On the influence of membrane inertia and shear deformation on the geometrically non-linear vibrations of open,cylindrical, laminated clamped shells

It is the objective of this work to verify the validity of different approximations in the analysis of geometrically non-linear vibrations of open, cylindrical, laminated, fully clamped shells. Namely, it is intended to verify if membrane inertia and transverse shear deformation can be neglected and when. p-version finite elements with hierarchic basis functions are employed to define the models. A simple comparison of the different stiffness and mass terms is carried out first, to assess the relative importance of membrane inertia and shear deformation. Then a few numerical tests are carried out in non-linear vibrations by solving the ordinary differential equations of motion by Newmark’s method. It is found that what are usually considered to be thin panels actually require the consideration of shear deformation.     

Three-dimensional free and transient vibration analysis of composite laminated and sandwich rectangular parallelepipeds: Beams,plates and solids

This paper presents an efficient method for predicting the free and transient vibrations of multilayered composite structures with parallelepiped shapes including beams, plates and solids. The exact three-dimensional elasticity theory combined with a multilevel partitioning hierarchy, viz., multilayered parallelepiped, individual layer and layer segment, is employed in the analysis. The continuity constraints on common interfaces of adjacent layer segments are imposed by a modified variational principle, and the displacement components of each layer segment are assumed in the form of orthogonal polynomials and/or trigonometric functions. Numerical studies are given for free vibrations of composite laminated and sandwich beams, plates, and solids. Some in-plane shear vibration modes missed in previous elasticity solutions for multilayered plates are examined. The natural frequencies derived from Reddy’s high-order shear deformation theory and layerwise theory for soft-core sandwich plates show significant deviation from elasticity solutions. Transient displacement and stress responses for angle-ply laminated and sandwich plates under four types of impulsive loads (including rectangular, triangular, half-sine and exponential pulses) are obtained by the Newmark integration procedure. The present solutions may serve as benchmark data for assessing the accuracy of advanced structural theories and new developments in numerical methods.     

Dynamic analysis of isotropic composite plates using a layerwise theory

The paper is concerned with small amplitude vibrations of moderately thick composite plates. Polygonal and simply supported plates of three asymmetrically arranged layers in perfect bond are considered. The homogeneous layers are assumed to consist of isotropic linear elastic materials. The Mindlin–Reissner kinematic assumptions are implemented separately to each layer, and as such model both the global and local elastic response of composite plates. The continuity of the transverse shear stress across the interfaces is prescribed according to Hooke's law. For plates composed of layers with a uniform ratio of mass density to shear modulus and a common Poisson's ratio the lateral deflection can be calculated independently from the in-plane displacements. It is shown that the lateral deflection is governed by a boundary value problem of the fourth order. Alternatively, by neglecting the longitudinal as well as rotary inertia the complex problem reduces to the simpler case of a homogenized shear-deformable plate with effective stiffness and corresponding set of boundary conditions. The theory is applied to statically and dynamically loaded rectangular simply supported composite plates with various dimensions and material properties.     

An analysis of nonlinear vibrations of coupled thickness-shear and flexural modes of quartz crystal plates with the homotopy analysis method

We investigated the nonlinear vibrations of the coupled thickness-shear and flexural modes of quartz crystal plates with the nonlinear Mindlin plate equations, taking into consideration the kinematic and material nonlinearities. The nonlinear Mindlin plate equations for strongly coupled thickness- shear and flexural modes have been established by following Mindlin with the nonlinear constitutive relations and approximation procedures. Based on the long thickness-shear wave approximation and aided by corresponding linear solutions, the nonlinear equation of thickness-shear vibrations of quartz crystal plate has been solved by the combination of the Galerkin and homotopy analysis methods. The amplitude frequency relation we obtained showed that the nonlinear frequency of thickness-shear vibrations depends on the vibration amplitude, thickness, and length of plate, which is significantly different from the linear case. Numerical results from this study also indicated that neither kinematic nor material nonlinearities are the main factors in frequency shifts and performance fluctuation of the quartz crystal resonators we have observed. These efforts will result in applicable solution techniques for further studies of nonlinear effects of quartz plates under bias fields for the precise analysis and design of quartz crystal resonators.     

