A new conjugate gradient method for solving multi-source dynamic load identification problem

In this paper, a new conjugate gradient method is suggested to estimate the multi-source dynamic loads acting on a surface of 3D plate. The algorithm reconstructs the distribution functions of the dynamic loads by the displacement responses at two receiving points. The dynamic Green’s function and response function of dynamic forces excitation are obtained by finite element method. Under linear and time-invariant supposition, the response at a receiving point on the structure can be expressed as a convolution integral of the desired unknown force and the corresponding Green’s function, where the continuous convolution functions are spatially and temporally discretized by trapezium formula. The preset method is suggested to identify the dynamic loads by minimizing the error functions. Finally, the multi-source dynamic loads are successfully identified by the proposed method, and the performance of numerical simulations is superior to Landweber iteration method, which demonstrates the efficiency and robustness of the proposed method.     

一种基于最优输出跟踪的多源动态载荷识别方法

基于最优输出跟踪的基本思想,提出了一种时域内多源动态载荷识别的方法。该方法从结构动力响应出发,设计一个最优输出跟踪器并构造性能指标,将载荷识别问题转变为最优输出跟踪问题。通过Adams法求解微分方程,实现了多源动态载荷的识别,并采用L曲线法确定了性能指标中的关键参数。数值算例表明,所述的载荷识别方法能够在响应数据含有噪声的情况下,有效稳定地实现多源动态载荷的重构,具有较强的抗噪能力。     

Numerical analysis of dynamic buckling of rectangular plates subjected to intermediate-velocity impact

This paper numerically investigates the dynamic buckling of thin imperfect rectangular plates subjected to intermediate-velocity impact loads. From numerical results obtained, a dynamic buckling and a dynamic yielding critical condition are defined, and the corresponding critical dynamic loads are estimated. Numerical model employed in the present study is validated by experimental data reported earlier. Results from parametric study indicate that initial imperfection and load duration have significant influence on the dynamic buckling of the plates. The smaller the initial imperfection and the load duration, the higher the dynamic buckling critical loads of the plates. Moreover, different hardening ratios of plate material also affect the elastic–plastic dynamic buckling properties of the plates. If the plate buckles plastically, the dynamic buckling load increases as the hardening ratio of plate material increases. Unlike thin plates under high-velocity impact that buckling always occur after load application, plates under intermediate-velocity impact analyzed in the present study all buckle during the loading phase.     

The dynamic analysis of a flat plate under a moving load by the finite element method

The object of this paper is to analyse by the finite element method the dynamic responses of a flat plate subjected to various moving loads. First of all the actual continuous flat plate was replaced by a discrete system composed of isoparametric rectangular plate elements. And then the elementary and overall stiffness and mass matrices were determined and the natural frequencies and mode shapes of the flat plate were solved. Next, the Newmark direct integration method was used to find the dynamic responses of the flat plate. The effects of eccentricity, acceleration and initial velocity of the moving load, and the effect of span length were the key points of study. The dynamic behaviour of a multi-span flat plate supported by the beam members of rigid plane frames and subjected to the action of a series of moving loads (in identical or opposite directions) were also investigated.     

Transient waves in plates of functionally graded materials

A numerical method is proposed for analysing transient waves in plates of functionally graded material (FGM) excited by impact loads. The material properties of the FGM plate have a gradient in the thickness direction and are anisotropic in the plane of the plate. In the present method, the FGM plate is divided into layer elements in the thickness direction. For an accurate modelling of the variation of the material property of FGM plates, it is expressed by second‐order polynomials in the thickness direction within an element. This can further reduce the number of elements to obtain more accurate results effectively. The principle of virtual work is used to develop approximate dynamic equilibrium equations. The displacement response is determined by employing the Fourier transformation and the modal analysis. As examples, the displacement response of FGM plates excited by line, point and distributed loads is calculated. The computations have shown the efficiency of the present method. Copyright © 2001 John Wiley & Sons, Ltd.     

