An efficient formulation for flexible multibody dynamics using a condensation of deformation coordinates

Multibody System Dynamics - In this work a new approach to deal with non-ideal operative aspects of spatial revolute joints by means of a three-dimensional finite element analysis (3D-FEA) is...     

On the use of absolute interface coordinates in the floating frame of reference formulation for flexible multibody dynamics

In this work a new formulation for flexible multibody systems is presented based on the floating frame formulation. In this method, the absolute interface coordinates are used as degrees of freedom. To this end, a coordinate transformation is established from the absolute floating frame coordinates and the local interface coordinates to the absolute interface coordinates. This is done by assuming linear theory of elasticity for a body’s local elastic deformation and by using the Craig–Bampton interface modes as local shape functions. In order to put this new method into perspective, relevant relations between inertial frame, corotational frame and floating frame formulations are explained. As such, this work provides a clear overview of how these three well-known and apparently different flexible multibody methods are related. An advantage of the method presented in this work is that the resulting equations of motion are of the differential rather than the differential-algebraic type. At the same time, it is possible to use well-developed model order reduction techniques on the flexible bodies locally. Hence, the method can be employed to construct superelements from arbitrarily shaped three dimensional elastic bodies, which can be used in a flexible multibody dynamics simulation. The method is validated by simulating the static and dynamic behavior of a number of flexible systems.     

A unified formulation of small-strain corotational finite elements: I. Theory

This paper presents a unified theoretical framework for the corotational (CR) formulation of finite elements in geometrically nonlinear structural analysis. The key assumptions behind CR are: (i) strains from a corotated configuration are small while (ii) the magnitude of rotations from a base configuration is not restricted. Following a historical outline the basic steps of the element independent CR formulation are presented. The element internal force and consistent tangent stiffness matrix are derived by taking variations of the internal energy with respect to nodal freedoms. It is shown that this framework permits the derivation of a set of CR variants through selective simplifications. This set includes some previously used by other investigators. The different variants are compared with respect to a set of desirable qualities, including self-equilibrium in the deformed configuration, tangent stiffness consistency, invariance, symmetrizability, and element independence. We discuss the main benefits of the CR formulation as well as its modeling limitations.     

Static form-finding analysis of a railway catenary using a dynamic equilibrium method based on flexible multibody system formulation with absolute nodal coordinates and controls

This paper proposes a dynamic equilibrium method for finding the initial equilibrium configuration of a railway catenary. In the proposed method, the catenary composed of flexible wires is modeled using two-node cable elements with absolute nodal coordinates based on a flexible multibody system formulation. Dynamic conditions that characterize the initial equilibrium configuration of the catenary are given and addressed as control processes in the form-finding procedure. The key feature of the proposed method is that the catenary configuration is continually evolved by dynamic simulation until characterization conditions are attained and an equivalent configuration of the centenary at static equilibrium is thus computed. It is validated using two examples and applied to the form-finding analysis of a two dimensional simple railway catenary. The obtained results are analyzed and discussed. It is general and can be applied to catenaries with complex configurations.     

Three-dimensional formulation of rigid-flexible multibody systems with flexible beam elements

Multibody systems generally contain solids with appreciable deformations and which decisively influence the dynamics of the system. These solids have to be modeled by means of special formulations for flexible solids. At the same time, other solids are of such a high stiffness that they may be considered rigid, which simplifies their modeling. For these reasons, for a rigid-flexible multibody system, two types of formulations coexist in the equations of the system. Among the different possibilities provided in the literature on the material, the formulation in natural coordinates and the formulation in absolute nodal coordinates are utilized in this paper to model the rigid and flexible solids, respectively. This paper contains a mixed formulation based on the possibility of sharing coordinates between a rigid solid and a flexible solid. The global mass matrix of the system is shown to be constant and, in addition, many of the constraint equations obtained upon utilizing these formulations are linear and can be eliminated.     

