Solution of three key problems for massive parallelization of multibody dynamics

This paper deals with the solution of three key problems for enabling the consideration of the massive parallelization for multibody dynamics. Instead of classical joints, the flexible joints with appropriate stiffness and damping are introduced in the multibody system, which enables to derive completely decoupled equations of motion and as a consequence to simulate them using massive parallel computing. Such formulation causes the uprise of high frequencies in the solution. Therefore, the heterogeneous multiscale method is used for numerical integration. However, three key problems had to be solved prior to such multibody simulation could be considered for further development. The problems are: the clear distinction of macro-model and micro-model in order to really reduce the eigenvalues of the integrated model, the completely decoupled procedure for estimation of reaction forces for each micro-integration restart, and the suitable choice of microintegration time length.     

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.     

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.     

Shock response analysis of hard disk drive using flexible multibody dynamics formulation

Recent development of the shock analysis on the HDD is briefly reviewed. A flexible multi-body dynamics formulation is developed to simulate the shock response of the HDD. If one component in the HDD is changed, only mode shapes and frequencies of that component should be re-calculated and then used to obtain the system’s response. Steady state Reynolds equation is solved to obtain the air pressure on the slider and disk for various slider positions. An air pressure table is formed and used to model the non-linear air bearing during the simulation. Responses of flying height for different direction and shock duration time are analyzed. Results show that the flying state of the slider is more sensitive to the shock with shorter duration time.     

An efficient formulation for flexible multibody dynamics using a condensation of deformation coordinates

Multibody System Dynamics - In this work a new approach to deal with non-ideal operative aspects of spatial revolute joints by means of a three-dimensional finite element analysis (3D-FEA) is...     

Shock analysis of non-operating hard disk drives based on a multibody dynamic formulation

Dropping, striking, or bouncing a hard disk drive (HDD) against a hard surface can damage it internally without external evidence of damage. Contact with a hard ground will lift the slider off the disk surface and then slap back on the surface. A drive that is subjected to this type of shock may fail on initial use or the reliability of the drive may degrade over time. Therefore, industry has a lot of interest on the shock conditions that cause a slider to lift off the disk surface. Finite element software such as ANSYS/LS-DYNA is often used to analyze this shock problem. However, this method consumes a great amount of time. It is also difficult to perform design parameter studies because it requires re-analysis of the model of the entire HDD system when certain design variables are changed. This paper presents a flexible multi-body dynamics formulation to analyze the shock problem of non-operating HDDs. Governing equations of motion of the voice coil motor (VCM)–actuators assembly and the disks–spindle system are derived using a Lagrangian formulation. By introducing constraint equations between the slider and the disk surface, the shock response of the whole HDD system has been obtained. Numerical results show that the method is reasonable and the acceleration amplitude which makes the slider lift off can be determined in a significantly shorter time than by the conventional approach. Finally, the effect of drive parameters on shock resistance, such as shock duration and slider resting location are analyzed.     

A superelement model based on Lagrangian coordinates for multibody system dynamics

This paper presents a general-purpose superelement model for computer-aided analysis of multibody systems subject to kinematic constraints. The superelement concept is defined and a method based on Lagrangian coordinates is developed to obtain superelement parameters. Differential equations of motion of a system consisting of superelements and other elements is developed. Constraints internal to the superelements do not appear in the resulting differential equations. This leads to a set of differential and algebraic equations that has fewer dimensions and that requires less CPU time than the existing general-purpose schemes. Two examples are presented to demonstrate the feasibility and efficiency of the model.     

A rotationless formulation of multibody dynamics: Modeling of screw joints and incorporation of control constraints

In the present work, a new energy-momentum conserving time-stepping scheme for multibody systems comprising screw joints is developed. In particular, it is shown that the underlying rotationless formulation of multibody dynamics along with a specific coordinate augmentation technique makes possible the energy-momentum discretization of the screw pair. In addition to that, control (or servo) constraints are treated within the rotationless framework of multibody dynamics. The control constraints are used to partially prescribe the motion of a multibody system. In particular, control constraints, in conjunction with the coordinate augmentation technique, make possible to solve inverse dynamics problems by applying the present simulation approach.     

Adjoint sensitivity analysis of hybrid multibody dynamical systems

Multibody System Dynamics - Sensitivity analysis computes the derivatives of general cost functions that depend on the system solution with respect to parameters or initial conditions. This work...     

Coupling multibody dynamics and computational fluid dynamics on 8192 processor cores

This paper describes a method for the fully resolved simulation of particle laden flows. For this purpose, we discuss the parallelization of large scale coupled fluid structure interaction with up to 37 million geometrically modeled moving objects incorporated in the flow. The simulation is performed using a 3D lattice Boltzmann solver for the fluid flow and a so-called rigid body physics engine for the treatment of the objects. The numerical algorithms and the parallelization are discussed in detail. Furthermore, performance results are presented for test cases on up to 8192 processor cores running on an SGI Altix supercomputer. The approach enables a detailed simulation of large scale particulate flows that are relevant for many industrial applications.     

