Simulation of friction and stiction in multibody dynamics model problems

A continuous model of Coulomb friction is used with a tangent space formulation of differential algebraic equations of motion for simulation of multibody dynamic model problems. Characteristics of the model problems studied are similar to those encountered in broad classes of multibody systems, without the associated geometric and analytical complexities. An implicit trapezoidal numerical solution algorithm is used to simulate dynamic response that includes the onset of stiction, its progression, and its termination, avoiding stiff behavior that is reported in the literature when index 3 formulations are used. Analytical criteria for stiction are derived for a three mass Coulomb friction model problem that defines the onset of and departure from stiction events with redundant equations of constraint. The tangent space formulation with implicit trapezoidal integration is applied to this analytical model to compute dynamic response, determine ranges of constraint forces that may occur during periods of stiction, and demonstrate that dynamic response is a discontinuous function of model parameters when stiction occurs. Accuracy of the continuous model of Coulomb friction is established, through comparison of results with those of the analytical model. Cartesian coordinate models of higher dimension are presented for three and four mass model problems that encounter a higher degree of redundancy in constraints during periods of stiction. Simulation of the Cartesian coordinate models, which have characteristics similar to more general multibody systems, yields accurate solutions, without any indication of stiffness in the tangent space equations of motion. Methods successfully demonstrated in model problems provide a foundation for simulation of spatial multibody dynamic systems with friction.     

Finite and impulsive motion of constrained mechanical systems via Jourdain's principle: discrete and hybrid parameter models

The main purpose of this paper is to present a unified analytical dynamics framework for the analysis of finite and impulsive motion of mechanical systems using Jourdain's principle. Emphasis is given to the general case when a mechanical system is described by a hybrid (discrete-distributed) parameter model. A large group of finite and impulsive, generally non-holonomic, constraints are analysed in detail and a so-called extended Appellian classification is presented for these constrained motion problems. The fundamental dynamic equation of constrained systems is developed in terms of velocity variations (Jourdain's principle). Based on this equation and the constraints, the methods of quasivelocities and Lagrangian multipliers are adopted and interpreted for the finite motion of hybrid parameter models of mechanical systems; and the methods of independent quasivelocity variations and Lagrangian multipliers are introduced for the analysis of impulsive motion of such models. To illustrate the proposed material, an example of a one-link flexible arm intercepting and capturing a moving target is considered.     

TURBULENT COHERENT STRUCTURE AND DYNAMIC SYSTEM

The main difficulties in the study of turbulence via dynamic system lie in how to relatecontinuum systems of infinite dimension with dynamic system in low dimension space andhow to depict its special structure.In this paper,we’ll give a comprehensive review onvarious methods to describe complex systems in low dimension space and new approaches tothe resolution of turbulence problems.     

复合柔性结构全局模态函数提取与状态空间模型构建

随着航空航天等领域中实际工程结构的大型化和柔性化,结构的非线性振动和主动振动控制问题越来越凸显.分析和处理此类结构出现的复杂振动问题的关键在于建立系统的非线性动力学模型与状态空间模型.对于由柔性部件、刚体、连接部件构成的复合柔性结构,由于各部件之间的振动耦合效应,单个柔性部件在悬臂、简支和自由等静定边界下的模态与结构的真实模态有较大差异.为此,本文提出复合柔性结构全局模态的解析提取方法,通过全局模态离散得到系统非线性动力学模型,从而构建状态空间模型.该方法采用笛卡尔坐标描述系统的运动,建立系统的运动方程;结合描述柔性部件的偏微分方程、刚体的常微分运动方程、连接界面处力、力矩、位移和转角的匹配条件以及系统的边界条件,利用分离变量法给出统一形式的频率方程,获取系统的固有频率和解析函数表征的全局模态.这里提出的全局模态提取方法不仅便于复合柔性结构固有频率和全局模态的参数化分析,而且为建立复合柔性结构低维非线性动力学模型和状态空间模型提供了有效的途径,对于推进这类结构的非线性动力学分析与主动振动控制研究具有重要意义.      