Interaction of flexible multibody systems with fluids analyzed by means of smoothed particle hydrodynamics

The present work deals with a computational approach to fluid-structure interaction (FSI) problems by coupling of flexible multibody system dynamics and fluid dynamics. Since the methods for the numerical modeling are well known, both for the structural and the fluid part, the focus of this work lies on the coupling formalism. Moreover, the applicability of the presented approach to arbitrary geometries and high structural stiffness is studied, as well as an easy model setup. No restriction should be made on the topology of the structure or the complexity of motion.For the fluid part a meshless method, known as smoothed particle hydrodynamics (SPH) is applied, which fulfills the above requirements. While an explicit time integration scheme in SPH provides a fast simulation of the fluid dynamics, advanced methods from flexible multibody dynamics provide a variety of benefits for the simulation of the solid part. Amongst these are specialized structural finite elements for both small and large deformation bodies, joints, stable implicit time-integration schemes, and model reduction techniques.A rule for the interaction between fluids and structures is derived from imposing a distributed potential over boundary segments of the structures, which the fluid particles respond to. The work is concluded by illustrative examples, demonstrating the successful coupling of flexible multibody systems with fluids.     

Partitioned formulation of internal and gravity waves interacting with flexible structures

This paper presents a partitioned modeling of internal and gravity fluid waves that interact with flexible structures. The governing interaction model consists of three completely partitioned entities: fluid model, structural model, and interface model that acts as an internal constraint on the fluid–structure interface boundary. Thus, the proposed partitioned multi-physics modeling can employ two completely modular fluid and structure software modules plus an interface solver, hence amenable to partitioned solution algorithms. The interface discretization can exploit the nonmatching interface algorithm previously developed via the method of localized Lagrange multipliers. Also noted is that the present fluid model can make use of widely available finite element software for standard Poisson-type problems.     

用于机器人柔性坐标测量系统的视觉传感器设计

视觉传感器作为机器人柔性视觉坐标测量系统最重要的组成部分之一,直接影响测量系统的精度、适应性.针对机器人柔性坐标测量系统的需要,提出了两步法测量空间形状的原理和数学模型,设计了一种线结构光单目视觉传感器,对其测量结果进行了误差分析,并用实验的方法证明了该传感器精度适中,体积小,重量轻,可以满足工业现场在线测量的需要.     

On the formulation and solution of shell problems by means of computerized algebraic manipulation

The basic equations of shell theory are conveniently formulated as tensor equations. However, the compactness and elegance of tensor notation are counteracted by the fact that the writing out of tensor equations for a given coordinate system results in tedious and cumbersome work in all but the simplest of cases.Use of the symbolic and algebraic manipulation language TENSOR FORMAC removes this limitation. It is shown that TENSOR FORMAC makes it possible to generate and handle, say, the equations of equilibrium expressed in the displacements, for a shell described by a non-trivial coordinate system with non-linear geometric effects taken into account.The solution of the system of equations generated manually or by TENSOR FORMAC can only be expressed in a few, exceptionally simple cases by elementary functions. But for axisymmetric deformations of shells of revolution is it possible to obtain the solution of the system of equations as a power series expansion utilizing a symbolic and algebraic manipulation language.     

Fluid formulation of high intensity laser beam propagation using lagrangian coordinates

The complete mathematical modeling of nonlinear light-matter interaction is presented in a hydrodynamic context. The field intensity and the phase gradient are the dependent variables of interest. The resulting governing equations are a generalization of the Navier-Stokes equations. This fluid formulation allows the insights and the methodologies which have been gained in solving hydrodynamics problems to be extended to nonlinear optics problems. To insure effective numerical treatment of the anticipated nonlinear self-lensing phenomena, a self-adjusted nonuniform redistribution, along the direction of propagation, of the computation points according to the actual local requirements of the physics must be used. As an alternative to the application of adaptive rezoning techniques in conjunction with Eulerian coordinates, Lagrangian variables are used to provide automatically the desired nonlinear mapping from the physical plane into the mathematical frame. In this paper we propose a method suitable for the solution of the described problemin one-dimensional cases as well as in two- dimensional cases with cylindrical symmetry. To overcome the numerical difficulties related to the inversion of the Jacobian, an analytical algorithm based on the paraxial approximation was developed.     

