Modeling Multibody Systems with Uncertainties. Part I: Theoretical and Computational Aspects

Data standardization is still an open research field in multibody system (MBS) dynamics, and the lack of a standard neutral data format to encode information about MBS models (topology, geometry, rigid and flexible body data, applied forces, type of analysis, …) hinders the exchange and share of models between users of MBS software. Therefore, the MBS research community should start to address the interoperability needs in this field as soon as possible. This article presents some foundations for that task: the requirements for a neutral data format are presented, the current interoperability state in MBS dynamics is evaluated, and two data modeling technologies (STEP and XML) are compared in order to develop a robust yet easy-to-use neutral data format for multibody systems. An XML-based prototype implementation of such a format is proposed, demonstrating the excellent aptitudes of XML for this task. Finally, guidelines for future standardization of multibody system data are given. In conclusion, both STEP and XML should be combined in the future to solve the interoperability problems in multibody system dynamics in an effective way.

In large, rigid multibody dynamics problems with friction and contact, encountered for instance in granular flows, one can witness distinctly different system-level dynamics. This contribution concentrates on the case of fluid-like behavior of large multibody dynamics systems such as granular materials, when the system experiences large strains. The results reported herein draw on computer simulation; on the one hand, we solve the Newton-Euler equations of motion, which govern the evolution of multibody dynamics system featuring frictional contact. On the other hand, we solve the Navier-Stokes equations which describe the time evolution of fluids. To demonstrate the similarities and differences between the multibody and fluid dynamics we consider three problems modeled and solved using different methods; (i) a compressibility test; (ii) the classical dam break problem, and (iii) the dam break simulation with an obstacle. These experiments provide insights into conditions under which on can expect similar characteristics from multibody and fluid dynamics systems governed by manifestly different equations of motion and solved by vastly different numerical solution methods. The models and simulation platform used are publicly available and part of an open source code called Chrono. Both the multibody and fluid dynamics simulations are carried out using GPU computing.

Redundant constraints in multibody system (MBS) models, reflected by a singular constraint Jacobian, impair the efficient dynamics simulation. In particular, kinematic loop constraints are often found to be permanently redundant. This problem is commonly attacked numerically by decomposing the constraint Jacobian either at every simulation time step or beforehand in an admissible assembly (assuming that the redundancy is permanent). This paper presents a method for the elimination of permanently redundant loop closure constraints, which, instead of numerically decomposing the constraints, relies on the geometric characterization of kinematic loops comprising lower kinematic pairs. In particular, the invariant vector space of velocities of a kinematic loop is taken into account, which can be determined as the sum of Lie (screw) algebras of two subchains of a kinematic loop. The actual reduction is achieved by restricting the constraints to this space. The presented method does not interfere with the actual generation of constraints but can be considered as a preprocessing step of MBS models. It is numerically robust and only uses a geometrically exact model. The method is able to completely eliminate redundant loop constraints for “nonparadoxical” single-loop mechanisms and applies conservatively to multiloop MBS. The presented method only requires information (vectors, matrices) that is readily available in any MBS simulation package. The only numerical operations involved are cross products and a singular value decomposition of a low dimensional matrix.

This paper describes the application of modeling of multibody systems to high-precision mechanical gears. Although the context of the paper is applied in nature, the objective is to present recent advances in computer-aided modeling of multibody systems with three-dimensional (3-D) contacts and to explore the limits of the technology, theory, and methods. Gear systems are traditionally modeled as single dimensional object with a simple transformation ratio. However, when we are interested in high precision gear systems, all joints and flexibility of gear shafts bearings are accounted for in the model. In this paper we discuss challenges in using recent advances in computer-aided multibody modeling (CAMM) with 3-D contacts applied to the design of a steering gear system. These advances in CAMM have made it possible to produce detailed dynamic simulation of constrained multibody motion in a much shorter time than the traditional Finite Element approach. Extracting position and velocity information from simulation of multibody models using CAMM is routine work today save for the most complex systems. However, extracting information about contact force between rigid bodies during impact subsequent rolling and sliding motions is a more challenging problem. This paper discusses these challenges and presents possible numerical and modeling solutions. Gears move in three-dimensional space, their surfaces undergoing continuous contact and separation; we distinguish between large dominant motion and microscopic motion along constraints axis. In an effort to understand motion of steering gear at the microscopic level, we modeled gears as multibody systems with their surfaces in a dynamic state of contact and separation. These models capture the dynamics of gears traveling distances of around 50 microns in less than 10 milliseconds. We applied newly developed analytic, numeric, and computer-aided tools to gain insight into gear forces, specifically, those forces responsible for the inception of mechanical knock in steering gear. The results provided a 3-D view of the sequence of events of gears with their surfaces in motion, in sudden contact, sliding, and the resulting force generation. The models include effect of friction on impact mechanics of the gears. First we briefly present a discussion of the state-of-knowledge in CAMM with three-dimensional surface contacts. Computer aided engineering requirements necessary to solve the problem are discussed together with computer hardware, software, and analysis methods that are necessary for successful modeling of multibody gear systems with 3-d contacts. We finally present a model of a generic gear system that demonstrates both the potential and the limitations of the current technology.

