Hybrid coordinate approach for the modelling and simulation of multibody systems

The main goal of the present work is to provide an add-on scheme for the formulation of multibody dynamics, based on natural coordinates, in regard to ideally balanced rigid bodies with high rotational spin, e.g. gyroscopes. The underlying aim of this approach is to achieve higher numerical accuracy whenever the preferred axis of rotation coincides with the balanced main axis of the body. This will be achieved by seperating the spin of the balanced rigid body along the denoted axis as an additional angular coordinate, whereas the other rotations will be covered by a carried frame, parameterized via natural coordinates. At the same time the carried frame provides a link to the existing modelling framework in terms of natural coordinates, enabling a straightforward implementation into existing multibody systems (e.g. rotary crane [2]). (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupled differential algebraic equations in the simulation of flexible multibody systems with hydrodynamic force elements

The mathematical modelling of elastohydrodynamic bearings in combustion engines leads to a coupled system of (partial) differential algebraic equations, which is represented by a flexible multibody system model of the engine including crankshaft and bearing and by the Reynolds equation that describes the non-linear effects in the fluid film. The hydrodynamic forces depend strongly on the position and the elastic deformation of crankshaft and bearing shell. We discuss the influence of the spatial discretization on accuracy and numerical effort. Since a fine spatial discretization substantially slows down the numerical solution, we propose a semi-analytical method based on singular perturbation theory to speed-up time integration. Numerical tests for a simplified benchmark problem illustrate the advantages of this approach. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parallel simulation of large scale multibody systems

Virtual prototyping plays an ever increasing role in the engineering disciplines. Nowadays, engineers can rely on powerful tools like object oriented modeling languages, e.g., Modelica. Models written in this language can be simulated by open source software as well as commercial tools. The advantage of this approach is that the engineers can concentrate themselves on modeling, whereas the numerical intricacies of the simulation are handled by the software. On the other hand the simulations are usually slower than implementations which are parallelized and optimized manually. This can lead to computation times which are infeasible in practice, e.g., when a real time simulation is necessary for a hardware-in-the-loop simulation. In this contribution we are concerned with speeding up such automated simulations by parallelization (on desktop hardware as well as HPC systems). We examine the parallelism across the system approaches. Additionally, the influence of the problem formulation on the simulation time is discussed. The implemented methods are demonstrated on engineering examples. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational integrators of higher order for flexible multibody systems

In this paper we present a time stepping scheme which is based on a variational integrator. This higher-order time stepping scheme includes constraints and a viscoelastic material formulation. A variational integrator is structure-preserving which results from using a discrete variational principle. Therefore, a variational integrator always takes the form of discrete EULER-LAGRANGE equations or the equivalent position-momentum equations. In this framework, we consider the motion of a flexible rope with non-holonomic constraints by the LAGRANGE-multiplier technique. The time stepping scheme is derived from a space-time discretization of HAMILTON's principle. The space discretization is based on one-dimensional linear LAGRANGE polynomials, whereas the time discretization is based on higher-order polynomials and higher-order quadrature rules. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A simulation approach to reliability analysis of weapon systems

We report a modeling simulation approach to analyse weapon systems reliability. The introduced functional diagram generalises the logic diagram allowing the replication on the functioning mode of system components. To handle the functional diagram, the availability and connection rules are also introduced. Based on the functional diagram, a simulation model is outlined and a case study, the propulsion system of a Mine Hunter, is included.     

A Petri net approach to the modelling and analysis of flexible manufacturing systems

In this paper we present an approach for modelling and analyzing flexible manufacturing systems (FMSs) using Petri nets. In this approach, we first build a Petri net model (PNM) of the given FMS in a bottom-up fashion and then analyze important qualitative aspects of FMS behaviour such as existence/absence of deadlocks and buffer overflows. The basis for our approach is a theorem we state and prove for computing the invariants of the union of a finite number of Petri nets when the invariants of the individual nets are known. We illustrate our approach using two typical manufacturing systems: an automated transfer line and a simple FMS.A shorter version of this paper was presented at the 1st ORSA/TIMS Special Interest Conference on FMSs, University of Michigan, Ann Arbor, August 1984.     

