Modelling of the forced motions of an elastic beam using the method of integrodifferential relations

A variational approach to the numerical modelling of forced lateral motions of an Euler–Bernoulli elastic beam is developed for a number of linear boundary conditions using the method of integrodifferential relations. A class of linear boundary actions is considered. A family of quadratic functionals, connecting the displacement field of points of the beam with the bending-moment functions in the cross section and the momentum density is proposed. Variational formulations of the original initial-boundary value problem on the motion of the beam are given and the necessary conditions for the functionals introduced to be stationary are analysed. The integral and local quality characteristics of the admissible approximate solutions are determined. The relation between the variational problems, formulated for the beam model, with the classical Hamilton–Ostrogradskii variational principles is demonstrated. An algorithm for constructing approximate systems of ordinary differential equations is developed, the solution of which yields stationary (minimum) values of the functionals introduced on a specified set of displacement fields, moments and momenta. Examples of calculations of the displacements for an elastic beam and an analysis of the quality of the numerical solutions obtained are presented.     

A variational approach to optimal control problems for elastic body motions

The initial-boundary problem for the linear theory of elasticity is considered. Based on the method of integrodifferential relations a new dynamical variational principle in which displacement, stress, and momentum functions are varied is proposed and discussed. To minimize the nonnegative functional under initial, boundary, and partial differential constraints arising in this approach a regular algorithm for approximation of the unknown functions is worked out. The algorithm gives us the possibility to estimate explicitly the local and integral quality of obtained numerical solutions. An effective numerical method for the optimization problems of controlled motions of elastic bodies with quadratic objective functionals is developed. As example, the 3D problems of optimal longitudinal motions of a rectilinear elastic prism with a quadratic cross section are considered for the terminal total mechanical energy to be minimized. The numerical results and their error estimates are presented and discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational analysis in dynamical problems of linear elasticity

The initial–boundary value problems in the linear theory of elasticity is considered. Based on the method of integrodifferential relations (MIDR) two dynamical variational principles is proposed and discussed. It is shown that the Hamilton principle as well as the corresponding complementary principle stated for dynamic boundary value problems follow out the variational formulations proposed. To minimize the nonnegative functional under algebraic and differential constraints a regular finite element algorithm is worked out. The algorithm allows us to estimate explicitly the local and integral quality of numerical solutions obtained. A 3D problem of lateral motions of a rectilinear elastic prism with a rectangular cross section are considered. The numerical results are presented and discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

非线性弹性体的弹性动力学变分原理

本文根据文献[1],对非线性应力应变关系的弹性体,导出了弹性动力学问题的变分原理和广义变分原理,提出了混合位移协调元和混合应力协调元的瞬时广义变分原理.     

Variational approach to modeling and control of dynamic systems with distributed parameters

Controlled motions of a thin elastic rod are studied. A variational statement of the initial-boundary value problem is given based on the integral formulation of constitutive relations. An approach relying on the Ritz method and finite element technique is proposed to design the boundary control that moves the rod with certain accuracy to a given final state and minimizes the mean energy. The results of numerical simulation and the estimates of solution quality are presented and discussed. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

弹性理论中各种变分原理的分类

本文按弹性理论中各种变分原理的约束条件的不同,对所有变分原理进行分类.我们在前文中业已指出,应力应变关系这样的约束条件是不能用拉氏乘子法解除的.剩下的可能约束条件共有四种:(1)平衡方程,(2)应变位移关系,(3)边界外力已知的边界条件,和(4)边界位移已知的边界条件.弹性理论的各种变分原理中,有的只有一种约束条件,有的有两种或三种,最多只能有四种约束条件.这样一共可能有15种变分原理,但是每种变分原理既可以用应变能A表示,又可以用余能B表示.这样,我们一共应有30种形式完全不同的变分原理,我们全部列出了这三十种形式的变分原理.     

A projection approach to optimal control of elastic beam dynamics

Controlled lateral motions of an elastic beam – extended body with a cuboid shape – are studied. A generalized statement of the initial-boundary value problem is given in the frame of 3D linear elasticity based on the integral formulation of constitutive relations. A numerical approach relying on the Fourier method and semi-discretization technique is proposed to design the optimal boundary control that moves the beam to a given final state and minimizes a quadratic cost function. Some results of numerical simulation and quality analysis are presented and discussed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

非线性弹性体的弹性力学变分原理

作者自1978年以后,曾发表了一系列有关弹性力学的变分原理和广义变分原理的文章如[1](1978),[6](1980),[2]、[3](1983),[4]、[5](1984),都是指线性应力应变关系的线性弹性体的.在1985年出版的广义变分原理中,初步推广至非线性弹性体,但并未进行较全面的探讨.本文特别讨论非线性应力应变关系的弹性体的变分原理和广义变分原理,这里有不少问题是值得注意的,有时,它对线性弹性体的变分原理,有指导意义.当应变很小,其高次项可以略去时,本文所得结论,都能近似地化简为通常线性理论的结果.     

