Efficient formulation of the Gibbs–Appell equations for constrained multibody systems

Multibody System Dynamics - In this study, we present explicit equations of motion for general mechanical systems exposed to holonomic and nonholonomic constraints based on the Gibbs-Appell...     

A rigorous proof for the equivalence of the projective Newton–Euler equations and the Lagrange equations of second kind for spatial rigid multibody systems

Multibody System Dynamics - It is well known that the projective Newton–Euler equation and the Lagrange equation of second kind lead to the same result when deriving the dynamical equations...     

A mixed formulation for the direct approximation of the control of minimal $$L^2$$-norm for linear type wave equations

This paper deals with the numerical computation of null controls for the wave equation with a potential. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. In [Cîndea, Fernández-Cara & Münch, Numerical controllability of the wave equation through primal methods and Carleman estimates, 2013], a so called primal method is described leading to a strongly convergent approximation of boundary controls: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality condition. In this work, we adapt the method to approximate the control of minimal square-integrable norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner controllability. For simplicity, we present the approach in the one dimensional case.     

On the choice of boundary conditions for mode shapes in flexible multibody systems

The modal representation of the deformation field is a widespread and efficient approach in the analysis of flexible multibody systems. However, it requires a pre-processing in advance to the actual multibody survey that includes the imposition of boundary conditions for the evaluation of the mode functions as an essential user input. Quite often the appropriateness of these boundary conditions is a point of discussion. Therefore the present paper reviews the theoretical background and the implications of this task. Then, a consistent and comprehensive proposal is made how these boundary conditions may be chosen. The suggestion is justified by theoretical considerations and compared to alternative approaches from the literature in a simulation study with three representative examples. It may be concluded that several approaches lead to reasonable results for a sufficient number of mode functions. However, the proposed approach turned out to be the most efficient one and provides a consistent framework.     

A methodology for the generation of planar models for multibody chain drives

To overcome the difficulty on building manually complex models of chain drives, this work proposes a comprehensive methodology to build multibody models of any general chain drive automatically from a minimal set of data. The proposed procedure also evaluates the initial positions and velocities of all components of the drive that are consistent with the kinematic joints or with the contact pairs used in the model. In this methodology, all links and sprockets are represented by rigid bodies connected to each other either by ideal or by clearance revolute joints. The clearance revolute joint contact is further extended to handle the contact between the chain rollers and the sprocket teeth exact profiles. A suitable cylindrical continuous contact law is applied to describe the interaction on all contact pairs. One of the complexities of the computational study of roller chain drives is the large number of bodies in the system and the dynamics of the successive engagement and disengagement of the rollers with the sprockets. Each time a roller engages or disengages with a sprocket tooth, the number of rigid bodies in contact changes. The search for the contact pairs is recognized as one of the most time consuming task in contact analysis. This work proposes a procedure to specify the contact pairs and their update during the dynamic analysis optimizing the computational efficiency of the contact search. The methodologies adopted result in a general computer program that is applied and demonstrated in a generic chain drive that can be used in industrial machines, vehicle engines or any other type of mechanical system.     

New efficient method for dynamic modeling and simulation of flexible multibody systems moving in plane

Efficient, precise dynamic analysis for general flexible multibody systems has become a research focus in the field of flexible multibody dynamics. In this paper, the finite element method and component mode synthesis are introduced to describe the deformations of the flexible components, and the dynamic equations of flexible bodies moving in plane are deduced. By combining the discrete time transfer matrix method of multibody system with these dynamic equations of flexible component, the transfer equations and transfer matrices of flexible bodies moving in plane are developed. Finally, a high-efficient dynamic modeling method and its algorithm are presented for high-speed computation of general flexible multibody dynamics. Compared with the ordinary dynamics methods, the proposed method combines the strengths of the transfer matrix method and finite element method. It does not need the global dynamic equations of system and has the low order of system matrix and high computational efficiency. This method can be applied to solve the dynamics problems of flexible multibody systems containing irregularly shaped flexible components. It has advantages for dynamic design of complex flexible multibody systems. Formulations as well as a numerical example of a multi-rigid-flexible-body system containing irregularly shaped flexible components are given to validate the method.     

Analysis of multibody systems with flexible plates using variational graph-theoretic methods

This work treats the problem of modelling multibody systems with structural flexibility. By combining linear graph theory with the principle of virtual work and finite elements, a dynamic formulation is obtained that extends graph-theoretic (GT) modelling methods to the analysis of thin flexible plates for multibody systems. The system is represented by a linear graph, in which nodes represent reference frames on flexible plates, and edges represent components that connect these frames. To generate the equations of motion with elastic deformations, the flexible plates are discretized using a triangular thin shell finite element based on the discrete Kirchhoff criterion and can be used to discretize bidirectional bodies such as satellite panels, flatbed trailers, and mechanisms with plates. Three flexible systems with plates are analyzed to illustrate the performance of this new variational graph-theoretic formulation and its ability to generate directly a set of motion equations for flexible multibody systems (FMS) without additional user input.     

