanobeams using semi-analytical differential transform method

In this paper nonlocal Euler–Bernoulli beam theory is employed for vibration analysis of functionally graded (FG) size-dependent nanobeams by using Navier-based analytical method and a semi analytical differential transform method. Two kinds of mathematical models, namely, power law and Mori-Tanaka models are considered. The nonlocal Eringen theory takes into account the effect of small size, which enables the present model to become effective in the analysis and design of nanosensors and nanoactuators. Governing equations are derived through Hamilton's principle and they are solved applying semi analytical differential transform method (DTM). It is demonstrated that the DTM has high precision and computational efficiency in the vibration analysis of FG nanobeams. The good agreement between the results of this article and those available in literature validated the presented approach. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as small scale effects, different material compositions, mode number and thickness ratio on the normalized natural frequencies of the FG nanobeams in detail. It is explicitly shown that the vibration of a FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.     

Free vibration analysis of moderately thick functionally graded plates by local Kriging meshless method

This paper mainly Presents free vibration analyses of metal and ceramic functionally graded plates with the local Kriging meshless method. The Kriging technique is employed to construct shape functions which possess Kronecker delta function property and thus make it easy to implement essential boundary conditions. The eigenvalue equations of free vibration problems are based on the first-order shear deformation theory and the local Petrov–Galerkin formulation. The cubic spline function is used as the weight function which vanishes on internal boundaries of local quadrature domains and hence simplifies the implementation. Convergence studies are conducted to examine the stability of the present method. Three types of functionally graded plates – square, skew and quadrilateral plates – are considered as numerical examples to demonstrate the versatility of the present method for free vibration analyses.     

Static analysis of functionally graded cylindrical shell with piezoelectric layers using differential quadrature method

Three-dimensional solution for static analysis of functionally graded (FG) cylindrical shell with bonded piezoelectric layers is presented using differential quadrature method (DQM) and state-space approach. Applying the DQM to the governing differential equations and to the edges boundary conditions, new state equations about state variables at discrete points are derived. The stress, displacement, and electric potential distributions are obtained by solving these state equations. The convergence and accuracy of the present method is validated by comparing numerical results for the hybrid FG cylindrical shell with simply-supported edges with the analytical solution that has been published in the literature. Both the direct and the inverse piezoelectric effects are investigated and the influence of piezoelectric layers and gradient index on the mechanical behavior of shell is studied.     

Free vibration analysis of sandwich beams using improved dynamic stiffness method

In this paper, free vibration of three-layered symmetric sandwich beam is investigated using dynamic stiffness and finite element methods. To determine the governing equations of motion by the present theory, the core density has been taken into consideration. The governing partial differential equations of motion for one element contained three layers are derived using Hamilton’s principle. This formulation leads to two partial differential equations which are coupled in axial and bending deformations. For the harmonic motion, these equations are combined to form one ordinary differential equation. Closed form analytical solution for this equation is determined. By applying the boundary conditions, the element dynamic stiffness matrix is developed. They are assembled and the boundary conditions of the beam are applied, so that the dynamic stiffness matrix of the beam is derived. Natural frequencies and mode shapes are computed by the use of numerical techniques and the known Wittrick–Williams algorithm. After validation of the present model, the effect of various parameters such as density, thickness and shear modulus of the core for various boundary conditions on the first natural frequency is studied.     

Free vibration analysis of rotating functionally graded cylindrical shells in thermal environment

The free vibration analysis of rotating functionally graded (FG) cylindrical shells subjected to thermal environment is investigated based on the first order shear deformation theory (FSDT) of shells. The formulation includes the centrifugal and Coriolis forces due to rotation of the shell. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The initial thermo-mechanical stresses are obtained by solving the thermoelastic equilibrium equations. The equations of motion and the related boundary conditions are derived using Hamilton’s principle. The differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to discretize the thermoelastic equilibrium equations and the equations of motion. The convergence behavior of the method is demonstrated and comparison studies with the available solutions in the literature are performed. Finally, the effects of angular velocity, Coriolis acceleration, temperature dependence of material properties, material property graded index and geometrical parameters on the frequency parameters of the FG cylindrical shells with different boundary conditions are investigated.     