A unified approach to controllability analysis for hybrid control systems

A unified approach is proposed for controllability analysis of a class of hybrid control systems, employing the controllability concept defined in [M. Tittus, B. Egardt, Control design for integrator hybrid systems, IEEE Transactions on Automatic Control 43 (4) (1998) 491–500; J.H. van Schuppen, A Sufficient Condition for Controllability of a Class of Hybrid Systems, in: LNCS, vol. 1386, Springer, 1998, pp. 374–383; Z. Yang, M. Verhaegen, Y.J. Wang, Z.J. Chen, Hybrid controllability of linear switched systems, in: Proceedings of 37th Control and Decision Conference, Tampa, FL, USA, December 1998, pp. 3920–3925]. The unified approach comprises global reachability analysis at the discrete event system level, local reachability analysis at the continuous-time dynamical system level and a merging of the results of these two methods using a discrete-path search algorithm. The proposed method is demonstrated useful for controllability analysis of complex hybrid control systems. The method is illustrated by analyzing the controllability of a linear switched system.     

Efficient integration of flexible multibody dynamics

The modelling of flexible multibody dynamics as finite dimensional Hamiltonian system subject to holonomic constraints constitutes a general framework for a unified treatment of rigid and elastic components. Internal constraints, which are associated with the kinematic assumptions of the underlying continuous theory, as well as external constraints, representing the interconnection of different bodies by joints, can be accounted for in a likewise systematic way. The discrete null space method developed in [0] provides an energy-momentum conserving integration scheme for the DAEs of motion of constrained mechanical systems. It relies on the elimination of the constraint forces from the discrete system along with a reparametrisation of the nodal unknowns. The resulting reduced scheme performs advantageously concerning different aspects: the constraints are fulfilled exactly, the condition number of the iteration matrix is independent of the time step and the dimension of the system is reduced to the minimal possible number saving computational costs. A six-body-linkage possessing a single degree of freedom is analysed as an example of a closed loop structure. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A nonlinear finite element framework for flexible multibody dynamics

We present a uniform treatment of rigid body dynamics and nonlinear structural dynamics. The advocated approach is based on a rotationless formulation of rigid bodies, nonlinear beams and shells. In this connection, the specific kinematic assumptions are taken into account by the explicit incorporation of holonomic constraints. This approach facilitates the straightforward extension to flexible multibody dynamics by including additional constraints due to the interconnection of rigid and flexible bodies. We further address the design of energy-momentum schemes for the stable numerical integration of the underlying finite-dimensional Hamiltonian systems. To demonstrate the superior numerical performance of the proposed methodology, the numerical examples deals with a multibody system containing both rigid and flexible bodies undergoing large deformations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new approach to the incorporation of servo constraints in multibody dynamics

A new index reduction approach is developed to solve the servo constraint problems [2] in the inverse dynamics simulation of underactuated mechanical systems. The servo constraint problem of underactuated systems is governed by differential algebraic equations (DAEs) with high index. The underlying equations of motion contain both holonomic constraints and servo constraints in which desired outputs (specified in time) are described in terms of state variables. The realization of servo constraints with the use of control forces can range from orthogonal to tangential [3]. Since the (differentiation) index of the DAEs is often higher than three for underactuated systems, in which the number of degrees of freedom is greater than the control outputs/inputs, we propose a new index reduction method [1] which makes possible the stable numerical integration of the DAEs. We apply the proposed method to differentially flat systems, such as cranes [1,4,5], and non-flat underactuated systems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of vibrations in large flexible hybrid systems

The mathematical model of a real flexible elastic system with distributed and discrete parameters is considered. It is a partial differential equation with non-classical boundary conditions. Complexity of the boundary conditions makes it impossible to find exact analytical solutions. To address the problem, we use the asymptotical method of small parameters together with the numerical method of normal fundamental systems of solutions. These methods allow us to investigate vibrations, and a technique for determination of complex eigenvalues of the considered boundary value problem is developed. The conditions, at which vibration processes of different characteristics take place, are defined. The dependence of the vibration frequencies on the physical parameters of the hybrid system is studied. We show that introduction of different feedbacks into the system allows one to control the frequency spectrum, in which excitation of vibrations is possible.     

