Dynamic analysis of circular and sector thick,layered plates

The finite element method is extended to the free vibration analysis of laminated thick plates with curved boundaries. Two elements are developed on the basis of Mindlin's thick plate theory in which the effects of thickness-shear deformation and rotary inertia are included. Both elements are derived in polar co-ordinates and can be joined together to handle annular as well as circular laminated anisotropic plate problems. Since axisymmetry has not been assumed, variations in material properties in the tangential direction can be dealt with. Numerical results are presented to demonstrate the influence of geometrical shape as well as that of thickness-shear deformation on the free vibrations of both homogeneous and layered plates. Comparisons between the numerical results obtained and those presented by other investigators confirm the accuracy of the new elements. The elements also can be used in the analysis of rectangular plates by assuming very large radii and very small subtended angle values.     

Vibration of moderately thick square orthotropic stepped thickness plates

Numerous studies that address the vibration of stepped thickness plates are reported in the literature. Predominately, classical plate theory has been used to formulate studies for both isotropic and anisotropic stepped plates. Mindlin plate theory has been employed to obtain results for thick isotropic stepped thickness plates. Exact solutions, Rayleigh-Ritz, differential quadrature and finite element methods have been employed to compute results for frequency of vibration. Results for frequency of vibration for thick orthotropic stepped thickness plates are presented here using orthorhombic material properties of aragonite. The finite element method has been used to compute frequencies and determine mode shapes for simply supported and clamped square Mindlin plates.     

Axisymmetric vibrations of polar orthotropic mindlin annular plates of variable thickness

Analysis and numerical results are presented for the axisymmetric vibrations of polar orthotropic annular plates with linear variation in thickness, according to Mindlin's shear theory of plates. A chebyshev collocation technique has been employed to obtain the frequency equations for the transverse motion of such plates, for three different boundary conditions. Frequencies, mode shapes and moments for the first three modes of vibration have been computed for different plate parameters. A comparison of frequencies with the corresponding values obtained by classical plate theory leads to some interesting conclusions.     

Axisymmetric vibrations of an isotropic elastic non-homogeneous circular plate of linearly varying thickness

Free axisymmetric vibrations of an isotropic, elastic, non-homogeneous circular plate of linearly varying thickness have been studied on the classical theory of plates. The non-homogeneity of the material of the plate is assumed to arise due to the variation of Young's modulus and density with the radius vector whereas Poisson's ratio is assumed to remain constant. The governing differential equation of motion is solved by the method of Frobenius. The transverse displacement of the plate has been expressed as a power series in terms of the radial co-ordinate. The frequency parameters corresponding to the first two modes of vibration have been computed for different values of the non-homogeneity parameter and taper constant and for clamped and simply supported edge conditions of the plate. A comparison between the numerical results for homogeneous and non-homogeneous material of the plate is also made.     

Effects of large amplitude and transverse shear on vibrations of triangular plates

In this paper an analytical investigation of large amplitude free flexural vibrations of isotropic and orthotropic moderately thick triangular plates is carried out. The governing equations are expressed in terms of the lateral displacement, w, and the stress function, F, and are based on an improved non-linear vibration theory which accounts for the effects of transverse shear deformation and rotatory inertia. Solutions to the governing equations are obtained by using a single-mode approximation for w, Galerkin's method and a numerical integration procedure. Numerical results are presented in terms of variations of non-linear frequency ratios with amplitudes of vibrations. The effects of transverse shear, rotatory inertia, material properties, aspect ratios, and thickness parameters are studied and compared with available solutions wherever possible. Present results are in close agreement with those reported for thin plates. It is believed that all of the results reported here that are applicable for moderately thick plates are new and therefore, no comparison is possible.     

A sector finite element for dynamic analysis of thick plates

A 24 degree of freedom sector finite element is developed for the static and dynamic analysis of thick circular plates. The element formulation is based on Reissner's thick plate theory. The convergence characteristic of the elements is first studied in a static example of an unsymmetrically loaded annular plate. The obvious advantageous effect of including the twist derivatives of deflection as degrees of freedom is shown. The elements are then used to analyze the natural frequencies of an annular plate with various ratios of inner to outer radius. The results are in good agreement with an alternative solution in which thick plate theory is used. The versatility of this finite element is finally demonstrated by performing free vibration analysis of an example of clamped sector plates with various thicknesses and different sectorial angles.     

