Dynamics of spatial structure-varying rigid multibody systems

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.

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.

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.

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.

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 numerical solution procedure for multibody dynamics (MBD) systems is termed a staggered MBD solution procedure that solves the generalized coordinates in a separate module from that for the constraint force. This requires a reformulation of the constraint conditions so that the constraint forces can also be integrated in time. A major advantage of such a partitioned solution procedure is that additional analysis capabilities such as active controller and design optimization modules can be easily interfaced without embedding them into a monolithic program. After introducing the basic equations of motion for MBD system in the second section, Section 3 briefly reviews some constraint handling techniques and introduces the staggered stabilized technique for the solution of the constraint forces as independent variables. The numerical direct time integration of the equations of motion is described in Section 4. As accurate damping treatment is important for the dynamics of space structures, we have employed the central difference method and the mid-point form of the trapezoidal rule since they engender no numerical damping. This is in contrast to the current practice in dynamic simulations of ground vehicles by employing a set of backward difference formulas. First, the equations of motion are partitioned according to the translational and the rotational coordinates. This sets the stage for an efficient treatment of the rotational motions via the singularity-free Euler parameters. The resulting partitioned equations of motion are then integrated via a two-stage explicit stabilized algorithm for updating both the translational coordinates and angular velocities. Once the angular velocities are obtained, the angular orientations are updated via the mid-point implicit formula employing the Euler parameters. When the two algorithms, namely, the two-stage explicit algorithm for the generalized coordinates and the implicit staggered procedure for the constraint Lagrange multipliers, are brought together in a staggered manner, they constitute a staggered explicit-implicit procedure which is summarized in Section 5. Section 6 presents some example problems and discussions concerning several salient features of the staggered MBD solution procedure are offered in Section 7.

The principle of virtual work is used to derive the Euler-Lagrange equations of motion in order to describe the dynamics of multibody android systems. The constrained variational equations are in fact differential-algebraic equations of high index and are cast as ordinary differential equations through differentiation of the constraint equations. The integration routine LSODAR and the fourth-order Runge-Kutta method are used to compute the generalized coordinates, their time derivatives and the body forces of two android models. The graphs of the constraint forces reveal the whiplash effect on the neck and that the stiffness of both multibody systems is due to large magnitude impulsive forces experienced by many bodies simultaneously.

Equations of motion for multibody systems with holonomic constraints are index-3 differential-algebraic equations (DAEs). Velocity constraint equations can be obtained by differentiating the position constraint equations with respect to time. When the velocity constraint equations are combined with index-3 equations of motion, the equations of motion in the form of index-2 over-determined DAEs can be obtained. HHT-SI2 method, which is extended from HHT (Hilbert-Hughs-Taylor) method in the community of structural dynamics and used for the numerical integration of equations of motion in the form of index-2 over-determined DAEs, is improved. Firstly, only the correction term of the HHT-SI2 method is improved, and two unknown vectors are introduced as before. The condition number of Jacobian matrix in Newton-Raphson method is decreased during the numerical solution of nonlinear equations from discretization, and then the limitation of step-size for the numerical integration method is reduced. Secondly, the form of the correction term in HHT-SI2 method is completely changed, and only one unknown vector is introduced. The quantity of equation in nonlinear equations form discretization is decreased, and the integration speed of the method is increased obviously. Then the validity of the two improved methods is verified by numerical experiments, and the improved methods are compared with HHT-SI2 method in the view of integration speed and step-size limitation. Numerical experiments also show that the two improved methods are second-order accuracy and numerical damping can be controlled. In the end, the two improved methods are also analyzed and compared with other existed similar methods in the conclusion section. The advantages of the new methods are pointed out.

Computational structural analysis of spatial multibody systems based on mobility criteria

Design optimization of dynamic flexible multibody systems using the discrete adjoint variable method

In this paper, we consider a new direction for generating and simultaneously solving equations of motion for dynamical systems, using automatic differentiation. We overview the current computational approaches for solving multibody dynamics problems and discuss several choices for equation of motion formulation. We present an operator-overloading method for generating equations of motion automatically via Lagrange’s Equations and solve them in a direct fashion. Several numerical examples are presented to demonstrate the accuracy and efficiency of the method for simulating the motion of multibody dynamical systems.