DYNAMIC ANALYSIS OF FLEXIBLE BODY WITH DEFINITE MOVING ATTITUTE

The nonlinear dynamic control equation of a flexible multi-body system with definite moving attitude is discussed.The motion of the aircraft in space is regarded as known and the influence of the flexible structural members in the aircraft on the motion and attitude of the aircraft is analyzed.By means of a hypothetical mode,the defor- mation of flexible members is regarded as composed of the line element vibration in the axial direction of rectangular coordinates in space.According to Kane’s method in dy- namics,a dynamic equation is established,which contains the structural stiffness matrix that represents the elastic deformation and the geometric stiffness matrix that represents the nonlinear deformation of the deformed body.Through simplification the dynamic equation of the influence of the planar flexible body with a windsurfboard structure on the spacecraft motion is obtained.The numerical solution for this kind of equation can be realized by a computer.     

张拉整体结构的动力学等效建模与实验验证

为了实现张拉整体结构高效动力学计算, 并考虑其大范围运动中柔性杆局部动态屈曲, 提出了一种受压细长杆动力学降阶模型, 采用五节点弹/扭簧集中质量离散模型等效连续杆的静力学和动力学特性. 首先, 通过静力学等效分析推导了弹簧拉压刚度和扭簧弯曲刚度表达式, 可准确预测杆件受压屈曲和近似预测其后屈曲行为. 第二, 通过动能等效分析推导了集中质量表达式, 可准确预测杆在线速度场下的运动. 第三, 通过弯曲振动固有模态等效分析确定弯曲刚度和节点质量的分布参数, 合适的分布参数取值组合可将降阶模型前两阶固有频率相对误差均降低至1%以内. 第四, 在全局坐标系下建立张拉整体结构瞬态动力学方程, 并利用静力凝聚法实现方程高效迭代求解. 最后, 分别对球形张拉整体结构准静态压缩、模态分析和碰撞动力学进行仿真和实验对比分析, 证明了提出的动力学降阶模型可有效预测张拉整体结构的静力学行为、固有振动特性及瞬态动力学响应, 并分析了结构参数变化对其力学特性的影响规律. 本文提出的动力学等效建模与计算方法, 可望用于软着陆行星探测器、大型可展开空间结构及点阵材料等复杂张拉整体系统的动力学分析与控制.      

漂浮基双臂空间机器人姿态与末端抓手惯性空间轨迹协调运动的模糊滑模控制

本文讨论了载体姿态受控、位置不受控制的双臂空间机器人系统的控制问题.利用拉格朗日方法并结合系统动量守恒关系,建立了双臂空间机器人系统的非线性系统动力学模型.以此为基础,考虑到空间机器人系统结构的复杂性及其某些参数的变动性,根据具有较强鲁棒性的变结构控制理论,设计了双臂空间机器人载体姿态与两机械臂末端抓手惯性空间轨迹协调运动的滑模变结构控制方案.为了克服滑模变结构控制器抖振的缺点,附加设计了一个模糊控制器,以便根据系统的输出来动态调节滑模变结构控制器等速趋近率的系数,从而既确保了系统的快速响应而又消除了原有的抖振.系统数值仿真,证明了上述控制方案良好的控制效果.     

Cyber-physical flexible wing for aeroelastic investigations of stall and classical flutter

The fluid–structure interactions of a finite aspect ratio, cantilevered, flexible wing were investigated using a cyber-physical system to virtually augment the torsional dynamics of the wing. Cyber-physical systems (CPS), which have recently been pursued by a small number of research groups, have proven to be a very useful mechanism to interrogate the fluid–structure interaction parameter space. The premise of a CPS is to use dynamic feedback control to make a system behave according to desired equations of motion. Systems are composed of embedded hardware and software coupled with real-time computing to give the user the flexibility to quickly explore a range of structural parameters. With the advancement of modern control theory, robotics and embedded systems, CPSs integrate both simulation and physical properties into a smart structure which can be used to push the boundaries of research investigations.The CPS in this work allows for the investigation of dynamic aeroelastic instabilities of a three-dimensional, flexible, rectangular planform wing. Two dynamic instability regimes are observed: first, stall flutter, in which the torsional or pitch mode is excited through the dynamic stall process, and second, coupled (or classical) flutter, in which the pitch mode couples with the bending mode. By varying the torsional stiffness and therefore the frequency of torsional versus bending oscillations of the wing, both of these regimes can be attained at the same aerodynamic conditions using the CPS.     

