Quadratic and Higher-Order Constraints in Energy-Conserving Formulations of Flexible Multibody Systems

The treatment of constraints is considered here within the framework ofenergy-momentum conserving formulations for flexible multibody systems.Constraint equations of various types are an inherent component of multibodysystems, their treatment being one of the key performance features ofmathematical formulations and numerical solution schemes.Here we employ rotation-free inertial Cartesian coordinates of points tocharacterise such systems, producing a formulation which easily couples rigidbody dynamics with nonlinear finite element techniques for the flexiblebodies. This gives rise to additional internal constraints in rigid bodies topreserve distances. Constraints are enforced via a penalty method, which givesrise to a simple yet powerful formulation. Energy-momentum time integrationschemes enable robust long term simulations for highly nonlinear dynamicproblems.The main contribution of this paper focuses on the integration of constraintequations within energy-momentum conserving numerical schemes. It is shownthat the solution for constraints which may be expressed directly in terms ofquadratic invariants is fairly straightforward. Higher-order constraints mayalso be solved, however in this case for exact conservation an iterativeprocedure is needed in the integration scheme. This approach, together withsome simplified alternatives, is discussed.Representative numerical simulations are presented, comparing the performanceof various integration procedures in long-term simulations of practicalmultibody systems.     

多柔体系统动力学研究进展与挑战

首先回顾多体系统动力学的学科发展和学术交流情况,然后系统概述了多柔体系统动力学方程数值算法、多柔体系统接触/碰撞动力学与柔性空间结构展开动力学三个方面的研究进展及值得关注的若干问题,最后给出了开展多柔体系统动力学研究的若干建议.     

Complex Flexible Multibody Systems with Application to Vehicle Dynamics

A formulation to describe the linear elastodynamics offlexible multibody systems is presented in this paper. By using a lumpedmass formulation the flexible body mass is represented by a collectionof point masses with rotational inertia. Furthermore, the bodydeformations are described with respect to a body-fixed coordinateframe. The coupling between the flexible body deformation and its rigidbody motion is completely preserved independently of the methods used todescribe the body flexibility. In particular, if the finite elementmethod is chosen for this purpose only the standard finite elementparameters obtained from any commercial finite element code are used inthe methodology. In this manner, not only the analyst can use any typeof finite elements in the multibody model but the same finite elementmodel can be used to evaluate the structural integrity of any systemcomponent also. To deal with complex-shaped structural models offlexible bodies it is necessary to reduce the number of generalizedcoordinates to a reasonable dimension. This is achieved with thecomponent mode synthesis at the cost of specializing the formulation toflexible multibody models experiencing linear elastic deformations only.Structural damping is introduced to achieve better numerical performancewithout compromising the quality of the results. The motions of therigid body and flexible body reference frames are described usingCartesian coordinates. The kinematic constraints between the differentsystem components are evaluated in terms of this set of generalizedcoordinates. The equations of motion of the flexible multibody systemare solved by using the augmented Lagrangean method and a sparse matrixsolver. Finally, the methodology is applied to model a vehicle with acomplex flexible chassis, simulated in typical handling scenarios. Theresults of the simulations are discussed in terms of their numericalprecision and efficiency.     

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.     

Relative Finite Element Coordinates in Multibody System Simulation

In the paper a numerical approach for deriving the nonlinear explicitform dynamic equations of rigid and flexible multibody systems ispresented. The dynamic equations are obtained as Ordinary DifferentialEquations for generalized coordinates and without algebraic constraints.The Finite Element Theory is applied for discretization of flexiblebodies. The minimal set of the generalized coordinates includesindependent joint motions, as well as independent small flexibledeflections of finite element nodes. The node deflections and stiffnessmatrices are calculated with respect to the moving relative coordinatesystems of the flexible bodies. The positions and orientations ofelement and substructure coordinate systems are updated according to thenode deflections. A major step of the numerical process is the kinematicanalysis and calculation of matrices of partial derivatives of thequasi-coordinates (dependent joint motions and coordinates of points andnodes) with respect to the generalized coordinates. The inertia terms inthe dynamic equations are obtained multiplying the matrices of thepartial derivatives by the mass matrices of the rigid and flexiblebodies. Stiffness properties of flexible bodies are presented in thedynamic equations by stiff forces that depend on the generalizedrelative flexible deflections only. Several examples of large motion ofbeam structures show the effectiveness of the algorithm.     

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.     