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.

Proposed is a time-marching al- gorithm to solve a nonlinear system of com- plementarity equations: Pi(xj) ≥ 0, Qi(xj) ≥ 0, Pi(xj)Qi(xj )= 0, i, j = 1,...,n, resulting from a discretization of nonlinear obstacle problem. We transform the above nonlinear complemen- tarity problem (NCP) into a nonlinear algebraic equations (NAEs) system: Fi(xj )= 0 with the aid of the Fischer-Burmeister NCP-function. Such NAEs are semi-smooth,highlynonlinearand usu- ally implicit, being hard to handle by the Newton- like method. Instead of, a first-order system of ODEs is derived through a fictitious time equa- tion. The time-stepping equations are obtained by applying a numerical integration on the resul- tant ODEs, which are derivative-free and do not need the inverse of any matrix. The computa- tional cost is thus greatly reduced. The numerical examples of Bratu, von Karman and other elliptic equationsareusedto demonstratethatthenewfic- titious time integration method (FTIM) is highly efficient to calculate the obstacle problems. Keyword: Nonlinear Obstacle Problem, Non- linear complementarity problem, Nonlinear alge- braic equations, Iterative method, Elliptic equa- tions, Fictitious time integration method (FTIM)

This research extends the generalized recursive formulas for the rigid body dynamics to the flexible body dynamics using the backward difference formula (BDF) and the relative generalized coordinate. When a new force or joint module is added to a general purpose program in the relative coordinate formulations, the modules for the rigid bodies are not reusable for the flexible bodies. Since the flexible body dynamics handles more degrees of freedom than the rigid body dynamics does, implementation of the flexible dynamics module is generally complicated and prone to coding mistakes. In order to overcome the implementation difficulties, a virtual rigid body is introduced at every joint and force reference frames. A virtual flexible body joint is introduced between two body reference frames of the virtual and original bodies. Since the multibody system dynamics are formulated by highly nonlinear algebraic and differential equations and there are many different types of joints, a tremendous amount of computer implementation is required to develop a general purpose dynamic analysis program using the relative coordinate formulation. The implementation burden is relieved by the generalized recursive formulas. The notationally compact velocity transformation method is used to derive the equations of motion in the joint space. The terms in the equations of motion which are related to the transformation matrix are classified into several categories each of which recursive formula is developed. Whenever one category of the terms is encountered, the corresponding recursive formula is invoked. Since computation time in a relative coordinate formulation is approximately proportional to the number of the relative coordinates, computational overhead due to the additional virtual bodies and joints is minor. Meanwhile, implementation convenience is dramatically improved.

This article is concerned with the numerical and experimental investigations of impact-induced vibrations in highly non-linear complex multibody system applications. In order to accurately model the impact response of such complex systems, a computational procedure based on the integration of finite element and multibody system algorithms is used. This procedure, which employs the finite element floating frame of reference formulation, allows for defining arbitrary joint constraint and forcing functions, for accurately capturing the coupling between the rigid body motion and the elastic deformation, and for modeling the impact forces between rigid and flexible bodies in the multibody system model. The system equations of motion are developed using the augmented Lagrangian formulation that allows for the use of sparse matrix techniques. The algorithm used in this study ensures that the kinematic constraint equations are satisfied at the position, velocity, and acceleration levels. The impact-induced vibration of a highly non-linear press machine is numerically and experimentally examined in this study. In order to accurately model the dynamic behavior of such a press machine, it is necessary to model the transfer feeder elastic deformations resulting from the impulsive impact forces. The deformations of the transfer press system as well as those of its frame are modeled in this investigation using the floating frame of reference formulation that allows for effectively reducing the number of degrees of freedom by filtering out very high-frequency modes of vibration using component mode synthesis techniques. The formulations of several relative angle and angular velocity kinematic constraints that are required to model this multibody system are developed in order to model transfer feeder complex rotational components. The numerical results obtained using the three-dimensional multibody system transfer press model developed in this study are validated using experimental results.