Numerical Analysis of Flexible Multibody Systems

Flexible multibody systems featurea coupled mathematical model with ODEs or DAEs governing the gross motion and PDEs describing the elastic deformation of particular bodies.Frequently, semidiscretization of elastic members leads to a stiff mechanical system with widely different time scales. Thispaper investigates the behavior of numerical time integration methodsat flexible multibody systems and gives some recommendations.Simulation results for a slider crank mechanism and a truck modelillustrate what can go wrong and how implicit methods like RADAU5 can beeffectively applied.     

First order sensitivity analysis of flexible multibody systems using absolute nodal coordinate formulation

Design sensitivity analysis of flexible multibody systems is important in optimizing the performance of mechanical systems. The choice of coordinates to describe the motion of multibody systems has a great influence on the efficiency and accuracy of both the dynamic and sensitivity analysis. In the flexible multibody system dynamics, both the floating frame of reference formulation (FFRF) and absolute nodal coordinate formulation (ANCF) are frequently utilized to describe flexibility, however, only the former has been used in design sensitivity analysis. In this article, ANCF, which has been recently developed and focuses on modeling of beams and plates in large deformation problems, is extended into design sensitivity analysis of flexible multibody systems. The Motion equations of a constrained flexible multibody system are expressed as a set of index-3 differential algebraic equations (DAEs), in which the element elastic forces are defined using nonlinear strain-displacement relations. Both the direct differentiation method and adjoint variable method are performed to do sensitivity analysis and the related dynamic and sensitivity equations are integrated with HHT-I3 algorithm. In this paper, a new method to deduce system sensitivity equations is proposed. With this approach, the system sensitivity equations are constructed by assembling the element sensitivity equations with the help of invariant matrices, which results in the advantage that the complex symbolic differentiation of the dynamic equations is avoided when the flexible multibody system model is changed. Besides that, the dynamic and sensitivity equations formed with the proposed method can be efficiently integrated using HHT-I3 method, which makes the efficiency of the direct differentiation method comparable to that of the adjoint variable method when the number of design variables is not extremely large. All these improvements greatly enhance the application value of the direct differentiation method in the engineering optimization of the ANCF-based flexible multibody systems.     

A new version of transfer matrix method for multibody systems

In order to avoid the global dynamics equations and increase the computational efficiency for multibody system dynamics (MSD), the transfer matrix method of multibody system (MSTMM) has been developed and applied very widely in research and engineering in recent 20 years. It differs from ordinary methods in multibody system dynamics with respect to the feature that there is no need for a global dynamics equation, and it uses low-order matrices for high computational efficiency. For linear systems, MSTMM is exact even if continuous elements like beams are involved. The discrete time MSTMM, however, has to use local linearization. In order to release the method from such approximations, a new version of MSTMM is presented in this paper where translational and angular accelerations, on the one hand, and internal forces and moments, on the other hand, are used as state variables. Already linear relationships among these quantities are utilized, which results in new element transfer matrices and algorithms making the study of multibody systems as simple as the study of single bodies. The proposed approach also allows combining MSTMM with any general numerical integration procedure. Some numerical examples of MSD are given to demonstrate the proposed method.     

Analysis of Nonlinear Multibody Systems with Elastic Couplings

This paper is concerned with the dynamic analysis of nonlinear multibody systems involving elastic members made of laminated, anisotropic composite materials. The analysis methodology can be viewed as a three-step procedure. First, the sectional properties of beams made of composite materials are determined based on an asymptotic procedure that involves a two-dimensional finite element analysis of the cross-section. Second, the dynamic response of nonlinear, flexible multibody systems is simulated within the framework of energy-preserving and energy-decaying time-integration schemes that provide unconditional stability for nonlinear systems. Finally, local three-dimensional stresses in the beams are recovered, based on the stress resultants predicted in the previous step. Numerical examples are presented and focus on the behavior of multibody systems involving members with elastic couplings.     