Over the years a considerable effort has been focused on the development of methods and computational algorithms for mathematical modelling and simulation of multibody systems. The methods are mainly concerned with effective formulation and numerical treatment of the dynamical equations of a general multibody system.

Stochastic constitutive modeling of elastic-plastic materials with uncertain properties

With the advent of multibody dynamics software (ADAMS) , it has become one of the main simulation techniques to build a multibody system (MBS) in order to evaluate the dynamics performance. The modeling of some component parts such as anti-roll bars and torsion beam rear suspensions is always difficult for the unique structural and non-linear characteristics.A joint force actuator based on multibody dynamics was introduced to represent the component within suspension systems. The kinematics analysis of the torsion beam rear suspension was carried out to validate accuracy and rationality by means of joint force actuators in ADAMS/Car. The result shows that the joint force actuator can be used in the MBS modeling and simulation analysis of non-linear characteristics conveniently.

Multibody systems in planar motion are modelled as two or more rigid components that are connected and can move relative to each other. The dynamics of such multibody systems in planar motion in a central gravitational force field is analysed. The equations of motion of the system include the equations for the orbital motion of the bodies, the orientation (attitude) of the assembly, and the relative orientation (shape) of the bodies with respect to each other. Dynamic coupling between these degrees of freedom gives rise to complex dynamical systems that are usually not integrable. Relative equilibria, corresponding to circular orbits of the multibody system, are obtained. The free dynamics has a symmetry due to a cyclic coordinate. Routh reduction is carried out to eliminate this coordinate and obtain the reduced dynamics. The stability of the relative equilibria is analysed using the Routh stability criterion when it is applicable; an expansion of the Hamiltonian in normal form is used otherwise. We apply the general results to a multibody system consisting of two hinged planar bodies, each modelled as a rigid massless link with a point mass at one end with their other ends connected by a hinge joint. We obtain the relative equilibria of this model, and carry out a stability analysis for the relative equilibria. Numerical simulations using a symplectic integrator are carried out for perturbations to these relative equilibria, to confirm their stability properties.

A new method of multibody system dynamics, named discrete time transfer matrix method of multibody system (MS-DT-TMM) is developed and widely used in engineering field in recent years. When using this method for the problems of multibody system dynamics, the global dynamics equations of the system are not needed and the orders of involved system matrices are always small. The advance in MS-DT-TMM in recent years has been studied in this paper as following: 1 Discrete time transfer matrix method of rigid-flexible multibody system is developed to study rigid-flexible multibody system dynamics. 2 Mixed method of MS-DT-TMM and finite element method is developed to study rigid-flexible multibody system dynamics. 3 Mixed method of MS-DT-TMM and multibody system dynamics method is developed to study rigid-flexible multibody system dynamics. 4 Riccati discrete time transfer matrix method of multibody system is developed to increase the computation speed and the computation stability. 5 Discrete time transfer matrix method of controlled multibody system is developed to study dynamics of controlled multibody system. Several corresponding numerical examples are done to validate these proposed methods. All this implies that the proposed methods provide a powerful tool for the study of multibody system dynamics.

