Vibration Analysis Of Sandwich Plates Research Articles

Postbuckling and postbuckled vibration analysis of sandwich plates under non-uniform mechanical edge loadings

Postbuckling and postbuckled vibration characteristics of sandwich plate subjected to non-uniform in-plane loadings is investigated in this article. The core compressibility effects are considered in the model by assuming fourth and fifth order expansions for the transverse and tangential displacement of the core. The exact stress distributions within the panel are determined by panel prebuckling analysis for the applied parabolic and partial edge loadings. Multi-term Galerkin's method is employed to reduce the governing partial differential equation to a set of non-linear ordinary differential equations in the case of dynamic analysis and to a set of non-linear algebraic equations in the case of static stability analysis. Newton-Raphson method in conjunction with Riks approach is used to trace the postbuckling path. Free vibration frequencies of the sandwich plate about a static equilibrium state are obtained by solving the associated linear eigenvalue problem. From numerical study it is observed that the postbuckling path becomes more softening type with increase of aspect ratio. Moreover, unloaded fixed edge of SCSC sandwich plate gives more rigidity than unloaded simply supported edge of CSCS. The prebuckling and postbuckling frequency increases with increase in edge restraint, and in postbuckling range, frequency increases with increase of in-plane load.

Free vibration analysis of sandwich plate with a transversely flexible core and FG-CNTs reinforced nanocomposite face sheets subjected to magnetic field and temperature-dependent material properties using SGT

In this research, the free vibration analysis of a sandwich plate with a transversely flexible core and functionally graded – carbon nanotubes (FG-CNTs) reinforced nanocomposite face sheets subjected to magnetic field and temperature-dependent material properties is presented based on high-order sandwich plate theory (HSAPT). The governing equations of motion are derived using Hamilton's principle. Classical plate theory (CPT) is used for modeling the face sheets and the effective properties of them are defined based on the extended mixture rule. Mechanical properties of the core such as Young's and shear modulli are assumed to be function of temperature. The influences of aspect and side ratios, temperature changes, core-to-face sheet thickness ratio, distribution types and volume fraction of carbon nanotubes are presented. The size-dependent mathematical formulation of the face sheets is developed based on the strain gradient theory (SGT). The results show that with increasing the aspect ratio, the non-dimensional natural frequency increases, however the side ratio has a reverse effect. As the temperature of the system increasing, the non-dimensional natural frequency decreases while by applying the magnetic field, the frequency parameter increases. Also it is concluded that the sandwich plate with X distribution type of carbon nanotubes in the face sheets, has higher frequency compared with the other types. Finally it is observed that by considering the size effect and introducing the material length scale parameters, the frequency of the sandwich plate increases. Employing FG-CNTs in face sheets, magnetic field, and size dependent effect leads to increase stiffness of nanostructures, thus the use of this sandwich plate has a prominent role in modern engineering applications.

High-order free vibration analysis of sandwich plates with both functionally graded face sheets and functionally graded flexible core

Free vibration analysis of functionally graded material sandwich plates is studied using a refined higher order sandwich panel theory. A new type of FGM sandwich plates, namely, both functionally graded face sheets and functionally graded flexible core are considered. The functionally graded material properties follow a power-law function. The first order shear deformation theory is used for the face sheets and a 3D-elasticity solution of weak core is employed for the core. On the basis of continuities of the displacements and transverse stresses at the interfaces of the face sheets and the core, equations of motion are obtained by using Hamilton’s principle. The accuracy of the present approach is validated by comparing the analytical results obtained for a degradation model (functionally graded face sheets and homogeneous flexible core) with ones published in the literatures, as well as the numerical results obtained by finite element method and good agreements are reached. Then, parametric study is conducted to investigate the effect of distribution of functionally graded material properties, thickness to side ratio on the vibration frequencies.

