具损伤压电智能层合板的非线性主动控制和损伤监测

基于Hamilton原理、高阶剪切变形板理论、von Krmn型几何非线性应变-位移关系以及应变能等效原理,考虑压电层的质量和刚度及复合材料层内的损伤效应,建立了具损伤压电智能层合板的非线性运动方程．采用耦合正、逆压电效应的负速度反馈控制原理,形成闭环控制回路,实现了对压电智能层合板的主动控制和损伤监测．数值计算中,以四边简支面内不可动的层合矩形板为例,讨论了压电层位置对振动控制的影响,以及损伤程度和损伤位置对传感层输出电压的影响,提出一种损伤监测的方法．     

Nonlinear dynamic characteristics of bi-graphene sheets/piezoelectric laminated films considering high order van der Walls force and scale effect

This paper presents an analytical method to investigate the nonlinear vibration characteristics of bi-graphene sheets/piezoelectric (BGP) laminated films subjected to electric loading based on a nonlocal continuum model, in which the two adjacent layers are coupled by van der Walls force. Utilizing von Kármán nonlinear geometric relation and nonlocal physical relation, the nonlinear dynamic equation of BGP laminated films under electric loading exerted on the piezoelectric layer is found, then the relation between the nonlinear resonant frequency and the nonlinear vibration amplitude for each layer of the BGP laminated films is obtained by using Galerkin method and harmonic-balance method. Results show that the nonlinear vibration amplitude for each layer of laminated films can be controlled by adjusting the electric potential exerted on piezoelectric layer, and the coupled effect of van der Walls force between graphene sheet and piezoelectric layer on the vibration amplitude of each layer depends on the order number of nonlinear resonant frequency and the mode shape.     

剪切变形对直线型正交异性层合圆板大幅度受迫振动的影响*

本文研究了计及横向剪切变形的直线型正交异性层合圆板在简谐载荷q₀cosωt作用下的非线性受迫振动问题.采用伽辽金方法得到强振频率与振幅关系的解析解.最后,分析了横向剪切对板振动的影响,并给出了板的非线性自由振动的非线性周期对线性周期的比值.     

Hamilton等参元与智能叠层板的半解析法

根据压电材料修正后的Hellinger-Reissner(H-R)变分原理,建立了各向异性压电材料4节点Hamilton等参元的一般形式．为智能叠层板自由振动问题和带有压电块的叠层悬臂梁的瞬态响应等问题提出了一种新的半解析法．数学模型的基本步骤:将压电层和主体层看成独立的三维体,在平面内离散各层,分别建立各层的方程;根据主体层和压电层在连接界面上广义应力和广义位移的连续条件,联立主体层和压电层的方程得到全结构的控制方程．等参元不限制智能板侧面的几何边界形状、板的厚度和层数,有广泛的应用领域．     

Free vibration analysis of piezoelectric coupled annular plates with variable thickness

This paper presents the free vibration analysis of piezoelectric coupled annular plates with variable thickness on the basis of the Mindlin plate theory. No work has yet been done on piezoelectric laminated plates while the thickness is variable. Two piezoelectric layers are embedded on the upper and lower surfaces of the host plate. The host plate thickness is linearly increased in the radial direction while the piezoelectric layers thicknesses remain constant along the radial direction. Different combinations of three types of boundary conditions i.e. clamped, simply supported, and free end conditions are considered at the inner and outer edges of plate. The Maxwell static electricity equation in piezoelectric layers is satisfied using a quadratic distribution of electric potential along the thickness. The natural frequencies are obtained utilizing a Rayleigh–Ritz energy approach and are validated by comparing with those obtained by finite element analysis. A good compliance is observed between numerical solution and finite element analysis. Convergence study is performed in order to verify the numerical stability of the present method. The effects of different geometrical parameters such as the thickness of piezoelectric layers and the angle of host plate on the natural frequencies of the assembly are investigated.     