Dynamic Bifurcation of Multibody Systems

In this paper, the process of loss of stability of multibody systems and structures is analyzed. A novel approach is presented and applied to the statically loaded spatial systems for the analysis of a dynamic response of systems imposed on impact, high velocity compulsive motion, or percussive forces. The analysis is based on the solution of the dynamic equations and eigenvalue problem of systems, and of the resultant motion simulation. The flexible systems are discretized using the finite element method. The dynamic equations are derived with respect to the relative coordinates of the finite elements. Large flexible deflections due to a loss of stability are simulated. The initial forms of the possible deformations are defined by the computed eigenvectors solving the eigenvalue problem for the system stiffness matrix. The critical forces and system deflections are then analyzed. Examples of bifurcation of beam and beam structure imposed on compulsive motion, percussive forces, and impact are presented.     

大型柔性航天器动力学与振动控制研究进展

随着航天重大工程的逐步实施,航天器正朝着超高速、超大尺度、多功能的方向发展,其面临的发射和运行环境也更加恶劣.航天器发射过程中的振动及其主/被动控制、在轨运行中大型柔性航天器动力学建模与动态响应分析、结构振动与飞行器姿态的混合控制等问题越来越复杂且难于处理;航天器结构的大型化和柔性化(如大阵面天线和太阳翼等)也对其地面试验和半实物仿真提出了挑战.本文着重介绍大型柔性航天器涉及到的动力学与振动控制问题,包括航天器发射过程中的整星隔振,大型柔性结构动力学建模与振动响应分析,大型柔性航天器的结构振动与姿轨控耦合动力学及其混合控制等.提炼出航天动力学与控制领域中亟待解决的若干基础科学问题,包括:多刚柔体系统动力学建模与模型降阶(涉及大变形柔性体动力学建模、多求解器合作仿真、模型降阶、组合结构动力学建模的解析方法等);复杂结构状态空间模型构建方法与能控性(涉及状态空间模型构建的理论与实验方法、复杂结构振动控制系统的能观性与能控性等);航天器姿态运动与大型柔性结构振动的混合控制律设计(涉及姿态机动与结构振动的鲁棒混合控制、执行机构与压电控制器的协同控制等).      

Modeling the motion of connected bodies

Spatial motion of mechanical systems consisting of jointed rigid bodies is considered. The methods previously developed to solve the dynamic equations of “carrier + loads” systems analytically for the accelerations of individual bodies are used to obtain the equations of motion of systems with complicated branched structure. The following practically important cases are analyzed: (i) a central carrier with peripheral loaded carriers and (ii) a chain and a ring consisting of loaded carriers. Numerical results are given for the stress-strain state of an elastic spacecraft represented as a chain of rigid bodies connected by elastic joints.     

Dynamics modeling and stability of robotic systems with discrete-time force control

This paper investigates the dynamic behavior of robotic echanical systems with discrete-time force control. Force control is associated with the constrained motion of a mechanical system. A novel approach is presented to analyze the stability and performance based on the separation of constrained and admissible motions. This results in a model representing the dynamics of the constrained motion of the system. The analysis connects the complex nonlinear model of a mechanical system to a set of abstract delayed oscillators. These oscillator models make it possible to perform a detailed closed-form mathematical analysis of the stability behavior. A planar two-degree-of-freedom (DoF) mechanism is presented as an example to illustrate the material. Results are illustrated by stability charts in the parameter space of mechanical parameters, control gains and the sampling rate.     

