Flexural wave scattering in a quarter-infinite thin plate with circular scatterers

Scattering of flexural waves by circular scatterers in a quarter-infinite thin plate is formulated using the wave expansion method together with the method of images. The scattered waves are expressed as a summation series of wave functions and the unknown scattering coefficients are determined by enforcing boundary conditions at the scatterers. Both holes and rigid scatterers are studied. Simply-supported and roller-supported boundary conditions on the quarter-infinite thin plate are also considered. The analysis can be used to determine the stress concentration caused by circular scatterers in quarter-infinite thin plates.     

Asymmetric bifurcations in a pressurised circular thin plate under initial tension

It has been known for some time that under certain circumstances the axisymmetric solution describing the deformation experienced by a stretched circular thin plate or membrane under sufficiently strong normal pressure does not represent an energy-minimum configuration. By using the method of adjacent equilibrium a set of coordinate-free bifurcation equations is derived here by adopting the Föppl–von Kármán plate theory. A particular class of asymmetric bifurcation solutions is then investigated by reduction to a system of ordinary differential equations with variable coefficients. The localised character of the eigenmodes is confirmed numerically and we also look briefly at the role played by the background tension on this phenomenon.     

Analysis of global dynamics in a parametrically excited thin plate

The global bifurcations and chaos of a simply supported rectangular thin plate with parametric excitation are analyzed. The formulas of the thin plate are derived by von Karman type equation and Galerkin's approach. The method of multiple scales is used to obtain the averaged equations. Based on the averaged equations, the theory of the normal form is used to give the explicit expressions of the normal form associated with a double zero and a pair of pure imaginary eigenvalues by Maple program. On the basis of the normal form, a global bifurcation analysis of the parametrically excited rectangular thin plate is given by the global perturbation method developed by Kovacic and Wiggins. The chaotic motion of thin plate is also found by numerical simulation. The project supported by the National Natural Science Foundation of China (10072004) and by the Natural Science Foundation of Beijing (3992004)     

On the jumping problems of a circular thin plate with initial deflection

In this paper, the jumping problems of a circular thin plate with initial deflection are studied by using the method of two variables^[3],[4] proposed by Jiang Fu-ru and the method of the normal perturbation (in this paper (1.1), (1.2)). We obtain Nth-order uniformly valid asymptotic expansion of the solution of this problem ((1.66), (1.67)). When the initial deflection vanishes the solution of a circular thinplate with initial deflection is reduced to the solution of the problems of the nonlinear bending of a circular thin plate^[6]. If the initial deflection is largish and the signs of the initial deflection with the intensity of the transverse load are opposite, when the intensity of the transverse load reaches a certain value, the circular thin plate with initial deflection should produce the jumping phenomenon^[8].     

双材料环扇形薄板弯曲问题的辛本征解

弹性力学辛对偶求解方法是通过引入原变量的对偶变量将问题导入辛空间,从而使得有效的数学物理方法,如分离变量和辛本征函数展开的方法得以实施并得出问题的解析解。本文通过引入弯矩函数和恰当的变换,首先建立了两侧边边界条件自由的双材料环扇形薄板弯曲问题的辛对偶体系。然后,讨论了弯矩函数表示的非齐次边界条件,并给出了三个有特定物理意义的解,其解在端部的力系是非自相平衡的。对双材料的楔形板而言,这三个解表示的就是在尖端有集中弯矩、集中扭矩、垂直集中力作用的解。最后,讨论了弯矩函数表示的齐次边界条件,并给出了辛本征值的超越方程以及辛本征解,所有这些解在端部的力系都是自相平衡的。本文的工作为相关问题的解析求解以及辛本征解的进一步应用研究奠定了基础。     

Non-linear dynamics of the sandwich double circular plate system

Multi-frequency vibrations of a system of two isotropic circular plates interconnected by a visco-elastic layer that has non-linear characteristics are considered. The considered physical system should be of interest to many researches from mechanical and civil engineering. The first asymptotic approximation of the solutions describing stationary and no stationary behavior, in the regions around the two coupled resonances, is the principal result of the authors. A series of the amplitude-frequency and phase-frequency curves of the two frequency like vibration regimes are presented. That curves present the evolution of the first asymptotic approximation of solutions for different non-linear harmonics obtained by changing external excitation frequencies through discrete as well as continuous values. System of the partial differential equations of the transversal oscillations of the sandwich double circular plate system with visco-non-linear elastic layer, excited by external, distributed, along plate surfaces, excitation are derived and approximately solved for various initial conditions and external excitation properties. System of differential equations of the first order with respect to the amplitudes and the corresponding number of the phases in the first asymptotic averaged approximation are derived for different corresponding multi-frequency non-linear vibration regimes. These equations are analytically and numerically considered in the light of the stationary and no stationary resonant regimes, as well as the multi-non-linear free and forced mode mutual interactions, number of the resonant jumps.     

