A symplectic analytical wave propagation model for damping and steady state forced vibration of orthotropic composite plate structure

An analytical wave propagation model is proposed in this paper for damping and steady state forced vibration of orthotropic composite plate structure by using the symplectic method. By solving an eigen-problem derived in the symplectic dual system of free bending vibration of orthotropic rectangular thin plates, the wave shape of plate is obtained in symplectic analytical form for any combination of simple boundary conditions along the plate edges. And then the specific damping capacity of wave mode is obtained symplectic analytically by using the strain energy theory. The steady state forced vibration of built-up plates structure is calculated by combining the wave propagation model and the finite element method. The vibration of the uniform plate domain of the built-up plates structure is described using symplectic analytical waves and the connector with discontinuous geometry or material is modeled using finite elements. In the numerical examples, the specific damping capacity of orthotropic rectangular thin plate with three different combinations of boundary condition is first calculated and analyzed. Comparisons of the present method results with respect to the results from the finite element method and from the Rayleigh–Ritz method validate the effectiveness of the present method. The relationship between the specific damping capacity of wave mode and that of modal mode is expounded. At last, the damped steady state forced vibration of a two plates system with a connector is calculated using the hybrid solution technique. The availability of the symplectic analytical wave propagation model is further validated by comparing the forced response from the present method with the results obtained using the finite element method.     

Numerical manifold method for vibration analysis of Kirchhoff's plates of arbitrary geometry

Many rectangular plate elements developed in the history of finite element method (FEM) have displayed excellent numerical properties, yet their applications have been limited due to inability to conform to the arbitrary geometry of plates and shells. Numerical manifold method (NMM), considered to be a generalization of FEM, can easily solve this issue by viewing a mesh made up of rectangular elements as mathematical cover. In this study, ACM element (Adini and Clough element from A. Adini, R.W. Clough, Analysis of plate bending by the finite element method, University of California, 1960), a typical rectangular plate element is first integrated in the framework of NMM. Then, vibration analysis of arbitrary shaped thin plates is conducted employing the tailored NMM. Using the definition of integral of scalar functions on manifolds, we developed a mathematically rigorous mass lumping scheme for creating a symmetric and positive definite lumped mass matrix that is easy to inverse. A series of numerical experiments have been studied and analyzed, including free and forced vibration of thin plates with various shapes, validating the proposed mass lumping scheme can supersede the consistent mass formulation in those cases.     

Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method

This paper presents a novel finite element formulation for static, free vibration and buckling analyses of laminated composite plates. The idea relies on a combination of node-based smoothing discrete shear gap method with the higher-order shear deformation plate theory (HSDT) to give a so-called NS-DSG3 element. The higher-order shear deformation plate theory (HSDT) is introduced in the present method to remove the shear correction factors and improve the accuracy of transverse shear stresses. The formulation uses only linear approximations and its implementation into finite element programs is quite simple and efficient. The numerical examples demonstrated that the present element is free of shear locking and shows high reliability and accuracy compared to other published solutions in the literature.     

Stress Analysis for a Thick Rectangular Plate of a Composite Material with a Spatially Periodically Curved Structure under Forced Vibration

Natural and forced vibrations of a thick rectangular plate fabricated from a composite material with a spatially periodically curved structure are investigated with the use of exact three-dimensional equations of motion of the theory of elastic anisotropic bodies. The investigations are carried out within the framework of the continuum approach developed by Akbarov and Guz'. It is supposed that the plate is clamped at all its edges and is loaded on the upper face with uniformly distributed normal forces periodically changing with time. The influence of curving parameters on the fundamental frequency of the plate and on the distribution of the normal stress acting in the thickness direction under forced vibration is studied. The corresponding boundary-value problems are solved numerically by employing the three- dimensional FEM modeling.     

The development of hybrid ES-FE-SEA method for mid-frequency vibration analysis of complex built-up structure

