具有内阻的旋转复合材料轴的非线性自由振动与稳定性

研究具有内阻的旋转复合材料轴的非线性自由振动和稳定性。采用大振幅不可伸缩旋转梁的假定,对复合材料轴进行非线性建模,模型引入非线性曲率和惯性的影响。根据复合材料的粘弹性耗散特性描述材料内阻。从复合材料本构关系、应变-位移关系基本方程出发,在导出动能、势能和内阻耗散能的基础上,基于扩展的Hamilton原理,建立旋转复合材料轴的弯-弯耦合非线性振动偏微分方程组。采用Galerkin法将偏微分方程离散化为常微分方程。为了预测旋转复合材料轴的稳定边界,对线性化方程进行特征值分析,给出了临界速度和失稳阈的表达式。采用四阶RungeKutta法对常微分方程组进行数值积分,获得位移时间响应图、相平面图和功率谱图,研究了铺层角、长径比和铺层方式对非线性振动响应和稳定性的影响。     

N自由度柔性机械臂通用的动力学建模方法研究

针对多自由度柔性机械臂动力学建模过程复杂的问题,提出一种适用于N自由度柔性臂刚柔耦合动力学建模的通用方法。该方法基于拉格朗日方程(Lagrange)和假设模态法(ASM),归纳推导得到了N自由度柔性机械臂动力学模型的最简通用符号表达式,根据该建模方法开发了"N-DOF柔性机械臂动力学方程符号计算软件";最后以两自由度柔性机械臂为例进行位置跟踪控制实验,用开发的软件自动生成了动力学模型,并搭建了实物模型进行对比验证,结果具有一致性,表明了该建模方法的正确性。与传统建模方法相比,该建模方法可减少90%以上计算时间,极大降低了建模过程的复杂性,同时对多自由度柔性臂动力学建模也具有通用性。     

研究发现碳纳米管的“多米诺”现象

纳米动力是实现纳米机械的重要条件，如何在纳米尺度上将其他能量转化为动能是当前纳米科技研究的热点。上海大学应用数学和力学研究所的张田忠教授通过计算机模拟，发现一定直径以上的碳纳米管经触发后会发生多米诺骨牌式的塌陷，释放相应的势能。     

粗糙界面法向接触振动响应与能量耗散特性研究

粗糙界面的法向接触振动与能量耗散特性对描述界面动力学机理具有重要的理论和实际意义。本文通过建立粗糙界面法向接触振动的动力学模型,提出了其法向接触振动响应特征量和振动能量耗散量的计算方法。基于粗糙表面的三维分形模型描述,构造了粗糙界面的接触力-变形关系式,并与Hertz接触模型的力-变形关系进行了对比分析;建立了粗糙界面接触振动系统的动力学方程,计算了不同表面形貌粗糙界面系统每周期的振动能量耗散率和累计能量耗散率;分析了粗糙界面法向接触振动的响应特征与能量耗散特性,从理论上对界面法向微动能量耗散的实验结果进行了解释,为描述接触界面的动力学机理提供了理论依据。     

材料内阻对旋转复合材料轴动力学稳定性的影响研究

由于复合材料与金属材料相比,具有更为突出的阻尼耗散能力,超临界旋转复合材料轴在材料内阻的作用下更容易产生不稳定自激振动。从复合材料本构关系、应变-位移关系基本方程出发,基于Bernoulli-Euler梁理论,并考虑复合材料的黏弹性阻尼耗散特性,在导出旋转复合材料轴的动能、势能和内阻耗散能的基础上,采用Hamilton原理建立了转子系统的运动微分方程,采用Galerkin法对复数形式的弯曲方程进行求解,导出转子系统的特征方程。通过数值分析得到固有频率-转速曲线和阻尼-转速曲线,求得了临界转速和失稳阈。研究了铺层角、长径比和铺层方式的影响。模型结果的正确性,通过与文献结果对比,得到了验证。     

基于节点里兹势能主自由度的结构动态缩减方法

为了提高结构动态分析效率,提出一种基于节点里兹势能主自由度的结构动态缩减方法。阐述改进缩减系统法的原理,分析里兹向量提取过程,定义节点里兹势能,并将其作为捕捉能精确反映结构动态特性主自由度的依据,给出节点里兹势能计算公式。通过计算分析圆柱曲板和曲轴2个实例,验证了该方法的可行性和优越性。研究结果表明,在里兹向量空间定义节点里兹势能更容易捕捉到高精度的动态特性,加权系数可提高高阶频率的精度,在结构缩减中取整体自由度数的约1/3主自由度数比较合适。     