Transfer matrix method for linear multibody system

A new method for linear hybrid multibody system dynamics is proposed in this paper. This method, named as transfer matrix method of linear multibody system (MSTMM), expands the advantages of the traditional transfer matrix method (TMM). The concepts of augmented eigenvector and equation of motion of linear hybrid multibody system are presented at first to find the orthogonality and to analyze the responses of the hybrid multibody system using modal method. If using this method, the global dynamics equation is not needed in the study of linear hybrid multibody system dynamics. The MSTMM has a small size of matrix and higher computational speed, and can be applied to linear multi-rigid-body system dynamics, linear multi-flexible-body system dynamics and linear hybrid multibody system dynamics. This method is simple, straightforward, practical, and provides a powerful tool for the study on linear hybrid multibody system dynamics. This method can be used in the following: (1) Solve the eigenvalue problem of linear hybrid multibody systems. (2) Obtain the orthogonality of eigenvectors of linear hybrid multibody systems. (3) Realize the accurate analysis of the dynamics response of linear hybrid multibody systems. (4) Find the connected parameters between bodies used in the computation of linear hybrid multibody systems. A practical engineering system is taken as an example of linear multi-rigid-flexible-body system, the dynamics model, the transfer equations and transfer matrices of various bodies and hinges; the overall transfer equation and overall transfer matrix of the system are developed. Numerical example shows that the results of the vibration characteristics and the response of the hybrid multibody system received by MSTMM and by experiment have good agreements. These validate the proposed method.     

Erratum to: New efficient method for dynamic modeling and simulation of flexible multibody systems moving in plane

Efficient, precise dynamic analysis for general flexible multibody systems has become a research focus in the field of flexible multibody dynamics. In this paper, the finite element method and component mode synthesis are introduced to describe the deformations of the flexible components, and the dynamic equations of flexible bodies moving in plane are deduced. By combining the discrete time transfer matrix method of multibody system with these dynamic equations of flexible component, the transfer equations and transfer matrices of flexible bodies moving in plane are developed. Finally, a high-efficient dynamic modeling method and its algorithm are presented for high-speed computation of general flexible multibody dynamics. Compared with the ordinary dynamics methods, the proposed method combines the strengths of the transfer matrix method and finite element method. It does not need the global dynamic equations of system and has the low order of system matrix and high computational efficiency. This method can be applied to solve the dynamics problems of flexible multibody systems containing irregularly shaped flexible components. It has advantages for dynamic design of complex flexible multibody systems. Formulations as well as a numerical example of a multi-rigid-flexible-body system containing irregularly shaped flexible components are given to validate the method.     

New efficient method for dynamic modeling and simulation of flexible multibody systems moving in plane

Efficient, precise dynamic analysis for general flexible multibody systems has become a research focus in the field of flexible multibody dynamics. In this paper, the finite element method and component mode synthesis are introduced to describe the deformations of the flexible components, and the dynamic equations of flexible bodies moving in plane are deduced. By combining the discrete time transfer matrix method of multibody system with these dynamic equations of flexible component, the transfer equations and transfer matrices of flexible bodies moving in plane are developed. Finally, a high-efficient dynamic modeling method and its algorithm are presented for high-speed computation of general flexible multibody dynamics. Compared with the ordinary dynamics methods, the proposed method combines the strengths of the transfer matrix method and finite element method. It does not need the global dynamic equations of system and has the low order of system matrix and high computational efficiency. This method can be applied to solve the dynamics problems of flexible multibody systems containing irregularly shaped flexible components. It has advantages for dynamic design of complex flexible multibody systems. Formulations as well as a numerical example of a multi-rigid-flexible-body system containing irregularly shaped flexible components are given to validate the method.     

Impacts in multibody systems: modeling and experiments

This paper outlines a novel approach to the modeling and analysis of impact involving multibody systems. This approach is based on an analysis of energy absorption and restitution during impact, using a decomposition of the kinetic energy, which decouples the parts associated with the spaces of admissible and constrained motions of the underlying unilateral constraints. Such a decomposition turns out to be useful in the analysis of energy dissipation during impact, and leads to a generalized definition of the energetic coefficient of restitution, which targets particularly collisions in multibody systems. The applicability of the approach reported is investigated by conducting an experimental study on a robotic testbed. It is shown that impact between multibody systems is considerably affected not only by the local dynamics characteristics of the interacting bodies, but also the configuration of the whole multibody system. The results reported here show that our decomposition can offer a sound characterization of impact in several problems of multibody systems.     

Nonholonomic Multibody Dynamics

The paper deals with the nonholonomic multibody system dynamics from apoint of view which is caused by some actual applications in high-tecareas like high-speed train technology or biomechanics of somedisciplines in high-performance sports. Obviously, looking at suchproblems, there are very close connections between classical analyticaldynamics, differential geometry and modern control theory. But theseconnections cannot be used to get new composed results in solvingcomplicated problems of multibody system dynamics because correspondingsoftware tools are not enough in tune with each other. This paper willgive some ideas for developing a unified basis for modeling, simulationand control of nonholonomic multibody systems.First, a derivative-free approach for generating Lagrangian motionequations of multibody systems with kinematical tree structure as wellas for constrained multibody systems is given. This has been done byusing differential-geometric concepts in a Riemannian space. Secondly,the well-known theorem of Frobenius is considered with respect to itsclassical interpretation by the so-called object of nonholonomy as wellas by its modern interpretation in the nonlinear control theory usingLie-brackets. The ideas are illustrated by the classical rollingcondition and edge condition on double-curved surfaces. Specialnumerical problems in simulation of multibody systems subject toadditional kinematic constraints are discussed. Finally threeapplications are given.     

