Numerical simulation of nonholonomic rigid-body systems

The paper proposes computer algebra system (CAS) algorithms for computer-assisted derivation of the equations of motion for systems of rigid bodies with holonomic and nonholonomic constraints that are linear with respect to the generalized velocities. The main advantages of using the D’Alembert-Lagrange principle for the CSA-based derivation of the equations of motion for nonholonomic systems of rigid bodies are demonstrated. Among them are universality, algorithmizability, computational efficiency, and simplicity of deriving equations for holonomic and nonholonomic systems in terms of generalized coordinates or pseudo-velocities __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 9, pp. 106–115, September 2006.     

A Coordinate Reduction Technique for Dynamic Analysis of Spatial Substructures with Large Angular Rotations∗

A substructuring technique is presented for transient dynamic analysis of systems composed of interconnected rigid and elastic bodies that undergo large angular displacements. Displacement of elastic bodies is represented by superposition of local linear elastic deformation on large displacement of body reference coordinate systems. Elastic bodies are thus represented by combined sets of reference and local elastic generalized coordinates. Modal analysis and substructuring of individual elastic components allow for elimination of insignificant modes. Equations of motion and constraint are formulated in terms of mixed sets of modal and reference generalized coordinates. Planar and spatial linkages with flexible elements are presented to illustrate use of the method developed.     

Dynamic Analysis of Planar Linkages: A Recursive Approach

In the present study, the equations of motion for generalized planar linkages that consist of a system of rigid bodies with all common types of kinematic joints are derived using a recursive approach. The system of rigid bodies is replaced by a dynamically equivalent constrained system of particles. Then for the resulting equivalent system of particles, the concepts of linear and angular momentums are used to generate the equations of motion without either introducing any rotational coordinates or distributing the external forces and force couples over the particles. For the open loop case, the equations of motion are generated recursively along the open chains. For the closed loop case, the system is transformed to open loops by cutting suitable kinematic joints and introducing cut-joints kinematic constraints. An example of a multi-branch closed-loop system is chosen to demonstrate the generality and simplicity of the proposed method.     

非连续变形计算力学模型的粘弹性分析方法Ⅰ:基本理论

基于粘弹性广义有限单元和接触力元，发展了适用于多体相互作用系统非连续变形分析的粘弹性数值分析方法，通过虚功原理，给出了其分区参变量最小势能原理，从而阐明了其理论基础。粘弹性广义有限单元的本构关系可由粘弹性退化为弹性或刚性，因此本文所提出的方法可对由刚体、弹性体和粘弹性体所构成的复杂多体系统在外荷载作用下的力学行为进行数值模拟，同时能够比本文精确地直接得到多体之间的接触应力。     

Dynamics of a Two-Pendulum Model with Impact Interaction and an Elastic Support

This paper examines the dynamic behavior of a double pendulummodel with impact interaction. One of the masses of the two pendulumsmay experience impacts against absolutely rigid container wallssupported by an elastic system forming an inverted pendulum restrainedby a torsional elastic spring. The system equations of motion arewritten in terms of a non-smooth set of coordinates proposed originallyby Zhuravlev. The advantage of non-smooth coordinates is that theyeliminate impact constraints. In terms of the new coordinates, thepotential energy field takes a cell-wise non-local structure, and theimpact events are treated geometrically as a crossing of boundariesbetween the cells. Based on a geometrical treatment of the problem,essential physical system parameters are established. It is found thatunder resonance parametric conditions of the linear normal modes thesystem's response can be either bounded or unbounded, depending on thesystem's parameters. The ability of the system to absorb energy from anexternal source essentially depends on the modal inclination angle,which is related to the principal coordinates.     

Chain of Physically Related Independent Mechanical Oscillators

A lumped-parameter mechanical system consisting of a chain of physically related rigid bodies, each of which has one rotational degree of freedom, is considered. It is shown that the inertialess elastic elements connecting the absolutely rigid bodies of the chain can be chosen so that the mechanical structure acquires the properties of a so-called absolute mechanical filter. The motion of any inertial element of this system is described by the equation of a classical harmonic oscillator with one degree of freedom. Using the system considered as an example, it is shown that there is a relationship between the models of classical and quantum mechanics. From the positions of modern classical mechanics, this lumped-parameter system confirms the well-known Einstein’s hypothesis theoretical physics that a rigid body is a system of independent oscillators.     

GAUSS PRINCIPLE OF LEAST CONSTRAINT IN GENERALIZED COORDINATES AND ITS GENERALIZATION

当采用广义坐标描述系统的运动时,相比质点形式的高斯最小拘束原理,通过广义坐标形式的高斯最小拘束原理来建立动力学优化模型,计算效率更高. 从高斯原理的变分形式出发推导了广义坐标形式的高斯最小拘束原理,并研究了非理想约束、单边约束及刚体碰撞情形下的高斯最小拘束原理的形式. 研究认为:对刚体碰撞情形下,高斯最小拘束原理不能取代碰撞恢复定律,碰撞恢复定律以碰撞后广义速度的约束方程形式起作用.     

