梁中非线性弯曲波传播特性的研究

研究了梁中的非线性弯曲波的传播特性，同时考虑了梁的大挠度引起的几何非线性效应和 梁的转动惯性导致的弥散效应，利用Hamilton变分法建立了梁中非线性弯曲波的波动方程. 对该方程进行了定性分析，在不同的条件下，该方程在相平面上存在同宿轨道或异宿轨道， 分别对应于方程的孤波解或冲击波解. 利用Jacobi椭圆函数展开法，对该非线性方程进行 求解，得到了非线性波动方程的准确周期解及相对应的孤波解和冲击波解，讨论了这些解存 在的必要条件，这与定性分析的结果完全相同. 利用约化摄动法从非线性弯曲波动方程中导 出了非线性Schr\"{o}dinger方程，从理论上证明了考虑梁的大挠度和转动惯性时梁中存在 包络孤立波.     

非圆截面杆中的非线性扭转波及其行波解

研究了非圆截面杆中非线性扭转波的传播特性.由于非圆截面杆的扭转运动会伴随有横截面的翘曲,这种翘曲运动将引起扭转波的弥散.如果同时考虑有限扭转变形和翘曲弥散的共同作用,将会得到非线性扭转波的方程.在相平面上,对非线性扭转波动方程进行定性分析,结果表明,在一定条件下方程存在同宿轨道或异宿轨道,分别相应于方程的孤波解或冲击波解.本文利用Jacobi椭圆函数展开法,对该非线性方程进行求解,得到了非线性波动方程的三类准确周期解及相应的孤波解和冲击波解,讨论了这些解存在的必要条件.这些条件与定性分析的结果相一致.     

摄动法在研究弹-粘塑性波问题中的应用

采用控制金属材料宏观塑性流动的两个无量纲物理参数作为小参数,将一维弹/粘-塑性问题的解摄动展开,从而,求解非线性波动方程的问题可以转化成求解相应的齐次或非齐次电报方程的问题,用Laplace积分变换或级数展开技术首先得到零次精确解。然后,用Riemann函数方法可获得一次和高次摄动解。与非线性问题的数值解比较,在恒应力或恒速度边界条件下,一次摄动解给出了波动问题的良好近似。这就表明,摄动技术在研究一类广泛的弹/粘-塑波问题中是有效的。     

含孔软铁磁材料Mindlin板中弹性波散射问题

基于考虑磁弹相互作用的Mindlin板弯曲波动方程，采用波函数展开法，分析研究 了含孔软铁磁材料Mindlin板中弹性波散射与动应力集中问题，给出了问题的分析 解和数值算例. 通过分析发现：磁感应强度对动弯矩集中系数和动剪力集中系数有 增加的作用，特别是在低频的情况下.     

含电学边界的压电层合梁的非线性弯曲波

非线性科学己成为近代科学发展的一个重要标志, 特别是非线性动力学和非线性波的研究对于解决自然科学各领域中遇到的复杂现象和问题有着极其重要的意义. 本文研究了含电学边界条件的压电层合梁的非线性弯曲波传播特性.首先, 考虑几何非线性效应和压电耦合效应, 利用哈密顿原理建立了一维无限长矩形压电层合梁弯曲波的非线性方程.其次, 采用Jacobi椭圆函数展开法对非线性弯曲波方程进行求解, 得到了非线性弯曲波动方程在近似情况下对应的冲击波解和孤波解.最后, 利用约化摄动法得到了非线性薛定谔方程, 进一步得到了亮孤子和暗孤子解.基于两种方法具体研究了外加电压、压电层厚度等参数对冲击波和孤立波以及亮孤子和暗孤子特性的影响. 研究结果表明, 在波速较小时, 外加电压对冲击波的影响较大, 波速较大时, 外加电压对孤立波影响减弱.通过调整作用在压电层合梁上的电压发现了存在亮孤子和暗孤子, 分析结果表明随着外加电压值的增大, 亮孤子和暗孤子的振幅都增大.      

非圆截面杆中非线性扭转波方程的解析求解

研究了非圆截面杆中非线性扭转波动方程的精确求解问题. 利用直接积分与微分变换相结合的方法,得到了该方程的隐式通解. 通过对积分常数和方程系数的不同情形的讨论, 给出了该方程的三角函数、双曲函数、椭圆函数、指数函数以及它们的组合形式的解,分别对应于的非线性扭转波的孤立波、周期波以及冲击波等多种传播形式.     

填隙幂率流体下两刚性圆球相对错移时的粘性阻力

湿颗粒离散元模型以两球作用时填隙流体定常流动解为基础,其中切向作用是难点,国外仅有Goldman的牛顿流体渐近解.基于Reynolds润滑理论导出了两刚性球切向错动时填隙幂律流体的压力方程,并利用傅立叶级数展开简化,通过数值解法得到相应的压力分布、黏性阻力及阻力矩.该方程的解较之作者先前对速度场附加假定的结果精确,而当幂指数为1时等价于Goldman的牛顿流体渐近解.     