Modeling large amplitude vibration of matrix cracked hybrid laminated plates containing CNTR-FG layers

This paper presents the mathematical modeling of the nonlinear vibration behavior of a hybrid laminated plate composed of carbon nanotube reinforced functionally graded (CNTR-FG) layers and conventional fiber reinforced composite (FRC) layers. Three type symmetric distributions of single walled carbon nanotubes (SWCNTs) through the thickness of layers are considered. The cracks are modeled as aligned slit cracks across the ply thickness and transverse to the laminate plane. The distribution of cracks is assumed to be statistically homogeneous corresponding to an average crack density. The obtained partial differential equations are solved by the element-free kp-Ritz method, and the iteration process is dealt with using the linearized updated mode method. Detailed parametric studies are conducted investigate the effects of matrix crack density, CNTs distributions, CNT volume fraction, plate aspect ratio and plate length-to-thickness ratio, boundary conditions and number of layers on the frequency-amplitude responses of hybrid laminated plates containing CNTR-FG layers.     

Compressive and thermal postbuckling behaviors of laminated plates with piezoelectric fiber reinforced composite actuators

This paper studied compressive postbuckling under thermal environments and thermal postbuckling due to a uniform temperature rise for a shear deformable laminated plate with piezoelectric fiber reinforced composite (PFRC) actuators based on a higher order shear deformation plate theory that includes thermo-piezoelectric effects. The material properties are assumed to be temperature-dependent and the initial geometric imperfection of the plate is considered. The compressive and thermal postbuckling behaviors of perfect, imperfect, symmetric cross-ply and antisymmetric angle-ply laminated plates with fully covered or embedded PFRC actuators are conducted under different sets of thermal and electric loading conditions. The results reveal that, the applied voltage usually has a small effect on the postbuckling load–deflection relationship of the plate with PFRC actuators in the compressive buckling case, whereas the effect of applied voltage is more pronounced for the plate with PFRC actuators, compared to the results of the same plate with monolithic piezoelectric actuators.     

Quasi-three-dimensional theory for solution of dynamic thermoelastic problem of laminated composite plates

An uncoupled dynamic thermoelastic problem for laminated composite plates has been considered. The hypotheses used take into account the nonlinear distribution of temperature and displacements over the thickness of a laminated plate. On the basis of these hypotheses a quasi-three-dimensional (layerwise) theory is constructed that makes it possible to investigate the internal thermal and stress-strain states, as well as the edge effects of the boundary layer type for laminated plates. Systems of the heat conduction and motion equations are derived using the variational method. The order of the equations depends on the number of layers and terms in expansions of temperature and displacements of each layer. An analytical solution of the dynamic thermoelastic problem is presented for a cross-ply laminated rectangular plate with simply supported edges. The reliability of the results is confirmed by a comparison with the known exact solutions. The results based on the proposed theory can be used for verifying various two-dimensional plate theories when solving the dynamic thermoelastic problems for laminated composite plates.     

Free Vibration Analysis of Laminated Composite Plates Resting on Elastic Foundations by Using a Refined Hyperbolic Shear Deformation Theory

The free vibration of laminated composite plates on elastic foundations is examined by using a refined hyperbolic shear deformation theory. This theory is based on the assumption that the transverse displacements consist of bending and shear components where the bending components do not contribute to shear forces, and likewise, the shear components do not contribute to bending moments. The most interesting feature of this theory is that it allows for parabolic distributions of transverse shear stresses across the plate thickness and satisfies the conditions of zero shear stresses at the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns in the present theory is four, as against five in other shear deformation theories. In the analysis, the foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second foundation parameter is zero. The equation of motion for simply supported thick laminated rectangular plates resting on an elastic foundation is obtained through the use of Hamilton’s principle. The numerical results found in the present analysis for free the vibration of cross-ply laminated plates on elastic foundations are presented and compared with those available in the literature. The theory proposed is not only accurate, but also efficient in predicting the natural frequencies of laminated composite plates.     

