圆薄板轴对称弯曲问题的基本方程讨论

首先通过级数展开和求极限运算的方法,确定了等厚度圆薄板轴对称大/小挠度问题的弯曲微分方程在圆心处的表达式.其次,根据轴对称问题的特点,推导出了实心圆薄板在圆心处应满足边界条件的数学表达式,使圆薄板轴对称大/小挠度问题的弯曲微分方程应满足的边界条件达到了应有的数量.本文工作进一步完善了圆薄板轴对称弯曲问题的微分方程形式和边界条件,从而使我们可以利用成熟的微分方程数值解法,对具有较复杂载荷的实心圆薄板轴对称弯曲微分方程进行数值求解.     

Burgers方程的小波精细积分算法

求解偏微分方程的常用方法包括有限差分法、有限元法等。近年来,小波分析在偏微分方程数值求解中的应用已引起很多学者的关注,例如采用Daubechies小波或shannon小波构造的小波配置方法已经取得较好的结果。钟万勰院士提出的偏微分方程的子域精细积分方法是一种半解析方法,方法简单,精度高。将小波方法和精细积分方法相结合应用于偏微分方程的数值求解中将有利于提高算法的精度和稳定性,为此本文以Burgers方程为例,提出了一种求解一维非线性抛物型偏微分方程的小波精积分方法。该方法用拟小波配点法对空间域进行离散,建立起对时间的常微分方程组,然后采用精细时程积分方法对该方程组求解。数值计算结果表明,该方法同其它方法相比,具有计算格式简单,数值稳定性和精度较高的优点。     

求解非线性振动问题的一种新方法

首先把描述非线性振动的微分方程归结为一非线性积分微分方程，然后把此积分微分方程的求解转化为一无穷阶的非线性代数方程组的求解。从理论上讲，可得到满足任何精度要求的周期解。本文用此方法对Ｄｕｆｉｎｇ系统进行了分析。     

基于神经网络的偏微分方程求解方法研究综述

神经网络作为一种强大的信息处理工具在计算机视觉,生物医学,油气工程领域得到广泛应用,引发多领域技术变革.深度学习网络具有非常强的学习能力,不仅能发现物理规律,还能求解偏微分方程.近年来基于深度学习的偏微分方程求解已是研究新热点.遵循于传统偏微分方程解析解、偏微分方程数值解术语,本文称用神经网络进行偏微分方程求解的方法为...     

直梁横弯问题精确微分方程数学规划解

本文利用微分方程最大值原理和非线性规划方法给出了直梁横向弯曲问题精确微分方程非线性规划近似解.     

层流边界层的有限差分计算

前言在解决边界层的问题时有两种方法,一种称为精确方法,对边界层基本微分方程(Prandtl方程)进行积分求解,另一种则称为近似方法,是从动量积分方程(Karman 方程)或能量积分方程来求解.在过去,用分析方法来解基本微分方程是颇困难而复杂的.Blasius 引入了速度剖面相似的假设使偏微分方程简化为常微分方...     

多柔体系统数值分析的模型降噪方法

多柔体系统的动力学方程通常是一组刚性微分方程, 目前普遍采用的刚性微分方程数值解法主要通过数值阻尼滤除系统响应中的高频分量, 其求解效率难以令人满意. 为了降低多柔体系统动力学方程的刚性, 从而可采用ODE45等常规微分方程求解器进行求解, 研究了在建模过程中滤除高频振荡分量的方法. 在以当前时刻为起点的短时间内对柔性体的应力进行均匀化, 用均匀化后的应力计算柔性体的变形虚功率, 由此得到的系统动力学方程的解中不含过高频率的弹性振动, 并且可以通过调节均匀化时间区间的长度参数控制滤波的范围. 数值算例表明: 这种模型降噪方法的计算效率和精度均不低于刚性微分方程求解器, 并且在刚性微分方程求解器失效的情况下模型降噪方法仍有良好的精度和效率. 本文所提的模型降噪方法可成为求解多柔体系统动力学方程的新途径.      

基于倒向随机微分方程的光纤陀螺稳定性评估方法

为了对卫星领域应用的光纤陀螺稳定性和寿命进行提前评估,提出了一种基于倒向微分方程理论的评估方法。首先,分析了倒向随机微分方程的理论基础和工程应用的结构形式;其次,针对光纤陀螺产品结构特点和零位稳定性指标要求,构建了光纤陀螺倒向微分方程模型;第三,通过对模型方程的推导,利用极大似然估计法对模型参数进行了最优估计;最后,利用陀螺零位稳定性测试数据,对模型的符合性进行了验证。验证结果表明:通过倒向随机微分方程模型得出的光纤陀螺零位预估值同陀螺实测值拟合度较好,其拟合优度可决系数R~2=0.7033,拟合误差的均方根值为0.0481 (°)/h,模型是适宜的,可用于工程实际,实现对陀螺零位稳定性指标的出厂控制。     

分数因子与分数阶完整力学系统的运动方程和循环积分

引入分数因子和分数增量,给出了分数阶微积分的定义和性质;基于分数阶导数的定义,证明了含有分数因子的等时变分与分数阶算子的交换关系;提出了分数阶完整保守和非保守系统的Hamilton原理;建立了分数阶完整保守系统和非保守系统的运动微分方程;得到了分数阶完整保守系统的循环积分;并利用分数阶循环积分导出分数阶罗兹方程．最后给出了两个例子．研究表明利用分数因子给出的分数阶微分方程是一个含有分数因子的通常的微分方程,那么分数阶系统运动微分方程的求解都可以采用通常微分方程的求解方法．     

具有幂率速度移动表面边界层问题解析近似解

利用微分方程相似变换,摄动渐进展开和Padé逼近方法对幂率速度移动表面边界层问题进行了研究,得到了问题的解析近似解,对相应的流动特性进行了探讨.     