横观各向同性饱和地基上无限板的稳态振动

研究了横观各向同性饱和土地基上无限板的稳态振动问题. 基于直角坐标系下横观各向同性饱和介质Biot波动方程的一般解，采用双重Fourier积分变换技术，建立了饱和地基与无限矩形板相互作用的动力方程，利用数值方法求解该方程，得到任意谐振荷载作用下饱和半空间体上无限板稳态响应的一般解. 数值结果表明，横观各向同性饱和地基上无限板的振动与各向同性饱和地基上的无限板的振动特性存在明显差异.     

有界弦的行波解及其与驻波解的关系

 有界弦的振动是介绍数学物理方程求解方法和了解 连续体振动、波动运动形态的一个典型例子. 给出了它的行波解, 并揭示了行波 解与驻波解的关系, 可以帮助加深对弦的运动形态的了解.     

含孔曲板弹性波散射与动应力分析

基于敞口浅柱壳弹性波动方程及摄动方法，对无限大含孔曲板弹性波散射及动应力问题进行了分析研究，将经典薄板弯曲波动问题的分析解作为本问题的主项，给出了在稳态波下孔洞附近散射波的零阶渐近解。建立了求解含孔曲板弹性波散射与动应力问题的边界积分方程法，利用积分方程法可获得问题的近似分析解。并给出了无限大曲板圆孔附近动应力集中系数的数值结果，且对计算结果进行了分析与讨论。     

NONLINEAR WAVES AND PERIODIC SOLUTION IN FINITE DEFORMATION ELASTIC ROD

A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.     

Nonlinear flexural waves and chaos behavior in finite-deflection Timoshenko beam

Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling wave method and integration skills, the nonlinear partial differential equations can be converted into an ordinary differential equation. The qualitative analysis indicates that the corresponding dynamic system has a heteroclinic orbit under a certain condition. An exact periodic solution of the nonlinear wave equation is obtained using the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function tends to one in the degenerate case, a shock wave solution is given. The small perturbations are further introduced, arising from the damping and the external load to an original Hamilton system, and the threshold condition of the existence of the transverse heteroclinic point is obtained using Melnikov＇s method. It is shown that the perturbed system has a chaotic property under the Smale horseshoe transform.     

Three kinds of nonlinear dispersive waves in elastic rods with finite deformation

On the basis of classical linear theory on longitudinal, torsional and flexural waves in thin elastic rods, and taking finite deformation and dispersive effects into consideration, three kinds of nonlinear evolution equations are derived. Qualitative analysis of three kinds of nonlinear equations are presented. It is shown that these equations have homoclinic or heteroclinic orbits on the phase plane, corresponding to solitary wave or shock wave solutions, respectively. Based on the principle of homogeneous balance, these equations are solved with the Jacobi elliptic function expansion method. Results show that existence of solitary wave solution and shock wave solution is possible under certain conditions. These conclusions are consistent with qualitative analysis.     

Forced dissipative Boussinesq equation for solitary waves excited by unstable topography

In the paper, the effects of topographic forcing and dissipation on solitary Rossby waves are studied. Special attention is given to solitary Rossby waves excited by unstable topography. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a forced dissipative Boussinesq equation. By using the modified Jacobi elliptic function expansion method and the pseudo-spectral method, the solutions of homogeneous and inhomogeneous dissipative Boussinesq equation are obtained, respectively. With the help of these solutions, the evolutional character of Rossby waves under the influence of dissipation and unstable topography is discussed.     

A class of coupled nonlinear Schr(o)dinger equations: Painlevé property,exact solutions,and application to atmospheric gravity waves

The Painlev(e) integrability and exact solutions to a coupled nonlinear Schrodinger (CNLS) equation applied in atmospheric dynamics are discussed. Some parametric restrictions of the CNLS equation are given to pass the Painleve test. Twenty periodic cnoidal wave solutions are obtained by applying the rational expansions of fundamental Jacobi elliptic functions. The exact solutions to the CNLS equation are used to explain the generation and propagation of atmospheric gravity waves.     

A class of coupled nonlinear Schrödinger equations: Painlevé property,exact solutions,and application to atmospheric gravity waves

The Painleve integrability and exact solutions to a coupled nonlinear Schrodinger （CNLS） equation applied in atmospheric dynamics are discussed. Some parametric restrictions of the CNLS equation are given to pass the Painleve test. Twenty periodic cnoidal wave solutions are obtained by applying the rational expansions of fundamental Jacobi elliptic functions. The exact solutions to the CNLS equation are used to explain the generation and propagation of atmospheric gravity waves.