Accurate free vibration analysis of thick laminated circular plates with attached rigid core

This paper deals with the free vibration behavior of laminated transversely isotropic circular plates with axisymmetric rigid core attached at the center. The governing equations of motion are obtained based on Mindlin's first-order shear deformation plate theory. Two possible categories of vibration modes related to up-down translation of the core and wobbly rotation of the core about a diameter are studied. Accurate natural frequencies hitherto not reported in the literature are presented for a wide range of thickness-to-radius ratio, inner-to-outer radius ratio, mass and moment of inertia ratios of the core and various boundary conditions at the outer edge of the plate. Numerical results are compared with those of a three-dimensional finite element method (3-D FEM) to demonstrate the high accuracy and reliability of the current analysis.     

Shear deformable versus classical theories for nonlinear vibrations of rectangular isotropic and laminated composite plates

In the present study, (i) the classical Von Kárman theory, (ii) the first-order shear deformation theory and (iii) the higher-order (third-order) shear deformation theory are compared for studying the nonlinear forced vibrations of isotropic and laminate composite rectangular plates. In particular, the harmonic response in the frequency neighborhood of the fundamental mode of rectangular plates is investigated and the response curves computed by using the three different theories are compared. The boundary conditions of the plates are simply supported with immovable edges. Geometric imperfections are taken into account. Calculations for isotropic and laminated composite plates are presented and results are discussed. For isotropic plates, the frequency-response curves for large-amplitude vibrations obtained by using the three theories are almost coincident. For laminated composite plates, differences arise for relatively thick plates (ratio between the thickness and the edge equal to 0.1), while for thin plates (ratio between the thickness and the edge equal to 0.01), no difference is obtained. For all cases, the first-order shear deformation (with shear correction factor ) and the higher-order shear deformation theories give practically coincident results and differences are observed with respect to the classical Von Kárman theory.     

Vibration of cylindrical shells of bimodulus composite materials

A theory is formulated for the small amplitude free vibration of thick, circular cylindrical shells laminated of bimodulus composite materials, which have different elastic properties depending upon whether the fiber-direction strain is tensile or compressive. The theory used is the dynamic, shear deformable (moderately thick shell) analog of the Sanders best first approximation thin shell theory. By means of tracers, the analysis can be reduced to that of various simpler shell theories, namely Love's first approximation, and Donnell's shallow shell theory. As an example of the application of the theory, a closed form solution is presented for a freely supported panel or complete shell. To validate the analysis, numerical results are compared with existing results for various special cases. Also, the effects of the various shell theories, thickness shear flexibility, and bimodulus action are investigated.     

The buckling and frequency of flexural vibration of rectangular isotropic and orthotropic plates using Rayleigh's method

A simple approximate formula for the natural frequencies of flexural vibration of isotropic plates, originally developed by Warburton using characteristic beam functions in Rayleigh's method, is modified to apply to specially orthotropic plates and extended to include the effect of uniform, direct inplane forces. The initial buckling problem is treated simply by equating the frequency expression to zero. The approach permits the ready determination of reasonably accurate natural frequencies and/or buckling loads for a given plate involving any combination of free, simply supported or clamped edges, without requiring the aid of a sophisticated calculating device or a knowledge of plate, vibration or buckling theory. To illustrate the applicability and accuracy of the approach, numerical results for a number of specific plate problems are presented.     

Study on laser-generated Lamb waves propagation in viscoelastic and anisotropic plate

The propagation characteristics of laser-generated Lamb waves in thin composite plates are theoretically studied. Taking the anisotropic and viscoelastic properties of the composite material into account, the finite element models for simulating laser-generated Lamb waves in the composite material are established in the frequency domain. Numerical results are calculated in purely elastic and viscoelastic transversely isotropic plates, respectively. The effects of the anisotropic and viscoelastic properties on the propagation of Lamb waves are analyzed in detail. The numerical results exhibit that the features of the laser-generated Lamb wave, including attenuation, velocity, frequency, and the dispersive nature, have a close relationship with the anisotropic and viscoelastic properties of the material.     

Sound radiation from shear deformable stiffened laminated plates

Sound radiation from shear deformable stiffened laminated plates is studied theoretically. The equations of motion for the composite laminated plate are derived on the basis of the first-order shear deformation plate theory. Two sets of parallel stiffeners interact with the laminated plate only through the normal line forces. By using the Fourier wavenumber transform and the stationary phase method, the far-field sound pressure is described analytically. Sound pressure given by the first-order shear deformation plate theory and the classical thin plate theory is compared, and the differences of sound pressure are shown in the high frequency range for an isotropic plate. Sound pressure and the transverse displacement spectra are presented to illustrate the effects of force location, stiffeners and angle-ply layers. Sound radiation from symmetric and antisymmetric composite plates with multiple loadings is also investigated.     

