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Many engineering structures involve some type of kinematic joints. Modelling such structures therefore represents a static or dynamic multi-body problem, be it deployment and retraction of roofs, domes, satellite aerials or rocket wings [, , , , , , , ], actuation of serial robot arms and parallel platforms and dynamics of rotating machinery [, , , , , , , ], operation of flexible mechanisms including suspension systems, steering mechanisms, railway collecting pantographs [, , , , ], or some other engineering multi-body problem.

Traditionally, these problems are analysed using specialist rigid-body mechanism codes, which often import flexibilities from finite element codes, but nowadays it is becoming exceedingly popular to employ non-linear finite element packages and enhance them so that they can handle this additional level of kinematic complexity. In this work, a particular technique within this approach has been applied to model a variety of simple and more complex joint types. The same method can be also employed to model general contact problems of elastic bodies, as a numerical example included at the end of this chapter demonstrates.

This paper, first of all, explains briefly some of the well-known and frequently used methods for contact treatment through finite element methods and multibody systems dynamics and gives a short description of their computational aspects in applications dealing with contact problems. Then, as the core of this paper, a formulation for considering unilateral contact of constrained and non-constrained planar deformable bodies in a multibody system leading to Linear Complementarity Problems (LCPs) is presented. In doing so, kinematic relationships governing the behavior of contact are formulated in a way that they consider the effect of deformations. As a general approach for flexible multibody systems, the moving frame of reference approach and modal coordinates are used to describe deformable bodies. In this formulation the effects of deformation are taken into account starting from the relative velocity of contact points in the normal and tangential direction and then the procedure is followed by introducing the relative acceleration of contact points which includes all the necessary terms needed for considering deformations. Then, the complementarity relations are reformulated following the same procedure as for rigid bodies contact. Therefore, the main difference of this algorithm compared to the rigid body case is in the formulation of the kinematics of contact. This formulation just considers the continual contact case of deformable bodies and for impact calculation this formulation has to be extended.

Several considerations are important if we try to carry out fast and precise simulations in multibody dynamics: the choice of modeling coordinates, the choice of dynamical formulations and the numerical integration scheme along with the numerical implementation. All these matters are very important in order to decide whether a specific method is good or not for a particular purpose.

A posteriori error estimators are useful tools in general purpose numerical computations because they provide an automatic, quantitative assessment of the accuracy of the results. Without some sort of error estimation the validity of any numerical results relies solely on the analyst experience and good judgment. While these are also necessary, they fail to be quantitative and are thus prone to mistakes.

The dynamic behavior and control of cranes executing prescribed motions of payloads are strongly affected by the underactuated nature of the robotic systems, in which the number of control inputs/outputs is smaller than the number of degrees-of-freedom. The outputs are specified in time load coordinates, which, expressed in terms of the system states, lead to servo-constraints on the system. The problem can then viewed from the perspective of constrained motion. It is noticed however that servo-constraints differ from passive constraints in several aspects. Mainly, they are enforced by means of control forces which may have any directions with respect to the servo-constraint manifold, and in the extreme (some of them) may be tangent. A specific methodology must be developed to solve the’ singular’ inverse dynamics problem. In this contribution, a theoretical background for the modeling of the partly specified/actuated motion is given. The initial governing equations, arising as index five differential-algebraic equations, are transformed to a more tractable index three form by projecting the dynamic equations into the orthogonal and tangent subspaces with respect to the servo-constraint manifold in the crane velocity space. A simple numerical code for solving the resultant differential-algebraic equations, based on backward Euler method, is then proposed. The feedforward control law obtained this way is enhanced by a closed-loop control strategy with feedback of the actual errors in load position to provide stable tracking of the required reference load trajectory in presence of perturbations. A rotary crane executing a load prescribed motion serves as an illustration. Some results of numerical experiments/simulations are reported.

We propose a methodology for extending the applicability of multibody-based comprehensive analysis codes to the maneuvering regime, with specific application to the flight of rotorcraft vehicles.

Maneuvers are here mathematically described in a concise yet completely general form as optimal control problems, each maneuver being defined by a specific form of the cost function and by suitable constraints on the vehicle states and controls. In principle, by solving the maneuver optimal control problem, one could determine the trajectory and the control time histories that steer the vehicle model, while minimizing the cost and satisfying the constraints. Unfortunately, optimal control problems are prohibitively expensive to solve for detailed comprehensive models of rotorcraft vehicles denoted by a large number of structural degrees of freedom and possibly sophisticated aerodynamics.

