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.     

机械系统中摩擦模型的研究进展

摩擦现象在机械系统中的作用日益突出, 合理地解决机械系统中摩擦环节尤其是非线性摩擦环节的制约问题 已成为当前研究的重点. 由于摩擦的复杂性, 很难从机理上获得其准确唯一的数学模型, 迄今已提出的摩擦模 型有数十种. 鉴于目前机械系统中摩擦建模的发展状况, 首先描述了几种重要的摩擦现象, 如库仑摩擦、黏性 摩擦、Stribeck效应、预滑动摩擦、可变的静态摩擦力和摩擦记忆效应等. 其次, 系统地介绍了几种较为重要的、 常用的摩擦模型, 包括6种静态摩擦模型和7种动态摩擦模型, 并对每一种模型的构成, 特点和适用范围等 进行了较为详细地论述. 比较而言, 静态摩擦模型结构简单, 参数辨识容易, 但是无法描述摩擦的动态特性, 动态摩擦模型能够比较全面的描述摩擦现象, 但结构复杂, 参数辨识难度较大. 再次, 简要概述了摩擦建模 对机械系统动力学行为的影响, 以及在高精度定位系统的控制中的作用. 最后, 针对当前机械系统中摩擦建 模方面存在的一些不足提出了几点展望. 为今后摩擦模型的选用和新摩擦模型的建立提供了参考.     

Hysteresis modelling and chaos prediction in one- and two-DOF hysteretic models

In the present work hysteresis is simulated by means of internal variables. The analytical models of different types of hysteresis loops allow the reproduction of major and minor loops and provide a high degree of correspondence with experimental data. In models of this type adding an external periodic excitation or increasing the number of dimensions can lead to the occurrence of chaotic behaviour. Using an effective algorithm based on numerical analysis of the wandering trajectories [1–7], the evolution of the chaotic behaviour regions of oscillators with hysteresis is presented in various parametric planes. The substantial influence of a hysteretic dissipation value on the form and location of these regions, as well as the restraining and generating effects of hysteretic dissipation on the occurrence of chaos, are ascertained. Conditions for pinched hysteresis are defined. Furthermore, autonomous coupled hysteretic oscillators under sliding friction are investigated. Conditions for the occurrence of chaotic behaviour in a two-degree-of-freedom (two-DOF) hysteretic system are found in the plane of maximal static friction forces of both oscillators versus belt velocity.     

双面约束多点摩擦多体系统的建模和数值方法

提出了一种建立具有固定双面约束多点摩擦的多体系统动力学方程的方法. 用笛卡尔坐标阵描述系统的位形,根据局部方法的递推关系建立系统的约束方程,应用第一类Lagrange方程建立该系统的动力学方程,使得具有摩擦的约束面的法向力与Lagrange乘子一一对应,便于摩擦力的分析与计算,并用矩阵形式给出了摩擦力的广义力的一般表达式. 应用增广法将微分-代数方程组转化为常微分方程组,并用分块矩阵的形式给出,以便于方程的编程与计算.给出了一种改进的试算法,可提高计算效率. 最后给出了一个算例,应用试算法和RK法对算例进行了数值仿真.      

Poincare型胞映射分析方法及其应用

本文用Poincare型胞映射方法对平衡及不平．衡轴承转子非线性动力系统的全局特性进行了分析研究,同时求得了一定状态空间内系统存在的周期解及其在各不同Poincare截面上的吸引域,得到了一些新的现象和规律,并通过对平衡及不平衡轴承转子系统的全局特性异同的比较,说明了要建立既适用于平衡轴承转子系统又适用于不平衡轴承转子系统的非线性稳定性准则应注意的几个问题     

Nonlinear dynamic analysis of a structure with a friction-based seismic base isolation system

Many dynamical systems are subject to some form of non-smooth or discontinuous nonlinearity. One eminent example of such a nonlinearity is friction. This is caused by the fact that friction always opposes the direction of movement, thus changing sign when the sliding velocity changes sign. In this paper, a structure with friction-based seismic base isolation is regarded. Seismic base isolation can be employed to decouple a superstructure from the potentially hazardous surrounding ground motion. As a result, the seismic resistance of the superstructure can be improved. In this case study, the base isolation system is composed of linear laminated rubber bearings and viscous dampers and nonlinear friction elements. The nonlinear dynamic modelling of the base-isolated structure with the aid of constraint equations, is elaborated. Furthermore, the influence of the dynamic characteristics of the superstructure and the nonlinear modelling of the isolation system, on the total system’s dynamic response, is examined. Hereto, the effects of various modelling approaches are considered. Furthermore, the dynamic performance of the system is studied in both nonlinear transient and steady-state analyses. It is shown that, next to (and in correlation with) transient analyses, steady-state analyses can provide valuable insight in the discontinuous dynamic behaviour of the system. This case study illustrates the importance and development of nonlinear modelling and nonlinear analysis tools for non-smooth dynamical systems.     