Bending and free vibration analysis of functionally graded sandwich plates using the Refined Zigzag Theory

The increasing application of functionally graded materials in the aerospace structures requires the adoption of accurate modeling strategies. The original formulation of the Refined Zigzag Theory is here extended to the bending and free vibration analysis of sandwich plates embedding functionally graded layers, either as faces or core. Extensive numerical investigation shows the superior predictive capabilities of Refined Zigzag Theory over both First-order Shear Deformation Theory and Third-order Shear Deformation Theory, if compared with the reference three-dimensional elasticity or high-fidelity FEM solutions. In virtue of its accuracy and of the C0–continuity requirement for shape functions, Refined Zigzag Theory can be adopted to formulate reliable and computationally efficient finite elements suited for large-scale analyses of both traditional and functionally graded structures.

Application of higher-order structural theory to bending and free vibration analysis of sandwich plates with CNT reinforced composite facesheets

In this paper, the bending and free flexural vibration behavior of sandwich plates with carbon nanotube (CNT) reinforced facesheets are investigated using QUAD-8 shear flexible element developed based on higher-order structural theory. This theory accounts for the realistic variation of the displacements through the thickness, and the possible discontinuity in slope at the interface, and the thickness stretch affecting the transverse deflection. The in-plane and rotary inertia terms are considered in the formulation. The governing equations obtained using Lagrange’s equation of motions are solved for static and dynamic analyses considering a sandwich plate with homogeneous core and CNT reinforced face sheets. The accuracy of the present formulation is tested considering the problems for which solutions are available. A detailed numerical study is carried out based on various higher-order models deduced from the present theory to examine the influence of the volume fraction of the CNT, core-to-face sheet thickness and the plate thickness ratio on the global/local response of different sandwich plates.

Analysis of sandwich plates with a new layerwise formulation

In the analysis of sandwich laminates, where core and skin materials are so different, layerwise formulations should be adopted.In this paper the static and free vibration analysis of sandwich plates by the use of collocation with radial basis functions and using a new layerwise theory with independent rotations in each layer and thickness stretching was performed. With this formulation, transverse normal and shear deformations and stresses are accurately computed.The equations of motion were automatically implemented via a Unified Formulation and interpolated with radial basis functions.Finally composite laminated plate and sandwich plate examples were tested and discussed.

A new higher-order shear and normal deformation theory for the static and free vibration analysis of sandwich plates with functionally graded isotropic face sheets

In this paper, a new higher-order shear and normal deformation theory for the bending and free vibration analysis of sandwich plates with functionally graded isotropic face sheets is developed. The number of unknown functions involved in the present theory is only five, as against six or more in case of other shear and normal deformation theories. The theory accounts for hyperbolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factor. The boundary conditions for the plate are assumed to be simply supported in all edges and in the static analysis, the plate is assumed to be subjected to a sinusoidally distributed load. Both symmetric and non-symmetric sandwich plates are considered. The equations of motion are obtained using Hamilton’s principle. Numerical results of present theory are compared with three-dimensional elasticity solutions and other higher-order theories reported in the literature. It can be concluded that the proposed theory is accurate and efficient in predicting the bending and free vibration responses of sandwich plates with functionally graded isotropic face sheets.

Free vibration analysis of sandwich plates with anisotropic face sheets in thermal environment by using the hierarchical trigonometric Ritz formulation

A free vibration analysis of sandwich plates with laminated anisotropic face sheets in thermal environment is carried out by means of the hierarchical trigonometric Ritz formulation (HTRF) built in the framework of Carrera’s unified formulation (CUF). CUF is presented in its generalized version allowing to produce hierarchically a wide range of advanced equivalent single layer, zig-zag and layer-wise plate theories including the capability to choose the desired expansion order for each unknown in the displacement field. The global–local free vibration response of the sandwich plates subjected to environment temperature variations is accurately described by the employed plate models accounting for through-the-thickness deformations. The effects of significant parameters such as length-to-thickness ratio, face-to-thickness ratio, number of layers and stacking sequence on the circular frequency parameters and on the modal displacements are examined.