Multibody System Dynamics: Roots and Perspectives

The paper reviews the roots, the state-of-the-art and perspectives of multibody system dynamics. Some historical remarks show that multibody system dynamics is based on classical mechanics and its engineering applications ranging from mechanisms, gyroscopes, satellites and robots to biomechanics. The state-of-the-art in rigid multibody systems is presented with reference to textbooks and proceedings. Multibody system dynamics is characterized by algorithms or formalisms, respectively, ready for computer implementation. As a result simulation and animation are most important. The state-of-the-art in flexible multibody systems is considered in a companion review by Shabana.Future research fields in multibody dynamics are identified as standardization of data, coupling with CAD systems, parameter identification, real-time animation, contact and impact problems, extension to control and mechatronic systems, optimal system design, strength analysis and interaction with fluids. Further, there is a strong interest on multibody systems in analytical and numerical mathematics resulting in reduction methods for rigorous treatment of simple models and special integration codes for ODE and DAE representations supporting the numerical efficiency. New software engineering tools with modular approaches promise improved efficiency still required for the more demanding needs in biomechanics, robotics and vehicle dynamics.     

Analysis of Flexible Multibody Systems with Intermittent Contacts

This paper is concerned with the dynamic analysis of flexible,nonlinear multibody systems undergoing intermittent contacts. Contact isassumed to be of finite duration, and the forces acting between thecontacting bodies which can be either rigid or deformable are explicitlycomputed during simulation. The modeling of contact consists of threeparts: a number of holonomic constraints that define the candidatecontact points on the bodies, a unilateral contact condition which istransformed into a holonomic constraint by the addition of a slackvariable, and a contact model which describes the relationship betweenthe contact force and the local deformation of the contacting bodies.This work is developed within the framework of energy preserving anddecaying time integration schemes that provide unconditional stabilityfor nonlinear, flexible multibody systems undergoing intermittentcontacts.     

多体系统动力学Lie 群微分⁃代数方程约束稳定方法

针对多体系统动力学微分-代数方程求解问题,研究基于Lie群表达的约束稳定方法.首先引入新的Lagrange乘子,结合位移约束、速度级约束和加速度级约束方程,构造了新的Lie群微分-代数方程.然后使用向后差商隐式方法和CG(Crouch-Grossman)方法,对微分–代数方程进行离散求解,得到精确度较高的动力学仿真结果.该方法在精确保持各级约束方程的同时,保持旋转矩阵的正交性,并且使系统总能量误差较小.     

Controllability and motion planning of a multibody Chaplygin's sphere and Chaplygin's top

This paper studies local configuration controllability of multibody systems with nonholonomic constraints. As a nontrivial example of the theory, we consider the dynamics and control of a multibody spherical robot. Internal rotors and sliders are used as the mechanisms for control. Our model is based on equations developed by the second author for certain mechanical systems with nonholonomic constraints, e.g. Chaplygin's sphere and Chaplygin's top in particular, and the multibody framework for unconstrained mechanical systems developed by the first and third authors. Recent methods for determining controllability and path planning for multibody systems with symmetry are extended to treat a class of mechanical systems with nonholonomic constraints. Specific results on the controllability and path planning of the spherical robot model are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Absolute nodal coordinate plane beam formulation for multibody systems dynamics

A new plane beam dynamic formulation for constrained multibody system dynamics is developed. Flexible multibody system dynamics includes rigid body dynamics and superimposed vibratory motions. The complexity of mechanical system dynamics originates from rotational kinematics, but the natural coordinate formulation does not use rotational coordinates, so that simple dynamic formulation is possible. These methods use only translational coordinates and simple algebraic constraints. A new formulation for plane flexible multibody systems are developed utilizing the curvature of a beam and point masses. Using absolute nodal coordinates, a constant mass matrix is obtained and the elastic force becomes a nonlinear function of the nodal coordinates. In this formulation, no infinitesimal or finite rotation assumptions are used and no assumption on the magnitude of the element rotations is made. The distributed body mass and applied forces are lumped to the point masses. Closed loop mechanical systems consisting of elastic beams can be modeled without constraints since the loop closure constraints can be substituted as beam longitudinal elasticity. A curved beam is modeled automatically. Several numerical examples are presented to show the effectiveness of this method.     

