弹性细杆弯曲的Kirchhoff方程的违约校正求解

采用Euler四元数表示的Kirchhoff方程来研究受力挤压作用下的弹性细杆的拓扑构形，进一 步研究弹性细杆的力学性质；将得到的微分方程与约束条件组成微分代数方程后再转化为微 分方程规范形式以便求解；为满足边界条件，应用数值打靶法求解边值条件，并将弹性细杆 在力作用下的拉压过程用Matlab仿真出来.同时对由于误差导致的违约现象进行处理，并针 对欧拉参数的特征，选取合适的修正系数以保持方程的稳定性. 关键词： DNA Euler四元数 Kirchhoff方程 弹性细杆 违约修正     

基于交互模式的柔性体接触碰撞动力学建模方法

目前对于任意形状的柔性体接触碰撞问题,一般采用有限元离散,通用的建模方法有两类:罚函数法和附加约束法.罚函数法将接触作用视为弹簧阻尼力元,无需求解约束方程,但依赖于碰撞力参数的选取;附加约束法可严格满足接触约束条件,但数值求解更为复杂.针对两类接触模型各自的优缺点,提出基于交互模式的建模方法.该方法将整个模型分为局部静力学模块和主体动力学模块,在每个积分步内,局部静力学模块求解接触力,主体动力学模块求解运动学变量,两个模型之间进行位移和力的交互.该方法综合了附加约束法和罚函数法各自的优点,既无需人为选取碰撞参数,又满足局部区域互不嵌入的约束条件,同时数值求解方便.通过杆-板碰撞的实验算例及滑块-滑槽多点碰撞的数值算例,验证了该方法的有效性.     

求解含有高阶导数偏微分方程的局部间断Petrov-Galerkin方法

构造一类求解三种类型偏微分方程的间断Petrov-Galerkin方法.求解的方程分别含有二阶、三阶和四阶偏导数,包括Burgers型方程、KdV型方程和双调和型方程.首先将高阶微分方程转化成为与之等价的一阶微分方程组,再将求解双曲守恒律的间断Petrov-Galerkin方法用于求解微分方程组.该方法具有四阶精度且具有间断Petrov-Galerkin方法的优点.数值实验表明该方法可以达到最优收敛阶而且可以模拟复杂波形相互作用,如孤立子的传播及相互碰撞等.     

求解含有高阶导数偏微分方程的局部间断Petrov-Galerkin方法（英文）

构造一类求解三种类型偏微分方程的间断Petrov-Galerkin方法.求解的方程分别含有二阶、三阶和四阶偏导数,包括Burgers型方程、KdV型方程和双调和型方程.首先将高阶微分方程转化成为与之等价的一阶微分方程组,再将求解双曲守恒律的间断Petrov-Galerkin方法用于求解微分方程组.该方法具有四阶精度且具有间断Petrov-Galerkin方法的优点.数值实验表明该方法可以达到最优收敛阶而且可以模拟复杂波形相互作用,如孤立子的传播及相互碰撞等.     

基于平面问题的位移压力混合配点法

引入压力变量,将弹性力学控制方程表达为位移和压力的耦合偏微分方程组,采用重心插值近似未知量,利用重心插值微分矩阵得到平面问题控制方程的矩阵形式离散表达式.采用重心插值离散位移和应力边界条件,采用附加法施加边界条件,得到求解平面弹性问题的过约束线性代数方程组,采用最小二乘法求解过约束方程组,得到平面问题位移数值解.数值算例验证了所提方法的有效性和计算精度.     

用三角函数法获得非线性Boussinesq方程的广义孤子解

找到一个合适的代换——三角函数法，将非线性Boussinesq微分方程转换为非线性代数方程组.用吴消元法求解该非线性代数方程组，从而获得一般形式Boussinesq微分方程的广义孤子解. 关键词： Boussinesq方程 吴消元法 非线性代数方程组 孤子解     

柔性杆与凸轮斜碰撞特性分析

在计入柔性杆横向变形及其二阶耦合量的条件下,利用Hamilton最小作用原理建立柔性杆与凸轮斜碰撞系统的动力学方程,提出了柔性杆与凸轮碰撞点的确定方法,实现了柔性杆自由下落后的碰撞前、碰撞过程和碰撞后3个阶段的动力学行为仿真.通过分析柔性杆的碰撞运动规律,发现杆的柔性、大范围运动和碰撞三者间存在耦合,碰撞后柔性杆的转角随时间波动变化,转角波动的幅值随时间增大总趋势在减小,但规律性较差.     

求解二维三温辐射扩散方程组的一种代数两层迭代方法

在二维三温辐射扩散方程离散代数方程组的求解中,由于光子、电子和离子温度之间存在耦合关系,而且三个温度在同种介质中有不同的扩散性质,使得经典的代数多重网格(AMG)方法难以直接应用.基于特殊粗化策略,在粗网格层解除了这种耦合关系,得到一种代数两层网格方法,而粗网格方程由经典AMG方法求解.将这一算法具体应用于JFNK(Jacobian自由的Newton-Krylov)框架中预处理方程的求解,并基于该框架求解二维三温辐射扩散方程组.数值结果显示了算法的可扩展性和健壮性.     