Dynamic stability analysis of finite element modeling of piezoelectric composite plates

The dynamic stability of negative-velocity feedback control of piezoelectric composite plates using a finite element model is investigated. Lyapunov’s energy functional based on the derived general governing equations of motion with active damping is used to carry out the stability analysis, where it is shown that the active damping matrix must be positive semi-definite to guarantee the dynamic stability. Through this formulation, it is found that imperfect collocation of piezoelectric sensor/actuator pairs is not sufficient for dynamic stability in general and that ignoring the in-plane displacements of the midplane of the composite plate with imperfectly collocated piezoelectric sensor/actuator pairs may cause significant numerical errors, leading to incorrect stability conclusions. This can be further confirmed by examining the complex eigenvalues of the transformed linear first-order state space equations of motion. To overcome the drawback of finding all the complex eigenvalues for large systems, a stable state feedback law that satisfies the second Lyapunov’s stability criteria strictly is proposed. Numerical results based on a cantilevered piezoelectric composite plate show that the feedback control system with an imperfectly collocated PZT sensor/actuator pair is unstable, but asymptotic stability can be achieved by either bonding the PZT sensor/actuator pair together or changing the ply stacking sequence of the composite substrate to be symmetric. The performance of the proposed stable controller is also demonstrated. The presented stability analysis is of practical importance for effective design of asymptotically stable control systems as well as for choosing an appropriate finite element model to accurately predict the dynamic response of smart piezoelectric composite plates.     

The pendulum-slosh problem: Simulation using a time-dependent conformal mapping

Suspending a rectangular vessel which is partially filled with fluid from a single rigid pivoting pole produces an interesting theoretical model with which to investigate the dynamic coupling between fluid motion and vessel rotation. The exact equations for this coupled system are derived with the fluid motion governed by the Euler equations relative to the moving frame of the vessel, and the vessel motion governed by a modified forced pendulum equation. The nonlinear equations of motion for the fluid are solved numerically via a time-dependent conformal mapping, which maps the physical domain to a rectangle in the computational domain with a time dependent conformal modulus. The numerical scheme expresses the implicit free-surface boundary conditions as two explicit partial differential equations which are then solved via a pseudo-spectral method in space. The coupled system is integrated in time with a fourth-order Runge–Kutta method. The starting point for the simulations is the linear neutral stability contour discovered by Turner et al. (2015, Journal of Fluid & Structures 52, 166–180). Near the contour the nonlinear results confirm the instability boundary, and far from the neutral curve (parameterized by longer pole lengths) nonlinearity is found to significantly alter the vessel response. Results are also presented for an initial condition given by a superposition of two sloshing modes with approximately the same frequency from the linear characteristic equation. In this case the fluid initial conditions generate large nonlinear vessel motions, which may have implications for systems designed to oscillate in a confined space or on the slosh-induced-rolling of a ship.     

Hamiltonian Structure and Dynamics of a Neutrally Buoyant Rigid Sphere Interacting with Thin Vortex Rings

In a previous paper, we presented a (noncanonical) Hamiltonian model for the dynamic interaction of a neutrally buoyant rigid body of arbitrary smooth shape with N closed vortex filaments of arbitrary smooth shape, modeled as curves, in an infinite ideal fluid in \mathbbR³\mathbb{R}^3. The setting of that paper was quite general, and the model abstract enough to make explicit conclusions regarding the dynamic behavior of such systems difficult to draw. In the present paper, we examine a restricted class of such systems for which the governing equations can be realized concretely and the dynamics examined computationally. We focus, in particular, on the case in which the body is a smooth sphere. The equations of motion and Hamiltonian structure of this dynamic system, which follow from the general model, are presented. Following this, we impose the constraint of axisymmetry on the entire system and look at the case in which the rings are all circles perpendicular to a common axis of symmetry passing through the center of the sphere. This axisymmetric model, in our idealized framework, is governed by ordinary differential equations and is, relatively speaking, easily integrated numerically. Finally, we present some plots of dynamic orbits of the axisymmetric system.     