Free and forced random vibration analysis of sandwich plates with thick viscoelastic cores

Free vibrations and the transverse response of sandwich plates with viscoelastic cores under wide-band random excitation is studied with special attention to the so-called pumping, thickness-shear and stretching modes. The quadratic displacement field is adopted for all displacement components of the core to accurately capture the higher modes excited by the wide-band excitation. The Love-Kirchhoff plate theory is used for the face layers. The viscoelastic behavior of the core is modeled by the Golla–Hughes–McTavish method. An analytical solution using the normal mode method is provided for the simply supported boundary conditions by including a different family of modes. The effects of some geometric and material properties on the frequencies, damping ratios and also root mean square responses are explored. The participation of the through-the-thickness deformation in the bending mode vibration of the top layer is also investigated, which is found to be mostly resulting from the second order term of the transverse displacement expansion in symmetric configurations. Moreover, the alteration of the response with the exclusion of a different family of modes from the solution is investigated.

Free vibration analysis of sandwich plates with functionally graded face sheets and temperature-dependent material properties: A new approach

Abstract Improved high-order sandwich plate theory is used to analyze the free vibration of sandwich plates with functionally graded (FG) face sheets in various thermal environments. The material properties of FG face sheets are assumed to be temperature-dependent by a third-order function of temperature and vary continuously through the thickness according to a power-law distribution in terms of the volume fractions of the constituents. Also, the material properties of the core are assumed to be temperature-dependent by a third-order function of temperature. The governing equations of motion in free natural vibration are derived using Hamilton's principle. A new approach is used to reduce the equations of motion and then solved them for both un-symmetric and symmetric sandwich plates. In-plane stresses of the core that usually are ignored in the vibration characteristics of the sandwich structures are considered in this formulation. The results show that the fundamental frequency parameter increases and decreases with increasing the volume fraction index for soft core and hard core sandwich plates, respectively. The results indicate that as the side-to-thickness ratio, the core-to-face sheet thickness ratio and the temperature are changed, a significant effect on the fundamental frequency parameter is observed. Good agreement is found between the theoretical predictions of the fundamental frequency parameters and the results obtained from other references for simply supported sandwich plates with functionally graded face sheets in the literature.

Dynamic behaviour of sandwich plates containing single/multiple debonding

Debonding is one of most severe defects associated with sandwich composite structures. The damage inflicted on debonding may lead to the significant degradation of the load carrying capacity of sandwich structures and affects their mechanical behaviours. Free vibration analyses of sandwich plates with honeycomb and PVC foam cores containing single/multiple debonding are carried out. Dynamic characteristics such as natural frequencies and corresponding mode shapes of sandwich plates with and without debonding are calculated using the commercial finite element code ABAQUS. Parametric studies over a wide range of size, location and number of debonding zones to examine the influence of these parameters on the overall dynamic behaviour of the damaged plates are performed. As a consequence, natural frequencies and mode shapes are being calculated as functions of these parameters that enable to draw conclusions concerning the debonding influence.

A generalized global–local high-order theory for bending and vibration analyses of sandwich plates subjected to thermo-mechanical loads

Although the global higher-order shear deformation theories may predict the gross responses of the sandwich plates sufficiently accurate, their results may show considerable errors in predicting the local effects. Layerwise and mixed layerwise theories are computationally expensive and generally, the interlaminar transverse stresses continuity conditions are not enforced in the former category of theories. Majority of the available zigzag and global–local theories suffer from the point that the transverse normal stress continuity that influences the transverse deformation significantly, especially in sandwich plates with soft-cores, is not satisfied at the layer interfaces. In the present paper, a generalized global–local theory that guarantees the continuity condition of all of the displacement and transverse stress components and considers the transverse flexibility under thermo-mechanical loads is introduced. One of the advantages of the present theory is that the number of unknown parameters is independent of the number of the layers. Furthermore, all stress components are considered in the formulations. Therefore, in contrast to the available works, the theory may be used for sandwich plates with stiff or soft cores. In contrast to the available global–local formulations, the present formulation is developed in a compact matrix form that makes it more desirable for computerized solutions. The present theory may be considered as a generalized layerwise theory with an optimized computational time. Compatible quadrilateral Hermitian elements are employed to further enhance the accuracy of the results. Validity, advantages, and efficiency of the present theory are investigated for different local and global behaviors of the layered composite and sandwich plates. Comparison of the present results with those of the three-dimensional theory of elasticity and the available plate theories confirms the efficiency and accuracy of the proposed theory. Results reveal that the global theories (e.g. the higher-order shear deformation theories) may encounter serious accuracy problems even in predicting the gross responses of the sandwich plates.