An application of a hybrid approach to modeling a flexible manufacturing system

Both analytic and simulation models were used to analyze the capabilities and requirements of an automated circuit card manufacturing system. Analytic models were used to determine the sensitivity of the measures of effectiveness (MOEs) to various design parameters. This analysis gave approximate results and bounded the range of input parameters for the simulation model. A detailed simulation model was required for use during both the design and production phases of the project. This simulation model incorporated only those variables to which the MOEs are most sensitive, and provided additional features to observe system behavior. The benefits and appropriate uses for each class of models are discussed.     

Computer system for kinematic and dynamic analysis synthesis of rigid and flexible multibody systems

In the paper an overview of a general numerical algorithm and program system library for deriving the kinematic constraint equations and dynamic equations of motion, as well as, computation of their first and second order partial derivatives with respect to kinematic parameters of motion, design parameters and mass and inertia characteristics for rigid and flexible multibody systems is presented. These are the main basic computational modules for implementation of kinematic and dynamic synthesis, optimization and design. The main theoretical basis consists in matrix methods for deriving the kinematic constraints and dynamic equations, as well as, the generalized Newton – Euler dynamic equations for rigid and flexible bodies, and finite element discretization in relative coordinates. Block–scheme of the computational procedures and problem oriented program compilation is presented. An example of kinematic synthesis of six–link path generating mechanism with singular points is presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient simulation of general stochastic hybrid systems

General Stochastic Hybrid System (SHS) are characterised by Stochastic Differential Equations (SDEs) with discontinuities and Poisson jump processes. SHS are useful in model based design of Cyber-Physical System (CPS) controllers under uncertainty. Industry standard model based design tools such as Simulink/Stateflow® are inefficient when simulating, testing, and validating SHS, because of dependence on fixed-step Euler–Maruyama (EM) integration and discontinuity detection. We present a novel efficient adaptive step-size simulation/integration technique for general SHSs modelled as a network of Stochastic Hybrid Automatons (SHAs). We propose a simulation algorithm where each SHA in the network executes synchronously with the other, at an integration step-size computed using adaptive step-size integration. Ito’ multi-dimensional lemma and the inverse sampling theorem are leveraged to compute the integration step-size by making the SDEs and Poisson jump rate integration dependent upon discontinuities. Existence and convergence analysis along with experimental results show that the proposed technique is substantially faster than Simulink/Stateflow®when simulating general SHSs.     

An automated modelling approach for dynamic performance evaluation of mechatronic multibody systems

An automated modelling approach of mechatronic multibody systems is presented in this paper. The proposed approach uses some object-oriented GUI modules to automatically generate the dynamic equations for different domains, solve them with numerical methods to obtain approximate solutions, and then evaluate the dynamic performances of the systems. By systematically defining an elementary linear graph and its general rules, the modules of mechanical parts and kinematic pairs can be modelled independently of special systems by the extensible elementary linear graph (EELG) method, and the member's dynamic equations can be derived by topology matrices operation. Some major advantages of this procedure are as follows: the combinations of mechanical components could be dealt with as an integrated member and directly assembled with other modules, the topology structure of individual members are described by elementary cutset and circuit matrices derived from the elementary linear graph, rotation vector is used to express angular variables for analysing rotation and translation with same linear graph; the function vertices, opening edges, and self-closed edges are first introduced to elementary linear graph of kinematic pairs modelling, multiport for special mechanical members and different ports for various energy domains are defined, and relation equations linking the ports are given for interdisciplinary domains, so that the modules could have the characteristics of reapplication and extensibility. For two typical cases, the approach carried out on a Modelica/Dymola software platform is proved feasible by comparing the results using the EELG method with those of the conventional approach.