一类弧形弹性杆大变形的多解性与分支

基于大变形理论建立弧形弹性杆大变形的数学模型,弹性杆的一端固定,另一端自由且在中间受一竖直向下的集中力,所建立的模型可变形为摆动方程的边值问题.利用流形法得到数学模型的分支图,进而分析弹性杆变形的多解性.     

Frequencies of Free Oscillations of a Truncated Spherical Sector Covered with a Thin Elastic Spherical Shell

An analytic solution of the problem concerning the frequencies and shapes of free axially symmetric oscillations of a truncated spherical sector filled with an ideal compressible fluid is constructed. A spherical wall of smaller radius and the radial wall of the sector are absolutely rigid. A thin elastic shell whose edge is clamped in the radial wall is located on a spherical boundary of larger radius. The outer surface of the shell borders vacuum. The phenomenon of an anomalous decrease in the fundamental frequency as the spherical walls approach each other is discovered. An approximate formula for determination of the lowest fundamental frequencies, which are approximately proportional to the square root of the difference of the radii of spherical walls for small values of this difference, is constructed and tested numerically. Bibliography: 13 titles.     

An Elastic Shell Model for the Human Tricuspid Valve

Acta Mathematicae Applicatae Sinica, English Series - The human tricuspid valve, one of the key cardiac structures, plays an important role in the circulatory system. However, there are few...     

一类半圆形弹性杆大变形的多解性与分支

研究一端固定,另一端自由且在其中间受一竖直向下的集中力,自由端受一水平向左的集中力的平面半圆形弹性杆的大变形.通过建立相应的数学模型,利用流形法画出模型解的分支图,进而分析系统的分支特性和多解性,并得到变形后弹性杆的形状.     

Variational Problem on Equilibrium of an Elastic Medium Located in Elastic Shell

The global solvability of the problem about equilibrium of an elastic medium located in an elastic shell is established by variational methods. In the case of one-parametric force field, the bifurcation problem is studied. As is shown, the standard necessary condition for bifurcation is also sufficient for the problem under consideration. Bibliography: 7 titles.     

一类弹性储液箱同液体耦合晃动问题的分岔行为分析

建立了弹性圆柱型储液箱同液体耦合系统在外激励下的非线性振动方程组．采用多尺度法、奇异性理论研究此非线性振动系统共振解的分岔行为,通过对其分岔行为的分析和讨论,得到了这一系统的多种转迁集和分岔图,建立了系统参数与其拓扑分岔解的联系,并且分析了不同参数下系统的分岔特性,为实现储液器参数的优化控制提供了理论依据．     

表面应力引起的弹性薄膜形状分叉

此工作致力于无约束各向同性弹性薄膜在表面应力发生变化时几何非线性变形分析。基于能量的考虑,导出了曲率与表面应力变化量的关系。与已有线弹性分析结果的显著区别是,当薄膜很薄而上下表面应力差足够大时,薄膜曲率的解不再唯一,而会发生分叉。     

Vibrations of Elastic Strings: Mathematical Aspects, Part One

This paper is divided into two parts. Part One Chapter 1 contains the deduction of the model for vertical vibrations of elastic strings with moving ends and its relations to the models of d'Alembert and Kirchhoff-Carrier. Chapter 2 contains the results of investigations related to Kirchhoff-Carrier model during the last twenty years and an almost complete bibliography.Part Two Chapter 3 of the paper is dedicated to the model with moving ends. It contains new results obtained by the authors during the last five years.     

Vibrations of Elastic Strings: Mathematical Aspects, Part Two

Dedicated to Professor Jacque-Louis Lions on the occasion of his 70th birthday We consider a mixed problem for the operator

The Acoustic Scattering of a Layered Elastic Shell Medium

This paper investigates the scattering of a three-dimensional elastic shell scatterer embedded in a layered unbounded structure. The background structure comprises a two-layer lossy media separated by an unbounded rough surface. The shell body is filled with an elastic material, the interior of which is vacuum. Given an incident acoustic point source, our aim is to determine the acoustic and elastic wave fields in the space. This problem is known as a fluid-solid interaction problem(FSIP). In th...     

矩形网格扁壳结构的非线性弹性理论

本文在基本假设的前题下,分析了矩形网格离散结构的变形以及变形量与内力之间的本构关系.在此基础上,通过建立等级模型,运用虚功原理推导出网格扁壳的作线性基本控制方程与边界条件.     

Dynamical and Static Analysis of the Deformation of a Fluid-Filled, Piecewise-Homogeneous Elastic Shell

The dynamical problem of the distribution of a pressure wave initiated at a certain distance from a junction of fluid-filled elastic shells having different properties is investigated. The static problem of two connected, empty (without the fluid) elastic shells having different properties is investigated as a special case of the dynamical problem and as a first step in the investigation of the stated problem. The results are subjected to a comparative analysis, and estimates are obtained for the solution of the dynamical problem in contrast with the static version.     