A linearized input–output representation of flexible multibody systems for control synthesis

In this paper, a linearized input–output representation of flexible multibody systems is proposed in which an arbitrary combination of positions, velocities, accelerations, and forces can be taken as input variables and as output variables. The formulation is based on a nonlinear finite element approach in which a multibody system is modeled as an assembly of rigid body elements interconnected by joint elements such as flexible hinges and beams. The proposed formulation is general in nature and can be applied for prototype modeling and control system analysis of mechatronic systems. Application of the theory is illustrated through a detailed model development of an active vibration isolation system for a metrology frame of a lithography machine.     

Energetically consistent simulation of simultaneous impacts and contacts in multibody systems with friction

This paper presents a methodology for treating energy consistency when considering simultaneous impacts and contacts with friction in the simulation of systems of interconnected bodies. Hard impact and contact is considered where deformation of the impacting surfaces is negligible. The proposed approach uses a discrete algebraic model of impact in conjunction with moment and tangential coefficients of restitution (CORs) to develop a general impact law for determining post-impact velocities. This process depends on impulse–momentum theory, the complementarity conditions, a principle of maximum dissipation, and the determination of contact forces and post-impact accelerations. The proposed methodology also uses an energy-modifying COR to directly control the system’s energy profile over time. The key result is that different energy profiles yield different results and thus energy consistency should be considered carefully in the development of dynamic simulations. The approach is illustrated on a double pendulum, considered to be a benchmark case, and a bicycle structure.     

Recursive formulations for multibody systems with frictional joints based on the interaction between bodies

In practice, the clearances of joints in a great number of mechanical systems are well under control. In these cases, some of the existing methods become unpractical because of the little differences in the order of magnitude between relative movements and computational errors. Assuming that the effects of impacts are negligible, we proved that both locations and forces of contacts in joints can be fully determined by parts of joint reaction forces. Based on this fact, a method particularly suited for multibody systems possessing frictional joints with tiny clearances is presented. In order to improve the efficiency of computation, recursive formulations are proposed based on the interactions between bodies. The proposed recursive formulations can improve the computation of joint reaction forces. With the methodology presented in this paper, not only the motion of bodies in a multibody system but also the details about the contacts in joints, such as forces of contacts and locations of contact points, can be obtained. Even with the assumption of impact free, the instants of possible impacts can be detected without relying upon any ambiguous parameters, as indicated by numerical examples in this paper.     

Development of generic multibody road vehicle models for crashworthiness

The crashworthiness analysis of road vehicles requires detailed data of the vehicles that the automotive manufacturers are, generally, unable to release due to commercial or legal restrictions. In the development of passive safety subsystems or substructures, the overall crash response of a vehicle model used to support it, must mimic that of the real vehicle; if this exists, regardless of any particular constructive detail of its structure provided that it is not located in the vicinity of such subsystem. This work proposes a methodology for the development of multibody models of road vehicles, for passive safety analysis, which include all general structural and mechanical features of real vehicles and start by exhibiting impact dynamic responses similar to the top of line vehicles. These vehicle models, designated as generic, do not require the knowledge of most of the particular details of the design of the real vehicle, which the manufacturers are unable to release, but can be adjusted to have crash responses similar to those of the real vehicle. Based on an existing finite element model of a car, which has all constructive features of vehicles of the chosen class, a multibody model is built applying the plastic hinge approach. By using a selected number of crash scenarios, defined in international standards such as the EuroNCAP, selected parameters of the vehicle multibody model are adjusted to ensure a good correlation between its impact responses and those of the finite element model. The crash responses are measured in terms of structural deformations, velocities and accelerations, occupant injury measures and structural energy absorption capabilities. Assuming that the plastic hinge constitutive equations of the multibody model are not exactly known, their parameters are used here as the multibody vehicle model that are adjusted. The methodology proposed is demonstrated by its application to the identification of the vehicle multibody model of a large family car for which the reference vehicle is available as a detailed finite element model.     

Erratum to: New efficient method for dynamic modeling and simulation of flexible multibody systems moving in plane

Efficient, precise dynamic analysis for general flexible multibody systems has become a research focus in the field of flexible multibody dynamics. In this paper, the finite element method and component mode synthesis are introduced to describe the deformations of the flexible components, and the dynamic equations of flexible bodies moving in plane are deduced. By combining the discrete time transfer matrix method of multibody system with these dynamic equations of flexible component, the transfer equations and transfer matrices of flexible bodies moving in plane are developed. Finally, a high-efficient dynamic modeling method and its algorithm are presented for high-speed computation of general flexible multibody dynamics. Compared with the ordinary dynamics methods, the proposed method combines the strengths of the transfer matrix method and finite element method. It does not need the global dynamic equations of system and has the low order of system matrix and high computational efficiency. This method can be applied to solve the dynamics problems of flexible multibody systems containing irregularly shaped flexible components. It has advantages for dynamic design of complex flexible multibody systems. Formulations as well as a numerical example of a multi-rigid-flexible-body system containing irregularly shaped flexible components are given to validate the method.     