随机参数结构的统计特征对

结合非正交多项式展式和传统的摄动技巧研究了随机参数结构的统计特征对问题,建立了和摄动法类似的一系列确定的递推方程,并用有限元方法进行了求解,得到了包括特征值和特征向量的特征对的统计值．最后,用算例验证了此方法的正确性．     

随机杆系结构几何非线性分析的递推求解方法

建立了随机静力作用下考虑几何非线性的随机杆系结构的随机非线性平衡方程. 将和 位移耦合的随机割线弹性模量以及随机响应量表示为非正交多项式展开式,运用传统的摄动方法获 得了关于非正交多项式展式的待定系数的确定性的递推方程. 在求解了待定系数后,利用非 正交多项式展开式和正交多项式展开式的关系矩阵,可以很方便地得到未知响应量的二阶统计矩. 两杆结构和平面桁架拱的算例结果表明,当随机量涨落较大时,递推随机有限元方法比基于 二阶泰勒展开的摄动随机有限元方法更逼近蒙特卡洛模拟结果,显示了该方法对几何非线性 随机问题求解的有效性.     

考虑场地介质随机特性的无限域波动分析

针对场地介质具有随机特性的无限域地震波动分析问题，在概率空间中将随机反应向量按随机介质场离散所得主导随机变量的正交多项式级数形式展开，使随机微分方程变换确定性的扩阶线性方程组，并在波动的元模拟技术的基础上，构造了扩阶透射人工边界公式，两者结合形成了求解无限域随机介质中波动传播问题的有限元分析方法，该方法不仅不受基于摄动思想各类方法的久期项的干扰，而且避免了采用模拟方法时人工边界区单元参数样本不均匀所引起的数值计算不稳定问题。     

Non-linear flexural-flexural-torsional-extensional dynamics of beams—I. Formulation

The non-linear differential equations of motion, and boundary conditions, for Euler-Bernoulli beams able to experience flexure along two principal directions (and, thus, flexure in any direction in space), torsion and extension are formulated. The beam's material is assumed to be Hookean but its properties may vary along its span. The nonlinearities present in the differential equations include contributions from the curvature expression and from inertia terms. A set of differential equations with polynomial nonlinearities to cubic order, suitable for a perturbation analysis of the motion, is also developed and the validity of the inextensional approximation is assessed. The equations developed here reduce to those for an inextensional beam. In Part II of this paper, a specific example of application is analyzed and the results obtained are compared with those available in the literature where several non-linear terms have been neglected a priori.     

特征值问题的边界形状灵敏度

研究连续系统振动特征值问题的边界形状灵敏度满足什么方程和边界条件,如何离散化作近似计算结果表明：如果采用相同的有限单元剖分模式,边界形状灵敏度方程和特征值问题方程具有相同的系数矩阵,但前者是非齐次方程,后者是齐次方程;前者需要施加非齐次边界条件,后者施加齐次边界条件。     

Delay equations with fluctuating delay related to the regenerative chatter

The mathematical models representing machine tool chatter dynamics have been cast as differential equations with delay. In this paper, non-linear delay differential equations with periodic delays which model the machine tool chatter with continuously modulated spindle speed are studied. The explicit time-dependent delay terms, due to spindle speed modulation, are replaced by state-dependent delay terms by augmenting the original equations. The augmented system of equations is autonomous and has two pairs of pure imaginary eigenvalues without resonance. The reduced bifurcation equation is obtained by making use of Lyapunov-Schmidt Reduction method. By using the reduced bifurcation equations, the periodic solutions are determined to analyze the tool motion. Analytical results show both modest increase of stability and existence of periodic solutions near the new stability boundary.     

基于侵入混沌多项式法的随机多孔介质内顺磁性流体热磁对流不确定度量化

目前流体流动与传热问题的研究大都基于确定性工况条件,而现实流体流动与传热问题中存在着大量不确定性因素,计算流体力学的不确定性量化提供了一种理解流体物性、边界条件与初始条件等不确定性因素对模拟结果影响的能力.为揭示随机多孔介质内顺磁性流体热磁对流的传播规律与演化特征,本文发展了一种基于侵入式多项式混沌展开法的热磁对流不确...     

Refined differential equations of deflections in axial symmetrical bending problems of spherical shell and their singular perturbation solutions

This paper deals with the research of accuracy of differential equations of deflections.The basic idea is as follows.Firstly,considering the boundary effect the meridianmidsurface displacement u=0,thus we derive the deflection differential equations;secondly we accurately prove that by use of the deflection differential equations or theoriginal differential equations the same inner forces solutions are obtained;finally,weaccurately prove that considering the boundary effect the meridian surface displacementu=0 is an exact solution.In this paper we give the singular perturbation solution of thedeflection differential equations.Finally we check the equilibrium condition and prove theinner forces solved by perturbation method and the outer load are fully equilibrated.Itshows that perturbation solution is accurate.On the other hand,it shows again that thedeflection differential equation is an exact equation.The features of the new differential equations are as follows:1.The accuracies of the new differentia     

Post-buckling analysis of slender elastic rods subjected to terminal forces

This paper presents formulation and solutions for the elastica of slender rods subjected to axial terminal forces and boundary conditions assumed hinged and elastically restrained with a rotational spring. The set of five first-order non-linear ordinary differential equations with boundary conditions specified at both ends constitutes a complex two-point boundary value problem. Solutions for buckling, initial post-buckling (perturbation), large loads (asymptotic) and numerical integration are developed. Results are presented in non-dimensional graphs for a range of rotational spring stiffness, tuning the analysis from double-hinged to hinged-built-in rods.     

Precise integration method for a class of singular two-point boundary value problems

In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations(ODEs) are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.