Free vibration analysis of variable thickness thin plates by two-dimensional differential transform method

This paper aims at extending the application of two-dimensional differential transform method (2D-DTM) to study the free vibration of thin plates with arbitrarily varying thickness. First, the differential equation of motion governing thin plates with varying thickness is derived using Hamilton’s principle. Afterward, the 2D-DTM, a numerical method which is capable of reducing the size of computational work and can be applied to various types of differential equations, has been applied to derive the natural frequencies of variable thickness thin plates with different boundary conditions. Several numerical examples have been carried out to demonstrate the applicability and accuracy of the present method in free vibration analysis of both uniform plates and plates with variable thickness.     

Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams

This paper investigates the nonlinear free vibration of functionally graded nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on Timoshenko beam theory and von Kármán geometric nonlinearity. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded in the thickness direction and estimated though the rule of mixture. The Ritz method is employed to derive the governing eigenvalue equation which is then solved by a direct iterative method to obtain the nonlinear vibration frequencies of FG-CNTRC beams with different end supports. A detailed parametric study is conducted to study the influences of nanotube volume fraction, vibration amplitude, slenderness ratio and end supports on the nonlinear free vibration characteristics of FG-CNTRC beams. The results for uniformly distributed carbon nanotube-reinforced composite (UD-CNTRC) beams are also provided for comparison. Numerical results are presented in both tabular and graphical forms to investigate the effects of nanotube volume fraction, vibration amplitude, slenderness ratio, end supports and CNT distribution on the nonlinear free vibration characteristics of FG-CNTRC beams.     

基于有限差分法的水平旋转梁的自由振动解析

在水平旋转梁模型的基础上,利用哈密尔顿原理建立了动力学微分方程。在悬臂梁边界条件下,运用二阶中心差分原理对欧拉梁进行有限差分离散,推导出系统模型的自由振动差分方程。运用MATLAB振动工具箱和一般阻尼振动理论对其进行了编程运算,得到了不同转速下水平梁的无量纲固有频率。相关文献的结果比对验证了有限差分方法的有效性,然后对旋转梁的自由振动特性进行了扩展分析和结果的优化处理。另外,对固支梁和自由梁的自由振动也进行了解析。     

A mixed Ritz-DQ method for forced vibration of functionally graded beams carrying moving loads

A mixed method is presented to study the dynamic behavior of functionally graded (FG) beams subjected to moving loads. The theoretical formulations are based on Euler–Bernoulli beam theory, and the governing equations of motion of the system are derived using the Lagrange equations. The Rayleigh–Ritz method is employed to discretize the spatial partial derivatives and a step-by-step differential quadrature method (DQM) is used for the discretization of temporal derivatives. It is shown that the proposed mixed method is very efficient and reliable. Also, compared to the single-step methods such as the Newmark and Wilson methods, the DQM gives better accuracy using larger time step sizes for the cases considered. Moreover, effects of material properties of the FG beam and inertia of the moving load on the dynamic behavior of the system are investigated and analyzed.     

3-D Free vibration analysis of thick functionally graded annular plates on Pasternak elastic foundation via differential quadrature method (DQM)

In this paper, the free vibration of functionally graded annular plates on elastic foundations, based on the three-dimensional theory of elasticity, using the differential quadrature method for different boundary conditions including simply supported–clamped, clamped–clamped and free–clamped ends is investigated. The foundation is described by the Pasternak or two-parameter model. A semi-analytical approach composed of differential quadrature method (DQM) and series solution are adopted to solve the equations of motions. The material properties change continuously through the thickness of the plate, which can vary according to power law, exponentially or any other formulations in this direction. The fast rate of convergence of the method is demonstrated, and comparison studies are carried out to establish its very high accuracy and versatility. Some new results for the natural frequencies of the plate are prepared, which include the effects of elastic coefficients of foundation, boundary conditions, material and geometrical parameters. The new results can be used as benchmark solutions for future researches.     

Vibration analysis of composite sandwich beams with viscoelastic core by using differential transform method

This paper deals with the vibration analysis of a three layered composite beam with a viscoelastic core. First, the equations of motion that govern the free vibrations of the sandwich beam are derived by applying Hamilton’s principle. Then, these equations are solved by using differential transform method (DTM) in the frequency domain. The variation of modal loss factor with system parameters is evaluated and presented graphically. Also, the results obtained with DTM are checked against the findings of previous studies and a good agreement is observed. It is the first time that DTM is used for the eigenvalue analysis of a sandwich structure.