低质量比舵系统流致振动特征研究

考虑到舵的展弦比对其振动特性可能造成影响,将经典两自由度系统动力方程进行参数修正。方程中代表惯性效应的项引入附加质量修正系数,而环量项引入环量修正系数,并通过MATLAB编程求解方程得到了不同来流速度时舵系统的时历曲线以及相轨线;随着来流速度的增加,相轨线的形状趋于扁平化,这意味着系统的动能可以转化为更大的势能;此外,利用有限体积法结合半隐式积分法对计算区域进行离散,并选择SIMPLE算法对离散后的控制方程进行求解,从而模拟了流体作用下舵系统的瞬态运动,分析了舵系统发生颤振时的绕流场变化。发现在舵的上表面,前缘和刚心之间的涡不断向后缘扩散,并在刚心和后缘之间形成了明显的涡层,此时舵的上表面形成一个完整的涡层,这与系统做衰减运动时有所不同;并且系统会出现大角度的俯仰运动,此时前缘和刚心之间出现较大的涡量。     

壳体的有限元线法分析（I）——基本理论

本文从退化壳理论^[6]出发构造了任意曲面壳体的四边形有限元线法^[1][2]单元。该单元满足C^0连续，为协调单元。对于所构造的单元，本文从最小势能原理出发推导出用该单元作壳体静力计算的控制微分方程的边界条件，得到一致的线法方程体系。全文共分两篇，此为上篇，主要介绍基本理论，数值算例将在下篇中给出。     

子结构界面连接刚度参数识别的一种直接方法

提出一种通过特征方程反问题识别子结构界面连接刚度参数的直接方法。新方法以动柔度矩阵特征方程为基础，建立通过界面位移求解界面内力的方程，由模态叠加法对可测自由度上不完全的振型拟合估计界面不可测自由度上的位移，适合于部分子结构不具备可测自由度的复杂组合结构。数值仿真算例和工程实例表明本方法具有良好的识别精度和数值稳定性。计算量和对计算机存贮量的要求也较小。     

岩石变形破坏过程中的能量传递和耗散研究

岩石变形破坏的过程是和外界产生能量交换的过程。从理论上分析了用能量方法研究岩石破坏问题的合理性,以及岩石在变形过程中弹性能、塑性能、表面能、辐射能、动能之间相互转化的过程、计算原理、以及对岩石破坏所起的不同作用。并分别从宏观和微观的角度研究了在不同的变形阶段中岩石能量耗散与释放问题。在宏观上,岩石变形前期以弹性应变能的方式存储外界提供的能量,同时又通过损伤演化等向外界耗散能量;变形的后期以剧烈的能量释放为主。微观上,存在多种引起岩石应变硬化和应变软化的机制,岩石存储能量还是向外界释放能量取决于这些微观机制竞争的最后结果,基于此推导了岩石变形中能量的传递方程,用试验研究了能量的转化和平衡,以及耗散能和释放能之间的比例关系。结果表明能量耗散导致岩石强度的降低,而能量释放是造成岩石灾变破坏的真正原因。从能量耗散与释放的观点研究岩石的破坏,可以从本质上把握岩石变形和破坏的物理机理,寻找岩石破坏的真正原因,为实际工程提供参考。     

板极限分析的一种改进方法

本文提出了以约束变尺度优化思想为基础的一种改进方法，分析板的极限荷载。在能量原理和平衡意义下建立相应的目标函数和约束方程。计算结果表明：这种方法与传统的确定方法相比具有概念清晰、算法可靠、实用等优点。     

Legendre函数法求解分数阶偏微分方程的数值解

分数阶偏微分方程作为一类常见的微分方程用以描述工程等实际问题．较传统的解析方法而言,本文提出的数值算法在计算精度及计算效率上有更大的优势．借助分数阶Legendre函数对待求方程中的二元函数进行级数展开,并结合算子矩阵将待求方程转化为非线性代数方程组,然后通过数学软件求解该方程组,获得原方程的数值解．本文介绍的分数阶Legendre函数法能更精确的模拟工程问题中一些复杂的数学现象,而且在函数推导及构造上都比较简单,很小的级数展开就能达到满意的数值精度．最后给出的误差分析也验证了该方法的收敛性．     

Finite difference energy techniques for arbitrary meshes applied to linear plate problems

A new energy based finite difference analytical technique is introduced. The method incorporates certain energy concepts and the ability to use arbitrary, irregular meshes within the framework of the Finite Difference Method. This formulation reduces any governing partial differential equations to a set of difference equations containing partial derivatives up to and including the second order. Further, certain strong similarities with the popular Finite Element Method are shown and the ability to solve problems with irregular boundaries is discussed. To demonstrate the Finite Difference Energy Method several plate bending problems are solved.     

A discussion on high-frequency wave energy behavior in viscoelastic media

A time-varying energy equation is proposed for general high-frequency elastoacoustic waves. Derivations are presented through a wave approach and transport theory. The assumption of uncorrelated waves is adopted in the wave approach and leads to a second-order partial differential equation. A similar process is also mathematically verified in the derivation by means of transport theory. Both derivations allow a simplified energy-based equation to be formulated. This simplified formulation can be of real interest in many engineering applications. Precisely, the recent mathematical results obtained for the high-frequency asymptotic of hyperbolic partial differential equations are used to ease the derivation. Some developments are introduced to extend the classic transport theory for bounded elastoacoustic systems with damping. It is shown that the behavior of high-frequency elastoacoustic energy, being diffusive or wavefront, depends on the correlation lengths of the inhomogeneities of media. The proposed energy equation is mathematically different from the equation applied in the conventional approaches, which are used extensively in recent study of time-varying energy in medium- and high-frequency domains, such as transient statistical energy analysis (SEA) or the time-varying conductivity approach.     

