Equilibrium analysis of multibody dynamic systems using genetic algorithm in comparison with constrained and unconstrained optimization techniques

The present paper describes a set of procedures for the solution of nonlinear equilibrium problems in complex multibody systems. To find the equilibrium position of the system, six different optimization algorithms are used to minimize the total potential energy (TPE) of the system and compared with respect to accuracy and efficiency. A computer program is developed to evaluate the equality constraints and objective function of a general multibody dynamic system to find the equilibrium condition. It is seen that the indirect methods have better results and converge faster. Also it is shown that the genetic algorithm (GA) results in a global optimum while the other methods converge to a local optimum.     

Impacts in multibody systems: modeling and experiments

This paper outlines a novel approach to the modeling and analysis of impact involving multibody systems. This approach is based on an analysis of energy absorption and restitution during impact, using a decomposition of the kinetic energy, which decouples the parts associated with the spaces of admissible and constrained motions of the underlying unilateral constraints. Such a decomposition turns out to be useful in the analysis of energy dissipation during impact, and leads to a generalized definition of the energetic coefficient of restitution, which targets particularly collisions in multibody systems. The applicability of the approach reported is investigated by conducting an experimental study on a robotic testbed. It is shown that impact between multibody systems is considerably affected not only by the local dynamics characteristics of the interacting bodies, but also the configuration of the whole multibody system. The results reported here show that our decomposition can offer a sound characterization of impact in several problems of multibody systems.     

Optimal design of controlled multibody dynamic systems for performance,robustness and tolerancing

The multidisciplinary design approach has gained increasing popularity in recent years due to its ability to deal with conflicting design requirements imposed by discipline-specific objectives. The traditional design process involving multiple disciplines is typically a sequential process where the design objectives are met one at a time in a sequence of designs. However, in doing so, unnecessary limitations are imposed on the design parameters and the final design is far from being optimal. The effectiveness of integrated design methodology has been proven and such designs are being obtained in many applications. However, most of the work in this area has been problem and/or system specific and does not address important manufacturing considerations, such as tolerance allocation, robustness with respect to machining tolerances, etc. The results presented in this paper are intended to contribution towards filling these gaps. In particular, the new approach will help designers avoid a common known pitfall of performance optimization, i.e. the fact that designs that are optimized for performance alone are notoriously sensitive to deviations from the nominal design. Thus, optimizing for performance alone leads to designs that fall below acceptable standards of robustness; they are also expensive to manufacture because the tolerances must be kept very tight to ensure acceptable performance. The approach presented here will allow the user to systematically tradeoff performance versus robustness and tolerancing concerns. A proof-of-concept example that was solved to evaluate this methodology is also presented in this paper. This example provides a convincing demonstration of the fact that small sacrifices in performance can yield huge benefits in the other areas, provided a methodology is available for making these tradeoffs in a systematic way. This especially can be used by designers in various fields such as automotive, aerospace, deployable structures, machine tools (including hexapods), robotic systems, precision machinery, etc.     

Rigid-elastic modeling of meshing gear wheels in multibody systems

In many applications in mechanical engineering, gear wheels are used to transmit power between rotating shafts and, therefore, the ability to incorporate them into multibody systems and to simulate contact between them has become an essential topic in multibody dynamics.However, in some applications gear wheels may not be considered as being perfectly rigid. Due to the effect of contact forces there occur relevant deformations in meshing teeth and it is required for a high quality of the analysis to introduce some elasticities in the model of meshing gear wheels. Therefore, in this work elastic elements between the teeth and the body of each gear wheel are considered. This approach is especially well suited for multibody systems since it is a compromise between a totally rigid model and a fully elastic model allowing the simulation of large motions with many revolutions while still important elasticities are considered. The teeth and the body of each gear wheel are still modelled as being rigid but they are connected to each other by elastic elements. In doing so, an efficient and physically motivated algorithm is described and implemented in order to find the effects of multi-tooth contact as well as backlash and left and right hand side contact of the meshing teeth. Some examples compare the simulation results of rigid, partially elastic and fully elastic models.     

Selection of modeling coordinates for forward dynamic multibody simulations

Modeling mechanical systems in a manner that allows the models to be simulated quickly is vital in many fields, such as real-time simulation and control. Modeling these systems using their symbolic equations, rather than the more widely-used numerical methods, generally produces faster solution times. However, the number, complexity, and computational efficiency of these equations is highly dependent upon which coordinate set was used to model the system. Most coordinate selection methods established thus far are based on the assumption that minimizing the number of modeling coordinates will produce models with faster simulation times. This paper will show that this technique is not always valid and proposes a new technique of selecting a system’s coordinates based on a series of heuristics. A large part of these heuristics will be established by closely analyzing a specific technique used to formulate a system’s equations, and the effect each step of this formulation process will have on the complexity of the final system equations.     