受热双层圆薄板的轴对称非线性振动

通过对双层薄板坐标参考面的选择,基于Von Kár mán理论和Hamilton原理,建立了受热双层圆板轴对称非线性振动控制方程的简化形式。采用Galerkin法进行时空变量分离后,由Lindstedt-Poincaré 摄动法获得了受热双层板非线性振动频率与振幅间的特征关系。文末以某热致微型泵的驱动膜片为例,研究了不同温度对其振动特性的影响。本文的方法极易推广到单层和多层受热薄板非线性振动的分析中。     

On vibrations of thin plates with one-dimensional periodic structure

A new approach to problems of thin plates with a periodic structure along one direction in the planes parallel to the plate midplane is proposed. The model is a certain generalisation of the length-scale models for periodic plates, which make it possible to take into account the effect of periodicity cell size on the dynamics of plates with two-dimensional periodic structure [J. J drysiak, C. Woźniak, J. Theoret. Appl. Mech. 33 (1995) 337–349; J. J drysiak, Eng. Trans. 46 (1998) 73–87]. In order to describe this effect in stationary processes, e.g., a plate stability, and non-stationary processes for one-dimensional periodic plates the generalised model is presented in this paper. In order to show differences between governing equations of the extended model and the model for plates with two-dimensional periodic structure or the homogenised model vibrations problems of one-dimensional periodic plates will be analysed.     

Effect of cold expansion on the fatigue life of Al 2024-T3 in double shear lap joints: Experimental and numerical investigations

In this paper the effect of cold expansion on fatigue life improvement of aluminum alloy 2024-T3 plates used in double shear lap joints is investigated experimentally by conducting fatigue tests and numerically by implementing finite element simulations. In the experimental part, fatigue tests were carried out on the plates with cold expansion levels of 0%, 1.5% and 4.7% which were used in double shear lap joints. In the numerical study, three-dimensional finite element models were employed to predict stress distributions in the cold expanded plates used in the double shear lap joint. The results obtained from finite element simulation, have been employed to explain the trends which were observed in the experimentally attained S–N data and the fatigue crack initiation location. The experimental and numerical results showed that cold expansion improves fatigue life at low load levels and the life enhancement is more for the bigger cold expansion size. However, the fatigue life improvement is smaller in double shear lap joints compared to a single cold expanded plate.     

Eigenstrain induced vibrations of composite plates

Summary The paper is concerned with eigenstrain induced small amplitude vibrations of moderately thick composite plates. In case of thermal shock loading such eigenstrains are imposed as thermal strains. Within a multiple field approach, thermal strains, piezoelectrically induced strains, as well as additional fields of strains that are due to inelastic deformations are interpreted as eigenstrains acting in the associated linear elastic background structure. Polygonal simply supported composite plates with three asymmetrically arranged layers in perfect bond are considered. The homogeneous layers consist of dissimilar linear elastic isotropic materials. The Mindlin-Reissner kinematic assumptions are implemented separately to each layer. The continuity of the transverse shear stress across the interfaces is prescribed according to Hooke's law. For plates composed of layers with a uniform ratio of mass density to shear modulus and a common Poisson's ratio the lateral deflection can be calculated independently from the in-plane displacements. It is shown that the lateral deflection is governed by a boundary value problem of the fourth order. Alternatively, by neglecting the longitudinal as well as rotatory inertia the complex problem reduces to the simpler case of a homogenized shear-deformable plate with effective stiffness, a corresponding set of boundary conditions and effective eigenstrain resultants. The theory is applied to rectangular simply supported composite plates of various dimensions.     