Supporting flexible regulation of crisis management by means of situated artificial institution

This paper highlights the use of situated artificial institution (SAI) within a hybrid, interactive, normative multi-agent system to regulate human collaboration in crisis management. Norms regulate the actions of human actors based on the dynamics of the environment in which they are situated. This dynamics results from both environment evolution and actors’ actions. Our objective is to situate norms in the environment in order to provide a context-aware crisis regulation. However, this coupling must be a loose one to keep both levels independent and easyto-change in order to face the complex and changing crisis situations. To that aim, we introduce a constitutive level between environmental and normative states providing a loose coupling of normative regulation with environment evolution. Norms are thus no more referring to environmental facts but to status functions, i.e., the institutional interpretation of environmental facts through constitutive rules. We present how this declarative and distinct SAI modelling succeeds in managing the crisis with a context-aware crisis regulation.     

Spatial joint constraints for the absolute nodal coordinate formulation using the non-generalized intermediate coordinates

In this investigation, a systematic procedure that can be used for modeling joint constraints for the absolute nodal coordinate formulation is developed. To this end, the non-generalized intermediate coordinates are introduced to derive a mapping between the generalized gradient coordinates and the non-generalized rotation parameters. With this mapping, a wide variety of joint constraints can be defined for the absolute nodal coordinate formulation in terms of the non-generalized reference coordinates and, therefore, existing well-developed constraint libraries formulated for the rigid body reference coordinates can be directly employed without significant modifications in existing codes. Furthermore, in order to define a rigid surface at the joint definition point, a set of orthonormality conditions is imposed on the gradient coordinates. This leads to not only accurate modeling of interface to mechanical joint, but also a simpler definition of the joint coordinate system obtained by the orthonormal gradient vectors. For this reason, a simpler form of constraint Jacobian and quadratic velocity vectors can be obtained as compared to those of the existing approach which requires the use of highly nonlinear joint coordinate system. A systematic procedure for eliminating the non-generalized coordinates and the dependent Lagrange multipliers associated with the coordinate mapping equations from the equations of motion is presented. As a result, a standard augmented form of the equations of motion can be obtained in terms of the generalized coordinates only. Several numerical examples are presented in order to demonstrate the use of the joint constraint formulation developed in this investigation.     

Improving retinal artery and vein classification by means of a minimal path approach

This paper describes a technique for the retinal vessel classification into artery and vein categories from fundus images within a framework to compute the arteriovenous ratio. This measure is used to assess the patient condition, mainly in hypertension and it is computed as the ratio between artery and vein widths. To this end, the vessels are segmented and measured in several circumferences concentric to the optic nerve. The resulting vessel segments at each radius are classified as artery or vein independently. After that, a tracking procedure joins vessel segments in different radii that belong to the same vessel. Finally, a voting system is applied to obtain the final class of the whole vessel. The methodology has been tested in a data set of 100 images labeled manually by two medical experts and a classification rate of over 87.68 % has been obtained.     

Modelling and control of a flexible rotary crane

We first derive a dynamical model for the control of a flexible rotary crane which carries out three kinds of motion (rotation, load hoisting, and boom hoisting) simultaneously. Only the joint between the boom and the jib is assumed to be flexible. The goal is to transfer a load to a desired place in such a way that at the end of the transfer the swing of the load decays as quickly as possible. We first apply an open-loop control input to the system such that the state of the system can be transferred to a neighbourhood of the equilibrium state. Then we apply a feedback control signal so that the stale of the system approaches the equilibrium state as quickly as possible.     

Modelling and control of non-slender flexible robot links

Based on Timoshenko's beam theory and Hamilton's principle, the dynamic model of a single non-slender flexible link rotating in a horizontal plane is derived. Spectral analysis is used to study the well-posedness of the dynamic control system. The tracking control problem is studied and a feedback control scheme, that controls the rigid-body motion and elastic behaviours simultaneously, is derived based on an n-modal model. The stabilization of the closed-loop system is studied analytically. Simulation results are presented     

Transient analysis of flexible multi-body systems. Part I: Dynamics of flexible bodies

The formulation for the dynamic analysis of undamped linear structural systems using the finite element method results in two element matrices; the mass and stiffness matrices, that describe the element inertia and stiffness properties. However, these matrices are not sufficient to describe the dynamics of structures that undergo large rigid-body motion. Other element matrices, in addition to the mass and stiffness matrices, are required to account for the inertia coupling between gross motion and elastic deformation. These matrices are time-invariant and can be generated and assembled in the same manner as the mass and stiffness matrices are assembled in linear structural dynamics. An inherent relation between these matrices and the deformable body mean axes exists. This paper is the first of two parts. It presents the two-dimensional and three-dimensional formulation of the system equations of motion of inertia-variant flexible bodies. In particular, Euler parameters are employed to describe the rotations of the body reference in the spatial analysis. In Part II [13], this formulation is applied to the impact analysis of a large-scale constrained flexible aircraft which are modeled as a multi-body system consisting of interconnected rigid and flexible components.     