人-板耦合系统动力特性研究

基于单自由度的人体模型,将分布人群简化成均匀连续分布的质量-弹簧-阻尼系统,并考虑楼板的阻尼,建立了有阻尼的分布人群-薄板耦合系统模型。分析了薄板与人体的频率比及人板质量比等参数对耦合系统动力特性的影响。以四边简支的均质矩形薄板为例,用有限元方法对该文结论进行了验证。研究成果可为人与结构相互作用及结构的人致振动分析提供参考。     

中厚矩形板的非线性动力稳定性分析

本文以Ｔｉｍｏｓｈｅｎｋｏ－Ｍｉｎｄｌｉｎ假设及Ｈａｍｉｌｔｏｎ原理为基础，建立了中厚板的非线性运动控制方程。应用参数增量法求解了四边简支矩形板在纵横谐载共同作用下的非线性动力稳定性问题，并对影响其动力稳定性的若干因素进行了讨论。     

Elastic stability of skew laminated composite plates subjected to biaxial follower forces

The present study investigates the elastic stability of skew laminated composite plates subjected to biaxial inplane follower forces by the finite element method. The plate is assumed to follow first-order shear deformation plate theory (FSDPT). The kinetic and strain energies of skew laminated composite plate and the work done by the biaxial inplane follower forces are derived by using tensor theory. Then, by Hamilton's principle, the dynamic mathematical model to describe the free vibration of this problem is formed. The finite element method and the isoparametric element are utilized to discretize the continuous system and to obtain the characteristic equations of the present problem. Finally, natural vibration frequencies, buckling loads (also the instability types) and their corresponding mode shapes are found by solving the characteristic equations. Numerical results are presented to demonstrate the effects of those parameters, such as various inplane force combinations, skew angle and lamination scheme, on the elastic stability of skew laminated composite plates subjected to biaxial inplane follower forces.     

机电耦合载荷下的压电层合板瞬态响应分析

针对压电层合板在机电耦合激振下的瞬态响应问题, 提出一种高效混合数值计算方法。经过位移场、 电势场在厚度方向的离散, 利用机电耦合理论和哈密顿原理, 推导出结构的运动方程。引入傅里叶变换, 得到波数域内运动控制方程。应用模态分析方法求解波数域内的位移场和电势场, 对结果进行傅里叶逆变换, 得到空间域内的瞬态响应。以PZT-5A/0° PVDF铺层两相材料复合压电层合板为算例, 分析了力、 电耦合线载荷激励下, 位移场和电势场的瞬态响应历程与分布规律, 计算结果给出了该结构的动力学基本特征。该方法结合了有限元法、 傅里叶变换和模态分析法, 计算高频载荷激振下的压电层合板瞬态响应较一般有限元法大幅减少了单元的划分。该方法可推广至分析任意机电载荷下的各类铺层材料压电层合板瞬态响应问题。      

Influence of the electric current waveform on the dynamic response of the electrified composites

The influence of the electric current waveform (DC, AC and pulsed currents) on the dynamic electromagnetic, thermal, and impact response of the composite plate is studied. The analysis includes solving Maxwell’s equations in the electrified composite plate to determine an electric-current induced magnetic field and heat transfer equation to estimate the electric-current-induced heating. In addition, the dynamic mechanical response of the electrified composite plate subjected to impact and various electromagnetic loads (DC, AC, pulsed electric currents and a constant magnetic field) is analyzed by solving a coupled system of equations of motion and Maxwell’s equations in the composite plate. The results show that the dynamic response of the plate is highly dependent on the characteristics of the electromagnetic field, and the pulsed electromagnetic fields are most effective in reducing vibrations caused by the application of dynamic mechanical loads.     

热环境中功能梯度圆板的非线性动力响应分析

研究了热环境中功能梯度圆板在横向简谐激励作用下的非线性动力响应和动应力问题。针对陶瓷-金属功能梯度圆板, 考虑几何非线性、材料物理性质参数随温度变化及材料组分沿厚度方向按幂律分布的情况, 应用虚功原理给出了热载荷与横向简谐载荷共同作用下的非线性振动偏微分方程。在固支无滑动的边界条件下, 利用伽辽金法得到了达芬型非线性强迫振动方程。通过数值算例, 给出了关于体积分数指数的分岔图, 相图、Poincare映射等响应图以及动应力变化规律图, 讨论了材料体积分数指数和温度场对功能梯度圆板非线性动力响应的影响。结果表明: 热环境中功能梯度圆板随体积分数指数的变化可使系统出现周期响应、倍周期响应和混沌响应。功能梯度圆板中心处动应力在系统发生分岔或出现混沌响应时出现大幅变化, 而且在混沌响应时具有不可预测性。      

Coupled horizontal and rocking vibrations of a rigid circular plate on a transversely isotropic bi-material interface