Optimum distribution of smart stiffeners in sandwich plates subjected to bending or forced vibrations

The problems of optimum distribution of active stiffeners manufactured from piezoelectric or shape memory alloy materials and bonded to or embedded within the facings of a sandwich plate are considered. The sandwich plate consists of thin composite or isotropic facings which are in the state of plane stress and a thick shear deformable core. The amplitude of forced vibrations of the plate is reduced using symmetric couples of piezoelectric stiffeners subjected to out-of-phase dynamic voltages. Shape memory alloy stiffeners are used to reduce bending deformations. In the latter case, a desirable effect is achieved by activating the stiffeners on one side of the middle surface. Optimum design is considered based on the requirement of minimal transverse static or dynamic deflections subject to a constraint on the volume of smart stiffeners. The variables employed in the process of optimization are the ratios of the cross-sectional areas of the stiffeners in each direction to their respective spacings. It is shown, that, dependent on the load, materials, and geometry, optimum design can significantly reduce deflections, i.e. enhance the strength, of sandwich plates.     

Axisymmetric parametric resonance of polar orthotropic sandwich annular plates

Parametric resonance of polar orthotropic sandwich annular plates with a viscoelastic core layer subjected to a periodic uniform radial stress is studied by the finite element method. The axisymmetric discrete layer annular element and Hamilton's principle are employed to derive the finite element equations of motion for a sandwich plate including the transverse shear effect. The viscoelastic material in the core layer is assumed to be incompressible, and the extensional and shear moduli are described by complex quantities. The regions of dynamic instability are determined by Bolotin's method. The effects of various parameters on the dynamic instability regions are investigated.     

Stability and bifurcations of the steady-state modes in nonlinear vibrations of plates and shells

A method is proposed for numerical investigation of stability and bifurcations of nonlinear vibrations of plates and shells under the action of periodic loads. A system of equations of motion are formed by the methods of finite elements and generalized coordinates by accounting for the geometrical nonlinearity of the objects. The obtained system of differential equations is solved by the parameter extension method. As a result, steady-state curves of the nonlinear vibrations are constructed. Branching solutions at the points of bifurcation are obtained by the method of projections. The proposed method makes it possible to determine the subharmonic solutions and investigate their stability over a wide range of load parameters. Nonlinear vibrations of a series of plates and shells are investigated.Kiev. Translated from Problemy Prochnosti, No. 12, pp. 97–102, December, 1989.     

Influence of transverse shear on stochastic instability of symmetric cross-ply laminated plates

The dynamic instability problem of symmetrically laminated cross-ply plates, compressed by time-dependent stochastic membrane forces, is investigated. The effect of shear deformation is included in the formulation. By using the direct Liapunov method, bounds of the almost sure instability of cross-ply plates are obtained. Furthermore, it is shown how the viscous damping coefficient, variances of the stochastic forces, ratio of the principal lamina stiffnesses, number of layers, plate aspect ratio, and cross-ply ratio influence the instability regions. A special attention is given to the proper choice of two shear correction factors of the laminated plate. Numerical calculations are performed for the Gaussian process with a zero mean and variance σ² as well as for the harmonic process with an amplitude A.     

法向应力和法向应变对复合材料层板自由振动基础频率的影响

一种包含法向应力,法向应变和横向剪切影响的复合材料层板自由振动的新位移模式在本文中提出.通过应用层板自由振动的Rayleigh-Ritz分析,比较了在简支边界条件下复合材料层板自由振动的YNS理论解[1]和本文新位移模式解,从而得出在简支边界条件下法向应力和法向应变对复合材料层板自由振动基础频率的影响.     

Asymmetric free vibrations of laminated annular cross-ply circular plates including the effects of shear deformation and rotary inertia: spline method

Asymmetric free vibrations of annular cross-ply circular plates are studied using spline function approximation. The governing equations are formulated including the effects of shear deformation and rotary inertia. Assumptions are made to study the cross-ply layered plates. A system of coupled differential equations are obtained in terms of displacement functions and rotational functions. These functions are approximated using Bickley- type spline functions of suitable order. Then the system is converted into the eigenvalue problem by applying the point collocation technique and suitable boundary conditions. Parametric studies have been made to investigate the effect of transverse shear deformation and rotary inertia on frequency parameter with respect to the circumferential node number, radii ratio and thickness to radius ratio for both symmetric and anti-symmetric cross-ply plates using various types of material properties.     

