更一般的杂交广义变分原理及有限元模型

本文根据钱伟长教授提出的更一般的广义变分原理,给出了适用于有限元法中的更广义杂交变分原理,并由此建立了新的广义杂交模理. 进一步以变厚度薄板弯曲单元为例,对基于各种不同的广义变分原理建立的各种杂交元做了比较.     

弹性厚板的分区广义变分原理

本文提出弹性厚板分区广义变分原理,其要点如下:1.各分区可任意定为势能区或余能区.分区势能、分区余能、分区混合变分原理是它的三种特殊形式.2.每个分区中独立变分变量的个数可任意规定.每个分区可定为单类变量区、二类变量区或三类变量区.3.每个交界线上的位移和力的连接条件可以放宽.这个原理为非协调元的厚板有限元法提供理论基础.各种厚板有限元模型可看作这个原理的特殊应用.特别是弹性厚板分区混合变分原理的提出为分区混合有限元法应用于厚板问题打下了基础.     

广义Ekeland变分原理的应用

研究了广义Ekeland变分原理在拟度量空间中的一些重要应用.利用广义Ekeland变分原理证明了函数f满足关于α的Takahashi ε-条件当且仅当f满足关于相同α的Hamel ε-条件.此外,利用关于α的Takahashi ε-条件得到了一些重要结论.     

一般力学中三类变量的广义变分原理

应用对合变换,将两类变量的广义变分原理的驻值条件变换为三类变量的基本方程.按照广义力和广义位移之间的对应关系,将各基本方程乘上相应的虚量,代数相加,然后积分,进而建立了完整系统的三类变量的广义变分原理.应用这种凑合法,建立了非完整系统的三类变量的广义变分原理.作为例子,将一般力学中的三类变量的广义变分原理和两类变量的广义变分原理推广应用于弹性动力学中.最后,讨论了有关的问题.     

改进的拉氏乘子法和多变量广义变分原理

拉氏乘子法是构造广义变分原理的重要途径 ,在识别拉氏乘子时 ,拉氏乘子是独立变分的 ,而识别后 ,它却是其他变量的函数 ,这是产生临界变分的原因 .本文对拉氏乘子法作了改进 ,提出了一种新的理论——凑合反推法 ,应用该方法可以方便地构造多变量的广义变分原理 ,并且不会出现临界变分现象     

大变形对称弹性理论的广义变分原理

本文以陈至达提出的变形几何非线性理论￣［１］为基础，应用Ｌａｇｒａｎｇｅ乘子法，对大变形对称弹性力学问题进行了研究，给出了相应的位能广义变分原理、余能广义变分原理，以及动力学问题的广义变分原理；同时，文中还证明了位能广义变分原理和余能广义变分原理的等价性。     

建立广义变分原理的一种方法

本文建议了一种根据问题的力学意义来建立广义变分原理的方法,本方法对于那些尚未建立起与之相应的变分原理的问题建立其相应的变分原理是有用的.文中不从最小势能原理的推广出发而从力学意义出发导出了弹性力学中的Hu-Washizu广义变分原理和胡海昌广义余能原理,给出了这两个广义变分原理的正确证明.本文并证明了,如果根据Hu-Washizu广义变分原理及胡海昌广义余能原理中含有σ_ij,e_ij和u_i三类变量,就认为这三类变量相互独立,就会导致错误.文中并阐明了这两个广义变分原理正确运用的条件.     

任意旋成面叶栅杂交型气动命题的变分原理与广义变分原理

本文建立了任意旋成面叶栅一种杂交型命题——在叶型周线的某段上给定叶型形状,而在其余段上则给定流速分布——的两族气动变分原理与广义变分原理,从而使正命题与反命题的处理完全统一起来,并加以推广.文中充分发挥了“自然边界条件”和“人工分界面”的有力作用,以简化各种复杂边界条件的处理,并提出了一个新的短函数.本文为在叶轮机叶栅的气动问题中引进和推广有限元法、变分-差分解法以及变分直接解法,提供一个更广泛、更完密的理论基础.     

矩形薄板的屈曲状态*

本文根据薄板的广义变分原理,用有限元法对单向受压矩形板的屈曲状态进行了讨论,并用延续算法获得了在不同边界条件下的屈曲状态.     

广义重调和算子及其在薄板弯曲中的应用

本文用δ－函数具体构造出广义重调和算子，建立相应的二次泛函表达式，并将其应用于弹性薄板的弯曲问题．结果表明．当自变量函数为广义函数时，变分泛函中的自变量函数自然就允许某种程度的不连续性，用Ｌａｇｒａｎｇｅ乘子法所得的修正变分原理实际上是文中给出的变分原理的特殊形式．     

匀速运动的线源荷载激励下无限长梁动力分析*

基于线性叠加原理,本文首先证明了广义Duhamel积分,把运动线源荷载作用下梁的动力问题转化为求解位置固定的线源荷载作用下梁的动力响应即线源脉冲响应函数。然后,利用Laplace变换和Fourier变换求解梁的动力方程,获得了线源脉冲响应函数,继而得到了运动线源荷载下梁的动力解答。对动力响应的进一步分析表明,其最大值总是发生在线源的中心并随荷载运动而运。动最后,定义了运动动力系数。     

非均匀Reissner板弯曲的精确元法

本文在阶梯折算法和精确解析法的基础上,提出构造有限元的新方法——精确元法.该方法不用变分原理,可适用于任意变系数正定和非正定偏微分方程.利用该方法,得到Reissuer板弯曲的一个非协调单元,它具有十五个自由度.由于节点位移参数仅含有挠度和转角,因此处理任意边界条件非常容易.文中给出证明,位移和内力均收敛于精确解.由精确元法所得到的单元不仅能用于厚板,也可用于薄板.文末给出四个算例.算例表明,利用本文的方法,可获得满意的结果,并有较高的数值精度.     