This paper investigates the coupled rocking and horizontal vibratory response of a rigid circular plate embedded in viscoelastic, transversely isotropic, three-dimensional unbounded media. The boundary-value problem corresponding to the case of distributed horizontal and rocking ring loads at a bi-material interface is solved to obtain the required influence functions for the solution of the present problem. The case of an embedded rigid plate is formulated in terms of a discretized integral equation, which couples the rigid body displacements of the plate with the tractions acting over its contact surface through a set of displacement influence functions. The system of resulting discretized integral equations is solved numerically. The solution results in the tractions over each disc element. This paper carefully takes into account the coupling of the rocking and horizontal responses of the plates that is typical of non-homogeneous interfaces, i.e., their horizontal displacements due to rocking moments and their rotations due to horizontal loads. The dynamic direct and cross compliances of the embedded plate are shown for different governing parameters such as frequency of excitation and bi-material configuration. The present results are useful to the study of dynamic response of deeply buried foundations and anchors in non-homogeneous soils.     

Analysis of functionally graded plates by meshless method: A purely analytical formulation

Functionally graded plates under static and dynamic loads are investigated by the local integral equation method (LIEM) in this paper. Plate bending problem is described by the Reissner moderate thick plate theory. The governing equations for the functionally graded material with respect to the neutral plane are presented in the Laplace transform domain and therefore the in-plane and bending problems are uncoupled. Both isotropic and orthotropic material properties are considered. The local integral equation method is developed with the locally supported radial basis function (RBF) interpolation. As the closed forms of the local boundary integrals are obtained, there are no domain or boundary integrals to be calculated numerically in this approach. The solutions of the nodal values for the entire plate are obtained by solving a set of linear algebraic equation system with certain boundary conditions. Details of numerical procedures are presented and the accuracy and convergence characteristics of the method are examined. Several examples are presented for the functionally graded plates under static and dynamic loads and the accuracy for proposed method has been observed compared with 3D analytical solutions.     

Differential quadrature analysis of functionally graded circular and annular sector plates on elastic foundation

The main purpose of this study is to investigate buckling and free vibration behaviors of radially functionally graded circular and annular sector thin plates subjected to uniform in-plane compressive loads and resting on the Pasternak elastic foundation. In-plane compressive loads may be applied to either radial, circumferential, or all edges of circular/annular sector plates. Based on the classical plate theory (CPT), critical buckling loads and fundamental frequencies of the circular/annular sector plates under simply-supported and clamped boundary conditions are obtained by using differential quadrature method (DQM). The inhomogeneity of the plate is characterized by taking exponential variation of Young’s modulus and mass density of the material along the radial direction whereas Poisson’s ratio is considered to be constant. Convergence study is carried out to demonstrate the stability of the present method. To confirm the excellent accuracy of the present approach, a few comparisons are made for limited cases between the present results and those available in literature. Critical buckling load and fundamental frequency parameters of the circular/annular sector thin plates are computed for different boundary conditions, various values of the material inhomogeneity constants, sector angles, and inner to outer radius ratios.     

Experimental investigation of response of monolithic and bilayer plates to impulsive loads

This article presents the results of a series of experiments performed to assess the dynamic response of circular monolithic steel and steel–polyurea bilayer plates to impulsive loads. A convenient technique to enhance the energy absorption capability of steel plates and to improve their resistance to fracturing in dynamic events, is to spray-cast a layer of polyurea onto the plates. Since polyurea readily adheres to metallic surfaces and has a short curing time, the technique may be used to retrofit existing metallic structures to improve their blast resistance. We have examined the effectiveness of this approach, focusing on the question of the significance of the relative position of the polyurea layer with respect to the loading direction; i.e., we have explored whether the polyurea layer cast on the front face (the impulse-receiving face) or on the back face of the steel plate would provide a more effective blast mitigating composite.The experimental results suggest that the polyurea layer can have a significant effect on the response of the steel plate to dynamic impulsive loads, both in terms of failure mitigation and energy absorption, if it is deposited on the back face of the plate. And, remarkably, when polyurea is placed on the front face of the plate, it may actually enhance the destructive effect of the blast, promoting (rather than mitigating) the failure of the steel plate, depending on the interface bonding strength between the polyurea and steel layers. These experimental results are supported by our computational simulations of the entire experiment, employing realistic physics-based constitutive models for the steel (DH-36, in the present work) and polyurea [Amini MR, Amirkhizi AV, Nemat-Nasser S. Numerical modeling of response of monolithic and bilayer plates to impulsive loads. Int J Impact Eng, submitted for publication].     