Stability analysis of constraints in flexible multibody systems dynamics

Automated algorithms for the dynamic analysis and simulation of constrained multibody systems assume that the constraint equations are linearly independent. During the motion, when the system is at a singular configuration, the constraint Jacobian matrix possesses less than full rank and hence it results in singularities. This occurs when the direction of a constraint coincides with the direction of the lost degree of freedom. In this paper the constraint equations for deformable bodies are modified for use in the neighborhood of the singular configuration to yield the system inertia matrix which is nonsingular and also to take the actual generalized constraint forces into account. The procedures developed are applicable to both the augmented approach and the coordinate reduction methods. For the modeling of the constrained flexible multibody systems, a general recursive formulation is developed using Kane's equations, finite element method and modal analysis techniques. The system may contain revolute, prismatic, spherical or other types of joints, as well as geometrical nonlinearities; the rotary inertia is also automatically included. Simulation of a two-link flexible manipulator is presented at a singular configuration to demonstrate the utility of the method.     

A quasi-one-dimensional asymptotic theory for non-linear water waves

Quasi-one-dimensional generalizations of different forms of the one-dimensional Boussinesq equations are derived asymptotically, then, from these quasi-one-dimentional Boussineq equations, a consistent and significant second-order KP equation is derived, according to the Kadomtsev-Petviashvili [1] limiting process, the asymptotic expansions in the derivation of the non-linear Schrödinger-Poisson (NLSP) system of two equations, obtained by Freeman and Davey [2], in the long-water-waves limit are also determined.Finally, I elucidate the influence of a bottom topography on the Boussinesq and KP equations.     

The Investigation of the Fractional-View Dynamics of Helmholtz Equations Within Caputo Operator

It is eminent that partial differential equations are extensively meaningful in physics, mathematics and engineering. Natural phenomena are formulated with partial differential equations and are solved analytically or numerically to interrogate the system’s dynamical behavior. In the present research, mathematical modeling is extended and the modeling solutions Helmholtz equations are discussed in the fractional view of derivatives. First, the Helmholtz equations are presented in Caputo’s fractional derivative. Then Natural transformation, along with the decomposition method, is used to attain the series form solutions of the suggested problems. For justification of the proposed technique, it is applied to several numerical examples. The graphical representation of the solutions shows that the suggested technique is an accurate and effective technique with a high convergence rate than other methods. The less calculation and higher rate of convergence have confirmed the present technique’s reliability and applicability to solve partial differential equations and their systems in a fractional framework.     

悬挂式层间减震结构地震反应的比较分析

在符拉索夫开口簿壁杆件理论平面假定的基础上,本文针对核筒,采用三次B样条有限条方法构造连续化模型,并分别计算其动、势能。针对悬挂部分,分别应用杆系有限元和悬吊体系理论,计算其动、势能。然后,由拉格朗日公式,得到悬挂结构体系的动力方程,并用wilson-θ法求解。最后,结合算例得到了具有实际意义的结论。     

A direct violation correction method in numerical simulation of constrained multibody systems

 A new direct violation correction method for constrained multibody systems is presented. It can correct the value of state variables of the systems directly so as to satisfy the constraint equations of motion. During the integration of the dynamic equations of constrained multibody systems, this method can efficiently control the violations of constraint equations within any given accuracy at each time-step. Compared to conventional indirect methods, especially Baumgarte's Constraint Violation Stabilization Method, this method has clear physical meaning, less calculation and obvious correction effect. Besides, this method has minor effect on the form of the dynamic equations of systems, so it is stable and highly accurate. A numerical example is provided to demonstrate the effectiveness of this method. Received: 17 December 1999     

A mathematical model for an upper convected Maxwell fluid with an elastic core: Study of a limiting case

In this paper we formulate a mathematical model for a continuum which behaves like an upper convected visco-elastic Maxwell fluid if the stress is above a certain threshold and like a neo-Hookean elastic solid if the stress is below that threshold. The constitutive equations for each phase are derived within the context of the theory of natural configurations and by means of the criterion of the maximization of the rate of dissipation [11]. We then focus on a limiting case in which the continuum becomes an elastic-rigid body. In this limiting case the constitutive relation of the material becomes implicit and, although there is no energy dissipation, it cannot be included in the class of hyperelastic (or Green) bodies. The stress indeed cannot be expressed as a function of the strain. This class of materials was first introduced by Rajagopal in [15] and is the subject of the forthcoming papers [3] and [4].