Nonlinear free and forced vibration of simply supported shear deformable laminated plates with piezoelectric actuators

This paper deals with the nonlinear vibration and dynamic response of simply supported shear deformable cross-ply laminated plates with piezoelectric actuators subjected to mechanical, electrical and thermal loads. The material properties are assumed to be independent of the temperature and electric field. Theoretical formulations are based on the higher order shear deformation plate theory and general von Kármán-type equation, which includes thermo-piezoelectric effects. Due to the bending and stretching coupling effects, a nonlinear static problem is first solved to determine the pre-vibration deformation caused by temperature field and control voltage. By adding an incremental dynamic state to the pre-vibration state, the equations of motion are solved by an improved perturbation technique to determine nonlinear frequencies and dynamic responses of hybrid laminated plates. The numerical illustrations concern nonlinear vibration characteristics of unsymmetric cross-ply laminated plates. The results presented show the effects of temperature rise, applied voltage and stacking sequence on the nonlinear vibration and dynamic response of the plates.     

Generic nonlinear behavior of antisymmetric angle-ply laminated plates

A generic presentation of the large deflection, postbuckling and nonlinear flexural vibration behavior of antisymmetric angle-ply laminated plates is investigated. A transformation between on-axis and off-axis stiffnesses is presented such that all the stiffnesses, including bending-stretching coupling, in the nonlinear governing equations of angle-ply laminates can be expressed in terms of bounded global constants in orthotropic plates. The results presented are applicable to a large variety of composite materials and show that a higher generalized rigidity ratio will lead to smaller deflection of plates, higher buckling loads and lower postbuckling deflection, and higher relative large-amplitude natural frequencies. The information presented should be helpful to anyone seeking a better understanding of the correlation between composite material properties and nonlinear behavior of laminated plates.     

Nonlinear free vibration of laminated composite plates on elastic foundation with random system properties

The present investigation is concerned with free vibration analysis of laminated composite plates resting on elastic foundation undergoing large amplitude oscillation with random system properties. The lamina material properties and foundation stiffness parameters are modeled as basic random variables for accurate prediction of the system behavior. The basic formulation of the problem is based on higher-order shear displacement theory including rotatory inertia effects and von Karman-type nonlinear strain displacement relations. A C⁰ finite element is used for descretization of the laminate. A direct iterative method in conjunction with first-order Taylor series based perturbation technique procedure is developed to solve random nonlinear generalized eigenvalue problem. The developed probabilistic procedure is successfully used for the nonlinear free vibration problem with a reasonable accuracy. Typical numerical results (second-order statistics) are obtained for the composite plates resting on Winkler and Pasternak elastic foundations with different support conditions, side-to-thickness ratio, aspect ratio, oscillation amplitude ratio, stacking sequences and foundation parameters for symmetric and anti-symmetric cross-ply and angle-ply laminates. The results are validated with existing available results and independent Monte Carlo simulation.     

Prestressed composite laminates featuring interlaminar imperfection

The third-order zigzag displacement model is improved to include inplane displacement jumps across each layer interface of composite laminates to enable interlaminar imperfection to be incorporated. The imperfection is characterized by a linear spring-layer model which includes perfect bonding as a special case. The principle of virtual work is used to derive a boundary value formulation for laminated composite plates initially in a prestress state. Bending, buckling and vibration problems are studied for the case of rectangular cross-ply laminated plates for illustrative purpose.     

Three-dimensional vibration of laminated composite plates and cylindrical panels with arbitrarily located lateral surfaces point supports

On the basis of fully three-dimensional elasticity considerations, this paper presents a free vibration analysis of simply supported, cross-ply laminated plates and cylindrical panels that are subjected to an arbitrary number of lateral surfaces point supports. The analysis is based on a recursive approach suitable for the vibration analysis of corresponding unconstrained structural elements. By means of dynamic equilibrium considerations, the reaction of the point supports is imposed by using the Lagrange multipliers method. This yields the eigendeterminant of a constrained panel by appropriately coupling the response of a suitably large number of natural vibration modes of the corresponding unconstrained structural element.     