空间机器人双臂捕获卫星力学分析及镇定控制

随着航天技术的发展,空间机器人要求具有对非合作卫星的在轨捕获能力. 双臂空间机器人与单臂空间机器人相比在这方面显然更具有优势. 然而由于太空环境的复杂性,使得空间机器人双臂捕获非合作卫星操作过程的动力学与控制问题表现出下述特点:非完整动力学约束,动量、动量矩与能量传递变化,捕获前后结构开、闭环变拓扑,与闭环接触几何、运动学约束多者共存. 因此空间机器人双臂捕获卫星技术相关动力学与控制问题变得极其复杂. 为此,讨论了双臂空间机器人捕获自旋卫星过程的动力学演化模拟,以及捕获操作后其不稳定闭链混合体系统的镇定控制问题. 首先,利用拉格朗日第二类方程建立了捕获操作前双臂空间机器人的开环系统动力学模型,利用牛顿-欧拉法建立了目标卫星的系统动力学模型;在此基础上基于动量守恒定律、力的传递规律,经过积分与简化处理分析、求解了双臂空间机器人捕获目标卫星后受到的碰撞冲击效应,给出了合适的捕获操作策略. 根据闭链系统的闭环约束几何及运动学关系获得了闭合链约束方程,推导了捕获操作后闭链混合体系统的动力学方程. 最后基于该动力学方程针对捕获操作结束后失稳的闭链混合体系统,设计了镇定运动模糊H_∞ 控制方案. 提出的方案利用模糊逻辑环节克服参数不确定影响,由H_∞ 鲁棒控制项消除逼近误差来保证系统控制精度;通过最小权值范数法分配各臂关节力矩,以保证两臂协同操作. 李雅普诺夫稳定性理论证明了系统的全局稳定性. 最后通过数值仿真实验模拟、分析了碰撞冲击响应,并验证了上述镇定运动控制方案的有效性.      

Experimental Observation of Chaotic Motion in a Rotor with Rubbing

This paper presents an application for chaotic motion identification in a measured signal obtained in an experiment. The method of state space reconstruction with delay co-ordinates with the dynamic evolution described by a map is used. Poincaré diagrams, correlation dimensions and Lyapunov exponents are obtained as tools for deciding about the existence of chaotic behaviour. The method is applied to measurements of the lateral displacement of a vertical rotor experiencing rubbing and in some signals chaos is observed. The work concludes that the possibility of chaotic motion is well determined with the observation of Poincaré diagrams and computation of Lyapunov exponents. Correlation dimensions computations, strongly influenced by noise, are not convenient tools for investigation of chaotic behaviour in signals generated by mechanical systems.     

刚-柔耦合动力学系统的建模理论研究

刚－柔耦合动力学系统的传统的混合坐标方法是零次近似方法,在建模过程中,直接套用的结构动力学的小变形假设,忽略了变形位移的高次耦合变形量．本文对柔性梁建立较零次近似更精确的高次耦合动力学模型,从连续介质力学理论出发,在变形位移中,计及横向位移引起的轴向缩短,导出变形位移的二次耦合量．用一致质量有限元方法对梁进行离散,基于Jourdain速度变分原理导出大范围运动为自由的柔性梁的刚－柔耦合动力学方程．计算了柔性重力摆的角速度和摆端点的横向变形,揭示零次近似模型和耦合模型的刚－柔耦合动力学性质的根本差异．     

空间网格结构动力分析的非线性模态方法

空间网格结构因自由度数多且无简化的力学模型,非线性动力分析通常要耗费大量时间．传统的非线性模态方法用于求解多高层结构的局部非线性问题已获得良好的效果,但对系统非线性问题的应用尚缺少研究．对比分析多高层结构和空间网格结构动力性能差异,指出网格结构动力非线性分析存在的问题．以主振型理论和切线刚度分离法为基础,将非线性模态方法用于几何非线性效应显著的空间网格结构动力分析．通过对运动方程的非线性恢复力进行拆分,形成线性表达形式,然后解耦到主振型所在的广义坐标系,以达到缩减自由度数量的目的．并通过实例验证非线性模态方法的高效性与适用性．