Multibody Dynamics

The pantograph–catenary system is still the most reliable form of collecting electric energy for running trains. This system should ideally run with relatively low contact forces, in order to minimize wear and damage of the contacting elements but without contact loss to avoid power supply interruption and electric arching. However, the quality of the pantograph–catenary contact may be affected by operational conditions, defects on the overhead equipment, environmental conditions or by the flexibility of the pantograph components. In this work a flexible multibody methodology based on the use of the mean-axis conditions, as reference conditions, mode component synthesis, as a form of reducing the number of generalized coordinates of the system and virtual bodies, as a methodology to allow the use of all kinematic joints available for multibody modeling and application of external forces, are used to allow building the flexible multibody pantograph models. The catenary model is built in a linear finite element code developed in a Matlab environment, which is co-simulated with the multibody code to represent the complete system interaction. A thorough description of rigid-flexible multibody pantograph models is presented in a way that the proposed methodology can be used. Several flexible multibody models of the pantograph are described and proposed and the quality of the pantograph–catenary contact is analyzed and discussed in face of the flexibility of the overhead components.

The present paper focuses on trajectory optimization problems for multibody vehicle models, accounting for the presence of pilot-in-the-loop effects and fast dynamic components in the solution. The trajectory optimal control problem is solved through a direct approach by means of a novel hybrid single–multiple shooting method. Specific focus of the present work is the inclusion of pilot models in the optimization process, in order to improve the fidelity of the solution by considering the entire coupled human-vehicle system. In particular we investigate a series of maneuvers flown with helicopters, quantifying the performance loss due to human limitations of the pilot-vehicle system with respect to the sole vehicle case.

This work addresses the optimization of flexible multibody systems based on the dynamic response of the full system with large amplitude motions and elastic deflections. The simulation model involves a nonlinear finite element formulation, a time integration scheme and a sensitivity analysis and it can be efficiently exploited in an optimization loop.In particular, the paper focuses on the topology optimization of structural components embedded in multibody systems. Generally, topology optimization techniques consider that the structural component is isolated from the rest of the mechanism and use simplified quasi-static load cases to mimic the complex loadings in service. In contrast, we show that an optimization directly based on the dynamic response of the flexible multibody system leads to a more integrated approach. The method is applied to truss structural components. Each truss is represented by a separate structural universe of beams with a topology design variable attached to each one. A SIMP model (or a variant of the power law) is used to penalize intermediate densities. The optimization formulation is stated as the minimization of the mean compliance over a time period or as the minimization of the mean tip deflection during a given trajectory, subject to a volume constraint. In order to illustrate the benefits of the integrated design approach, the case of a two degrees-of-freedom robot arm is developed.

Wind-turbines represent an important means to extract energy from the environment in a ‘green’ manner. The concept of extracting energy from the wind dates back thousands of years, including not only power generation (e.g. mills, water pumps) but also direct locomotion (e.g. sailing). Modern wind-energy technology relies on efficient aerodynamic design and durable mechanical systems.

In the field of multibody dynamics, structural components, such as beams or plates, have been discretized in different ways, according to special requirements of certain problem configurations. In literature, models, which follow the same mechanical theories but a different numerical discretization technique, such as the absolute nodal coordinate formulation (ANCF) and the floating frame of reference formulation (FFRF), have been calculated for comparison. In existing examples, the solutions of these calculations do not always coincide very accurately. Therefore, in the present contribution, which is an extension of a former work of the authors, standard static and dynamic problems in the large deformation regime are treated. Special emphasis is laid on converged solutions, using an analytical reference value in the static case. For dynamic examples a reference value based on the strain energy is provided, in order to simplify the comparison of the different formulations and to provide a reference value, similar to the static case, for future studies. For both formulations planar finite elements based on the Bernoulli–Euler theory are utilized. In case of the ANCF the finite element consists of two position and two slope coordinates in each node only. In the FFRF beam finite element, as usual, two sets of coordinates are used to describe the actual configuration. The first set of coordinates defines the location and orientation of the body reference frame. The second set of coordinates describes small superimposed transverse and axial deflections relative to the body frame. The transverse deflections are approximated by means of two static modes for the rotation at the boundary and a user-defined number of eigenmodes of the clamped-clamped beam. The axial deflection is represented by a linear approach. In numerical studies, the accuracy of the two formulations is compared for two example problems, a cantilever beam with a singular force at the free end and a slider-crank mechanism. It turns out that both formulations have comparable performance and that the results coincide in the converged case.