The beam-plate structures and its complex composite structures are widely used in practical engineering. Recently, a hybrid FE-SEA method was applied for the vibration analysis of beam-plate structures in mid-frequency regime. However, the accuracy of prediction is still low, due to “overly-stiff” feature of embedded conventional FEM. In this work, a hybrid ES-FE-SEA model of a beam-plate built-up structure is developed for response prediction in vibration testing, in which, the edge-based smoothed technique is applied in the 3D beam structure FEM model to soften the whole system stiffness. Then, combined with SEA, this ES-FEM is embedded to achieve a hybrid ES-FE-SEA framework, to improve the accuracy of mid-frequency response predictions of the complex built-up structure. In hybrid ES-FE-SEA model, the plate structure which has a higher model density is considered as a statistical subsystem and modeled statistically using statistical energy analysis (SEA). The beam structure with a relative lower model density is modeled deterministically using edge-based smooth finite element method (ES-FEM). The coupling between these two different types of subsystems is achieved through the diffuse field reciprocity relation. The acceleration loads are applied to the model. The results obtained by the ES-FE-SEA and FE-SEA are compared. It is found that the hybrid ES-FE-SEA method is reliable for the mid-frequency vibration problems of the beam-plate built-up structure. The proposed ES-FE-SEA is verified by various numerical examples.     

Time dependent uncertain free vibration analysis of composite CFST structure with spatially dependent creep effects

In this study, the time dependent free vibration analysis of composite concrete-filled steel tubular (CFST) arches with various uncertainties is thoroughly investigated within a non-stochastic framework. From the practical inspiration, both uncertain material properties and mercurial creep effect associated with such composite materials are simultaneously incorporated. Unlike traditional non-probabilistic schemes, both spatially independent (i.e., conventional interval models) and dependent (i.e., interval fields) interval system parameters can be comprised within a unified uncertain free vibration analysis framework for CFST arches. For the purpose of achieving a robust framework of the time-dependent uncertain free vibration analysis, a new computational approach, which has been developed within the scheme of the finite element method (FEM), has been proposed for determining the extreme bounds of the natural frequencies of practically motivated CFST arches. Consequently, by successfully solving two eigenvalue problems, the upper and lower bounds of the natural frequencies of such composite structures with various uncertainties can be rigorously secured. The unique advantage of the proposed approach is that it can be effectively integrated within commercial FEM software with preserved sharp bounds on natural frequencies for any interval field discretisation. The competence of the proposed computational analysis framework has been thoroughly demonstrated through investigations on both 2D and3D engineering structures.     

An energy flow model for high-frequency vibration analysis of two-dimensional panels in supersonic airflow

Many energy flow models have been proposed for high-frequency forced vibration analysis of structures. In this paper, a novel energy flow model is developed to predict the high-frequency vibration response of panels in supersonic airflow and quantify the effects of supersonic airflow on high-frequency forced vibration characteristics. The additional damping due to supersonic airflow is derived from the motion equation of a two-dimensional panel. The relationship between the wavenumber and the group velocity is introduced to describe the energy transmission property considering the effects of supersonic airflow. Then the energy density governing equation (i.e. energy flow model) is established and solved by the energy flow analysis (EFA) and the energy finite element method (EFEM). Finally, comparing the vibration responses obtained by the present energy flow model with the corresponding exact analytical solutions, the developed energy flow model is verified to be effective for high-frequency vibration analysis of panels in supersonic airflow. Furthermore, the numerical simulations indicate that supersonic airflow can affect the equivalent damping of the propagating elastic waves, and thus change the energy density distribution of the panel.     

A Coupled FE-BE Analysis of Smart Lightweight Structures for Active Noise and Vibration Control

Over the past years much research and development has been done in the area of active control in order to improve the acoustical and vibrational properties of thin–walled lightweight structures. An efficient technique for actively reducing the structural vibration and sound radiation is the application of smart structures. In smart structures piezoelectric materials are often used as actuators and sensors. The design of smart structures requires fast and reliable simulation tools. Therefore, the purpose of this paper is to present a coupled finite element–boundary element formulation, which enables the modeling of piezoelectric smart lightweight structures. The paper describes the theoretical background of the coupled approach in which the finite element method (FEM) is applied for the modeling of the passive vibrating shell structure as well as the surface attached piezoelectric actuators and sensors. The boundary element method (BEM) is used to characterize the corresponding sound field. In order to derive a coupled FE–BE formulation additional coupling conditions are introduced at the fluid–structure interface. Since the resulting overall model contains a large number of degrees of freedom, the mode superposition method is employed to reduce the size of the FE submodel. To validate the accuracy of the proposed approach, numerical simulations are carried out in the frequency domain and the results are compared with analytical reference solutions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A wavelet-based stochastic finite element method of thin plate bending

A wavelet-based stochastic finite element method is presented for the bending analysis of thin plates. The wavelet scaling functions of spline wavelets are selected to construct the displacement interpolation functions of a rectangular thin plate element and the displacement shape functions are expressed by the spline wavelets. A new wavelet-based finite element formulation of thin plate bending is developed by using the virtual work principle. A wavelet-based stochastic finite element method that combines the proposed wavelet-based finite element method with Monte Carlo method is further formulated. With the aid of the wavelet-based stochastic finite element method, the present paper can deal with the problem of thin plate response variability resulting from the spatial variability of the material properties when it is subjected to static loads of uncertain nature. Numerical examples of thin plate bending have demonstrated that the proposed wavelet-based stochastic finite element method can achieve a high numerical accuracy and converges fast.     