Free Vibration Analysis of Sandwich Plates with a Uniformly Distributed Attached Mass, Flexible Core, and Different Boundary Conditions

The free vibration analysis of rectangular sandwich plates with a flexible core both with and without a uniformly distributed attached mass on the top facesheet is carried out using the finite element method (FEM) through ANSYS parametric design language, which is validated by several literatures through some numerical examples. The model uses the 8-node shell 99 element to model the composite laminates of the top and bottom facesheets of the sandwich plate and 20-node high-order solid 95 element in order to model the flexible PVC core. The validated finite element model is then used to study the parametric effects of geometry such as aspect ratio, length-to-thickness ratio, core thickness-to-plate thickness ratio, the size and the stiffness of the attached mass on the natural frequencies of the sandwich plate as well as the normal and shear stresses. The use of FEM also allows studying the effect of different boundary conditions for both the top and the bottom facesheets of sandwich plates with or without distributed attached mass. Numerical results that hitherto not reported in the literature have been presented in this article. The results presented in this investigation could be useful to acquire a better insight into the behavior of sandwich laminates carrying attached mass for engineering designers of sandwich structures. The results are presented and compared with the latest Numerical results found in literature.

Free Vibration of Sandwich Plates and Shells by Using Zig-Zag Function

This paper analyses the free vibration response of sandwich curved and flat panels by introducing the zig-zag function (—1)kζk(ZZF) in the displacement models of classical and higher order two-dimensional shell theories. The main advantage of ZZF is the introduction of a discontinuity in the first derivative, zig-zag effect, of the displacements distribution with correspondence to the core/faces interfaces. Results including and discarding ZZF are compared. Several values of face-to-core stiffness ratio (FCSR) and geometrical plate/shell parameters have been analyzed. Both fundamental vibration modes and those corresponding to high wave numbers are considered in the analysis. It is concluded that: (1) ZZF is highly recommended in the free vibration analysis of sandwich plates and shells; (2) the use of ZZF makes the error almost independent by FCSR parameter; (3) ZZF is easy to implement and its use should be preferred with respect to other `more cumbersome' refined theories.

Free vibration of sandwich plates and shells by using Zig-Zag function

This paper analyses the free vibration response of sandwich curved and flat panels by introducing the zig-zag function (—1)k ζ k (ZZF) in the displacement models of classical and higher order two-dimensional shell theories. The main advantage of ZZF is the introduction of a discontinuity in the first derivative, zig-zag effect, of the displacements distribution with correspondence to the core/faces interfaces. Results including and discarding ZZF are compared. Several values of face-to-core stiffness ratio (FCSR) and geometrical plate/shell parameters have been analyzed. Both fundamental vibration modes and those corresponding to high wave numbers are considered in the analysis. It is concluded that: (1) ZZF is highly recommended in the free vibration analysis of sandwich plates and shells; (2) the use of ZZF makes the error almost independent by FCSR parameter; (3) ZZF is easy to implement and its use should be preferred with respect to other `more cumbersome' refined theories.