The main purpose of this paper is to discuss a method for a dynamic modeling and analysis of multibody systems with translational clearance joints. The method is based on the nonsmooth dynamics approach, in which the interaction of the elements that constitute a translational clearance joint is modeled with multiple frictional unilateral constraints. In the following, the most fundamental issues of the nonsmooth dynamics theory are revised. The dynamics of rigid multibody systems are stated as an equality of measures, which are formulated at the velocity-impulse level. The equations of motion are complemented with constitutive laws for the normal and tangential directions. In this work, the unilateral constraints are described by a set-valued force law of the type of Signorini’s condition, while the frictional contacts are characterized by a set-valued force law of the type of Coulomb’s law for dry friction. The resulting contact-impact problem is formulated and solved as a linear complementarity problem, which is embedded in the Moreau’s time-stepping method. Finally, the classical slider-crank mechanism is considered as a demonstrative application example and numerical results are presented. The obtained results show that the existence of clearance joints in the modeling of multibody systems influences their dynamics response.

In this chapter robotic applications of general multibody system (MBS) simulation methods, based on absolute coordinates formalism, are presented. Three typical problems, often encountered in robotics, are discussed: kinematic analysis with singular configuration detection, simulation of parallel robot dynamics investigated jointly with the robot control systems properties, and finally, simulation of a robot with flexibility effects taken into account. In case of singular configuration detection simplest types of singular configurations are analyzed – turning point and bifurcation point. The second case of MBS application is an example of parallel robot dynamic analysis when model based control is taken into account. The last part of the chapter is devoted to the analysis of complex, flexible power transmission mechanism carried out with general MBS formalism.

There are many difficulties involved in the numerical integration of index-3 Differential Algebraic Equations (DAEs), mainly related to stability, in the context of mechanical systems. An integrator that exactly enforces the constraint at position level may produce a discrete solution that departs from the velocity and/or acceleration constraint manifolds (invariants). This behaviour affects the stability of the numerical scheme, resulting in the use of stabilization techniques based on enforcing the invariants. A coordinate projection is a poststabilization technique where the solution obtained by a suitable DAE integrator is forced back to the invariant manifolds. This paper analyzes the energy balance of a velocity projection, providing an alternative interpretation of its effect on the stability and a practical criterion for the projection matrix selection.

This paper discusses the optimal trajectory planning problem of multibody systems. The aim of this study is to develop a general purpose optimal trajectory planning algorithm to be applied to arbitrary multibody systems. Multibody systems may be divided into two groups, i.e. open loop systems and closed loop systems [8]. In [11] an optimal trajectory planning algorithm for open loop systems was presented. In this paper, optimal trajectory planning algorithms for closed loop systems are proposed by extending the algorithm for open loop systems. Two types of methods are presented based on the dynamic analysis by computational algorithms for closed loop systems. The first method uses generalized coordinate partitioning and embedding techniques. The second method is based on an augmented formulation with Lagrange multipliers. The first method is easily applicable to non-redundant actuation systems, while the second method considers redundant actuation. The validity of these methods for optimal trajectory planning is confirmed by computational results and their features are compared.

For the virtual engine development, testing and calibration, it is advantageous to use the same physical model on different platforms. Due to the complexity of the model and its evaluation one has to coop with serve evaluation restrictions on the realtime platform. For coupled problems which includes an electrical system, the equilibrium conditions include an algebraic constraints. Hence it is not sufficient to use only an explicit time integration scheme. We extend an explicit scheme to a mixed scheme such that the overall performance per time step still is below the timing constraint of the realtime platform for reasonable complex model with electrical system.