A systematic procedure for writing and solving equations of sampled-data systems†

A method is presented for choosing variables and writing and solving equations of sampled-data systems. The need for such a procedure is evident from attempts to pulse-transform equations involving unfiltered sampled variables. Such operations are equivalent to having two ideal samplers in succession, the output of which is undefined. The proposed method eliminates the occurrence of such situations and demonstrates a straightforward solution procedure     

Convergence of Craig’s method for linear algebraic systems

Craig??s iterative method is designed to solve linear algebraic systems with an asymmetric (or even rectangular) matrix. A simple representation is constructed for the method. Test examples are used to study iterative convergence and compare the method with the conjugate gradient method. Although round-off errors in Craig??s method proved to slow down iterative convergence significantly, they allowed attaining high accuracy (if the matrix was well-conditioned). An efficient iteration-stopping criterion is found.     

Dynamic motion planning of overarm throw for a biped human multibody system

We propose a motion planning formulation of overarm throw for a 55-degree-of-freedom biped human multibody system. The unique characteristics of the throwing task—highly redundant, highly nonlinear, and highly dynamic—make the throwing motion simulation challenging in the literature and are addressed within the framework of multibody dynamics and optimization. To generate physically feasible throwing motions in a fully predictive method without input reference from motion capture or animation, rigorous dynamics modeling, such as dynamic balance based on Zero-Moment Point (ZMP) and ground reaction loads, is associated with the constraints. Given the target location and the object mass, the algorithm outputs the motion, required actuator torques, release conditions, and projectile and flight time of the object. Realistic human-like motions of throwing are generated for different input parameters, which demonstrate valid cause–effect relations in terms of both kinematic and kinetic outputs.     

Simulation process of flexible multibody systems with non-modal model order reduction techniques

One important issue for the simulation of flexible multibody systems is the reduction of the flexible body’s degrees of freedom. In this work, nonmodal model reduction techniques for flexible multibody systems within the floating frame of reference framework are considered. While traditionally in the multibody community modal techniques in many different forms are used, here other methods from system dynamics and mathematics are in the focus. For the reduction process, finite element data and user inputs are necessary. Prior to the reduction process, the user first needs to choose boundary conditions fitting the chosen reference frame before defining the appropriate in- and outputs. In this work, four different possibilities of modeling appropriate interface points to reduce the number of inputs and outputs are presented.     

On generalized Melvin’s solutions for Lie algebras of rank 2

We consider a class of solutions in multidimensional gravity which generalize Melvin’s well-known cylindrically symmetric solution, originally describing the gravitational field of a magnetic flux tube. The solutions considered contain the metric, two Abelian 2-forms and two scalar fields, and are governed by two moduli functions H₁(z) and H₂(z) (z = ρ², ρ is a radial coordinate) which have a polynomial structure and obey two differential (Toda-like) master equations with certain boundary conditions. These equations are governed by a certain matrix A which is a Cartan matrix for some Lie algebra. The models for rank-2 Lie algebras A₂, C₂ and G₂ are considered. We study a number of physical and geometric properties of these models. In particular, duality identities are proved, which reveal a certain behavior of the solutions under the transformation ρ → 1/ρ; asymptotic relations for the solutions at large distances are obtained; 2-form flux integrals over 2-dimensional regions and the corresponding Wilson loop factors are calculated, and their convergence is demonstrated. These properties make the solutions potentially applicable in the context of some dual holographic models. The duality identities can also be understood in terms of the Z₂ symmetry on vertices of the Dynkin diagram for the corresponding Lie algebra.     

Simulation of multibody systems with the use of coupling techniques: a case study

Simulation coupling (or cosimulation) techniques provide a framework for the analysis of decomposed dynamical systems with the use of independent numerical procedures for decomposed subsystems. These methods are often seen as very promising because they enable the utilization of the existing software for subsystem analysis and usually are easy to parallelize, and run in a distributed environment. For example, in the domain of multibody systems dynamics, a general setup for “Gluing Algorithms” was proposed by Wang et al. It was intended to provide a basis for multilevel distributed simulation environments. The authors presented an example where Newton’s method was used to synchronize the responses of subsystem simulators.     

On multidimensional analogs of Melvin’s solution for classical series of Lie algebras

A multidimensional generalization of Melvin’s solution for an arbitrary simple Lie algebra is presented. The gravitational model contains n 2-forms and l ≥ n scalar fields, where n is the rank of . The solution is governed by a set of n functions obeying n ordinary differential equations with certain boundary conditions. It was conjectured earlier that these functions should be polynomials (the so-called fluxbrane polynomials). A program (in Maple) for calculating these polynomials for classical series of Lie algebras is suggested. The polynomials corresponding to the Lie algebra D ₄ are obtained. It is conjectured that the polynomials for A _n -, B _n - and C _n -series may be obtained from polynomials for D _n+1-series by using certain reduction formulas. Talk given at the International Conference RUSGRAV-13, June 23–28, 2008, PRUR, Moscow.