Natural frequencies of symmetrically laminated elliptical and circular plates

Elliptical and circular fibre reinforced composite plates are important structural elements in modern engineering structures. Vibration analysis of these elements are of interest to structural designers. The present paper deals with the free transverse vibration analysis of symmetrically laminated solid and annular elliptic and circular plates with several complicating effects.The approach developed is based on the Rayleigh–Ritz method where the deflection of the plate is approximated by a general shape function of polynomial type.The analysis includes several complicating effects, such as the presence of an internal hole, an internal ring support, several concentrated masses and the boundary elastically restrained against rotation and translation.Several examples are solved and some results which correspond to particular cases are compared with existing values in the literature. New results are also presented for cross-ply and angle-ply elliptical and circular laminates with different boundary conditions.The algorithm developed can be applied to a wide range of elastic restraint conditions, to any aspect ratio and to higher modes. The effect of the restraint parameters along the boundary on the natural frequencies for plates with these complicating effects is considered.     

Thermal buckling analysis of cross-ply laminated rectangular plates under nonuniform temperature distribution: A differential quadrature approach

Differential quadrature method (DQM) is implemented for analyzing the thermal buckling behavior of the symmetric cross-ply laminated rectangular thin plates subjected to uniform and/or non-uniform temperature fields. The approach includes two steps: (1) solving the problem of in-plane thermo-elasticity to obtain the in-plane force resultants and (2) solving the buckling problem under the force distribution obtained in the previous step. Solution procedures are numerically performed by discretizing the governing differential equations and boundary conditions using DQM method. Applying the developed DQ formulation, the buckling loads are obtained for several sample plates. The numerical results compared well with those available in the literature as well as those obtained by ABAQUS. Parametric studies are conducted to investigate the influence of some important parameters including the plate aspect ratio, cross-ply ratio, and stiffness ratio on the critical temperature and mode shape of buckling.     

Free vibration of laminated composite plates using two variable refined plate theory

Free vibration of laminated composite plates using two variable refined plate theory is presented in this paper. The theory accounts for parabolic distribution of the transverse shear strains through the plate thickness, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton's principle. The Navier technique is employed to obtain the closed-form solutions of antisymmetric cross-ply and angle-ply laminates. Numerical results obtained using present theory are compared with three-dimensional elasticity solutions and those computed using the first-order and the other higher-order theories. It can be concluded that the proposed theory is not only accurate but also efficient in predicting the natural frequencies of laminated composite plates.     

叠层平面圆环的振动行为

为了研究叠层平面圆环的振动行为和叠层对其振动影响,用有限元法对系列均质平面圆环进行振动分析以获得试验比较参照,对系列均质圆环和不同夹紧压力下的系列叠层圆环进行大量试验。理论分析和试验结果表明,叠层平面圆环的振动行为是由组成该叠层圆环的单片圆环的振动行为所支配,但叠层对圆环的振动有着重要影响,且这种影响与模态类型、夹紧压力和叠层层数有很大关系。随着叠层层数和夹紧力的增加,圆环纵向振动模态将很快消失,而平面模态得以保留并支配叠层平面圆环的振动。     

Free vibration of laminated cross-ply plates including shear deformation by spline method

Spline function approximation technique is used to analyze the free vibration of symmetric and anti-symmetric cross-ply plates under shear deformation theory. The equations of motion of the plate are derived using YNS theory. A system of coupled differential equations in terms of displacement and rotational functions are obtained by assuming the solution in a separable form. These functions are approximated using Bickley-type splines of suitable orders. A generalized eigenvalue problem is obtained on applying the process of point collocation with suitable boundary conditions. Parametric studies have been made to investigate the frequency response of the plates with reference to the material properties, number of layers, fiber orientation, side-to-thickness ratio, aspect ratio and relative layer thickness. Some results are compared with existing solution obtained by FEM.     

