Structure-preserving Analysis on Folding and Unfolding Process of Undercarriage

The main idea of the structure-preserving method is to preserve the intrinsic geometric properties of the continuous system as much as possible in numerical algorithm design. The geometric constraint in the multi-body systems, one of the difficulties in the numerical methods that are proposed for the multi-body systems, can also be regarded as a geometric property of the multi-body systems. Based on this idea, the symplectic precise integration method is applied in this paper to analyze the kinematics problem of folding and unfolding process of nose undercarriage. The Lagrange governing equation is established for the folding and unfolding process of nose undercarriage with the generalized defined displacements firstly. And then, the constrained Hamiltonian canonical form is derived from the Lagrange governing equation based on the Hamiltonian variational principle. Finally, the symplectic precise integration scheme is used to simulate the kinematics process of nose undercarriage during folding and unfolding described by the constrained Hamiltonian canonical formulation. From the numerical results, it can be concluded that the geometric constraint of the undercarriage system can be preserved well during the numerical simulation on the folding and unfolding process of undercarriage using the symplectic precise integration method.     

TIME PRECISE INTEGRATION METHOD FOR CONSTRAINED NONLINEAR CONTROL SYSTEM

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DYNAMIC ANALYSIS OF UNDERGROUND COMPOSITE STRUCTURES UNDER EXPLOSION LOADING

In selecting rational types of underground structures resisting explosion, in order to improve stress states of the structural section and make full use of material strength of each part of the section, the research method of composite structures is presented.Adopting the analysis method of micro-section free body, equilibrium equations, constraint equations and deformation coordination equations are given. Making use of the concept of generalized work and directly introducing Lagrange multiplier specific in physical meaning,the validity of the constructed generalized functional is proved by using variation method.The rational rigidity matching relationship of composite structure section is presentedthrough example calculations.     

非线性最优控制系统的时程精细计算研究

针对非线性最优控制问题 ,通过一阶 Taylor级数展开 ,得到线性化的动力学方程 ,进而在方程原变量的基础上 ,引入对偶向量 (Lagrange乘子向量 ) ,将动力学方程从 Lagrange体系引入到了 Hamilton体系 ,在全状态下 ,从一个新的角度对非线性最优控制问题进行了描述 ,进一步基于时程精细积分理论 ,对其方程进行了有效的精细求解 ,并通过算例说明了文中方法的有效性     

Method of high-order lagrange multiplier and generalized variational principles of elasticity with more general forms of functionals

It is known[1]that the minimum principles of potential energy andcomplementary energy are the conditional variation principles underrespective conditions of constraints.By means of the method of La-grange multipliers,we are able to reduce the functionals of condi-tional variation principles to new functionals of non-conditionalvariation principles.This method can be described as follows:Mul-tiply undetermined Lagrange multipliers by various constraints,andadd these products to the original functionals.Considering these un-determined Lagrange multipliers and the original variables in thesenew functionals as independent variables of variation,we can see thatthe stationary conditions of these functionals give these undeter-mined Lagrange multipliers in terms of original variables.The sub-stitutions of these results for Lagrange multipliers into the abovefunctionals lead to the functionals of these non-conditional varia-tion principles.However,in certain cases,some of the undetermined Lagrangemultipliers ma     

Fer’s expansion for generalized Hamiltonian system based on Lie transformation technique

In the present paper, we present a new method for integrating the ordinary differential equation, especially for the ordinary differential equation derived from explicitly time-dependent generalized Hamiltonian dynamic system, which is based on taking a factorization of the evolution operator as an infinite product of the exponentials of Lie operators. The above process is a Lie group (algebraic) method that retains the structural intrinsic properties of the exact solution when truncated and is used to analyze the main features of the so-called Fer’s expansion. The numerical examples are presented at the end of this paper.     

Reduced augmented Lagrangian bi-conjugate gradient method for impact-contact problems

To avoid the numerical oscillation of the penalty method and non-compatibility with explicit operators of conventional Lagrange multiplier methods used in transient contact problems to enforce surface contact conditions, a new approach to enforcing surface contact constraints for the transient nonlinear finite element problems, referred to as “the reduced augmented Lagrangian bi-conjugate gradient method (ALCG)”, is developed in this paper. Based on the nonlinear constrained optimization theory and is compatible with the explicit time integration scheme, this approach can also be used in implicit scheme naturally. The new surface contact constraint method presented has significant advantages over the widely adopted penalty function methods and the conventional Lagrangian multiplier methods. The surface contact constraints are satisfied more accurately for each step by the algorithm, so the oscillation of numerical solution for the explicit scheme is depressed. Through the development of new iteration strategy for solving nonlinear equations, ALCG method improves the computational efficiency greatly. Project supported by State Education Commission Doctoral Foundation and Natural Science Foundation of Liaoning Province.     