High-order free vibration of sandwich panels with a flexible core

Free vibration analysis of sandwich panels with a flexible core based on the high-order sandwich panel theory approach is presented. The mathematical formulation uses the Hamilton principle and includes derivation of the governing equations along with the appropriate boundary conditions. The formulation embodies a rigorous approach for the free vibration analysis of sandwich plates with a general construction, having high-order effects owing to the non-linear patterns of the in-plane and the vertical displacements of the core through its height. As such, it improves on the available classical and high-order theories. The formulation uses the classical thin plate theory for the face sheets and a three-dimensional elasticity theory or equivalent one for the core. The analyses are valid for any type of loading scheme, localized as well as distributed, and distinguish between loads applied at the upper or the lower face. It can also deal with any type of boundary conditions that may be different at the upper and the lower face sheets at the same edge. The effects of the rotary inertia of the various constituents of the sandwich construction are included. Two types of computational models are considered. The first model uses the vertical shear stresses in the core in addition to the displacements of the upper and the lower face sheets as its unknowns. The second model assumes a polynomial description of the displacement fields in the core that is based on the displacement fields of the first model. In this case the unknowns are the coefficients of these polynomials in addition to the displacements of the various face sheets. The two computational models have been validated numerically through a very good comparison with the well known classical and high-order plate theories. The numerical study consists of free vibration eigenmodes of two typical simply-supported panels, including higher modes that cannot be detected by other high-order computational models, and a parametric study that compares the results of the various computational models and the first-order shear deformable results.

Sandwich-Plate Vibration Analysis: Three-Layer Quasi-Three-Dimensional Finite Element Model

A three-layer finite element model for the vibration analysis of sandwich plates with laminated composite face sheets is evaluated. In the model the face sheets are represented as Reissner-Mindlin plates, and the core is modeled as a three-dimensional continuum. This representation allows accurate modeling for a wide range of core types. The three-dimensional problem is reduced to two dimensions by analytical through-thickness integration of the energy expressions for the evaluation of mass and stiffness matrices. The results from this model are compared to finite element results based on solid elements, classical sandwich analysis, and classical plate theory. The objective in the work is to compare natural frequency and mode shape predictions using these models for a broad range of core stiffness. When large differences between face sheet and core stiffness are present, it is illustrated that traditional laminate theories yield significant inaccuracy. Moreover, unlike plate models, the present theory is also capable of representing a variety of three-dimensional boundary conditions. Furthermore, compared to solid models, the present laminated model avoids numerical problems as a result of three-dimensional element aspect ratio. Therefore, the present model provides a powerful general tool for the analysis of natural modes and frequencies of sandwich plates.

Vibration Analysis of Arbitrarily Shaped Sandwich Plates via Ritz Method

This article presents the Ritz method for the vibration analysis of sandwich plates having an orthotropic core and laminated facings. The planform of the plate may take on any arbitrary shape. On the basis of the Mindlin plate theory and the Ritz method, the governing eigenvalue equation for determining the natural frequencies was derived. The Ritz method was automated and made computationally effective for general-shaped plates with any boundary conditions by (1) adopting the product of polynomial functions and boundary equations that were raised to appropriate powers and (2) applying Green's theorem to transform the integration over the general-shaped domain into a closed line integration. The Ritz formulation and software were verified by the close agreement with vibration frequencies obtained by previous researchers for a wide range of subset plate problems involving isotropic, laminated, and sandwich plates of various shapes. Moreover, sample natural frequencies of sandwich plates with laminated facings are presented for some quadrilateral plate shapes. These frequencies should be useful as reference results to researchers who are developing new methods or software for vibration analysis of sandwich plates.

Free vibration analysis of sandwich plates with laminated faces using spline finite point method

The free vibration problem of unsymmetrical sandwich plates with laminated composite faces and orthotropic core is analysed in the consideration of the effects of bending, transverse shear, plane inertia, rotatory inertia, and coupling inertia. The spline finite point method (SFPM) is selected to solve the present problem and to establish the basic formulae for a computer programme that can be run on microcomputer. A lot of numerical examples of the cross-ply face sandwich plates, with various boundary conditions, are obtained by the present method to make clear for coupling effects of composite material plates and to seek the optimum ply design of engineering structures.

Plane finite element for static and free vibration analysis of sandwich plates

Triangular sandwich finite elements (MPLW30) with 30 DOF have been developed for the static and free vibration analysis of sandwich plates with thin faces and low core shear modules. The results of numerical examples presented here demonstrate the accuracy and suitability of the formulations for the analysis of sandwich plates with kH > 25 and kE > 200.