Flexible Multibody Systems Models Using Composite Materials Components

The use of a multibody methodology to describe the large motion of complex systems that experience structural deformations enables to represent the complete system motion, the relative kinematics between the components involved, the deformation of the structural members and the inertia coupling between the large rigid body motion and the system elastodynamics. In this work, the flexible multibody dynamics formulations of complex models are extended to include elastic components made of composite materials, which may be laminated and anisotropic. The deformation of any structural member must be elastic and linear, when described in a coordinate frame fixed to one or more material points of its domain, regardless of the complexity of its geometry. To achieve the proposed flexible multibody formulation, a finite element model for each flexible body is used. For the beam composite material elements, the sections properties are found using an asymptotic procedure that involves a two-dimensional finite element analysis of their cross-section. The equations of motion of the flexible multibody system are solved using an augmented Lagrangian formulation and the accelerations and velocities are integrated in time using a multi-step multi-order integration algorithm based on the Gear method.     

Penalty,Semi-Recursive and Hybrid Methods for MBS Real-Time Dynamics in the Context of Structural Integrators

In a previous work, the authors presented a new real-time formulation for the dynamics of multibody systems, which encompasses high ranks of efficiency, accuracy, robustness and easiness of implementation. The new method, called hybrid, was obtained as a combination of a topological semi-recursive formulation based on velocities transformation, and a global penalty formulation for closed-loops consideration. It was proven to be more robust and efficient that its predecessors for large problems. For the three methods compared, the implicit, single-step trapezoidal rule was used as numerical integrator. In this paper, the influence of the integration scheme on the performance of the three mentioned methods is studied. Since the hybrid formulation becomes competitive for large multibody systems, a rather demanding simulation of the full model of a car vehicle is selected as benchmark problem. Computer codes implementing the three dynamic formulations in combination with different structural integrators, like Newmark, HHT and Generalized- algorithms, are used to run the simulation, so that the performance of each couple dynamic-formulation/numerical-integrator can be appraised. The example is also analyzed through a commercial tool, so as to provide the readers with a well-known reference for comparison.     

An Efficient Implementation of the Recursive Approach to Flexible Multibody Dynamics

This paper deals with the efficient extension of the recursive formalism (articulated body inertia) for flexible multibody systems. Present recursive formalisms for flexible multibody systems require the inversion of the mass matrix with the dimension equal to the number of flexible degrees of freedom of particular bodies. This is completely removed. The paper describes the derivation of equations of motion expressed in the local coordinate system attached to the body, then the discretization of these equations of motion based on component mode synthesis and FEM shape functions and, finally, two versions of the new recursive formalism.     

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.     

Finite Element Method in Dynamics of Flexible Multibody Systems: Modeling of Holonomic Constraints and Energy Conserving Integration Schemes

In this work we discuss an application of the finite elementmethod to modeling of flexible multibody systems employing geometricallyexact structural elements. Two different approaches to handleconstraints, one based on the Lagrange multiplier procedure and anotherbased on the use of release degrees of freedom, are examined in detail.The energy conserving time stepping scheme, which is proved to be wellsuited for integrating stiff differential equations, gouverning themotion of a single flexible link is appropriately modified and extendedto nonlinear dynamics of multibody systems.     

变拓扑柔性多体系统接触碰撞动力学研究

在实际工程领域中存在着大量接触碰撞等非连续动力学问题,现有的解决柔性多体系统连续动力学过程的建模理论与方法,已经无法解决或无法很好解决这些问题.本文基于变拓扑思想,提出了附加接触约束的柔性多体系统碰撞动力学建模理论；通过设计柔性圆柱杆接触碰撞实验,验证了所提出附加约束接触碰撞模型的有效性；针对柔性多体系统全局动力学仿真面临时间和空间的多尺度问题,提出多变量的离散方法,从而提高了柔性多体系统非连续动力学的仿真效率.     

A DAE Approach to Flexible Multibody Dynamics

The present work deals with the dynamics of multibody systems consisting ofrigid bodies and beams. Nonlinear finite element methods are used to devise a frame-indifferent spacediscretization of the underlying geometrically exact beam theory. Both rigid bodies and semi-discrete beams are viewed as finite-dimensional dynamical systems with holonomic constraints. The equations of motion pertaining to the constrained mechanical systems under considerationtake the form of Differential Algebraic Equations (DAEs).The DAEs are discretized directly by applying a Galerkin-based method.It is shown that the proposed DAE approach provides a unified framework for the integration of flexible multibody dynamics.     