Fractional conservation laws in optimal control theory

Using the recent formulation of Noether’s theorem for the problems of the calculus of variations with fractional derivatives, the Lagrange multiplier technique, and the fractional Euler–Lagrange equations, we prove a Noether-like theorem to the more general context of the fractional optimal control. As a corollary, it follows that in the fractional case the autonomous Hamiltonian does not define anymore a conservation law. Instead, it is proved that the fractional conservation law adds to the Hamiltonian a new term which depends on the fractional-order of differentiation, the generalized momentum and the fractional derivative of the state variable. Partially presented at FDA ’06—2nd IFAC Workshop on Fractional Differentiation and its Applications, 19–21 July 2006, Porto, Portugal.     

Note on the critical variational state in elasticity theory

Recently Prof. Chien Wei-zang pointed out that in certain cases, by means of ordinary Lagrange multiplier method, some of undetermined Lagrange multipliers may turn out to be zero during variation. This is a critical state of variation. In this critical state, the corresponding variational constraint can not be eliminated by means of simple Lagrange multiplier method. This is indeed the case when one tries to eliminate the constraint condition of strain-stress relation in variational principle of minimum complementary energy by the method of Lagrange multiplier.By means of Lagrange multiplier method, one can only derive, from minimum complementary energy principle, the Hellinger-Reissner Principle, in which only two type of in-dependent variables, stresses and displacements, exist in the new functional. Hence Prof. Chien introduced the high-order Lagrang multiplier method bu adding the quadratic terms.to original functions. The purpose of this paper is to show that by adding to original functionals one     

NONLINEAR DYNAMIC ANALYSIS OF FLEXIBLE MULTIBODY SYSTEM

The nonlinear dynamic equations of a multibody system composed of ?exible beams are derived by using the Lagrange multiplier method. The nonlinear Euler beam theory with inclusion of axial deformation e?ect is employed and its deformation ?eld is …     

Semi-inverse method and generalized variational principles with multi-variables in elasticity

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A General Formulation and Solution Scheme for Fractional Optimal Control Problems

Accurate modeling of many dynamic systems leads to a set of Fractional Differential Equations (FDEs). This paper presents a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The fractional derivative is described in the Riemann–Liouville sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic system. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An approach similar to a variational virtual work coupled with the Lagrange multiplier technique is presented to find the approximate numerical solution of the resulting equations. Numerical solutions for two fractional systems, a time-invariant and a time-varying, are presented to demonstrate the feasibility of the method. It is shown that (1) the solutions converge as the number of approximating terms increase, and (2) the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs. It is hoped that the simplicity of this formulation will initiate a new interest in the area of optimal control of fractional systems.     

A General Formulation and Solution Scheme for Fractional Optimal Control Problems

Accurate modeling of many dynamic systems leads to a set of Fractional Differential Equations (FDEs). This paper presents a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The fractional derivative is described in the Riemann–Liouville sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic system. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An approach similar to a variational virtual work coupled with the Lagrange multiplier technique is presented to find the approximate numerical solution of the resulting equations. Numerical solutions for two fractional systems, a time-invariant and a time-varying, are presented to demonstrate the feasibility of the method. It is shown that (1) the solutions converge as the number of approximating terms increase, and (2) the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs. It is hoped that the simplicity of this formulation will initiate a new interest in the area of optimal control of fractional systems.     

Fractional generalized Hamiltonian mechanics and Poisson conservation law in terms of combined Riesz derivatives

In this paper, we present a new kind of fractional dynamical equations, i.e., the fractional generalized Hamiltonian equations in terms of combined Riesz derivatives, and it is proved that the fractional generalized Hamiltonian system possesses consistent algebraic structure and Lie algebraic structure, and the Poisson conservation law of the fractional generalized Hamiltonian system is investigated. Then the conditions, which a fractional generalized Hamiltonian system can be reduced to a generalized Hamiltonian system, a fractional Hamiltonian system and a Hamiltonian system are given. Further, the conserved quantities of a fractional dynamical system are given to illustrate the method and results of the application. At last, a new fractional Volterra model of the three species groups is presented and its conserved quantities are obtained, by using the method of this paper.     

An improved precise integration method for nonlinear dynamic system

In the present paper, based on the precise integration method established in linear dynamic system, an improved precise integration method is presented for nonlinear dynamic system. Firstly, the nonlinear dynamic system is converted into an augmented Lie type dynamic system. Then the precise integration method is improved for solving the above augmented equation and preserving its group structure in the meantime. Finally, two numerical examples are presented to demonstrate the validity and effectiveness of the proposed method.     