An integrated system for computer aided design and analysis of multibody systems

An integrated system for design and analysis of multibody systems has been developed and is described in this paper. The use of the system is demonstrated through the example provided. The system consists of a commercially available CAD program and a multibody system analysis code developed at the Royal Institute of Technology in Stockholm, These tools are integrated using a relational database, the structure of which was developed at Luleå University of Technology, in order to get a complete system for design and analysis of multibody systems. The main gains from the integrated system are the possibilities of using calculated component data like mass and moment of inertia from the CAD program in the simulation models, the automatic formulation of input files for the analysis code, and finally the visualization of simulation results using the surface data of the solid models. The interactive structured query language (ISQL) of the database management system provides the possibility of examining the components of the multibody system during the design work and before any simulation is performed.     

Dynamic simulation of crankshaft multibody systems

General multibody system approaches are often not sufficient for specific situations in applications to yield an efficient and accurate solution. We concentrate on the simulation of the crankshaft dynamics which is characterized by flexible bodies and force laws describing the interaction between the bodies. The use of the floating frame of reference approach in our model leads to an index-2 DAE system. The algebraic constraints originate from the reference conditions and the normalization equation for the quaternions. For the time integration of this system, two aspects have to be taken into account: firstly, for efficiency exploiting the structure of the system and using parallelization. Secondly, consistent initial values also with respect to a related index-3 system have to be computed in order to compute missing initial velocities and to reduce transient phenomena. The work of C.B. Drab, J.R. Haslinger, and R.U. Pfau is supported by the “Bundesministerium für Wirtschaft und Arbeit” and by the government of Upper Austria within the framework “Industrielle Kompetenzzentren und Netzwerke”.     

Interoperability and neutral data formats in multibody system simulation

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. Commemorative Contribution.     

real-time motion planning for multibody systems

The solution of constrained motion planning is an important task in a wide number of application fields. The real-time solution of such a problem, formulated in the framework of optimal control theory, is a challenging issue. We prove that a real-time solution of the constrained motion planning problem for multibody systems is possible for practical real-life applications on standard personal computers. The proposed method is based on an indirect approach that eliminates the inequalities via penalty formulation and solves the boundary value problem by a combination of finite differences and Newton–Broyden algorithm. Two application examples are presented to validate the method and for performance comparisons. Numerical results show that the approach is real-time capable if the correct penalty formulation and settings are chosen.     

Impulsive motion of multibody systems

This article uses the piecewise model and Kane’s method to present a procedure for studying impulsive motion of multibody systems. Impulsive motion occurs when the system is subject to either impulsive forces or impulsive constraints, or when subjected to both simultaneously. The Appellian classification of impulsive constraints and the corresponding equations of impulsive motion of the multibody system are discussed. The governing equations are derived based upon multibody formulation procedures developed by Huston. Constraint impulses associated with finite and impulsive constraints are incorporated into impact dynamical equations through the impulsive Lagrange multipliers. The kinetic energy change of the scleronomic multibody system due to the impact is derived. Newton’s impact law is treated as an impulsive constraint equation to study single-point frictionless collision between two multibody systems. Several examples are used to demonstrate and validate the procedure.     

Linearization and parametric vibration analysis of some applied problems in multibody systems

The paper deals with the application of the Runge–Kutta method for calculating steady-state periodic vibrations of the parametric vibration systems governed by linearized differential equations. The numerical calculation is also demonstrated by two models of multibody systems and measurements on real objects. Good agreement is obtained between the numerical and experimental results. Consequently, the obtained results can also be applicable to investigate other complicated models of multibody systems which perform the steady-state motions.     

Structural topology optimization of multibody systems

Flexible multibody dynamics (FMD) has found many applications in control, analysis and design of mechanical systems. FMD together with the theory of structural optimization can be used for designing multibody systems with bodies which are lighter, but stronger. Topology optimization of static structures is an active research topic in structural mechanics. However, the extension to the dynamic case is less investigated as one has to face serious numerical difficulties. One way of extending static structural topology optimization to topology optimization of dynamic flexible multibody system with large rotational and transitional motion is investigated in this paper. The optimization can be performed simultaneously on all flexible bodies. The simulation part of optimization is based on an FEM approach together with modal reduction. The resulting nonlinear differential-algebraic systems are solved with the error controlled integrator IDA (Sundials) wrapped into Python environment by Assimulo (Andersson et al. in Math. Comput. Simul. 116(0):26–43, 2015). A modified formulation of solid isotropic material with penalization (SIMP) method is suggested to avoid numerical instabilities and convergence failures of the optimizer. Sensitivity analysis is central in structural optimization. The sensitivities are approximated to circumvent the expensive calculations. The provided examples show that the method is indeed suitable for optimizing a wide range of multibody systems. Standard SIMP method in structural topology optimization suggests stiffness penalization. To overcome the problem of instabilities and mesh distortion in the dynamic case we consider here additionally element mass penalization.     