Three-dimensional layerwise-finite element free vibration analysis of thick laminated annular plates on elastic foundation

Using a three-dimensional layerwise-finite element method, the free vibration of thick laminated circular and annular plates supported on the elastic foundation is studied. The Pasternak-type formulation is employed to model the interaction between the plate and the elastic foundation. The discretized governing equations are derived using the Hamilton’s principle in conjunction with the layerwise theory in the thickness direction, the finite element (FE) in the radial direction and trigonometric function in the circumferential direction, respectively. The fast rate of convergence of the method is demonstrated and to verify its accuracy, comparison studies with the available solutions in the literature are performed. The effects of the geometrical parameters, the material properties and the elastic foundation parameters on the natural frequency parameters of the laminated thick circular and annular plates subjected to various boundary conditions are presented.     

初始缺陷和拉-弯耦合对于叠层板的振动、屈曲和非线性动力稳定性的影响

研究了复合材料叠层板的初始缺陷和拉伸＿弯曲耦合对于其振动、屈曲和非线性动力稳定性的影响·推导了控制方程·这是一个修正的非线性Ｍａｔｈｉｅｕ方程·进行了５种典型复合材料的数值计算,它们是玻璃环氧Ｓｃｏｔｃｈ＿１００２,芳纶环氧Ｋｅｖｌａｒ＿４９,硼环氧Ｂ４／５５０５,石墨环氧Ｔ３００／５２０８,和ＡＳ／３５０１·结果表明：由于初始缺陷耦合效应的存在,使叠层板对于进入参数共振更加敏感,并且其振幅大于无初始缺陷或者无耦合效应的叠层板·对于不同复合材料的叠层板,尤其是层数较少的板,耦合效应是不相同的·在板结构的屈曲和动力稳定性设计中,如果忽略了耦合效应的影响,其不安全性将超过１０％以上     

Post buckling response of laminated composite plate on elastic foundation with random system properties

This paper deals with second order statistics of post buckling load of shear deformable laminated composite plates resting on linear elastic foundation with random system properties. The formulation is based on higher order shear deformation plate theory in general von Karman sense, which includes foundation effect using two-parameter Pasternak model. The random system equations are derived using the principal of virtual work. A finite element method is used for spatial descretization of the laminate with a reasonable accuracy. A perturbation technique has been the first time successfully combined with direct iterative technique by neglecting the changes in nonlinear stiffness matrix due to random variation of transverse displacements during iteration. The numerical results for the second order statistics of post buckling loads are obtained. A detailed study is carried out to highlight the characteristics of the random response and its sensitivity to different foundation parameters, the plate thickness ratio, the plate aspect ratio, the support condition, the stacking sequence and the lamination angle on the post buckling response of the laminate. The results have been compared with existing results and an independent Monte Carlo simulation.     

Large amplitudes free vibrations and post-buckling analysis of unsymmetrically laminated composite beams on nonlinear elastic foundation

The purpose of this paper is to present efficient and accurate analytical expressions for large amplitude free vibration and post-buckling analysis of unsymmetrically laminated composite beams on elastic foundation. Geometric nonlinearity is considered using Von Karman’s strain–displacement relations. Besides, the elastic foundation has cubic nonlinearity with shearing layer. The nonlinear governing equation is solved by employing the variational iteration method (VIM). This study shows that the third-order approximation of the VIM leads to highly accurate solutions which are valid for a wide range of vibration amplitudes. The effects of different parameters on the ratio of nonlinear to linear natural frequency of beams and the post-buckling load–deflection relation are studied.     

Optimization of laminated composite plates for maximum fundamental frequency using Elitist-Genetic algorithm and finite strip method

In the present paper, fundamental frequency optimization of symmetrically laminated composite plates is studied using the combination of Elitist-Genetic algorithm (E-GA) and finite strip method (FSM). The design variables are the number of layers, the fiber orientation angles, edge conditions and plate length/width ratios. The classical laminated plate theory is used to calculate the natural frequencies and the fitness function is computed with a semi-analytical finite strip method which has been developed on the basis of full energy methods. To improve the speed of the optimization process, the elitist strategy is used in the Genetic algorithm. The performance of the E-GA is also compared with the simple genetic algorithm and shows the good efficiency of the E-GA algorithm. A multi-objective optimization strategy for optimal stacking sequence of laminated box structure is also presented, with respect to the first natural frequency and critical buckling load, using the weighted summation method to demonstrate the effectiveness of the E-GA. Results are corroborated by comparing with other optimum solutions available in the literature, wherever possible.     

