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1、太原理工大学 硕士学位论文 非线性固体结构中的孤立波与混沌 姓名：张涛 申请学位级别：硕士 专业： 指导教师：张善元 20100501 太原理工大学硕士研究生学位论文 I 非线性固体结构中的孤立波与混沌 摘 要 20 世纪 60 年代，自然科学的各个学科分支出现了非线性问题的研究热 潮，孤子、湍流、混沌、分形及复杂系统等新的物理现象被揭示，表明非 线性科学已经成为现代科学发展的一个重要标志。在这一热潮推动下，固 体结构中的非线性波的传播和混沌运动的研究也取得了很大进展。本文在 综述已有研究的基础上，研究了几类典型结构元件中孤立波的传播特征和 混沌行为，主要工作和成果如下: 1. 在 Berno
2、ulli-Euler 梁、Rayleigh 修正梁和 Timoshenko 梁三种经典梁 理论的基本方程中，引入有限挠度和轴向惯性，导出了相应的支配弯曲波 传播的非线性偏微分方程组。对这些方程进行了定性分析，并采用 Jacobi 椭圆函数展开法进行求解， 给出了精确的周期解及模数 m1 退化情况下的 孤立波解和冲击波解。 2. 在上述三类有限挠度梁的运动方程中引入外加载荷和阻尼对系统的 摄动，利用 Melnikov 方法给出了出现 Smale 马蹄意义下混沌的临界条件， 揭示了孤立波与混沌两大类非线性现象之间的联系。 3. 研究了埋置于弹性地基内充液压力管道中非线性波的传播。假定管 壁材料是线
3、弹性的，管中流体为不可压理想流体，地基反力采用 Winkle 线 性地基模型，建立了地基、管壁与流体耦合作用的非线性运动方程组，借 助约化摄动法(RPT)得到 KdV 方程，表征着系统有孤波解。 4. 研究了充有压力流体的粘弹性管中孤立波的传播特性。管壁是由 Kelvin-Voigt 模型描述的粘弹性材料，流体的运动为一维无粘流动，利用约 化摄动法(RPT)从支配耦合系统运动的非线性偏微分方程组得到了 太原理工大学硕士研究生学位论文 II KdV-Burgers 方程。 根据粘性大小的不同， 系统有振荡的孤波解或冲击波解， 并利用数值解给出其传播的图象。 5. 考虑血液流动的对流项及血管壁的大
4、变形，采用二维情况下 Hilmi Demiray 建议的管壁材料的应变能函数，研究了动脉血管中非线性压力波 的传播。在长波近似情况下，借助约化摄动法(RPT)得到具有孤子解的 KdV 方程。从临床角度讨论了参数对解的影响。 6. 对于轴压圆柱壳经受轴向和横向扰动时的非线性振动，分别采用 Donnell-Krmn 大挠度理论和环向对数应变建立了两种非线性运动方程。 借助 Bubnov-Galerkin法将它们分别转化为含有三次和二次非线性的常微分 方程。利用次谐轨道和同宿轨道的 Melnikov 函数给出了前屈曲和后屈曲情 况下发生 Smale 马蹄混沌的临界条件。使用 Matlab 软件计算了
5、分岔图、相 图、时程曲线和 poincar 映射，给出了混沌运动的数字特征。 关键词：大挠度梁， 充液压力管道， 轴压圆柱壳， 孤立波， 混沌运动， Jacobi 椭圆函数展开法，约化摄动法 基金项目：本项研究为国家自然科学基金(10772129)资助项目。 太原理工大学硕士研究生学位论文 III THE SOLITARY WAVE AND CHAOS IN NONLINEAR SOLID STRUCTURE ABSTRACT In 1960s the research upsurge of the nonlinear problems appeared simultaneously in m
6、any branches of natural science. Many new physical phenomena, such as soliton， turbulence， chaos， fractal and complex system etc. were discovered, which indicates that the nonlinear science has been an important symbol in the development of modern science. Owing to the promotion of this upsurge, the
7、 investigations on the propagation of nonlinear wave and chaotic motion in solid structure also have made great progress. In this dissertation, on the basis of summarizing the existing research achievement, the propagation property of solitary wave and chaotic behavior in several kinds of typical st
8、ructural elements are studied. Main works and the important results are as follows: 1. On the basis of three classic theories of Bernoulli-Euler，Rayleigh and Timoshenko beam, taking finite-deflection and axial inertia into consideration, the nonlinear partial differential equations governing the pro
9、pagation of nonlinear flexural wave are derived. The qualitative analyses are carried out, and the exact periodic solutions, the shock wave and solitary wave solution when the modulus m1 are obtained by means of Jacobi elliptic function expansion method. 2. In above three kinds of equilibrium equati
10、on, the external load and damping are viewed as small perturbation into the system and the threshold condition of the existence of Smale horseshoe chaos are obtained by Melnikovs method, which further reveal the interrelationship between the two kinds of 太原理工大学硕士研究生学位论文 IV nonlinear phenomenon solit
11、ary wave and chaos. 3. The propagation of nonlinear wave in a fluid-filled elastic thin tube buried inside elastic foundation is studied. In the analysis, the material of the tube is assumed to be linear elastic, the reaction of foundation is calculated by Winkler model, and the fluid is incompressi
12、ble and inviscid. The solid-liquid coupled equations is obtained by the mass conservation and balance of linear momentum. Employing the reductive perturbation method(RPT) the KdV equation is derived, which indicates the system admits a solitary wave solution. 4. The propagation property of nonlinear
13、 waves in a viscoelastic thin tube filled with incompressible inviscid fluid is studied. The tube is considered to be made of an incompressible isotropic viscoelastic material described by Kelvin-Voigt model. In the long-wave approximation the nonlinear solid-liquid coupled equations can be derived
14、employing the reductive perturbation technique(RPT). According to the size of viscous effects, the system admits a oscillating solitary wave solution or shock-wave solution and their propagation graph are obtained by numerical calculation. 5. Taking the convective term of blood flow and the large de
15、formation of blood vessels into account, the propagation of nonlinear pressure wave in arterial blood vessels is studied by means of strain energy function about soft tissue materials proposed by Hilmi Demiray. Employing the reductive perturbation method(RPT) the KdV equation with soliton solution i
16、s derived in the long wave approximate. The effects of system parameters on solution are discussed from a clinical point of view. 6. With regard to the nonlinear vibration of an axially compressed cylindrical shell subjected to axial or radial disturbance, two kinds of nonlinear motion equations of cylindrical shell are obtained by adopting Donnell-Krmn large deflection equations or logarithmic circumferential strain definitions, respectively. By means of Bubnov-Galerkin approach two