Normal mode expansion method for laser-generated ultrasonic Lamb waves in orthotropic thin plates

A quantitative theory for modeling the laser-generated transient ultrasonic Lamb waves, which propagates along arbitrary directions in orthotropic plates, is presented by employing an expansion method of generalized Lamb wave modes. The displacement field is expressed by a summation of the symmetric and antisymmetric modes in the surface stress-free orthotropic plate, and therefore the theory is particularly appropriate for waveform analyses of Lamb waves in thin plates because one needs only to evaluate several lower modes. The transient waveforms excited by the thermoelastic expansion and the oil-coating evaporation are analyzed for a transversely isotropic thin plate. The results show that the theory provides a quantitative analysis to characterize anisotropic elastic stiffness properties of orthotropic plates by laser-generated Lamb wave detection.     

VIBRATION CHARACTERISTICS AND TRANSIENT RESPONSE OF SHEAR-DEFORMABLE FUNCTIONALLY GRADED PLATES IN THERMAL ENVIRONMENTS

Free and forced vibration analyses for initially stressed functionally graded plates in thermal environment are presented. Material properties are assumed to be temperature dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Theoretical formulations are based on Reddy's higher order shear deformation plate theory and include the thermal effects due to uniform temperature variation. The plate is assumed to be clamped on two opposite edges with the remaining two others either free, simply supported or clamped. One-dimensional differential quadrature technique, Galerkin approach, and the modal superposition method are used to determine the transient response of the plate subjected to lateral dynamic loads. Comprehensive numerical results for silicon nitride/stainless-steel rectangular plates are presented in dimensionless tabular and graphical forms. The roles played by the constituent volume fraction index, temperature rise, shape and duration of dynamic loads, initial membrane stresses as well as the character of boundary conditions are studied. The results reveal that, when thermal effects are considered, functionally graded plates with material properties intermediate to those of isotropic ones do not necessarily have intermediate natural frequencies and dynamic responses.     

Dynamic responses of Reissner–Mindlin plates with free edges resting on tensionless elastic foundations

Dynamic response analysis is presented for a Reissner–Mindlin plate with four free edges resting on a tensionless elastic foundation of the Winkler-type and Pasternak-type. The mechanical loads consist of transverse partially distributed impulsive loads and in-plane static edge loads while the temperature field is assumed to exhibit a linear variation through the thickness of the plate. The material properties are assumed to be independent of temperature. The two cases of initially compressed plates and of initially heated plates are considered. The formulations are based on Reissner–Mindlin first-order shear deformation plate theory and include the plate–foundation interaction and thermal effects. A set of admissible functions is developed for the dynamic response analysis of moderately thick plates with four free edges. The Galerkin method, the Gauss–Legendre quadrature procedure and the Runge–Kutta technique are employed in conjunction with this set of admissible functions to determine the deflection-time and bending moment–time curves, as well as shape mode curves. An iterative scheme is developed to obtain numerical results without using any assumption on the shape of the contact region. The numerical illustrations concern moderately thick plates with four free edges resting on tensionless elastic foundations of the Winkler-type and Pasternak-type, from which results for conventional elastic foundations are obtained as comparators. The results confirm that the plate will have stronger dynamic behavior than its counterpart when it is supported by a tensionless elastic foundation.     

不同铺设角度下层合板结构参数对声功率影响*

本文研究了不同纤维铺设角度的层合板结构参数对声功率的影响,从而为层合板的低噪声设计提供理论依据。通过分层有限元理论获得层合板结构动力学响应,基于声辐射模态概念分析不同铺设角度下层合板不同铺设方式、宽厚比和弹性模量比对其声功率影响。结果表明,层合板的铺设角度和宽厚比对复合材料层合板结构的声辐射功率影响较大。首先相同铺设角度的层合板,改变弹性模量比,声功率变化不明显;其次改变不同的铺设角度,宽厚比较小的层合板声功率下降的空间更大,更易于声功率的降低。层合板作为结构件时,从降低声功率角度而言总体上对称铺设结构比单向铺设层合板结构有优势,并且相同铺设角度下,反对称铺设层合板可获得更小的辐射声功率。     