树形多体系统非线性动力学的数值分析方法

研究了树形多体系统大线性动力学分析的数值方法,利用多体系统的正则方程及其线性化程,给出了多体系统Ｌｙａｐｕｎｏｖ指数和Ｐｏｉｎｃａｒｅ映射的计算方法,该算法具有较好的计算精度和通用性,既适用于说明该算法的有效性,并对该系统的动力学行为进行分析,最后用算例说明该算法的有效性,并对该系统的动力学特征（周期解、准周期解、分岔、混沌以及通往混沌的道路等）进行了分析。     

A controller for 2-DOF underactuated mechanical systems with discontinuous friction

We propose a controller for a class of 2-DOF underactuated mechanical systems with discontinuous friction in the unactuated joint. The control objective is the regulation of the unactuated variable while the position and speed of the actuated joint remain bounded. The unactuated joint is considered as a mechanical system with discontinuous friction but continuous, artificial control input given by a term depending on the actuated positions and velocities. The proposed controller guarantees the convergence of the position error of the unactuated joint to zero, and it is robust with respect to some uncertainty in the discontinuous friction coefficients. We illustrate the technique with its application to two systems.     

非光滑多体系统动力学数值算法的研究进展

非光滑多体系统动力学数值计算方法是多体系统动力学研究的重要内容之一. 本文介绍了近年来含摩擦与碰撞的非光滑多体系统动力学数值算法方面的研究进展. 首先, 讨论了库仑摩擦模型和修正的库仑摩擦模型, 以及具有单边和双边约束的多体系统中法向约束力的特点. 其次, 回顾了基于连续模型和非连续模型的多体系统动力学方程的数值计算方法, 详细介绍了基于互补概念的非光滑多体系统动力学的事件驱动法和时间步进法, 分析比较了相关的数值算法. 最后, 指出了一些需要进一步研究的问题.     

A parametrically excited pendulum with irrational nonlinearity

In this paper, we propose a parametrically excited pendulum with irrational nonlinearity which comprises a simple pendulum linked by a linear spring under base excitation. This parametric vibration system exhibits bistable state and discontinuous characteristics due to the geometry configuration. For small oscillations, this system can be described by Mathieu equation coupled with SD (Smooth and Discontinuous) oscillator whose dynamic response is examined analytically by using the averaging method in both smooth and discontinuous case. Numerical simulations are carried out to demonstrate the complicated dynamic behavior of multiple periodic motions and different types of chaotic motions.     

Discontinuous fold bifurcations in mechanical systems

Summary  This paper treats discontinuous fold bifurcations of periodic solutions of discontinuous systems. It is shown how jumps in the fundamental solution matrix lead to jumps of the Floquet multipliers of periodic solutions. A Floquet multiplier of a discontinuous system can jump through the unit circle, causing a discontinuous bifurcation. Numerical examples are treated, which show discontinuous fold bifurcations. A discontinuous fold bifurcation can connect stable branches to branches with infinitely unstable solutions. Received 20 September 2000; accepted for publication 26 June 2001     

带有摩擦耗能元件的框架结构动力分析

本文提出粘性屈服模型来模拟摩擦耗能元件的力-速度关系,该模型是连续变化的,克服了库仑摩擦模型不连续导致数值计算复杂的缺点,在进行摩擦耗能体系的动力分析中,采用缩减自由度技术,并作适当的变换,则带有摩擦耗能元件体系的动力分析归结为求解微分代数方程,本文采用增量型Rosenbrock二级三阶半隐式Runge-Kutta法求解该方程,以考虑框架和支撑的材料和几何非线性。对带有摩擦耗能元件的钢框架进行了弹塑性动力分析,研究了支撑刚度与结构层刚度的比值、摩擦力的大小以及地震波类型等参数对体系的影响。     

Nonlinear Oscillations in a System with Dry Friction

The case is examined where the right-hand side of the equations of motion is discontinuous. Attraction only in the stick domain ensures existence of periodic oscillations. Sufficient stability conditions for the periodic solution of a nonlinear system with dry friction are established__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 110–116, April 2005.     