热环境中功能梯度圆形薄板的混沌运动

针对陶瓷-金属功能梯度圆板,同时考虑几何非线性、材料物性参数随温度变化且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,通过引入位移函数,利用伽辽金方法得到了达芬型非线性动力学方程。利用Melnikov方法,给出了热环境中功能梯度圆板可能发生混沌运动的临界条件。通过数值算例,给出了不同体积分数指数和温度的同宿分岔曲线,平面相图和庞加莱映射图,讨论其对临界条件的影响,证实了系统混沌运动的存在。通过分岔图和与其相对应的最大李雅普诺夫指数图,分析了激励频率和激励幅值对倍周期分岔的影响及变化规律,发现系统可出现周期、倍周期和混沌等复杂动力学响应。     

Multi-frequency analysis of the double circular plate system non-linear dynamics

This paper presents a multi-frequency analysis of non-linear dynamics in a double circular plate system. The original series of the amplitude–frequency and phase–frequency graphs as well as eigen forced time functions–frequency graphs are obtained and analyzed for stationary resonant states. The series of the frequency characteristic of the forced time non-linear harmonics are presented first. The analyses identify the presence of singularities and triggers of coupled singularities, as well as resonant jumps.     

Unsymmetrical nonlinear bending problem of circular thin plate with variable thickness

Introduction Manystructuralelements(pole,plate,shell)withunevenandvariablethicknessarewidely usedinallkindsofengineeringfields.Engineerscansavematerialswhentheydesignbecause theseelementshavebetteroptimizedshapeofstructuralfeature,butthisaddsdifficultytotheanalysisoftheirmechanicalcapability.Manypreviouspapers[1-4]havesolvedtheproblemof symmetricalaxis,butnobodyhassolvedtheunsymmetricalnonlineardeformationproblemof circularthinplatewithvariablethicknessandunsymmetricalaxisuptonow,afewworkonly …     

周边嵌入弹性圆薄板非线性自由振动直接摄动解法

采用弹性圆薄板中心无量纲振幅和板厚与半径的比值为参数。将挠度、应力函数对半径的导数以及自由振动频率展开为双参数的幂级数。用直接摄动法获得各级递推线性偏微分方程。应用变分法求得各级递推方程的近似解，从而给出弹性圆薄板非线性自由振动频率的基本公式。     

Axisymmetric static and dynamics snap-through phenomena in a thermally postbuckled temperature-dependent FGM circular plate

Thermal postbuckling analysis and the axisymmetric static and dynamic snap-through phenomena due to static/sudden uniform lateral pressure in a thermally postbuckled functionally graded material circular plate are performed in this research. Plate is formulated using the first order shear deformation plate theory. Thermo-mechanical properties of the plate are assumed to be temperature dependent where dependency is described according to the higher order Touloukian representation. Two types of temperature loading are considered. Uniform temperature rise and heat conduction across the thickness direction. The one dimensional heat conduction equation in the thickness direction is obtained and discreted via the central finite difference method. The obtained system of equations is nonlinear since the thermal conductivity itself is a function of the unknown nodal temperatures. Using the von-Kármán assumptions, the governing equations of the plate are obtained in a matrix representation with the aid of the conventional Ritz method whose shape functions are developed using the Gram-Schmidt process. At first thermal postbuckling analysis is performed which is a nonlinear problem with respect to both temperature and displacements. Afterwards, response of the bulged thermally postbuckled plate is obtained under the static and dynamic uniform pressure. Snap-through phenomenon may be observed in both static and dynamic loading cases, due to the immovability of the edge of the plate and the initial deflection caused by postbuckling deflection. To capture the snapping phenomenon and trace the path beyond the limit loads, cylindrical arch-length technique is used. In dynamic snap-through analysis, the effect of structural damping is also included. Numerical results of this study reveal that the structure is sensitive to the initial deflection caused by thermal postbuckling load. Increasing the temperature prior to mechanical loads enhances the snap-through intensity and also increases both the upper and lower limit loads. As shown, dynamic snap-through loads are lower than the static ones, however dynamic snap-through intensity is more than the static snap-though intensity. Furthermore, structural damping enhances the dynamic buckling loads of the plate and decreases the dynamic postbuckling deflection of the plate.     

Nonlinear analysis of a thin circular functionally graded plate and large deflection effects on the forces and moments

The dynamic von Karman equations are used for nonlinear analysis of a thin circular plate made of a functionally graded material. The thickness of the plate is constant and the properties of the functionally graded material depend on temperature and vary throughout the thickness. It is assumed that the plate oscillates with large amplitudes. The forces and moments in the plate are determined in solving the equations for harmonic vibrations. Relevant results are obtained in the case of stead-state free vibrations. These results indicate that the volume fraction has a strong effect on the forces, moments, and material properties Published in Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 134–144, June 2008. An erratum to this article can be found at     

Dissipative particle dynamics simulation of flow through periodic arrays of circular micropillar

Flow through arrays of micropillar embedded inside microfluidic chip systems is important for various microfluidic devices. It is critical to accurately predict the mass flow rate through pillar arrays based on the pillar design. This work presents a dissipative particle dynamics (DPD) model to simulate a problem of flow across periodic arrays of circular micropillar and investigates the permeability of two types of micropillar arrays. The flow fields including horizontal and vertical velocity fields, the number density field, and the streamline of the flow are analyzed. The predicted solid volumes by the presented DPD simulation of both types of arrays are quite close to the actual counterparts. These quantitative agreements show usefulness and effectiveness of the DPD model in simulating arrays of micropillar. By comparing two types of micropillar arrangement patterns, we find that the arrangement pattern of micropillar does not have significant influence on the permeability of the array.