The development of a sliding joint for very flexible multibody dynamics using absolute nodal coordinate formulation

In this paper, a formulation for a spatial sliding joint is derived using absolute nodal coordinates and non-generalized coordinate and it allows a general multibody move along a very flexible cable. The large deformable motion of a spatial cable is presented using absolute nodal coordinate formulation, which is based on the finite element procedures and the general continuum mechanics theory to represent the elastic forces. And the nongeneralized coordinate, which is related to neither the inertia forces nor the external forces, is used to describe an arbitrary position along the centerline of a very flexible cable. Hereby, the non-generalized coordinate represents the arc-length parameter. The constraint equations for the sliding joint are expressed in terms of generalized coordinate and nongeneralized coordinate. In the constraint equations for the sliding joint, one constraint equation can be systematically eliminated. There are two independent Lagrange multipliers in the final system equations of motion associated with the sliding joint. The development of this sliding joint is important to analyze many mechanical systems such as pulley systems and pantograph-catenary systems for high speed-trains.     

Energy-momentum conserving integration of multibody dynamics

A rotationless formulation of multibody dynamics is presented, which is especially beneficial to the design of energy-momentum conserving integration schemes. The proposed approach facilitates the stable numerical integration of the differential algebraic equations governing the motion of both open-loop and closed-loop multibody systems. A coordinate augmentation technique for the incorporation of rotational degrees of freedom and associated torques is newly proposed. Subsequent to the discretization, size-reductions are performed to lower the computational costs and improve the numerical conditioning. In this connection, a new approach to the systematic design of discrete null space matrices for closed-loop systems is presented. Two numerical examples are given to evaluate the numerical properties of the proposed algorithms.     

Graph theoretic foundations of multibody dynamics

This second part of a two part paper uses concepts from graph theory to obtain a deeper understanding of the mathematical foundations of multibody dynamics. The first part (Jain in Graph theoretic foundations of multibody dynamics. Part I. Structural properties, 2010) established the block-weighted adjacency (BWA) matrix structure of spatial operators associated with serial- and tree-topology multibody system dynamics, and introduced the notions of spatial kernel operators (SKO) and spatial propagation operators (SPO). This paper builds upon these connections to show that key analytical results and computational algorithms are a direct consequence of these structural properties and require minimal assumptions about the specific nature of the underlying multibody system. We formalize this notion by introducing the notion of SKO models for general tree-topology multibody systems. We show that key analytical results, including mass-matrix factorization, inversion, and decomposition hold for all SKO models. It is also shown that key low-order scatter/gather recursive computational algorithms follow directly from these abstract-level analytical results. Application examples to illustrate the concrete application of these general results are provided. The paper also describes a general recipe for developing SKO models. The abstract nature of SKO models allows for the application of these techniques to a very broad class of multibody systems.     

Graph theoretic foundations of multibody dynamics

This is the first part of two papers that use concepts from graph theory to obtain a deeper understanding of the mathematical foundations of multibody dynamics. The key contribution is the development of a unifying framework that shows that key analytical results and computational algorithms in multibody dynamics are a direct consequence of structural properties and require minimal assumptions about the specific nature of the underlying multibody system. This first part focuses on identifying the abstract graph theoretic structural properties of spatial operator techniques in multibody dynamics. The second part paper exploits these structural properties to develop a broad spectrum of analytical results and computational algorithms.Towards this, we begin with the notion of graph adjacency matrices and generalize it to define block-weighted adjacency (BWA) matrices and their 1-resolvents. Previously developed spatial operators are shown to be special cases of such BWA matrices and their 1-resolvents. These properties are shown to hold broadly for serial and tree topology multibody systems. Specializations of the BWA and 1-resolvent matrices are referred to as spatial kernel operators (SKO) and spatial propagation operators (SPO). These operators and their special properties provide the foundation for the analytical and algorithmic techniques developed in the companion paper.We also use the graph theory concepts to study the topology induced sparsity structure of these operators and the system mass matrix. Similarity transformations of these operators are also studied. While the detailed development is done for the case of rigid-link multibody systems, the extension of these techniques to a broader class of systems (e.g. deformable links) are illustrated.     

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.     

On the use of absolute interface coordinates in the floating frame of reference formulation for flexible multibody dynamics

In this work a new formulation for flexible multibody systems is presented based on the floating frame formulation. In this method, the absolute interface coordinates are used as degrees of freedom. To this end, a coordinate transformation is established from the absolute floating frame coordinates and the local interface coordinates to the absolute interface coordinates. This is done by assuming linear theory of elasticity for a body’s local elastic deformation and by using the Craig–Bampton interface modes as local shape functions. In order to put this new method into perspective, relevant relations between inertial frame, corotational frame and floating frame formulations are explained. As such, this work provides a clear overview of how these three well-known and apparently different flexible multibody methods are related. An advantage of the method presented in this work is that the resulting equations of motion are of the differential rather than the differential-algebraic type. At the same time, it is possible to use well-developed model order reduction techniques on the flexible bodies locally. Hence, the method can be employed to construct superelements from arbitrarily shaped three dimensional elastic bodies, which can be used in a flexible multibody dynamics simulation. The method is validated by simulating the static and dynamic behavior of a number of flexible systems.