Hygrothermal effects on the postbuckling of shear deformable laminated plates

The influence of hygrothermal effects on the postbuckling of shear deformable laminated plates subjected to a uniaxial compression is investigated using a micro-to-macro-mechanical analytical model. The material properties of the composite are affected by the variation of temperature and moisture, and are based on a micro-mechanical model of a laminate. The governing equations of a laminated plate are based on Reddy's higher-order shear deformation plate theory that includes hygrothermal effects. The initial geometric imperfection of the plate is taken into account. Two cases of the in-plane boundary conditions are considered. A perturbation technique is employed to determine buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, antisymmetric angle-ply and symmetric cross-ply laminated plates under different sets of environmental conditions. The influences played by temperature rise, the degree of moisture concentration, the character of in-plane boundary conditions, transverse shear deformation, plate aspect ratio, total number of plies, fiber orientation, fiber volume fraction and initial geometric imperfections are studied.     

Free vibration analysis of moderately thick antisymmetric cross-ply laminated rectangular plates with elastic edge constraints

This study presents a simple formulation for studying the free vibration of shear-deformable antisymmetric cross-ply laminated rectangular plates having translational as well as rotational edge constraints. The aim is to fill the void in the available literature with respect to the free vibration results of antisymmetric cross-ply laminated rectangular plates. The spatial discretization of the resulting mathematical model in five field variables is carried out using the two-dimensional Differential Quadrature Method (DQM). Several combinations of translational and rotational elastic edge constraints are considered. Convergence study with respect to the number of nodes has been carried out and the results are compared with those from past investigations available only for simpler problems. Effects of stiffness parameters, geometrical features, moduli ratio and lamination schemes on the natural frequencies are studied.     

Modified stiffness formulation of unbalanced anisotropic sandwich plates

An approximate formulation is presented for the analysis of sandwich plates consisting of an orthotropic core and two unequal thickness anisotropic face plates. The method uncouples the membrane and bending actions, thereby significantly reducing the effort involved in accurately predicting displacements and stresses. Defining modified stiffnesses, the formulation is valid for relatively thick anisotropic laminated face plates and can be combined with a variety of analysis techniques. Using a series solution, results are generated herein for simply supported sandwich plates with unbalanced cross-ply and angle-ply faces.     

A refined nonlinear vibration absorber

Presented here is a theoretical study on how to use saturation phenomena to design nonlinear vibration absorbers and how to improve their stability and effective frequency bandwidth. The so-called original saturation control method uses 2 : 1 internal resonances and saturation phenomena to suppress steady-state vibrations of a dynamical system by connecting it to a second-order controller using quadratic position coupling terms, which do not really suppress vibration to zero as a linear vibration absorber does. However, a linear vibration absorber uses direct position feedbacks and splits one natural frequency of the original system into two and hence spill-over effects exist when the system is subjected to broad-band and/or transient excitations. Although a saturation controller does not split one natural frequency into two, one large-amplitude nonlinear solution coexists with a small-amplitude linear solution outside of the resonance area. Hence, the existence of spillover effects depends on initial conditions. A refined nonlinear vibration absorber is designed by using a quadratic velocity coupling term in the controller and adding a negative velocity feedback to the system. It is shown that the quadratic velocity coupling term enables a saturation controller to suppress system vibrations to zero. Moreover, the linear velocity feedback enhances the capability of suppressing transient vibrations and prevents the system from having the large-amplitude nonlinear response. Two equations describing the first-mode vibration of a stainless-steel beam and a saturation controller from the authors’ previous experimental work are used in this theoretical study. Both perturbation and direct numerical integration solutions are presented. Guidelines for designing nonlinear vibration absorbers are derived.     

Natural frequencies of composite plates with random material properties using higher-order shear deformation theory

Composites are known to display a considerable amount of scatter in their material properties due to large number of parameters associated with the manufacturing and fabrication processes. In the present work, the material properties have been taken as random variables for accurate prediction of the system behavior. Higher order shear theory including rotatory inertia effects has been accounted for in the system dynamic equations. A first order perturbation technique has been employed to obtain the solution of the governing equations. An approach has been outlined for obtaining closed form expressions for the variances of eigen solutions. The effects of side to thickness ratio and variation in standard deviation of the material properties have been investigated for cross-ply symmetric and anti-symmetric laminates. The mean and standard deviations of the first five natural frequencies have been worked out for laminated rectangular plates with all edges simply supported. The higher order shear deformation theory results have been validated with Monte Carlo simulation results and compared with the results based on classical laminate and first order shear deformation theories.     