Behavior of laterally and vertically loaded piles in multi-layered transversely isotropic soils

In this paper, a coupled approach of the finite element method (FEM) and the analytical layer-element method (ALEM) is proposed to conduct a research on vertically and laterally loaded piles. The FEM is used to model the pile, and the ALEM is utilized to solve the multi-layered transversely isotropic soils. Then with the assumption of force equilibrium and deformation compatibility, the interaction equation of pile and soils is obtained. Finally, the behavior of piles simultaneously subjected to lateral and vertical loads in layered transversely isotropic soils is investigated by considering the influence of lateral–vertical loads interaction and soft soil stratum.     

New exact series solutions for transverse vibration of rotationally-restrained orthotropic plates

The exact series solutions of plates with general boundary conditions have been derived by using various methods such as Fourier series expansion, improved Fourier series method, improved superposition method and finite integral transform method. Although the procedures of the methods are different, they are all Fourier-series based analytical methods. In present study, the foregoing analytical methods are reviewed first. Then, an exact series solution of vibration of orthotropic thin plate with rotationally restrained edges is obtained by applying the method of finite integral transform. Although the method of finite integral transform has been applied for vibration analysis of orthotropic plates, the existing formulation requires of solving a highly non-linear equation and the accuracy of the corresponding numerical results can be questionable. For that reason, an alternative formulation was proposed to resolve the issue. The accuracy and convergence of the proposed method were studied by comparing the results with other exact solutions as well as approximate solutions. Discussions were made for the application of the method of finite integral transform for vibration analysis of orthotropic thin plates.     

Vibration and stability analysis of thick orthotropic plates using hybrid-Trefftz elements

This paper discusses the vibration and stability analysis of thick orthotropic plate structures using finite elements based on the hybrid-Trefftz formulation. While the formulation can be used for elements of arbitrary geometry, the paper concentrates on the use of a simple and robust triangular element. The key feature of the formulation is to use element interpolations that are consistent for all values of the plate thickness, including the limit when it goes to zero. This eliminates the locking problem automatically and ensures a robust approximation for thick and thin plates. Results for various problems are included to demonstrate the accuracy and efficiency of the element.     

Nonlinear transient thermal stress and elastic wave propagation analyses of thick temperature-dependent FGM cylinders,using a second-order point-collocation method

Nonlinear transient thermal stress and elastic wave propagation analyses are developed for hollow thick temperature-dependent FGM cylinders subjected to dynamic thermomechanical loads. Stress wave propagation, wave shape distortion, and speed variation under impulsive mechanical loads in thermal environments are also investigated. In contrast to researches accomplished so far, a second-order formulation rather than a first-order one is employed to improve the accuracy. The FDM method (as a point-collocation FEM method) is used. It is known that other FEM methods cannot show the actual trend jumps due to distributing the abrupt changes in the quantities as the numerical errors and the residuals of the governing equations among the nodal results. Furthermore, the required computational time and allocated computer memory are much reduced by the present solution algorithm. The cylinder is not divided into isotropic sub-cylinders. Therefore, artificial wave reflections from the hard interfaces are avoided. Time variations of the temperatures, displacements, and stresses due to the dynamic or impulsive loads are determined by solving the resulted highly nonlinear governing equations using an iterative updating solution scheme. A sensitivity analysis includes effects of the volume fraction indices, dimensions, and temperature-dependency of the material properties is performed. Results reveal the significant effect of the temperature-dependency of the material properties on the thermoelastic stresses and present some interesting characteristics of the thermoelastic and wave propagation behaviors.     