Remarks on variational methods and lower-upper solutions

The presence of lower and upper solutions to semilinear elliptic equations is linked to some variational properties of the associated action functionals. We exploit this variational characterization to find existence and multiplicity of solutions when multiple lower and upper solutions are given in various ordering relations. Received July 1998     

弹性正交索网结构动力学的各类非传统Hamilton型变分原理

根据古典阴阳互补和现代对偶互补的基本思想,通过罗恩早已提出的一条简单而统一的新途径,系统地建立了正交索网结构几何非线性弹性动力学的各类非传统Hamilton型变分原理．这种新的非传统Hamilton型变分原理能反映这种动力学初值-边值问题的全部特征．文中首先给出正交索网结构几何非线性动力学的广义虚功原理的表式,然后从该式出发,不仅能得到正交索网结构几何非线性动力学的虚功原理,而且通过所给出的一系列广义Legendre变换,还能系统地成对导出正交索网结构几何非线性弹性动力学的5类变量、4类变量、3类变量和2类变量非传统Hamilton型变分原理的互补泛函、以及相空间非传统Hamilton型变分原理的泛函与1类变量非传统Hamilton型变分原理势能形式的泛函．同时,通过这条新途径还能清楚地阐明这些原理的内在联系．     

Unconventional Hamilton-type variational principles for nonlinear coupled thermoelastodynamics

According to the basic idea of classical yin-yang complementarity and modern dual-complementarity, in a simple and unified new way proposed by Luo, the unconventional Hamilton-type variational principles for geometrically nonlinear coupled thermoelastodynamics can be established systematically. The new unconventional Hamilton-type variational principle can fully characterize the initial-boundaty-value problem of this dynamics. In this paper, an important integral relation is given, which can be considered as the expression of the generalized principle of virtual work for geometrically nonlinear coupled thermodynamics. Based on this relation, it is possible not only to obtain the principle of virtual work in geometrically nonlinear coupled thermodynamics, but also to derive systematically the complementary functionals for eight-field, six-field, four-field and two-field unconventional Hamilton-type variational principles by the generalized Legendre transformations given in this paper. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly.     

Variational formulations of certain problems of the theory of the flow of rigid-plastic media

The mathematical content of variational problems of the mechanics of rigid-plastic media reduces to minimizing convex functionals in non-reflexive spaces of solenoidal vector fields. Different formulations are presented in /1/. Problems are examined there in which discontinuous velocity fields occur. The initial functional is undetermined in such fields. In this connection, the problem is posed of constructing that extension of the original set of kinematically allowable velocity fields as would contain all their possible discontinuities allowed by the mechanics of rigidly plastic media, and of continuing the original functional into the extension obtained. This problem is solved in this paper for the case when the velocity field is given on the whole surface of the rigid-plastic body.     

平面弹性第二基本问题的新提法*

本文给出了平面弹性第二基本问题的一般提法,这里在多连通弹性区域的各个闭边界围线上给出的位移只是相对的,各允许有一不同的刚性移动.在这种情况下,为了解的唯一性,证明了必须另外给出每一边界围线上外力的主矢量与主力矩.还给出了求解的方法以及一些例题.     

Method of Integro-Differential Relations for Optimal Beam Control

An approach to modelling and optimization of controlled dynamical systems with distributed elastic and inertial parameters are considered. The general method of integro-differential relations (IDR) for solving a wide class of boundary value problems is developed and criteria of solution quality are proposed [1]. A numerical algorithm for discrete approximation of controlled motions is worked out [2] and applied to design the optimal control low steering an elastic system to the terminal position and minimizing the given objective function [3]. The polynomial control of plane motions of a homogeneous cantilever beam is investigated. The optimal control problem of beam transportation from the initial rest position to given terminal states, in which the full mechanical energy of the system reaches its minimal value, is considered. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The method of integrodifferential relations for analysing the natural oscillations of membranes

An approach to solving of problems of the free oscillations of membranes, based on the method of integrodifferential relations, is proposed. The variational properties of the integrodifferential problems are investigated and their relation to classical variational principles is demonstrated. Integral and local criteria for the quality of the approximate solutions are proposed. A numerical-analytical algorithm is developed for finding the characteristic frequencies and modes of oscillations. Problems involving the free transverse motions of circular and elliptic membranes are considered as an example.     

Reissner夹层板动力学的非传统Hamilton型变分原理

根据古典阴阳互补和现代对偶互补的基本思想,通过早已提出的一条简单而统一的新途径,系统地建立了Reissner夹层板动力学的各类非传统Hamilton型变分原理．这种新的非传统Hamilton型变分原理能反映这种动力学初值-边值问题的全部特征．文中首先给出一个Reissner夹层板广义虚功原理的表式．然后从该式出发,不仅能得到Reissner夹层板动力学的虚功原理,而且通过所给出的一系列广义Legendre变换, 还能系统地成对导出五类变量、 二类变量和一类变量非传统Hamilton型变分原理的互补泛函．同时,通过这条新途径还能清楚地阐明这些原理的内在联系．     

A flexible approach to the simulation of hybrid multibody systems

The present works deals with the incorporation of both flexible beam and shell structures into the realm of flexible multibody dynamics. Geometrically exact beam formulations based on classical Simo-Reissner kinematics are suitable for modelling beam-type flexible components in the context of finite-deformation multibody dynamics. So geometrically exact shell formulations are based on Reissner-Mindlin kinematics. In [2], a flexible framework for dealing with flexible structural elements in a multibody context is described. A specific isoparametric finite element discretization of a shell formulation leads to semi-discrete equations of motions assuming the form of differential-algebraic equations (DAEs). A compatible isoparametric formulation of beams has already been developed in [1]. The uniform DAE framework makes possible the incorporation of alternative finite element formulations. In addition to that, various time-stepping schemes such as energy-momentum methods or variational integrators can be applied. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solving the problem of contact equilibrium of a punch and an elastic body

The variational problem of contact equilibrium of a punch and an elastic body is considered. An equivalent formulation of the problem is given in variational inequality form. Existence and uniqueness of the solution is investigated in a particular case. A penalty method is proposed for approximate solution of the problem.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 97–103, 1985.