Modal characteristics of symmetrically laminated composite plates flexibly restrained at different locations

A method for determining modal characteristics (natural frequencies and mode shapes) of symmetrically laminated composite plates restrained by elastic supports at different locations in the interior and on the edges of the plates is presented. The classical lamination theory together with an appropriate set of characteristic functions are used in the Rayleigh-Ritz method to formulate the eigenvalue problem for determining the modal characteristics of the flexibly supported laminated composite plates. Sweep-sine vibration testing of several laminated composite plates flexibly restrained at different locations on the plates is performed to measure their natural frequencies. The close agreement between the experimental and theoretical natural frequencies of the plates has verified the accuracy of the proposed method. The effects of elastic restraint locations on the modal characteristics of flexibly supported laminated composite plates with different lamination arrangements and aspect ratios are studied using the present method. The usefulness of the results obtained for predicting sound radiation behavior of flexibly supported laminated composite plates is discussed.     

Vibration analysis of cross-ply laminated beams with general boundary conditions by Ritz method

The present study is concerned with the vibration analysis of cross-ply laminated beams subjected to different sets of boundary conditions. The analysis is based on a three-degree-of-freedom shear deformable beam theory. The continuity conditions between layers of symmetric cross-ply laminated beams are satisfied by the use of the shape function incorporated into the theory which also unifies the 1D shear deformable beam theories developed previously. The governing equations are obtained by means of Hamilton's principle. Six different combinations of free, clamped and simply supported edge boundary conditions are considered. The free vibration frequencies are obtained by applying the Ritz method where the three displacement components are expressed in a series of simple algebraic polynomials. The numerical results obtained for different length-to-thickness ratios and lay-ups are presented and compared with results available in the literature.     

Analysis of sandwich plates with unbalanced cross-ply faces

The analysis of rectangular sandwich plates constructed of an orthotropic core and unbalanced cross-ply face plates is presented. A double Fourier series approach is used for simply supported sandwich plates under lateral loads. Results are compared to the corresponding sandwich plate results with orthotropic faces. The results indicate that the effect of bending-membrane coupling depends mainly upon the relative thicknesses of the core and faces; other factors include the shear stiffnesses of the core, the degree of anisotropy of the individual plies, the total number and lay-up of plies in the faces and the aspect ratio of the plate.     

Shear deformation in thermal vibration analysis of laminated plates by the GDQ method

The interlaminar stresses and deflections in a laminated rectangular plate under thermal vibration were determined by using the generalized differential quadrature (GDQ) method involving the effect of shear deformation. The approximate stress and deflection solutions are obtained under the vibration of sinusoidal temperature of thermal load for layer in cross-ply laminate. Numerical results show that the maximum deflection and stresses at center position of laminate increasing with the side-to-thickness ratio a/h^* value decreasing, especially in the region a/h^*20. The GDQ method provides an efficient method for calculating the interlaminar stresses and deflections in a multi-layered plate of cross-ply laminate subjected to thermal vibration of sinusoidal temperature including shear deformation with a few grid points.     

Uniformly loaded, clamped, cross-ply laminated, elliptic plates—An initial failure study

An exact polynomial solution for a uniformly loaded, antisymmetric cross-ply laminated, clamped elliptic plate is combined with the Tsai-Hill failure criterion for an initial flexural failure analysis of GFRP and CFRP plates. Non-dimensionalised initial failure loads and associated plate centre deflections—both of which are significantly dependent on the direction of the applied loading—are presented graphically for the complete range of aspect ratios for plates consisting of varying numbers of laminae. It is also demonstrated that changes in the in-plane boundary conditions produce negligible changes in the initial failure response of the plate.