Best domain for an elliptic problem in cartesian coordinates by means of shape‐measure

In (ZAA J. Anal. Appl., Vol. 16, No. 1, pp. 143–155) we introduced a method to determine the optimal domains for elliptic optimal‐shape design problems in polar coordinates. However, the same problem in cartesian coordinates, which are more applicable, is found to be much harder, therefore we had to develop a new approach for these designs. Herein, the unknown domain is divided into a fixed and a variable part and the optimal pair of the domain and its optimal control, is characterized in two stages. Firstly, the optimal control for the each given domain is determined by changing the problem into a measure‐theoretical one, replacing this with an infinite dimensional linear programming problem and approximating schemes; then the nearly optimal control function is characterized. Therefore a function that offers the optimal value of the objective function for a given domain, is defined. In the second stage, by applying a standard optimization method, the global minimizer pair will be obtained. Some numerical examples are also given. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

An approach for modeling long flexible bodies with application to railroad dynamics

Considering track flexibility in railroad vehicle simulations can lead to improved results. In modeling a railroad vehicle as a multibody system, track flexibility can be incorporated by using the floating frame of reference formulation (FFRF), which describes rail deformations in terms of shape functions defined in the track body frame of reference. However, the FFRF method is subject to two serious shortcomings, namely: it uses unreal track boundary conditions to calculate shape functions and requires a large number of functions to describe deformation. These shortcomings can be circumvented by defining shape functions in the trajectory frame of reference. Based on this notion, a new form of FFRF that can be used to describe the dynamics of long bodies subjected to moving loads (cable cars, zip-lines, elevator guides, pantograph catenary mechanism, etc.) was developed here. The shape functions selected in this work are based on the steady deformation exhibited by a beam on a Winkler foundation under the action of a moving load. However, other sets of shape functions more appropriate for transient dynamics are suggested. The definition of the deformation shape functions in a frame that moves with respect to the flexible body produces new terms in the generalized inertia forces of the flexible track. The proposed approach was applied to an unsuspended wheelset traveling on a tangent track supported on an elastic foundation. The results thus obtained under variable foundation stiffness conditions are discussed and comparisons made with the case of a rigid track. The new approach is also compared with the moving mass problem as solved with the mode superposition method.     

On the formulation of cost functions for torque-optimized control of rigid bodies

In the context of controlling the attitude of a rigid body, this communique uses recent results on representations of torques (moments) to establish cost functions. The resulting cost functions are both properly invariant under whatever choice of Euler angles is used to parameterize the rotation of the rigid body and have transparent physical interpretations. The function is related to existing works in geometric control theory and applications of optimal control theory to biomechanical systems.     

Dynamic analysis of flexible multi-body systems using mixed modal and tangent coordinates

This paper presents a mixed modal and tangent coordinate technique for computer aided analysis of flexible mechanical systems whose components undergo large translations and large rotations. In this model the configuration of a flexible component is identified by using two sets of generalized coordinates, namely rigid body and elastic coordinates. The rigid body coordinates define the location and orientation of a body axis, whereas the elastic coordinates define the displacement field of a component with respect to its body axis. The elastic coordinates are introduced by using finite element discretization to model flexible components with complex geometries. A modal analysis technique is used to identify the elastic mode shapes and to eliminate insignificant higher frequency modes. An orthonormalization of constraint Jacobian matrix associated with rigid body coordinates is used to identify the rigid body tangent coordinates. The resulting modal and tangent coordinates are used to develop an automated numerical integration scheme to solve the system differential and algebraic equations. Two numerical examples are considered to show the feasibility of dynamic analysis of flexible mechanical systems using this scheme.