In order to make the problem computationally tractable, our formulation makes use of two models of the same vehicle. A coarse level flight mechanics model is used for solving the trajectory optimal control problem. Being based on a reduced model of the vehicle with only a few degrees of freedom, the resulting non-linear multi-point boundary value problem is computationally feasible. Next, the fine scale comprehensive model is steered in closed loop, tracking the trajectory computed at the flight mechanics level using a receding horizon model predictive controller. This amounts to a standard time marching problem for the comprehensive model, which is therefore also computationally feasible. The flight mechanics model is iteratively updated for ensuring close matching of the trajectories flown by the two models, by resorting to a neural adaptive element. This two-level procedure enables the simulation using comprehensive models of arbitrary complexity of maneuvers of possibly long duration, with general constraints on the vehicle inputs and outputs.

The new procedures are demonstrated with the help of numerical applications.

It is often perceived that one key issue in the development of modern, high-performance numerical software is the need to find a good trade-off between modularity, extensibility and performance requirements. This paper discusses how the need to add real-time simulation capabilities to an existing general-purpose multibody analysis software, and the resulting need for performance improvements, pushed an overall performance improvement and capability extension within an existing modular generic programming environment.

This chapter of the book is devoted to teaching the discipline of Multibody Dynamics. This discipline concerns young students (baccalaurean) as well as more senior students (master’s degree, and even PhD degree). The first part of this chapter is strongly inspired by various opinions, which are also those of the authors of this chapter. In particular, some of the comments already expressed in [] are rigorously reproduced in this part, as well as various comments which have also already been published [, ]. The content of this first part of the chapter also greatly benefits from the opinions expressed at the occasion of the round-table during the session on “Education of multibody dynamics” of the International ECCOMAS Thematic Conference on Advances in Computational Multibody Dynamics (Universidad Politecnica de Madrid, Spain, June 21–24, 2005). The continuation of the chapter gives an example of the experiment lived at the Université Catholique de Louvain (Belgium) [, ] with undergraduate students, experience which has also been successively reproduced in Equator (University of Quito) [] with graduate students.

Passive safety systems in vehicles, which are directly responsible for the optimal passenger protection in the case of an accident, are critical factors in contemporary vehicle development. Rollover maneuvers, e.g. rides over an embankment or a ramp, are considered to be especially hazardous, as they always involve a high risk of injury or even death and of vehicle damage. Today, for reasons of cost and development time, computer simulations play an important role in the development of such safety systems.

For simulating the vehicle dynamics, the vehicle is realized as a complex multibody system, as a base for the mechanical part, and which is combined with additional non-mechanical components like hydraulics, sensors, driver and environment into an overall mechatronical system. This concept is realized within the three-dimensional vehicle dynamic simulation environment _C++. The central issue in this paper is the modeling of the vehicle ground contact, which may occur between the underbody and the ground during embankment and ramp maneuvers. Finally, the results from test drives for a cabriolet, which has been selected by the car manufacturer from simulations with _C++, show a good coincidence between simulations and experiment.

The paper presents three case studies of dynamic analysis of aircraft during landing manoeuvre using two basic formalisms encountered in rigid and flexible multibody system (MBS) modelling. In the first case a formulation in natural coordinates has been used to analyze the dynamics of a medium size aircraft. Equations of motion have been formulated and solved using velocity transformation method. The aircraft has been modelled as consisting of rigid bodies connected by universal joints with springs. Aerodynamic forces have been taken into account by applying the Vortex Lattice Method (VLM) to the calculations performed. The effect of ground proximity on the results (ground effect) has been analyzed. In the second case, a dynamic analysis of a glider during the landing manoeuvre has been carried out from the point of view of stress recovery by means of various methods. Body positions and orientations have been written in absolute coordinates with floating frame approach for flexible bodies. Finite element method (FEM) and component mode synthesis has been used to model the flexibility of the bodies. A comparison of stress results obtained for different computation methods has been carried out. In the third analysis a MBS model of the Su-22 military airplane main landing gear has been presented. The absolute coordinates and the differential algebraic equations (DAE) formulations were used in all calculations. The whole landing gear model includes individual models of hydraulic actuators, shock absorber, flexible tire and contacts between some landing gear parts. Several types of simulations like landing gear extension and selected ground manoeuvres were performed. On that basis values of the forces which will allow to assess fatigue and durability of landing gear in future experiments were obtained. The received results were compared to the experimental measurements which were carried out on a real military airplane. The key issues of that comparison and general remarks were formulated. In the final part of the paper general conclusions regarding application of various computation MBS methods to dynamical analyses of aircrafts have been presented.