Modelling of Wave Propagation using Spectral Finite Elements

For the analysis of wave propagation at high frequencies, the spectral finite element method (SFEM) is under investigation. In contrast to the conventional finite element method high-order shape functions are used. They are composed of Lagrange polynomials with nodes at the Gauß-Lobatto-Legendre points. The Gauß-Lobatto-Legendre integration scheme is applied in order to obtain a diagonal mass matrix. So, the resulting system equations can be solved efficiently. In the numerical examples, spectral finite elements with shape functions of different order are applied to a plane strain problem. The numerical examples cover structures without and with stiffness discontinuities. It is shown that the results agree well with analytical solutions. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stress analysis for a rectangular thick plate of a composite material with a spatially locally curved structure under forced vibration

Natural and forced vibrations of a thick rectangular plate fabricated from a composite material with a spatially locally curved structure are investigated with the use of exact three-dimensional equations of motion of the theory of elastic anisotropic bodies. The investigations are carried out within the framework of the continuum approach developed by Akbarov and Guz. It is supposed that the plate is clamped at all its edges and is loaded on the upper face with uniformly distributed normal forces periodically changing with time. The influence of the parameters of local curving on the fundamental frequency of the plate and on the distribution of the normal stress acting in the thickness direction under forced vibration is studied. The corresponding boundary-value problems are solved numerically by employing the three-dimensional FEM modelling.Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 779–790, November–December, 2004.     

One-way and two-way couplings of CFD and structural models and application to the wake-body interaction

The current study focuses on the wake-body interaction of a circular cylinder, whose transverse free vibration is modeled by a mass-spring-damper system coupled to a computational fluid dynamics (CFD) model for the flow and wake. We first simulate the free vibration of the elastically-mounted cylinder and the wake, and analyze the transverse load it exerts on the cylinder and its phase with the vibration. We vary the damping by three orders of magnitude and examine the difference in the wake-body interaction for slightly-damped and highly-damped systems. We then use the spectral properties of the free vibration and use them to construct two different types of forced vibrations: one consists only of the fundamental component of the free vibration, and the other accounts for all spectral properties of it. We compare the wake load for each type to that corresponding to the free vibration. The forced vibrations correspond to a one-way coupling and the information is communicated from the CFD model to the structural model, whereas the free vibration corresponds to a two-way coupling of the models. By comparing the spectral properties of the wake load, including the phase relation of its components with the vibration, which we obtained for the free vibration and for the equivalent forced vibration, we identify the effects of the wake feedback. The findings show that a forced vibration does not reproduce exactly the wake load at small and intermediate levels of structural damping. As the damping increases, the vibration changes from being in-phase with the wake load to being 90° out-of-phase with it, corresponding to two different wake states, and the forced vibration gives wake load that is very close to the one occurring in the case of full wake-body interaction.     

Multiscale computational method for thermoelastic problems of composite materials with orthogonal periodic configurations

This study develops a novel multiscale computational method for thermoelastic problems of composite materials with orthogonal periodic configurations. Firstly, the multiscale asymptotic analysis for these multiscale problems is given successfully, and the formal second-order two-scale approximate solutions for these multiscale problems are constructed based on the above-mentioned analysis. Then, the error estimates for the second-order two-scale (SOTS) solutions are obtained. Furthermore, the corresponding SOTS numerical algorithm based on finite element method (FEM) is brought forward in details. Finally, some numerical examples are presented to verify the feasibility and effectiveness of our multiscale computational method. Moreover, our multiscale computational method can accurately capture the local thermoelastic responses in composite block structure, plate, cylindrical and doubly-curved shallow shells.     

Spectral finite element analysis of elastically connected double-beam systems

A spectral finite element method for two parallel beams connected to each other by the vertical springs uniformly distributed along the beam length is introduced in this paper. The effects of the shear deformation and rotary inertia of the beams are accounted for. The coupled equations of motion are derived by using Hamilton's principle and the spectral element matrix is established based on the exact solutions of the governing equations. The use of the proposed spectral element formulation to investigate the free vibration characteristics of the particular double-beam systems is demonstrated by applying the Muller root search algorithm. Once the natural frequencies and mode shapes are obtained, a spectral element based normal mode method is introduced to compute the dynamic response of the double-beam systems subjected to various kinds of concentrated and distributed loads. Numerical results of the present method are verified by comparing with those available in the literature.     

混凝土断裂力学虚拟裂缝模型的半解析有限元法

利用平面扇形域哈密顿体系的方程,通过分离变量法及共轭辛本征函数向量展开法,以解析的方法推导出基于混凝土断裂力学中虚拟裂缝模型的平面裂纹解析元列式．将该解析元与有限元相结合,构成半解析的有限元法,可求解任意几何形状和荷载混凝土平面裂纹的虚拟裂缝模型计算问题．数值计算结果表明方法对该类问题的求解是十分有效的,并有较高的精度．