Today, the accurate assessment of muscle forces performed by the human body in motion is still expected for many clinical applications and studies. However, as most of the joints are overactuated by several muscles, any non-invasive muscle force quantification needs to solve a redundancy problem. Consequently, the aim of this study is to propose a non-invasive method to assess muscle forces in the human body during motion, using a multibody model-based optimization process that attempts to solve the agonistic and antagonistic muscle overactuation. The main originality of the proposed method is the cautious using of Electromyographic (EMG) data information, known by all to be noisy-corrupted, via a protocol divided into two main steps:

In this chapter, the process is applied to a benchmark case: the force quantification of the elbow flexor and extensor muscle sets of subjects engaged in weightlifting and performing cycles of forearm flexion/extension. A statistical validation of this method shows a good inter-test reproducibility and a very good correlation between a. the net joint torques resulting from the obtained muscle forces and b. the net joint torques given by inverse dynamics.Consequently, since the method is able to consider measured information on the actual muscle activation, it becomes a promising alternative to methods based on preset strategies, usually presented in literature, such as the strategy that maximizes endurance defined by Crowninshield et al.

This document discusses computing time reduction possibilities for the dynamic simulation of large nonsmooth multibody systems. Exemplary regarding the model of a pushbelt continuously variable transmission known from literature, simplifications are derived to examine the underlying problems. Usually one has to deal with the calculation of an adequate initial state and the reduction of the computational effort during integration. In three sections stationary belt models for the determination of an initial value are proposed, time step size enlargements due to implicit time integration schemes are analysed and the distribution of the main effort per time step on several central processing units by shared memory parallelisation is investigated.

An underactuated multibody system has less control inputs than degrees of freedom, e.g. due to passive joints or body flexibility. The analysis of the mechanical design of these kind of underactuated multibody systems might show that they are non-minimum phase, i.e. they have an internal dynamic which is not asymptotically stable. Therefore, feedback linearization is not possible, and also feed-forward control design for output trajectory tracking becomes a very challenging task. In this paper it is shown that through the use of an optimization procedure underactuated multibody systems can be designed in such a way that they are minimum phase. Thus feed-forward control design is significantly simplified and also feedback linearization of the underactuated multibody system is possible.

This work reports on advances in large-scale multibody dynamics simulation facilitated by the use of the Graphics Processing Unit (GPU). A description of the GPU execution model along with its memory spaces is provided to illustrate its potential parallel scientific computing. The equations of motion associated with the dynamics of large system of rigid bodies are introduced and a solution method is presented. The solution method is designed to map well on the parallel hardware, which is demonstrated by an order of magnitude reductions in simulation time for large systems that concern the dynamics of granular material. One of the salient attributes of the solution method is its linear scaling with the dimension of the problem. This is due to efficient algorithms that handle in linear time both the collision detection and the solution of the nonlinear complementarity problem associated with the proposed approach. The current implementation supports the simulation of systems with more than one million bodies on commodity desktops. Efforts are under way to extend this number to hundreds of millions of bodies on small affordable clusters.

The classical approach to simulate contacts between gears is to use rigid body models coupled with a parallel spring damper combination. However, these models had been developed for properly meshing gears with smooth contacts and cannot cover wave propagation caused by hard contacts or impacts. Moreover, as they are based on the assumption of rigidness, often light weight designs, resulting in very compliable gear bodies, cannot be considered appropriately. To evaluate how appropriate these rigid body models are to simulate impact forces, a very detailed finite element model is used to simulate several impacts and the results are compared to simulations with a rigid body model. The results reveal that for compliable gear bodies, there exist dynamic effects that considerably affect contact forces and motion and that these effects cannot be covered by rigid body models at all. Hence, a flexible model is imperative to precisely simulate impact forces. To reduce integration time, we present a modally reduced elastic multibody model including contact that allows very precise simulations in reasonable time. For the contact calculations a node-to-segment penalty formulation is introduced and is integrated using central differences. Even though the elastic model is a reduced model, it is still of huge size, as any node on any flank is a potential contact node. Also, the transformation data between modal and nodal coordinates must be accessible during integration. To reduce the required amount of memory a coarse collision detection is introduced that allows to dynamically reload only the transformation required in the current integration step. This approach allows very precise simulations of contacts between gears with integration times about 400 times faster than for associated finite element simulations. At the same time the model is robust and fast enough to allow the simulation of many contacts and many revolutions. To validate this approach basic experimental investigations with simple impact bodies have been carried out. The results from these experiments and related simulations agree very well.