Boundary value problems occurring in plate deflection theory

The paper deals with several iterative methods for solving some boundary value problems occurring in plate deflection theory, making it possible to avoid the lengthy problem preparation which is required to solve these problems by finite difference methods.     

An inverse spectral problem for complex Jacobi matrices

We introduce the concept of generalized spectral function for finite order complex Jacobi matrices and solve the inverse problem with respect to the generalized spectral function. The results obtained can be used for solving of initial-boundary value problems for finite nonlinear Toda lattices with the complex-valued initial conditions by means of the inverse spectral problem method.     

翼板变厚度箱形梁的剪力滞效应分析

针对翼板厚度沿截面宽度方向线性变化的混凝土箱形梁，利用势能变分原理对其进行剪力滞效应分析.选取剪力滞效应引起的附加挠度作为描述剪力滞变形状态的广义位移，并考虑轴力平衡条件对剪力滞效应的影响.将简支箱梁在均布荷载和集中荷载作用下的理论计算值与有限元值进行对比，结果表明:采用该文分析方法得到的计算值与有限元值吻合良好，证实了该文分析方法的正确性.与将翼板变厚度箱梁简化为等厚度箱梁的计算方法相比，考虑翼板厚度变化的计算方法可提高计算精度，误差减小量最大达到了5.65%.     

A New approach to finding the control that transports a system from one phase state to another

In their previous papers, the authors have considered the possibility of applying the theory of motion for nonholonomic systems with high-order constraints to solving one of the main problems of the control theory. This is a problem of transporting a mechanical system with a finite number of degrees of freedom from a given phase state to another given phase state during a fixed time. It was shown that, when solving such a problem using the Pontryagin maximum principle with minimization of the integral of the control force squared, a nonholonomic high-order constraint is realized continuously during the motion of the system. However, in this case, one can also apply a generalized Gauss principle, which is commonly used in the motion of nonholonomic systems with high-order constraints. It is essential that the latter principle makes it possible to find the control as a polynomial, while the use of the Pontryagin maximum principle yields the control containing harmonics with natural frequencies of the system. The latter fact determines increasing the amplitude of oscillation of the system if the time of motion is long. Besides this, a generalized Gauss principle allows us to formulate and solve extended boundary problems in which along with the conditions for generalized coordinates and velocities at the beginning and at the end of motion, the values of any-order derivatives of the coordinates are introduced at the same time instants. This makes it possible to find the control without jumps at the beginning and at the end of motion. The theory presented has been demonstrated when solving the problem of the control of horizontal motion of a trolley with pendulums. A similar problem can be considered as a model, since when the parameters are chosen correspondingly it becomes equivalent to the problem of suppression of oscillations of a given elastic body some cross-section of which should move by a given distance in a fixed time. The equivalence of these problems significantly widens the range of possible applications of the problem of a trolley with pendulums. The previous solution of the problem has been reduced to the selection of a horizontal force that is a solution to the formulated problem. In the present paper, it is offered to seek an acceleration of a trolley with which it moves by a given distance in a fixed time, as a time function but not a force applied to the trolley, while the velocities and accelerations are equal to zero at the beginning and end of motion. In this new problem, the rotation angles of pendulums are the principal coordinates. This makes it possible to find a sought acceleration of a trolley on the basis of a generalized Gauss principle according to the technique developed before. Knowing the motion of a trolley and pendulums it is easy to determine the required control force. The results of numerical calculations are presented.     

圆形弹性薄板非轴对称大挠度问题的一个解法

本文建议一个求解圆形弹性薄板非轴对称大挠度问题的方法.本文以周边固定受非轴对称载荷作用下圆形薄板的大挠度问题为例阐述所述方法的原理和解题步骤.文中所述方法可以用以求解其他边界及载荷作用下圆形薄板的非轴对称大挠度问题.     

Direct method for solving optimal control problems with kinks

In this paper, we introduce a direct solution concept applicable to a slightly restricted class of optimal control problems with kinks. The method treats the optimal trajectory when it lies on a kink for an interval of time (a kink arc) in a special manner. The nonkink arc must satisfy the classical set of necessary conditions of the maximum principle, excluding from that set terminal equations on the kink terminal (the junction between this nonkink arc and a kink arc). At this junction, certain conditions, which compensate for the missing terminal conditions, have to be satisfied. In this paper, the method is applied to solve a fairly realistic model for an important inventory problem, namely, the problem of scheduling the production of a commodity.     

钱氏定理在有限变形极矩弹性力学广义变分原理的应用

应用Lagrange乘子法和钱伟长证明的两类广义变分原理的等价定理,在本文中导出有限变形极矩弹性力学的广义变分原理.文中采用了在拖带坐标系描述法建立的有限变形应变张量(称为Biot有限变形应变定义的准确形式)和应变速率定义与拖带系应力张量构成完整的数学描述.