Arrays of dynamic circular loads moving on an infinite plate

Steady‐state dynamic responses of a Kirchhoff plate resting on a viscoelastic Winkler foundation subject to arrays of moving constant and harmonic loads with uniform circular distribution is studied in this paper using Fourier transform and generalized Duhamel's integral. A computational schema is constructed based on a two‐fold fast Fourier transform in a moving co‐ordinate system. Numerical case studies using a single moving load as well as four moving loads mimicking four wheel‐loads of a moving vehicle are conducted. Parameters of the plate are typical values of structural and material properties of highway pavements. Displacement responses of the plate to these five loading speeds seem to be very close to each other, though the vertical velocity and acceleration responses show considerable differences. The effect of load frequency on amplitude of response of plate is insignificant for parameters used in this study. Copyright © 2006 John Wiley & Sons, Ltd.     

Mesh‐free radial point interpolation method for the buckling analysis of Mindlin plates subjected to in‐plane point loads

In this paper, a mesh‐free approach is employed for buckling analysis of Mindlin plates that are subjected to in‐plane point loads. The radial point interpolation method (RPIM) is used to approximate displacements based on nodes. Variational forms of the system equations are established. Two‐step solution procedures are implemented. The non‐uniform pre‐stress distribution of plate is first obtained using the RPIM based on a two‐dimensional (2D) elastic plane stress problem. This predetermined non‐uniform pre‐stress distribution is then used to compute buckling loads of plate using the RPIM based on Mindlin's plate assumption. The RPIM can easily handle any number and location of nodes in the plate domain for a desired computational accuracy without major difficulties in solving the initial stresses and buckling loads. Numerical examples considered here include circular and rectangular Mindlin plates that are subjected to in‐plane uniform and point loads with different aspect ratios and boundary conditions. The present results are validated against the available analytical and numerical solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Flexural response to impulsive loads of a fluid-saturated thin poroelastic circular plate

Biot's poroelastic theory is used with classic plate theory and plane stress theory to determine the constitutive relationships for a thin poroelastic plate. The dynamic equations for the thin poroelastic plate are derived from the extended Hamilton's principle. The dynamic equations are then transformed to frequency domain and Galerkin's finite element method is used to derive the stiffness matrix of a triangular plate element. When impulsive loads and elastic boundary conditions are applied, the finite element frequency domain analysis for the thin poroelastic plates is achieved. Vibration behavior of thin elastic and poroelastic circular plates is accurately predicted.     

A successive bounding method to find the exact eigenvalues of transcendental stiffness matrix formulations

An alternative algorithm for finding exact natural frequencies or buckling loads from the transcendental, e.g. dynamic, stiffness matrix method is presented in this paper and evaluated by using the plate assembly testbed program VICONOPT. The method is based on the bounding properties of the eigenvalues provided by either linear or quadratic matrix pencils on the exact solutions of the transcendental eigenvalue problem. The procedure presented has five stages, including two accuracy checking stages which prevent unnecessary calculations. Numerical tests on buckling of general anisotropic plate assemblies show that significant time savings can be achieved compared with an earlier multiple determinant parabolic interpolation method.     

Effect of cross-sectional warping of anisotropic sandwich laminates due to dynamic loads using a refined theory and C° finite elements

An attempt has been made to study the effect of cross-sectional warping in the symmetrically laminated anisotropic composite sandwich plates for transient loads. A higher-order shear deformation theory (HOST) is used in conjunction with the simple displacement based C° finite element method (FEM). As is well-known, the classical first-order theories hitherto considered were inadequate to describe the propagation of waves in the highly orthotropic sandwich laminates. The present theory, which is more accurate than the Reissner-Mindlin theory, is applied herein, for the evaluation of plate response to different types of dynamic loads. An explicit central difference scheme is employed for the integration of dynamic equations of equilibrium with a diagonalized mass matrix obtained by a special procedure applicable to quadrilateral isoparametric elements. The numerical results of the present investigation have been compared with the first-order shear deformation theory (FOST) and the differences between HOST and FOST are examined. The results presented here should be useful in obtaining better correlation between theory and experiment, and to numerical analysts in verifying their results.