Non-linear vibrations of variable stiffness composite laminated plates

It is the objective of this work to analyze vibrations of variable stiffness composite laminated plates (VSCL), and investigate the differences between the oscillations of these plates and traditional laminates. The analysis is based on numerical experiments and a new p-version finite element with hierarchic basis functions, which follows first order shear deformation theory and considers geometrical non-linearity, is derived. Considering first linear oscillations, the natural frequencies and mode shapes of different VSCL are computed and compared with the ones of constant stiffness laminates. The linear natural frequencies of the present model are also compared with the ones computed using a recently developed higher-order model for VSCL. After, numerical tests are carried out in the time domain and, for the first time in VSCL, taking geometric non-linearity into account, to investigate the response to external forces. The non-linear ordinary differential equations of motion are solved by Newmark’s method. It is verified that the variation of the fibre orientation can lead to significant differences in the amplitudes of the non-linear response.     

Nonlinear vibrations of thick plates using mindlin plate elements

Large amplitude vibrations of Mindlin plates are studies using Lagrangian, isoparametric, quadrilateral elements with selective integration. Square, rectangular, circular and elliptical plates with clamped and simply supported boundary conditions are considered.     

Numerical algorithms and results for SC-cut quartz plates vibrating at the third harmonic overtone of thickness shear

Finite element matrix equations, derived from two-dimensional piezoelectric high frequency plate theory are solved to study the vibrational behavior of the third overtone of thickness shear in square and circular SC-cut quartz resonators. The mass-loading and electric effects of electrodes are included. A perturbation method which reduces the memory requirements and computational time significantly is employed to calculate the piezoelectric resonant frequencies. A new storage scheme is introduced which reduces memory requirements for mass matrix by about 90% over that of the envelope storage scheme. Substructure techniques are used in eigenvalue calculation to save storage. Resonant frequency and the mode shapes of the harmonic third overtone thickness shear vibrations for square and circular plates are calculated. A predominant third overtone thickness shear displacement, coupled with the third overtone of thickness stretch and thickness twist, is observed. Weak coupling between the third order thickness shear displacement and the zeroth-, first-, and second-order displacements is noted. The magnitudes of the lower order displacements are found to be about two orders smaller than that of the third overtone thickness shear displacement.     

Modal validation of a set of C0-compatible composite shell finite elements

This paper presents efficient C⁰-compatible finite elements for modelling laminated composite shells under free vibrations. Derived from the first-order shear deformation theory (equivalent single-layer laminate model), the elements are well adapted for evaluating the global dynamic response (natural frequencies and mode shapes) of moderately thick multilayered shells. The components of their structural matrices are based on an exact integration per layer, which results in a higher solution accuracy than with standard explicit through-the-thickness schemes. The described finite element formulation, which can be easily implemented in commercial finite element codes, is next validated by means of several experimental modal test cases on thin to relatively thick plates or shells.     

A Reissner-Mindlin-type plate theory including the direct piezoelectric and the pyroelectric effect

Summary This paper is concerned with flexural vibrations of composite plates, where piezoelastic layers are used to generate distributed actuation or to perform distributed sensing of strains in the plate. Special emphasis is given to the coupling between mechanical, electrical and thermal fields due to the direct piezoelectric effect and the pyroelectric effect. Moderately thick plates are considered, where the influence of shear and rotatory inertia is taken into account according to the kinematic approximations introduced by Mindlin. An equivalent single-layer theory is thus derived for the composite plates. It is shown that coupling can be taken into account by means of effective stiffness parameters and an effective thermal loading. Polygonal plates with simply supported edges are treated in some detail, where quasi-static thermal bending as well as free, forced and actuated vibrations are studied.     

Test plant for the analysis of deformation vibrations of models of structures by means of a method of contactless excitation

A plant for the analysis of the strain vibrations of a periodic shelf structure by a method of low-frequency, contactless excitation is created and tested. The method involves varying the amplitude of a stepwise force, recording the strain response of the structure at sites where ferroelectric gauges are located, and Fourier analysis of the dynamic strain. The presence of stochastic and high-Q strain vibrations in the model structure is confirmed. It is established that, in general, both natural as well as induced (including stochastic) vibrations may be recorded and distinguished by means of two or three ferroelectric strain gauges.