Bifurcations in Nonlinear Discontinuous Systems

This paper treats bifurcations of periodic solutions indiscontinuous systems of the Filippov type. Furthermore, bifurcations offixed points in non-smooth continuous systems are addressed. Filippov'stheory for the definition of solutions of discontinuous systems issurveyed and jumps in fundamental solution matrices are discussed. It isshown how jumps in the fundamental solution matrix lead to jumps of theFloquet multipliers of periodic solutions. The Floquet multipliers canjump through the unit circle causing discontinuous bifurcations.Numerical examples are treated which show various discontinuousbifurcations. Also infinitely unstable periodic solutions are addressed.     

On periodic steady state response and stability of Filippov-type mechanical models

In the first part of this study, the basic steps of a methodology are presented, leading to a long time response of a class of periodically excited mechanical models with contact and dry friction. In particular, the models examined belong to the special class of Filippov-type dynamical systems, which possess continuous displacements and velocities, but exhibit discontinuities in their accelerations. The direct determination of periodic steady state response of this class of models is achieved by combining suitable numerical integration of the equations of motion with an appropriate technique yielding the corresponding monodromy matrix. This matrix, which arises from a linearization of the motion around a located periodic solution, involves saltations (jumps) and is also useful in predicting its stability properties. The analytical part is complemented by a suitable continuation procedure, enabling evaluation of complete branches of periodic motions. In the second part of the study, the effectiveness of the methodology developed is confirmed by presenting representative sets of numerical results obtained for selected examples. The first two of them are single degree of freedom oscillators. Besides investigating some interesting aspects of regular periodic response, some cases involving rich dynamics of the class of the system examined are also studied in a systematic way. The last example is a more involved and challenging model, related to the function of an engine valve and characterized by large numerical stiffness.     

Continuation of the Periodic Orbits for the Differential Equation with Discontinuous Right Hand Side

In this paper the discontinuous system with one parameter perturbation is considered. Here the unperturbed system is supposed to have at least either one periodic orbit or a limit cycle. The aim is to prove the continuation of the periodic orbits under perturbation by means of the bifurcation map and the zeroes of this map imply the periodic orbits for the perturbed system. The tools for this problem are jumps of fundamental matrix solutions and the Poincare map for discontinuous systems. Therefore, we develop the Diliberto theorem and variation lemma for the system with discontinuous right hand side. At the end, as application of our method, the effect of discontinuous damping on Van der pol equation, and the effect of small force on the discontinuous linear oscillator with add a ·sgn(x) are considered.     

Qualitative criteria in non-linear dynamic buckling and stability of autonomous dissipative systems

A general qualitative approach for dynamic buckling and stability of autonomous dissipative structural systems is comprehensively presented. Attention is focused on systems which under the same statically applied loading exhibit a limit point instability or an unstable branching point instability with a non-linear fundamental path. Using the total energy equation, the theory of point and periodic attractors of the basin of attraction of a stable equilibrium point, of local and global bifurcations, of the inset and outset manifolds of a saddle and of the geometry of the channel of motion, the stability of the fundamental equilibrium path and the mechanism of dynamic buckling are thoroughly discussed. This allows us to establish useful qualitative criteria leading to exact, approximate and upper/lower bound buckling estimates without integrating the highly non-linear initial-value problem. The individual and coupling effect of geometric and material non-linearities of damping and mass distribution on the dynamic buckling load are also examined. A comparison of the results of the above qualitative analysis with those obtained via numerical simulation is performed on several two- and three-degree-of-freedom models of engineering importance.     

Uniformly ultimate boundedness for discontinuous systems with time-delay

The uniformly ultimate boundedness of discontinuous systems with time-delay in the sense of Filippov solutions is discussed. Based on the Lyapunov-Krasovskii functional, the Lyapunov theorem for the globally strongly uniformly ultimate boundedness of retarded discontinuous systems is presented. Furthermore, the result is applied to a class of mechanical systems with a retarded discontinuous friction item.     

Modeling of unilateral contact conditions with application to aerospace systems involving backlash, freeplay and friction

This paper describes an analysis procedure for the modeling of backlash, freeplay and friction in flexible multibody systems. The first two effects are formulated in a general manner as unilateral contact conditions in multibody dynamics. The incorporation of the effects of friction in joint elements is also discussed, together with an effective computational strategy. These non-standard effects are formulated within the framework of finite element based multibody dynamics that allows the analysis of complex, flexible systems of arbitrary topology. The versatility and generality of the approach are demonstrated by presenting applications to aerospace systems: the flutter analysis of a wing-aileron system with freeplay, the impact of an articulated rotor blade on its doop stop during engagement operation in high wind conditions, and the dynamic response of a space antenna featuring joints with friction.