扁锥面网壳非线性动力分岔与混沌运动

对曲面为正三角形网格的3向扁锥面单层网壳,用拟壳法建立了轴对称非线性动力学方程．在几何非线性范围内给出了协调方程．网壳在周边固定条件下,通过Galerkin作用得到一个含2次、3次的非线性微分方程,通过求Floquet指数讨论了分岔问题．为了研究混沌运动,对一类非线性动力系统的自由振动方程进行了求解,继之给出了单层扁锥面网壳非线性自由振动微分方程的准确解,通过求Melnikov函数,给出了发生混沌的临界条件,通过数值仿真也证实了混沌运动的存在．     

关于非均质变截面弹性直杆的纵向自主振动

本文以集合木制螺旋桨作为启示的出发点,从而想使弹性直杆振动问题能初步臻于一个比较更一般的处理.文中把弹性模量、截面面积和单位杆长的质量施以依指数函数变化的规律之后,推出了一个通解,其中三个参数看为影响非均质变截面弹性直杆纵向自主振动频率的主要因素.     

简支梯形底扁球壳自由振动问题的准Green函数方法

以简支梯形底扁球壳的自由振动问题为例,详细阐明了准Green函数方法的思想.即利用问题的基本解和边界方程构造一个准Green函数,此函数满足了问题的齐次边界条件,采用Green公式,将简支梯形底扁球壳自由振动问题的振形控制微分方程化为两个耦合的第二类Fredholm积分方程.边界方程有多种选择,在选定一种边界方程的基础上,可以通过建立一个新的边界方程来表示问题的边界,以克服积分核的奇异性.最后由积分方程的离散化方程组有非平凡解的条件,求得固有频率.数值结果表明,该方法具有较高的精度.     

Wrinkling Behavior of Highly Stretched Rectangular Elastic Films via Parametric Global Bifurcation

We consider the wrinkling of highly stretched, thin rectangular sheets—a problem that has attracted the attention of several investigators in recent years, nearly all of which employ the classical Föppl–von Kármán (F–K) theory of plates. We first propose a rational model that correctly accounts for large mid-plane strain. We then carefully perform a numerical bifurcation/continuation analysis, identifying stable solutions (local energy minimizers). Our results in comparison to those from the F–K theory (also obtained herewith) show: (i) For a given fine thickness, only a certain range of aspect ratios admit stable wrinkling; for a fixed length (in the highly stretched direction), wrinkling does not occur if the width is too large or too small. In contrast, the F–K model erroneously predicts wrinkling in those very same regimes for sufficiently large applied macroscopic strain. (ii) When stable wrinkling emerges as the applied macroscopic strain is steadily increased, the amplitude first increases, reaches a maximum, decreases, and then returns to zero again. In contrast, the F–K model predicts an ever-increasing wrinkling amplitude as the macroscopic strain is increased. We identify (i) and (ii) as global isola-center bifurcations—in terms of both the macroscopic-strain parameter and an aspect-ratio parameter. (iii) When stable wrinkling occurs, for fixed parameters, the transverse pattern admits an entire orbit of neutrally stable (equally likely) possibilities: These include reflection symmetric solutions about the mid-plane, anti-symmetric solutions about the mid-line (a rotation by π radians about the mid-line leaves the wrinkled shape unchanged) and a continuously evolving family of shapes “in-between”, say, parametrized by an arbitrary phase angle, each profile of which is neither reflection symmetric nor anti-symmetric.     

非线性弹性地基上的圆薄板的分岔与混沌问题

根据非线性弹性地基上圆薄板大幅度方程,弹性抗力有线性项,三次非线性项和抗弯曲弹性项。在周边固定的条件下,利用Galerkin法得到了一个非线性振动方程。在无外激励情况下,求出在平衡点处的Floquet指数。分析了其稳定性与可能发生的分岔条件。在外激励条件下,用Melnikov方法分析研究了可能发生的混沌振动。通过数字仿真给出了各种地基参数下混沌区域的临界曲线和相平面图。     

弹性扁壳由移动质量引起的强迫振动

本文用变分法讨论在弹性扁壳上有移动的质量所引起的壳体的强迫振动.文中讨论了关于强迫振动、共振条件及临界速度等一系列问题.     

Transformation of a Pressure Pulse in a Fluid-Filled Cylindrical Elastic Shell

The influence of a tubular elastic insert of finite length in a fluid-filled elastic shell on the propagation of a pressure pulse generated at a certain distance from the insert is investigated. The Laplace integral transform is used to solve the corresponding mathematical problem. Analytical solutions are first obtained in each separate domain of the hydroelastic system, subject to the coupling boundary conditions, and then the Laplace transforms are inverted numerically. The influence of the geometrical parameters of the insert and the shell on the pressure distribution in the fluid, the radial displacement, the bending moments, and the shearing forces are analyzed in detail both quantitatively and qualitatively. It is shown that the transmission of the pressure pulse in the junctions of the insert with the shell is accompanied by highly localized stress concentration there, which is several orders of magnitude greater than in the homogeneous shell.