A method for controller parameter estimation based on perturbations

Simulation and prediction of eigenfrequencies and mode shapes for active flexible multibody systems is an important task in disciplines such as robotics and aerospace engineering. A challenge is to accurately include both controller effects and flexible body dynamics in a multidisciplinary system model appropriate for modal analysis. A method for performing modal analyses of such systems in a finite element environment was recently developed by the authors. On issue is, however, that for engineers working in a finite element environment, the controller properties are not always explicitly available prior to modal analyses. The authors encountered this problem when working with the design of a particular offshore windmill. The controller for the windmill was delivered in the form of a dynamic link library (dll) from a third party provider, and when performing virtual testing of the windmill design, it was of great importance to use the “real” controller in the form of the provided dll, rather than re-model it in for instance Simulink or EASY5. This paper presents a method for estimating the controller parameters of PID-type controllers when solving the closed-loop eigenvalue problem for active flexible multibody systems in a finite element environment. The method is based on applying incremental changes, perturbations, to relevant system variables while recording reactions from other system variables. In this work, the theory of the method is derived and the method is tested through several numerical examples.     

Modal analysis of active flexible multibody systems

When performing modal analyses of active flexible multibody systems, both controller effects and flexible body dynamics should be included in a multidisciplinary system model. This paper deals with the theory of solving the closed-loop eigenvalue problem for active flexible multibody systems with multiple-input multiple-output proportional-integral-derivative (PID) type feedback controllers and multiple degrees of freedom finite element models. Modal analyses are performed on both a simple and complex active flexible multibody system in order to illustrate the difference between current modal analysis method for such systems and the proposed theory derived in this paper.     

Convergence of generalized-$\boldsymbol{\alpha}$ time integration for nonlinear systems with stiff potential forces

Generalized-\(\alpha\) time integration schemes, originally developed for application in structural dynamics, are increasingly popular throughout many branches of multibody system simulation. Their simple implementation and the opportunity to control the numerical dissipation make them highly appealing for use in broad fields of application.Initially introduced for the solution of linear ordinary differential equations, there have been several extensions to nonlinear structural dynamics and constrained multibody systems in various formulations.In the present paper, we consider the application to systems with very stiff potential forces (singular singularly perturbed systems) whose solution approaches in the limit case that of a constrained system (index-3 differential–algebraic equation). We give a convergence analysis comparing the highly oscillatory solutions of the stiff system to those of the associated constrained one and show that the classical second order convergence result holds for position coordinates as well as for appropriately projected errors on the velocity level.The theoretical results are verified by numerical experiments for a simple test example.     

Non-linear dynamics of flexible multibody systems

In this work we set to examine several important issues pertinent to currently very active research area of the finite element modeling of flexible multibody system dynamics. To that end, we first briefly introduce three different model problems in non-linear dynamics of flexible 3D solid, a rigid body and 3D geometrically exact beam, which covers the vast majority of representative models for the particular components of a multibody system. The finite element semi-discretization for these models is presented along with the time-discretization performed by the mid-point scheme. In extending the proposed methodology to modeling of flexible multibody systems, we also present how to build a systematic representation of any kind of joint connecting two multibody components, a typical case of holonomic contraint, as a linear superposition of elementary constraints. We also indicate by a chosen model of rolling contact, an example of non-holonomic constraint, that the latter can also be included within the proposed framework. An important aspect regarding the reduction of computational cost while retaining the consistency of the model is also addressed in terms of systematic use of the rigid component hypothesis, mass lumping and the appropriate application of the explicit-implicit time-integration scheme to the problem on hand. Several numerical simulations dealing with non-linear dynamics of flexible multibody systems undergoing large overall motion are presented to further illustrate the potential of presented methodology. Closing remarks are given to summarize the recent achievements and point out several directions for future research.     

多体系统动力学数值解法

多体系统动力学研究的主要内容动力学建模与数值解法是多体系统动力学研究的主要内容之一。对多体系统动力学方程及其动力学数值解法的研究成果进行了较为全面的阐述。多体系统动力学及动力学方程进行了简单的归纳和总结,多体系统动力学数值求解,特别是刚柔耦合多体系统微分／代数方程的数值解法等研究热点进行了详细的阐述,并简要展望了多体系统动力学数值解法今后的发展趋势,为多体系统动力学计算机仿真奠定了基础。