DQM large amplitude vibration of composite beams on nonlinear elastic foundations with restrained edges

This paper presents an efficient and accurate differential quadrature (DQ) large amplitude free vibration analysis of laminated composite thin beams on nonlinear elastic foundation. Beams under consideration have elastically restrained against rotation and in-plane immovable edges. Elastic foundation has cubic nonlinearity with shearing layer. We impose the boundary conditions directly into the governing equations in spite of the conventional DQ method and without any extra efforts. A direct iterative method is used to solve the nonlinear eigenvalue system of equations after transforming the governing equations into the frequency domain. The fast rate of convergence of the method is shown and their accuracy is demonstrated by comparing the results with those for limit cases, i.e. beams with classical boundary conditions, available in the literature. Besides, we develop a finite element program to verify the results of the presented DQ approach and to show its high computational efficiency. The effects of different parameters on the ratio of nonlinear to linear natural frequency of beams are studied.     

考虑地基耦合效应含裂纹中厚矩形板的非线性振动分析

基于Reissner板理论和Hamilton变分原理,建立了双参数地基上具有表面横向贯穿裂纹的中厚矩形板的非线性运动控制方程.在周边自由的条件下,提出了一组满足问题全部边界条件和裂纹处连续条件的试函数.且利用Galerkin法和谐波平衡法对方程进行求解,分析了考虑地基耦合效应的中厚矩形裂纹板的非线性振动特性.数值计算中,讨论了不同裂纹位置、裂纹深度、板的结构参数和地基物理参数对弹性地基上具裂纹的四边自由中厚矩形板的非线性幅频响应的影响.     

Nonlinear free vibration analysis of simply supported piezo-laminated plates with random actuation electric potential difference and material properties

Studies are made on nonlinear free vibrations of simply supported piezo-laminated rectangular plates with immovable edges utilizing Kirchoff’s hypothesis and von Kármán strain–displacement relations. The effect of random material properties of the base structure and actuation electric potential difference on the nonlinear free vibration of the plate is examined. The study is confined to linear-induced strain in the piezoelectric layer applicable to low electric fields. The von Kármán’s large deflection equations for generally laminated elastic plates are derived in terms of stress function and transverse deflection function. A deflection function satisfying the simply supported boundary conditions is assumed and a stress function is then obtained after solving the compatibility equation. Applying the modified Galerkin’s method to the governing nonlinear partial differential equations, a modal equation of Duffing’s type is obtained. It is solved by exact integration. Monte Carlo simulation has been carried out to examine the response statistics considering the material properties and actuation electric potential difference of the piezoelectric layer as random variables. The extremal values of response are also evaluated utilizing the Convex model as well as the Multivariate method. Results obtained through the different statistical approaches are found to be in good agreement with each other.     

3D free vibration analysis of laminated cylindrical shell integrated piezoelectric layers using the differential quadrature method

This paper addresses the free vibration problem of multilayered shells with embedded piezoelectric layers. Based on the three-dimensional theory of elasticity, an approach combining the state space method and the differential quadrature method (DQM) is used. The shell has arbitrary end boundary conditions. For the simply supported boundary conditions closed-form solution is given by making the use of Fourier series expansion. Applying the differential quadrature method to the state space formulations along the axial direction, new state equations about state variables at discrete points are obtained for the other cases such as clamped or free end conditions. Natural frequencies of the hybrid laminated shell are presented by solving the eigenfrequency equation which can be obtained by using edges boundary condition in this state equation. Accuracy and convergence of the present approach is verified by comparing the natural frequencies with the results obtained in the literatures. Finally, the effect of edges conditions, mid-radius to thickness ratio, length to mid-radius ratio and the piezoelectric thickness on vibration behaviour of shell are investigated.