An index 0 Differential-Algebraic equation formulation for multibody dynamics: Holonomic constraints

The Lagrange multiplier form of index 3 differential-algebraic equations of motion for holonomically constrained multibody systems is transformed using tangent space generalized coordinates to an index 0 form that is equivalent to an ordinary differential equation. The index 0 formulation includes embedded tolerances that assure satisfaction of position, velocity, and acceleration constraints and is solved using established explicit and implicit numerical integration methods. Numerical experiments with two spatial applications show that the formulation accurately satisfies constraints, preserves invariants due to conservation laws, and behaves as if applied to an ordinary differential equation.     

岩质圆形隧洞围岩应力场弹塑性新解

针对动态接触问题的有限元并行计算，提出了一种新的接触算法. 新算法引入局部拉氏 乘子技术来计算接触力. 由于同时考虑了无穿透的接触约束条件和相邻接触对的相互影响， 较之广泛使用的罚参数法，新算法使接触约束条件和系统平衡方程得到更充分的满足. 虽然 为提高接触计算精度而在局部采用了迭代技术，但算法仍然具有较高的效率，且与显式时间 积分方案完全相容. 此外，通过构造专门的区域分解方案，实现了将现有为串行程序开发的 搜索算法平滑移植到并行环境的目标. 数值算例表明，所提出的接触算法具有很好的并行性， 在保证了接触问题并行计算精度的同时，取得了满意的并行效率.     

爬壁机器人系统的Noether对称性和守恒量

爬壁机器人的运动是一种模仿壁虎爬行的运动, 爬壁机器人的运动可分解为四肢带动身体的运动, 先前的研究都是基于牛顿力学的方法. 本文采用Lagrange 力学的方法建立爬壁机器人系统的运动方程, 并运用Lie群分析方法建立该系统的Noether对称性理论, 得出爬壁机器人的运动规律. 首先, 给出非完整爬壁机器人系统的动能、势能和Lagrange函数以及所受的非完整约束, 从而建立了非完整爬壁机器人系统的Lagrange方程; 其次, 引入关于时间和广义坐标的无限小变换, 提出了非完整爬壁机器人系统的Hamilton作用量和Hamilton作用量的基本变分公式; 第三, 给出爬壁机器人系统 Noether对称性变换和广义准对称变换的定义, 判据和存在的Noether守恒量, 并提出了非保守完整系统和非保守非完整爬壁机器人系统的Noether定理; 最后, 以圆锥面上爬壁机器人为例, 对给出的守恒量直接进行积分给出圆锥面上爬壁机器人整体运动的精确解和四肢运动的数值解, 发现了该爬壁机器人的运动规律, 很好地验证了非完整爬壁机器人系统的Noether对称性理论. 本文的研究为Lie群分析方法应用于其他复杂的机器人系统以及柔性机器人系统的对称性求解提出了一种新的对称性求解方法.      

一种基于Hamel形式的无条件稳定动力学积分算法

Time integration algorithm is a key issue in solving dynamical system. An unconditionally stable Hamel generalized α method is proposed to solve the instability issue arising in the time integration of dynamic equations and to eliminate the pseudo high order harmonics incurred by the spatial discretization of finite element simultaneously. Therefore, the development of numerical integration algorithm to solve the above-mentioned problems has important theoretical and application value. The algorithm proposed in this paper is developed based on the moving frame method and Hamel’s field variational integrators along with the strategy to construct an unconditionally stable Hamel generalized α method. It is shown that a new numerical formalism with higher accuracy can be derived under the same framework of the unconditional stable algorithm established through a special variational formalism and variational integrators. The above-mentioned formalism can be extended from general linear space to Lie group by utilizing the moving frame method and the Lie group formalism of the Hamel generalized α method has been obtained. Both the convergence and stability of the algorithm are discussed, and some numerical examples are presented to verify the conclusion. It is demonstrated by the theoretical analysis that the Hamel generalized α method proposed in the paper is unconditionally stable, second-order accurate and can quickly filter out pseudo high-frequency harmonics. Both conventional and proposed methods have been applied to numerical examples respectively. Comparisons between results of numerical examples show that the aforementioned advantages of the proposed method in terms of accuracy, dissipation and stability are tested and verified. At the same time, it can be developed that new numerical integration algorithms with even higher order accuracy. The scheme can also be proposed, which is suitable for both general linear space and Lie group space. A new way for constructing variational integrators is also obtained in this paper.      

空间展开折叠桁架结构动力学分析研究

本文以笛卡尔坐标系下节点自然坐标为未知量,建立了桁架结构系的基本运动力学方程,并首次推导出桁架结构中常用节点附加几何约束方程,相应约束Jacobi矩阵及其导数矩阵,采用奇异值分解法求约束Jacobi矩阵的零空间基和M-P广义逆,并由矩阵缩减法建立了带约束桁架体系的运动力学方程和求解方法。数值算例表明该方法适于可展折叠桁架结构运动力学分析。