Optimization of Vehicle Suspension Systems for Improved Comfort of Road Vehicles Using Flexible Multibody Dynamics

Methods that account for the flexibility of multibody systems extend the range of applications to areas such as flexible robots, precision machinery, vehicle dynamics or space satellites. The method proposed here for flexible multibody models allows for the representation of complex-shaped bodies using general finite-element discretizations which deform during the dynamic loading of the system, while the gross rigid body motion of these bodies is still captured using fixed-body coordinate frames. Components of the system for which the deformations are relatively unimportant are represented with rigid bodies. This method is applied to a road vehicle where flexibility plays an important role in its ride and handling dynamic behavior. Therefore, for the study of the limit behavior of the vehicles, the use of flexible multibody models is of high importance. The design process of these vehicles, very often based on intuition and experience, can be greatly enhanced through the use of generalized optimization techniques concurrently with multibody codes. The use of sparse matrix system solvers and modal superposition, to reduce the number of flexible coordinates, in a computer simulation, assures a fast and reliable analysis tool for the optimization process. The optimum design of the vehicle is achieved through the use of an optimization algorithm with finite-differencesensitivities, where the characteristics of the vehicle components are the design variables on which appropriate constraints are imposed. The ride optimization is achieved by finding the optimum of a ride index that results from a metric that accounts for the acceleration in several key points in the vehicle properly weighted in face of their importance for the comfort of the occupant. Simulations with different road profiles are performed for different speeds to account for diverse ride situations. The results are presented and discussed in view of the different methods usedwith emphasis on models and algorithms.     

柔性机械臂动力学方程单向递推组集方法

本文基于Jourdain变分原理提出一种柔性机械臂动力学方程的单向递推组集方法。用规则标号法描述系统中物体和铰的邻接关系;用铰相对坐标和模态坐标分别描述物体的大位移运动和弹性变形。文末以三连杆机器人操作手为例说明本文建模的过程。     

含刚性体可展开结构的展开过程动力学分析方法

在可展结构中设置刚性体可以有效提高展开效率,保证形状精度,因此含刚性体可展结构具有较高的实用价值。刚性体按需要可为刚性曲板等,一般形状复杂且形式多样,含任意形状刚性体可展结构的展开过程动力学分析比较困难。本文以笛卡尔坐标系下节点坐标及位移为变量,利用广义逆矩阵建立了一种通用的刚性体动力学方程,给出了利用Newmark—...     

完全笛卡尔坐标描述的多体系统动力学

用完全笛卡尔坐标描述多体系统的运动学和动力学在提高计算效率方面有突出优点．导出用完全笛卡尔坐标表示的刚体及多体系统的动量和动量矩的解析式，给出与之对应的广义惯量矩阵概念，建立无力矩状态下用完全笛卡尔坐标描述的多体系统动力学的一阶微分方程组，用于多体航天器的姿态运动分析     

On the numerical solution of tracked vehicle dynamic equations

In this investigation, the solution of the nonlinear dynamic equations of the multibody tracked vehicle systems are obtained using different procedures. In the first technique, which is based on the augmented formulation that employes the absolute Cartesian coordinates and Lagrange multipliers, the generalized coordinate partitioning of the constraint Jacobian matrix is used to determine the independent coordinates and the associated independent differential equations. An iterative Newton-Raphson algorithm is used to solve the nonlinear constraint equations for the dependent variables. The numerical problems encountered when one set of independent coordinates is used during the simulation of large scale tracked vehicle systems are demonstrated and their relationship to the track dynamics is discussed. The second approach employed in this investigation is the velocity transformation technique. One of the versions of this technique is discussed in this paper and the numerical problems that arise from the use of inconsistent system of kinematic equations are reported. In the velocity transformation technique, the tracked vehicle system is assumed to consist of two kinematically decoupled subsystems; the first subsystem consists of the chassis, the rollers, the sprocket and the idler, while the second subsystem consists of the track which is represented as a closed kinematic chain that consists of rigid links connected by revolute joints. It is demonstrated that the use of one set of recursive equations leads to numerical difficulties because of the change in the track configuration. Singular configurations can be avoided by repeated changes in the recursive equations. The sensitivity of the predictor-corrector multistep numerical integration schemes to the method of formulating the state equations is demonstrated. The numerical results presented in this investigation are obtained using a planner tracked vehicle model that consists of fifty four rigid bodies.     

