On the constraints violation in forward dynamics of multibody systems

It is known that the dynamic equations of motion for constrained mechanical multibody systems are frequently formulated using the Newton–Euler’s approach, which is augmented with the acceleration constraint equations. This formulation results in the establishment of a mixed set of partial differential and algebraic equations, which are solved in order to predict the dynamic behavior of general multibody systems. The classical solution of the equations of motion is highly prone to constraints violation because the position and velocity constraint equations are not fulfilled. In this work, a general and comprehensive methodology to eliminate the constraints violation at the position and velocity levels is offered. The basic idea of the described approach is to add corrective terms to the position and velocity vectors with the intent to satisfy the corresponding kinematic constraint equations. These corrective terms are evaluated as a function of the Moore–Penrose generalized inverse of the Jacobian matrix and of the kinematic constraint equations. The described methodology is embedded in the standard method to solve the equations of motion based on the technique of Lagrange multipliers. Finally, the effectiveness of the described methodology is demonstrated through the dynamic modeling and simulation of different planar and spatial multibody systems. The outcomes in terms of constraints violation at the position and velocity levels, conservation of the total energy and computational efficiency are analyzed and compared with those obtained with the standard Lagrange multipliers method, the Baumgarte stabilization method, the augmented Lagrangian formulation, the index-1 augmented Lagrangian, and the coordinate partitioning method.     

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.     

Linear dynamics of flexible multibody systems

We consider mechanical systems where the dynamics are partially constrained to prescribed trajectories. An example for such a system is a building crane with a load and the requirement that the load moves on a certain path.Enforcing this condition directly in form of a servo constraint leads to differential-algebraic equations (DAEs) of arbitrarily high index. Typically, the model equations are of index 5, which already poses high regularity conditions. If we relax the servo constraints and consider the system from an optimal control point of view, the strong regularity conditions vanish, and the solution can be obtained by standard techniques.By means of the well-known \(n\)-car example and an overhead crane, the theoretical and expected numerical difficulties of the direct DAE and the alternative modeling approach are illustrated. We show how the formulation of the problem in an optimal control context works and address the solvability of the optimal control system. We discuss that the problematic DAE behavior is still inherent in the optimal control system and show how its evidences depend on the regularization parameters of the optimization.     

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.     

Nonsmooth spatial frictional contact dynamics of multibody systems

Multibody System Dynamics - Computational prediction of 3D crutch-assisted walking patterns is a challenging problem that could be applied to study different biomechanical aspects of crutch walking...     

Efficient numerical solution of constrained multibody dynamics systems

A numerical algorithm for conducting coupled system dynamical simulation is presented. The interconnected system, comprising numerous modules, is treated as a constrained multibody dynamics system. Of particular focus is the efficient solution of coupled system simulation without sacrificing the independence of the separate dynamical modules. The proposed algorithm, Maggi’s equations with perturbed iteration (MEPI) emanates from numerical methods for differential-algebraic equations. Separate treatment of the constraint equations from the resolution of subsystem dynamical responses marks MEPI’s main characteristic.     

Towards dynamics of controlled multibody systems with magnetostrictive transducers

The paper addresses mechatronic issue of multibody systems comprising giant magnetostrictive material based transducers (sensors and/or actuators). Interaction between dynamics and control in multibody system with smart material based transducers makes it possible to change system properties and functionality substantially as a response to applied electric, magnetic or temperature fields. To use this interaction in an optimal way, the proper mathematical models of controlled electro-magneto-elastic multibody systems need to be developed. In the paper, a general mathematical model of multibody systems with magnetostrictive transducers is presented. The model consists of the constitutive equations of magnetoelastic behavior of transducers, standard formulae of electromagnetism for induced voltage and current in the pick-up coil due to variation of magnetic field intensity, and finally, the equations of motion of multibody system itself. The last one can be derived using one of the well-known multibody dynamics formalisms. General model has been developed in detail for linearized dynamics of magnetostrictive transducers and implemented virtually for two practically important cases of interaction of hosting multibody system with transducers, namely for systems with displacement driven transducers and for systems with force driven transducers. Physical prototype of magnetostrictive transducer and test rig (hosting multibody system) have been built and used successfully for verification of developed models.     

Improved bushing models for general multibody systems and vehicle dynamics

The development and computational implementation, on a multibody dynamics environment, of a constitutive relation to model bushing elements associated with mechanical joints used in the models of road and rail vehicles is presented here. These elements are used to eliminate vibrations in vehicles, due to road irregularities, to allow small misalignment of axes, to reduce noise from the transmission, or to decrease wear of the mechanical joints. Bushings are made of a special rubber, used generally in energy dissipation, which presents a nonlinear viscoelastic relationship between the forces and moments and their corresponding displacements and rotations. In the methodology proposed here a finite element model of the bushing is developed in the framework of the finite element code ABAQUS to obtain the constitutive relations of displacement/rotation versus force/moment for different loading cases. The bushing is modeled in a multibody code as a nonlinear restrain that relates the relative displacements between the bodies connected with the joint reaction forces, and it is represented by a matrix constitutive relation. The basic ingredients of the multibody model are the same vectors and points relations used to define kinematic constraints in any multibody formulation. One particular, and relevant, characteristic of the formulation now presented is its ability to represent standard kinematic joints, clearance, and bushing joints just by defining appropriate constitutive relations. Spherical, revolution, cylindrical, and translational bushing joints are modeled, implemented, and demonstrated through the simulation of two multibody models of a road vehicle, one with perfect kinematic joints for the suspension sub-systems, and other with bushing joints. The tests conducted include an obstacle avoidance maneuver and a vehicle riding over bumps. It is shown that the bushing models for vehicle multibody models proposed here are accurate and computationally efficient so that they can be included in the vehicle models leading reliable simulations.     