一类复代数方程组的高阶PCG法

对二维非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程离散后的复代数方程组将高阶预处理技术与双ＣＧ法相结合给出高阶ＰＣＧ法,同时,将Ｍ阶复代数方程组化成２Ｍ阶非对称代数方程组,给出０阶、１阶和２阶近似ＬＵ分解的公式,并高阶ＰＣＧ法求解。计算结果表明,高阶ＰＣＧ法可以在０阶ＰＣＧ法的基础上将计算效率提高近一倍。     

低能重离子碰撞过程中的壳效应

利用耗散动力学方程与主方程耦合求解并在考虑壳修正的情况下计算了单闭壳原子核碰撞系统和非满壳原子核碰撞系统的电荷、质量的一次矩、二次矩和中子质子关联函数.对于两类不同的碰撞系统而言,电荷、质量的二次矩和中子质子关联函数均存在系统而明显的差别.将理论计算结果与实验值比较,基本一致.表明壳效应对于低能重离子碰撞中的电荷、质量二次矩和中子质子关联函数的影响是明显的.     

On dual Schur domain decomposition method for linear first-order transient problems

This paper addresses some numerical and theoretical aspects of dual Schur domain decomposition methods for linear first-order transient partial differential equations. The spatially discrete system of equations resulting from a dual Schur domain decomposition method can be expressed as a system of differential-algebraic equations (DAEs). In this work, we consider the trapezoidal family of schemes for integrating the ordinary differential equations (ODEs) for each subdomain and present four different coupling methods, corresponding to different algebraic constraints, for enforcing kinematic continuity on the interface between the subdomains. Unlike the continuous formulation, the discretized formulation of the transient problem is unable to enforce simultaneously the continuity of both the primary variable and its rate along the subdomain interface (except for the backward Euler method).Method 1 (d-continuity) is based on the conventional approach using continuity of the primary variable and we show that this method is unstable for a lot of commonly used time integrators including the mid-point rule. To alleviate this difficulty, we propose a new Method 2 (modified d-continuity) and prove its stability for coupling all time integrators in the trapezoidal family (except the forward Euler). Method 3 (v-continuity) is based on enforcing the continuity of the time derivative of the primary variable. However, this constraint introduces a drift in the primary variable on the interface. We present Method 4 (Baumgarte stabilized) which uses Baumgarte stabilization to limit this drift and we derive bounds for the stabilization parameter to ensure stability. Our stability analysis is based on the “energy” method, and one of the main contributions of this paper is the extension of the energy method (which was previously introduced in the context of numerical methods for ODEs) to assess the stability of numerical formulations for index-2 differential-algebraic equations (DAEs). Finally, we present numerical examples to corroborate our theoretical predictions.     

Dynamics modelling of a flexible hub-beam system with a tip mass

This paper presents a finite-element model for a flexible hub-beam system with a tip mass. Both viscous damping and air drag force are introduced into this model. The complete coupling between the system rigid and flexible degrees of freedom is allowed since the start of the formulation and developing the system kinematic variables. Based on deformation theory and geometric constraints, a second order approximation for the displacement field is proposed and the dynamic stiffening is accounted for. Hamilton's principle is utilized in deriving the equations of motion. The corresponding dynamics models of the tip mass and damping forces are developed in a consistent manner through formulating their energy expressions and applying Hamilton's principle. The finite element method is employed for spatial discretization due to its versatility, high accuracy and convergence. Numerical simulations show that the second order term in deformation field can have significant effect on dynamics behavior of flexible multibody systems. It is also shown that the traditional linear model cannot account for dynamic stiffening and may lead to erroneous result in some high-speed systems because the deformation field commonly used in structural dynamics is straight employed in this model. In contrast, the developed model (CCM) based on the second order deformation field can predict valid results. The effects of tip mass and damping on dynamics behavior of the hub-beam system are also discussed.     

Viscoelastic analysis of multi-body systems using the finite element method

A study of the effect of viscoelastic material damping on the dynamic response of multibody systems, consisting of interconnected rigid, elastic and viscoelastic components, is presented. The motion of each elastic or viscoelastic body is identified by using three sets of modes: rigid body, reference and normal modes. Rigid body modes describe translation and large angular rotation of a body reference. Reference modes are the result of imposing the body-axis conditions. Normal modes define the deformation of the body relative to the body reference. Constraints between different components are formulated by using a set of non-linear algebraic equations that can be introduced to the dynamic formulation by using a Lagrange multiplier technique or can be utilized to eliminate dependent co-ordinates by partitioning the constraint Jacobian matrix. In developing the system equations of motion of the viscoelastic component, an assumption of a linear viscoelastic model is made. A Kelvin-Voigt model is employed, wherein the stress is assumed to be proportional to the strain and its time derivative. The formulation yields a constant damping matrix and the damping forces depend only on the local deformation; thus, no additional coupling between the reference and elastic co-ordinates appears in the formulation when considering the viscoelastic effects. It is demonstrated, by a numerical example, that the viscoelastic material damping can have a significant effect on the dynamic response of multibody systems.     