Combining analytical modeling,realistic simulation and real experimentation for the optimization of Monte-Carlo applications on the European Grid Infrastructure

Evaluating the performance of distributed systems through real experimentation is resource-consuming and by essence very difficult to reproduce. Conversely, analytical modeling and simulation facilitate investigation, but their level of realism needs to be evaluated to avoid misinterpretation. In this paper, we combine production experiments and realistic simulation for performance modeling and optimization of application workflows deployed on the European Grid Infrastructure (EGI), one of the largest distributed systems in the world. We use a validated simulator to (i) exhaustively evaluate an analytical model of the application makespan and (ii) study the influence and calibrate the parameters of the application workflow, in particular the checkpointing period. Experimental results show that the model fits the simulated makespan with a relative error of at most 15%, and that simulation allows us to validate analytical models in a more exhaustive manner than what is possible with production experiments. Results also show that, provided that the simulator is correctly validated and instantiated, simulation can be safely used for exhaustive parameter studies, allowing for a quick and fine tuning of sensitive application parameters.     

A multi-agent methodology for multi-level modeling of mechatronic systems

Mechatronic design aims to integrate the models developed during the mechatronic design process, in order to be able to optimize the overall mechatronic system performance. A lot of work has been done in the last few years by researchers and software developers to achieve this objective. However, the level of integration does not yet meet the purposes of mechatronic system designers, particularly when dealing with modeling changes. Therefore, new methodologies are required to manage the multi-view complexity of mechatronic design. In this paper, we propose a multi-agent methodology for the multi-abstraction modeling issue of mechatronic systems. The major contribution deals with proposing a new method for the decomposition of the multi-level design into agents linked with relationships. Each agent is representing an abstraction level and both agent and relationships are managed with rules. By considering an application to a piezoelectric energy harvesting system, we show how we associate agents, rules and inter-level relationships to multi-abstraction modeling. We also show how modeling errors are identified using this approach.     

Graph theoretic modeling and analysis of multibody planarmechanical systems

A methodology of modeling and analysis of planar mechanical systems is developed based on graph theoretic methods, with improvements in component models. The system model based on cutset and circuit topologies is used to derive a new hybrid cutset-circuit method of formulation of the equations of motion for planar systems. Computer-aided formulation is based on analysis of the substitution procedure mandated by the hybrid cutset-circuit formulation. A new graphical representation of the formulation process is introduced: substitution graphs. No special programming is needed for computer-aided formulation which can be achieved in a symbolic form using the off the shelf Maple symbolic mathematics system. Symbolic formulation requires only inputting the systems equations in an order and form as derived from the analysis of the hybrid formulation. An algorithm for symbolic formulation using Maple is given. A compact set of differential-algebraic equations results, which can be solved numerically. Some simple systems will result in closed-form solutions. A number of examples are given to illustrate the modeling and formulation. Numerical solutions are also given to demonstrate the effectiveness and correctness of the formulation procedure     

Nonlinear control of planar multibody systems in shape space

Modeling, reduction, and nonlinear control of planar multibody systems motivated by the classicalcat-fall problem and the practical problem of reorientation of free-floating multibody satellites with rotational joints using angular-momentum-preserving controls is studied. The system model considered is reduced by the first integral (the system angular momentum) resulting in a Hamiltonian system with a configuration space of relative joint angles (shape space). Reconstruction of dynamics is applied to modify the shape-space model and track the phase shift of the absolute angles. An important reachability result is then proved in the unreduced configuration space. Control synthesis can then be found in a feedback form, solving the reorientation problem completely. Surprisingly, the reachability result breaks down in the case of the planar coupled two-body system with zero angular momentum, proving that the cat-fall phenomenon is definitely nonplanar.     

From particle-mass to multibody systems: graph-theoretic modeling

The distinction between geometry and dynamic interactions is fundamental for the consistent dynamic analysis of physical systems. A unified treatment of such systems is possible when we adopt a hierarchical mathematical model with a consistent set of embedded abstractions. This new view is adopted in the general formulation strategy for obtaining a simplified dynamics model of mechanical systems. We show that there exists a consistent general extension from the model of constrained particle-mass systems (PMS) to the model of multibody systems (MBS) based entirely on graph-theoretic concepts     

Performance of automatic differentiation tools in the dynamic simulation of multibody systems

Within the multibody systems literature, few attempts have been made to use automatic differentiation for solving forward multibody dynamics and evaluating its computational efficiency. The most relevant implementations are found in the sensitivity analysis field, but they rarely address automatic differentiation issues in depth. This paper presents a thorough analysis of automatic differentiation tools in the time integration of multibody systems. To that end, a penalty formulation is implemented. First, open-chain generalized positions and velocities are computed recursively, while using Cartesian coordinates to define local geometry. Second, the equations of motion are implicitly integrated by using the trapezoidal rule and a Newton–Raphson iteration. Third, velocity and acceleration projections are carried out to enforce kinematic constraints. For the computation of Newton–Raphson’s tangent matrix, instead of using numerical or analytical differentiation, automatic differentiation is implemented here. Specifically, the source-to-source transformation tool ADIC2 and the operator overloading tool ADOL-C are employed, in both dense and sparse modes. The theoretical approach is backed with the numerical analysis of a 1-DOF spatial four-bar mechanism, three different configurations of a 15-DOF multiple four-bar linkage, and a 16-DOF coach maneuver. Numerical and automatic differentiation are compared in terms of their computational efficiency and accuracy. Overall, we provide a global perspective of the efficiency of automatic differentiation in the field of multibody system dynamics.