DETERMINATION OF ELASTIC CONSTANTS OF ANISOTROPIC LAMINATED PLATES USING ELASTIC WAVES AND A PROGRESSIVE NEURAL NETWORK

In this paper, a procedure is suggested to inversely determine the elastic constants of anisotropic laminated plates using a progressive neural network (NN). The surface displacement responses are used as the inputs for the NN model. The outputs of the NN are the elastic constants of anisotropic laminated plates. The hybrid numerical method (HNM) is used to calculate the displacement responses of laminated plates to an incident wave for given elastic constants. The NN model is trained using the results from the HNM. A modified back-propagation learning algorithm with a dynamically adjusted learning rate and an additional jump factor is developed to tackle the possible saturation of the sigmoid function and to speed up the training process for the NN model. The concept of orthogonal array was adopted to generate the representative combinations of elastic constants, which reduces significantly the number of training data while maintaining its data completeness. Once trained, the NN model can be used for on-line determination of the elastic constants if the dynamic displacement responses on the surface of the laminated plate can be obtained. The determined elastic constants are then used in the HNM to calculate the displacement responses. The NN model would go through a progressive retraining process until the calculated displacement responses using the determined results are sufficiently close to the actual responses. This procedure is examined for an actual glass/epoxy laminated plate. It is found that the present procedure is very robust and efficient for determining the elastic constants of anisotropic laminated plates.     

EXPERIMENTAL AND THEORETICAL INVESTIGATION OF THE LINEAR AND NON-LINEAR DYNAMIC BEHAVIOUR OF A GLARE 3 HYBRID COMPOSITE PANEL

A theoretical model based on Hamilton's principle and spectral analysis is used to study the non-linear free vibration of hybrid composite plates made of Glare 3, a new aircraft structural material. It consists of alternating layers of metal- and fibre-reinforced composites. In previous work, the theoretical model has been used to calculate the first non-linear mode of fully clamped rectangular composite fibre-reinforced plastic (CFRP) laminated plates. This study concerns determination of the linear dynamic properties of the Glare 3 hybrid composite rectangular panel (G3HCRP) such as natural frequencies and mode shapes. The theoretical model is used to calculate the fundamental non-linear mode shape and associated flexural behaviour of the fully clamped G3HCRP. A series of experimental investigations have been conducted using a G3HCRP in order to determine linear dynamic properties. The response due to random excitation was investigated and the experimental measurements are analyzed and discussed. Comparisons are made with finite element predictions and response estimates given by the ESDU method, the latter being a “design guide” approach used by industry. Concerning the non-linear analysis, the results are given for various plate aspect ratios and vibration amplitudes, showing a higher increase of the induced bending stress near the clamps at large deflections. Comparisons between the dynamic behaviour of an isotropic plate and G3HCRP at large vibration amplitudes are presented and good results are obtained.     

Energy trapping in high-frequency vibrations of piezoelectric plates with partial mass layers under lateral electric field excitation

We study coupled face-shear (FS) and thickness-twist (TT) motions of a piezoelectric plate of monoclinic crystals with mass layers on the central parts of the plate surfaces. The plate is driven by a lateral electric field. Mindlin’s first-order theory of piezoelectric plates is used. An analytical solution is obtained. Numerical results are presented for an AT-cut quartz plate, including the motional capacitance of the plate as a resonator and the vibration modes trapped under the mass layers in the central portion of the plate. The relationship between the dimension of the mass layers and the number of trapped modes is examined.     

A variable kinematic Ritz formulation for vibration study of quadrilateral plates with arbitrary thickness

A new variable kinematic Ritz method applied to free vibration analysis of arbitrary quadrilateral thin and thick isotropic plates is presented. Carrera's unified formulation and the versatile pb-2 Ritz method are properly combined to build a powerful yet simple modeling and solution framework. The proposed technique allows to generate arbitrarily accurate Ritz solutions from a large variety of refined two-dimensional plate theories by expanding so-called Ritz fundamental nuclei of the plate mass and stiffness matrices. Theoretical development of the present methodology is described in detail. Convergence and accuracy of the method are examined through several examples on thin, moderately thick, and very thick plates of rectangular, skew, trapezoidal and general quadrilateral shapes, with an arbitrary combination of clamped, free and simply supported edges. Present results are compared with existing three-dimensional solutions from open literature. Maximum and average differences of various higher-order plate theories and three-dimensional results are also presented with the aim of providing useful guidelines on the choice of appropriate plate theory to get a desired accuracy on frequency parameters.