基于无网格点插值法的旋转悬臂梁的动力学分析

将基于多项式点插值的无网格方法用于旋转悬臂梁的动力学分析. 利用无网格点插值方法对柔性梁的变形场进行离散, 考虑梁的纵向拉伸变形和横向弯曲变形, 并计入横向弯曲变形引起的纵向缩短, 即非线性耦合项, 运用第二类Lagrange方程推导得到系统刚柔耦合动力学方程. 与有限元法相比, 该方法只需节点信息, 无需定义单元, 具有前处理简单的优势; 构造的形函数采用更多的节点插值, 具有高阶连续性. 将无网格点插值方法的仿真结果与有限元和假设模态法进行比较分析, 验证了该方法的正确性, 并表明其作为一种柔性体离散方法在刚柔耦合多体系统动力学的研究中具有可推广性.     

一类弹性和阻尼双分段非线性约束系统周期响应特性研究

考虑动态条件下的两种典型分段非线性约束, 根据广义耗散Lagrange原理建立一类具有弹性和阻尼双分段非线性约束系统动力学模型. 采用平均法求解得到系统在周期激励下的幅频响应关系. 分别比较系统在不同分段非线性约束条件下的时域响应、分岔响应和幅频响应, 得到受分段非线性约束的系统响应特性以及约束条件变化时系统响应的变化规律. 对比两种约束条件下的幅频响应, 研究得到系统稳定性受不同分段非线性因素影响及两种分段非线性约束之间的相互影响规律.     

Element-free Galerkin (EFG) method for analysis of the time-fractional partial differential equations

The present paper deals with the numerical solution of time-fractional partial differential equations using the element-free Galerkin (EFG) method, which is based on the moving least-square approximation. Compared with numerical methods based on meshes, the EFG method for time-fractional partial differential equations needs only scattered nodes instead of meshing the domain of the problem. It neither requires element connectivity nor suffers much degradation in accuracy when nodal arrangements are very irregular. In this method, the first-order time derivative is replaced by the Caputo fractional derivative of order α (0<α ≤1). The Galerkin weak form is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Several numerical examples are presented and the results we obtained are in good agreement with the exact solutions.     

Vibration analysis of the tension shadow mask with wire impact damping

Non-linear vibration of the CRT shadow mask with impact damping wires is analyzed in consideration of the mask tension distribution and the effect of wire impact damping. A reduced order FEM model of the shadow mask is obtained from dynamic condensation of the mass and stiffness matrices, and damping wire is modelled using the lumped parameter method to effectively describe its contact interactions with the shadow mask. The non-linear contact-impact model is composed of spring and damper elements, of which parameters are determined from the Hertzian contact theory and the restitution coefficient, respectively. The analysis model of the shadow mask with damping wires is experimentally verified through impact tests of shadow masks performed in a vacuum chamber. Using the validated analysis model of the shadow mask with damping wires, the ‘design of experiments’ technique is applied to search for the optimal damping wire configuration so that the vibration attenuation of the shadow mask is maximized.     

Integrated design of piezoelectric damping system for flexible structure

This paper aims at developing an integrated design method of the active/passive hybrid type of piezoelectric damping system for reducing the dynamic response of the flexible structures due to external dynamic loads. The design method is based on the numerical optimization technique whose objective function is a control effort of the active damping. A vibration suppression performance, which is evaluated by the maximum value of the gain of the frequency response function of the structure, is constrained. In order to demonstrate the structural damping capability of the hybrid type of piezoelectric damping system designed by proposed method, numerical simulation and laboratory experiment will be done using a three-story flexible structure model equipped with 12 surface bonded PZT tiles pairs. Both numerical and experimental results indicate that the optimally designed hybrid piezoelectric damping system can be successfully achieving excellent performance as compared to a conventional purely active piezoelectric damping system.     

A HYBRID FINITE SEGMENT/FINITE ELEMENT MODELLING AND EXPERIMENT ON A FLEXIBLE DEPLOYMENT SYSTEM

This paper focuses on the dynamic responses of a flexible deployment system that has a central rigid body and four articulated flexible beams and undergoes locking impact. A hybrid finite segment/finite element model and an experiment are presented for the deploy-ment system. The flexible beam components in the system are modelled with the finite segments connected by massless beam elements, wherein the finite segments describe the inertia of the large rotation flexible beam and the massless elastic elements describe the elas-ticity of the flexible beam by taking the advantage of small deformation in the relative co-ordinate system. To model the internal impacts in the articulate joints due to clearances, a continuous contact force model of locking joint is also proposed. The governing differential-algebraic equations of the system are established by the Newton-Euler method with Lagrange multipliers and are solved with the method of generalized co-ordinate partitioning. To accelerate the numerical integration, a “longitudinal constraint” is suggested to alleviate the stiff problem of the dynamic equations. In addition, a physical model of the deployment system is constructed. The deployment is released by the compressed springs in the joints. A position measuring system of linear CCD cameras is used to measure the large displacement of the system. Correlations between the mathematical model and the experiments are also presented. Reasonable results are obtained.