多刚体系统分离策略及释放动力学研究

紧密连接的多刚体系统可在脱离运载航天器后在轨自主分离,无需多次利用航天器发射装置或在航天器中安装多个发射装置进行分离释放,从而有效提高运载航天器空间利用率, 简化分离释放操作和降低碰撞风险.本文针对多刚体系统的在轨分离释放问题, 研究在轨分离策略及释放过程动力学.首先, 考虑刚体相对运动及姿态变化,基于虚功原理及自然坐标方法建立单个刚体的动力学模型.考虑多刚体系统在轨分离释放阶段的轨道运动和连接约束变化,计入分离时刚体间的相互作用,利用拉格朗日乘子法获得含连接约束的非线性动力学模型. 考虑到实际工程应用,在多刚体系统分离释放阶段,通过安装在刚体间每个接触表面4个角上的弹射装置实现自主分离. 其次,为保证分离过程中刚体之间无碰撞发生, 规划了多刚体系统的分离时序,并基于不同弹射方向及分离顺序设计了两种分离释放方案. 最后,通过算例研究分析了在轨分离释放过程中刚体的非线性动力学行为,验证了分离释放方案的有效性.      

Unilaterality and Dry Friction: A Geometric Formulation for Two-Dimensional Rigid Body Dynamics

The dynamics of a rigid body simply supported on a moving rigid ground in the presence of dry friction is investigated. Rigid body kinematics, in terms of generalized coordinates, and features of contact are discussed. According to the contact laws, a variational formulation is adopted to describe the dynamics and to analyze, in the case of contact, the connection between dynamically possible motions and actual evolution of the system. A geometric method is then proposed which allows the dynamic evolution to be determined without any direct evaluation of the contact forces. Though different situations are possible, depending on the instantaneous values of relative position, velocity and active forces, a unique solution is identified. Examples illustrating applications of the theory are presented.     

Motion of a Sphere in a Liquid Caused by Vibrations of Another Sphere

The motion of an absolutely rigid sphere in a nonuniformly vibrating, ideal, incompressible liquid is considered. The liquid vibrates under the action of an absolutely rigid sphere vibrating in a specified manner. Refined conditions are obtained under which the inclusion sphere recedes from the vibrating sphere, approaches it, and does not perform mean motion. It is found that under nonuniform vibrations of the liquid, the motion of the inclusion can depend on the geometrical parameters of the hydromechanical system.     

An Augmented Formulation for Mechanical Systems with Non-Generalized Coordinates: Application to Rigid Body Contact Problems

In computational multibody algorithms, the kinematic constraintequations that describe mechanical joints and specified motiontrajectories must be satisfied at the position, velocity andacceleration levels. For most commonly used constraint equations, onlyfirst and second partial derivatives of position vectors with respect tothe generalized coordinates are required in order to define theconstraint Jacobian matrix and the first and second derivatives of theconstraints with respect to time. When the kinematic and dynamicequations of the multibody systems are formulated in terms of a mixedset of generalized and non-generalized coordinates, higher partialderivatives with respect to these non-generalized coordinates arerequired, and the neglect of these derivatives can lead to significanterrors. In this paper, the implementation of a contact model in generalmultibody algorithms is presented as an example of mechanical systemswith non-generalized coordinates. The kinematic equations that describethe contact between two surfaces of two bodies in the multibody systemare formulated in terms of the system generalized coordinates and thesurface parameters. Each contact surface is defined using twoindependent parameters that completely define the tangent and normalvectors at an arbitrary point on the body surface. In the contact modeldeveloped in this study, the points of contact are searched for on lineduring the dynamic simulation by solving the nonlinear differential andalgebraic equations of the constrained multibody system. It isdemonstrated in this paper that in the case of a point contact andregular surfaces, there is only one independent generalized contactconstraint force despite the fact that five constraint equations areused to enforce the contact conditions.     

A Pressure Field in an Acoustic Medium Containing a Cylindrical Solid and a Sphere Oscillating in a Predetermined Manner

A problem on the interaction of a spherical body oscillating in a predetermined fashion and a rigid cylinder is formulated. The bodies do not intersect, are immersed into an ideal compressible liquid, and their centers are in one plane. The solution is based on the possibility of representing the partial solution of the Helmholtz equation, written in cylindrical coordinates, in terms of partial solutions in spherical coordinates, and vice versa. An infinite system of linear algebraic equations is obtained by satisfying the boundary conditions on the sphere and cylinder surfaces. The system is intended for determining the coefficients of the expansion of the velocity potential into a series in terms of spherical and trigonometric functions. The system obtained is solved by the reduction method. The appropriateness of this method is substantiated. The hydrodynamic characteristics of the liquid surrounding the spherical and cylindrical bodies are determined. A comparison is made with the problem on a sphere oscillating in an infinite incompressible liquid that contains also a cylinder and in a compressible liquid that contains nothing more. Two types of motion of the sphere — pulsation and oscillation — are considered