Modelling of clearances and joint flexibility effects in multibody systems dynamics

An effective method for modelling the dynamic response of multibody systems with flexible joints is presented. The method combines the use of finite element method with Kane's equations to present an algorithm strictly in terms of the generalized coordinates of the system. The procedures developed outline the automatic incorporation of the joint flexibility in the equations of motion, hence a more accurate mathematical model is developed. One other advantage to the method presented lies in the explicit forms of the coefficients needed in the analysis where they are readily expressed in a form suited for computer implementation. A discussion on possible applications is also presented.     

Galerkin meshfree methods applied to the nonlinear dynamics of flexible multibody systems

Meshfree Galerkin methods have been developed recently for the simulation of complex mechanical problems involving large strains of structures, crack propagation, or high velocity impact dynamics. At the present time, the application of these methods to multibody dynamics has not been made despite their great advantage in some situations over standard finite element techniques.     

Characterization of the contribution of inertial parameters to the dynamics of multibody systems

In this paper, an approach is developed for sensitivity analysis of inertial parameters. It is based on the reformed equations of motion linear in terms of the inertial parameters, which are transformed to dimensionless inertial parameters. Based on an eigenvalue analysis, directions of the parameter space are found along which maximum/minimum changes are achieved on the generalized forces, i.e., on the dynamics of the system. Directional derivatives of the generalized force vector in these directions characterize how each inertial parameter contributes to the system dynamics.     

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.     

On the optimal scaling of index three DAEs in multibody dynamics

We propose a preconditioning strategy for the governing equations of multibody systems in index-3 differential-algebraic form. The method eliminates the amplification of errors and the ill-conditioning which affect numerical solutions of high index differential algebraic equations for small time steps. We develop a new theoretical analysis of the perturbation problem and we apply it to the derivation of preconditioners for the Newmark family of integration schemes. The theoretical results are confirmed by numerical experiments.     

A parallel Hamiltonian formulation for forward dynamics of closed-loop multibody systems

This paper presents a novel recursive divide-and-conquer formulation for the simulation of complex constrained multibody system dynamics based on Hamilton’s canonical equations (HDCA). The systems under consideration are subjected to holonomic, independent constraints and may include serial chains, tree chains, or closed-loop topologies. Although Hamilton’s canonical equations exhibit many advantageous features compared to their acceleration based counterparts, it appears that there is a lack of dedicated parallel algorithms for multi-rigid-body system dynamics based on the Hamiltonian formulation. The developed HDCA formulation leads to a two-stage procedure. In the first phase, the approach utilizes the divide and conquer scheme, i.e., a hierarchic assembly–disassembly process to traverse the multibody system topology in a binary tree manner. The purpose of this step is to evaluate the joint velocities and constraint force impulses. The process exhibits linear \(O(n)\) (\(n\) – number of bodies) and logarithmic \(O(\log_{2}{n})\) numerical cost, in serial and parallel implementations, respectively. The time derivatives of the total momenta are directly evaluated in the second parallelizable step of the algorithm. Sample closed-loop test cases indicate very small constraint violation errors at the position and velocity level as well as marginal energy drift without any additional form of constraint stabilization techniques involved in the solution process. The results are comparatively set against more standard acceleration based Featherstone’s DCA approach to indicate the performance of the HDCA algorithm.     

空间多体机械系统耦合动力学的键合图法

为提高多能域耦合空间多体系统动力学分析的效率及可靠性,提出了键合图法．从能量守恒的基本原理出发,讨论了键合图中的多通口元件MTF所具有的特性．以此为基础阐述了建立空间多体机械系统键合图模型的一般方法及其动力学原理．将运动副约束反力视做未知势源加在系统键合图模型相应的0-结处,有效地解决了微分因果关系及非线性结型结构所带来的十分困难的代数问题,推导出便于计算机自动生成的系统状态方程及运动副约束反力方程的统一公式,实现了计算机自动建模与仿真．实际算例表明了所述方法的有效性．     

Port-Hamiltonian flexible multibody dynamics

Multibody System Dynamics - A new formulation for the modular construction of flexible multibody systems is presented. By rearranging the equations for a flexible floating body and introducing the...     

Extension of the divide-and-conquer algorithm for the efficient inverse dynamics analysis of multibody systems

This paper presents a new mathematical framework to extend the Generalized Divide-and-Conquer Algorithm (GDCA) for the inverse dynamics analysis of fully actuated constrained multibody systems. Inverse-GDCA (iGDCA) is a highly parallelizable method which does not create the mass and Jacobian matrices of the entire system. In this technique, generalized driving forces and constraint loads due to kinematic pairs are clearly and separately differentiated from each other in the equations of motion. As such, it can be easily used for control scheme purposes. iGDCA works based on a series of recursive assembly and disassembly passes to form and solve the equations governing the inverse dynamics of the system. Herein, the mathematical formulations to efficiently combine the dynamics of consecutive bodies in the assembly pass for the purpose of inverse dynamics analysis are presented. This is followed by generating the disassembly pass algorithm to efficiently compute generalized actuating forces. Furthermore, this paper presents necessary mathematical formulations to efficiently treat the inverse dynamics of multibody systems involving kinematic loops with various active and passive boundary conditions. This is followed by the design of a new strategy to efficiently perform the assembly–disassembly pass in these complex systems while avoiding unnecessary computations. Finally, the presented method is applied to selected open-chain and closed-chain multibody systems.     

On design sensitivities in the structural analysis and optimization of flexible multibody systems

Multibody System Dynamics - The structural analysis and optimization of flexible multibody systems become more and more popular due to the ability to efficiently compute gradients using...