Fuzzy Expected Value Analysis of an Industrial Grinding Process

Developing the analytical model of a multibody system is often the initial step in control and optimization. The analytical model (equations of motion) describes a system’s time evolution under specified forcing conditions. Although developing these equations is easy for simple systems, this process becomes more complex for systems composed of multiple bodies. Deriving equations of motion for complex multibody systems requires specialized expertise in multibody dynamics, is time-consuming, and is susceptible to error. To address this issue, this paper presents an open-source, easy-to-use, systematic framework to derive symbolic equations of motion in both Python and MATLAB using the joint coordinate formulation. This formulation results in a set of ordinary differential equations that use the minimum set of coordinates needed to model a system. The symbolic representation provides better insight into the influence of design parameters on system performance, facilitates sensitivity analysis and parameter studies, and supports direct implementation of control and optimization routines. The tool enables numerical simulation for specified parameter sets, is modular for straightforward integration with other tools and libraries, and allows incorporation of hydrodynamics, mooring, and other external forces. The result is a reproducible, extensible pipeline for modeling, simulation, and design of complex multibody systems. The proposed tool is versatile and can be applied to domains such as robotics, control, and design. In addition, we integrated external libraries that provide capabilities for modeling offshore systems such as underwater robots and marine energy converters.

A multiresolution approach is presented to study the uncertainty propagation problem in dynamic systems. The proposed approach is an extension of the generalized polynomial chaos and involves a separation of random variables from deterministic ones in the solution algorithm for a stochastic dierential equation. The random variables are expanded in various local polynomial expansions which are further blended together to obtain a global approximation for the random state variables. These polynomials are associated with the assumed probability density functions for the input variables. Galerkin projection is used to generate a system of deterministic dierential equations for the expansion coecients. We rst present this method to linear systems with parametric uncertainties, and subsequently we generalize it to nonlinear systems with parametric and initial condition uncertainties. The eectiveness of the proposed method is demonstrated by considering dierent numerical examples.

It is an exceptional success when multibody dynamics researchers Multibody System Dynamics journal one of the most highly ranked journals in the last 10 years. In the inaugural issue, Professor Schiehlen wrote an interesting article explaining the roots and perspectives of multibody system dynamics. Professor Shabana also wrote an interesting article to review developments in flexible multibody dynamics. The application possibilities of multibody system dynamics have grown wider and deeper, with many application examples being introduced with multibody techniques in the past 10 years. In this paper, the development of multibody dynamics is briefly reviewed and several applications of multibody dynamics are described according to the author’s research results. Simulation examples are compared to physical experiments, which show reasonableness and accuracy of the multibody formulation applied to real problems. Computer simulations using the absolute nodal coordinate formulation (ANCF) were also compared to physical experiments; therefore, the validity of ANCF for large-displacement and large-deformation problems was shown. Physical experiments for large deformation problems include beam, plate, chain, and strip. Other research topics currently being carried out in the author’s laboratory are also briefly explained.

The dynamics of a holonomically constrained rigid body can be modeled by Newton-Euler equations subjected to geometric constraints. This is frequently formulated as a differential-algebraic equation (DAE) system of index 1. In multibody system (MBS) dynamics it is common (1) to numerically solve this system by means of integration schemes for ordinary differential equations, and (2) to treat the rigid body motion on the direct product Lie group $$\textit{SO}\,(3) \times {\mathbb {R}}^{3}$$ , although rigid body motions form the semidirect product Lie group $$\textit{SE}\,(3) $$ . It is has been observed that the constraint satisfaction depends on which Lie group is used as configuration space (c-space). In this paper the problem is considered from a geometric perspective. It is shown that the constraints are exactly satisfied by a numerical integration scheme if they define a subgroup of the c-space. The subgroups of $$\textit{SE}\,(3) $$ have a significance for modeling mechanical systems, including lower kinematic (Reuleaux) pairs and are implicitly used in MBS modeling. It is concluded that $$\textit{SE}\,(3) $$ is the appropriate c-space for numerical DAE modeling of a constrained rigid body. This result does not immediately apply to MBS, however.

This text is a classical and complete book on rigid multibody dynamics. Some of the topics covered include: various angular orientation descriptions, including Euler angles, Bryan angles, and Euler-Rodrigues parameters; quaternion algebra; virtual power; graph theory; and vibration analysis of chains of bodies. The textbook is an excellent reference, containing valuable material for graduate students and researchers. The chapters have an average of six example and problem sets, for which solutions are provided. Aside from the lack of problems, the book can easily be used as a text for a graduate course on the dynamics of rigid multibody systems.