Multibody Dynamics of Very Flexible Damped Systems

An efficient multibody dynamics formulation is presented for simulating the forward dynamics of open and closed loop mechanical systems comprised of rigid and flexible bodies interconnected by revolute, prismatic, free, and fixed joints. Geometrically nonlinear deformation of flexible bodies is included and the formulation does not impose restrictions on the representation of material damping within flexible bodies.The approach is based on Kane's equation without multipliers and the resulting formulation generates 2ndof+m first order ordinary differential equations directly where ndof is the smallest number of system degrees of freedom that can completely describe the system configuration and m is the number of loop closure velocity constraint equations. The equations are integrated numerically in the time domain to propagate the solution.Flexible bodies are discretized using a finite element approach. The mass and stiffness matrices for a six-degree-of-freedom planar beam element are developed including mass coupling terms, rotary inertia, centripetal and Coriolis forces, and geometric stiffening terms.The formulation is implemented in the general purpose multibody dynamics computer program flxdyn. Extensive validation of the formulation and corresponding computer program is accomplished by comparing results with analytically derived equations, alternative approximate solutions, and benchmark problems selected from the literature. The formulation is found to perform well in terms of accuracy and solution efficiency.This article develops the formulation and presents a set of validation problems including a sliding pendulum, seven link mechanism, flexible beam spin-up problem, and flexible slider crank mechanism.     

Optimization of flexible components of multibody systems via equivalent static loads

An optimization methodology that iteratively links the results of multibody dynamics and structural analysis software to an optimization method is presented to design flexible multibody systems under dynamic loading conditions. In particular, rigid multibody dynamic analysis is utilized to calculate dynamic loads of a multibody system and a structural optimization algorithm using equivalent static loads transformed from the dynamic loads are used to design the flexible components in the multibody dynamic system. The equivalent static loads, which are derived from equations of motion, are used as multiple loading conditions of linear structural optimization. A simple example is solved to verify the proposed methodology and the pelvis part of the biped humanoid, a complex multibody system which consists of many bodies and joints, is redesigned using the proposed methodology.     

具有奇异位置的多体系统动力学方程的改进算法

多体系统进行数值仿真时,很多选择了微分代数混合方程作为多体系统动力学数学模型.本文在现有的约束稳定化理论基础上,提出了针对具有奇异位置的多体系统动力学方程的改进算法.算法通过修正速度违约和控制稳定项,讨论了具有奇异位置的微分代数混合方程的数值仿真问题并给出了稳定项中相关系数的建议值,从而有效克服了求解混合方程时因为构型奇异给计算造成的困难.算例分别采用改进算法与ADAMS软件进行仿真,计算结果的比较表明了改进算法的有效性.本文给出的基于能量守恒的能量差曲线也证明了改进算法的有效性.     

Multidisciplinary Optimization of Multibody Systems with Application to the Design of Rail Vehicles

A methodology for the design optimization of multibody systems is presented. The methodology has the following features: (1) multibody dynamics is employed to model and simulate complex systems; (2) multidisciplinary optimization (MDO) methods are used to combine multibody systems and additional systems in a synergistic manner; (3) using genetic algorithms (GAs) and other effective search algorithms, the mechanical and other design variables are optimized simultaneously. The methodology is shown to handle the conflicting requirements of rail vehicle design, i.e., lateral stability, curving performance, and ride quality, in an effective manner. By coordinating these conflicting requirements at the system level, three multibody models corresponding to each of these requirements for a rail vehicle are optimized simultaneously.     

Three Approaches for Elastodynamic Contact in Multibody Systems

In machine dynamics impacts may occur by interaction of solid bodies. There is no doubt that the method of multibody systems is most efficient for the dynamical analysis of the overall motion. However, during impact energy is lost macromechanically measured by the coefficient of restitution. This coefficient has to be estimated from experiments and experience but cannot be computed within the multibody system approach. The impacts, on the other hand, are generating waves in the bodies which are propagating until they are vanishing due to material damping. These high frequency phenomena are analyzed using wave propagation, modal approach and finite elements. The results of the simulation on the fast time scale are used to compute the coefficient of restitution which is then fed back to the multibody system equations and the solution continues on the related slow time scale. The efficiency of the approach presented is shown for the impact of a steel sphere on four different shaped aluminum bodies of comparable mass. The simulation results are verified with experiments performed on different time scales, too. Using the impact of a double pendulum on a stop as example, the application of the multiscale approach to a multibody system is shown.