Reduction of Physical and Constraint Degrees-of-Freedom of Redundant Formulated Multibody Systems

2015 ◽  
Vol 11 (3) ◽  
Author(s):  
Daniel Stadlmayr ◽  
Wolfgang Witteveen ◽  
Wolfgang Steiner
Keyword(s):  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Multibody System ◽  
A Priori ◽  
Plane Motion ◽  
Critical Discussion ◽  
Galerkin Projection ◽  
Differential Algebraic Equation ◽  
Numerical Examples ◽  
Constraint Equations

Commercial multibody system simulation (MBS) tools commonly use a redundant coordinate formulation as part of their modeling strategy. Such multibody systems subject to holonomic constraints result in second-order d-index three differential algebraic equation (DAE) systems. Due to the redundant formulation and a priori estimation of possible flexible body coordinates, the model size increases rapidly with the number of bodies. Typically, a considerable number of constraint equations (and physical degrees-of-freedom (DOF)) are not necessary for the structure's motion but are necessary for its stability like out-of-plane constraints (and DOFs) in case of pure in-plane motion. We suggest a combination of both, physical DOF and constraint DOF reduction, based on proper orthogonal decomposition (POD) using DOF-type sensitive velocity snapshot matrices. After a brief introduction to the redundant multibody system, a modified flat Galerkin projection and its application to index-reduced systems in combination with POD are presented. The POD basis is then used as an identification tool pointing out reducible constraint equations. The methods are applied to one academic and one high-dimensional practical example. Finally, it can be reported that for the numerical examples provided in this work, more than 90% of the physical DOFs and up to 60% of the constraint equations can be omitted. Detailed results of the numerical examples and a critical discussion conclude the paper.

An Efficient Method for the Kinematics of Multibody Systems That Works in Singular Positions

1992 ◽  
Author(s):  
A. Avello ◽  
E. Bayo
Keyword(s):  
Efficient Method ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Augmented Lagrangian ◽  
Multibody System ◽  
Jacobian Matrix ◽  
Numerical Examples ◽  
Constraint Equations ◽  
The Stability ◽  
Augmented Lagrangian Formulation

Abstract When a multibody system reaches a singular position, one or more degrees of freedom appear instantaneously and the jacobian matrix of the constraint equations becomes rank-deficient. The classical kinematic formulation is based on the factorization of the jacobian and, therefore, fails in singular positions. In this paper we develop an efficient method, which uses a penalty and an augmented Lagrangian formulation, and successfully handles singular positions. This formulation automatically copes with redundant incompatible constraints and guarantees the stability of the constraints during numerical integration. Critical numerical examples are shown which corroborate these findings.

EQUATIONS OF MOTION OF COMPLEX MULTIBODY SYSTEMS USING KINEMATICAL DIFFERENTIALS

1989 ◽  
Vol 13 (4) ◽  
pp. 113-121 ◽  
Author(s):  
M. HILLER ◽  
A. KECSKEMETHY
Keyword(s):  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Multibody System ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Nonholonomic Systems ◽  
Automatic Generation ◽  
Minimal Order ◽  
Closed Form Solutions ◽  
Constraint Equations

In complex multibody systems the motion of the bodies may depend on only a few degrees of freedom. For these systems, the equations of motion of minimal order, although more difficult to obtain, give a very efficient formulation. The present paper describes an approach for the automatic generation of these equations, which avoids the use of LAGRANGE-multipliers. By a particular concept, designated “kinematical differentials”, the problem of determining the partial derivatives required to state the equations of motion is reduced to a simple re-evaluation of the kinematics. These cover the solution of the global position, velocity and acceleration problems, i.e. the motion of all bodies is determined for given generalized (independent) coordinates. For their formulation and solution, the multibody system is mapped to a network of nonlinear transformation elements which are connected by linear equations. Each transformation element, designated “kinematical transformer”, corresponds to an independent multibody loop. This mapping of the constraint equations makes it possible to find closed-form solutions to the kinematics for a wide variety of technical applications, and (via kinematical differentials) leads also to an efficient formulation of the dynamics. The equations are derived for holonomic, scleronomic systems, but can also be extended to general nonholonomic systems.

A Vector Bond Graph Method of Kineto-Static Analysis for Spatial Multibody Systems

2013 ◽  
Vol 321-324 ◽  
pp. 1725-1729 ◽  
Author(s):  
Zhong Shuang Wang ◽  
Yang Yang Tao ◽  
Quan Yi Wen
Keyword(s):  
Static Analysis ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Multibody System ◽  
Graph Model ◽  
Bond Graph ◽  
Constraint Forces ◽  
Decoupling Method ◽  
Complete Form ◽  
Three Degrees Of Freedom

In order to increase the reliability and efficiency of the kineto-static analysis of complex multibody systems, the corresponding vector bond graph procedure is proposed. By the kinematic constraint condition, spatial multibody systems can be modeled by vector bond graph. For the algebraic difficulties brought by differential causality in system automatic kineto-static analysis, the effective decoupling method is proposed, thus the differential causalities in system vector bond graph model can be eliminated. In the case of considering EJS, the unified formulae of driving moment and constraint forces at joints are derived based on vector bond graph, which are easily derived on a computer in a complete form and very suitable for spatial multibody systems. As a result, the automatic kineto-static analysis of spatial multibody system on a computer is realized, its validity is illustrated by the spatial multibody system with three degrees of freedom.

Approximate End-Effector Tracking Control of Flexible Multibody Systems Using Singular Perturbations

2013 ◽  
Vol 9 (1) ◽  
Author(s):  
Thomas Gorius ◽  
Robert Seifried ◽  
Peter Eberhard
Keyword(s):  
Tracking Control ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Multibody System ◽  
Feedforward Control ◽  
Flexible Multibody Systems ◽  
Flexible Manipulator ◽  
End Effector ◽  
Nonminimum Phase ◽  
Flexible Multibody

In many cases, the design of a tracking controller can be significantly simplified by the use of a 2-degrees of freedom (DOF) control structure, including a feedforward control (i.e., the inversion of the nominal system dynamics). Unfortunately, the computation of this feedforward control is not easy if the system is nonminimum-phase. Important examples of such systems are flexible multibody systems, such as lightweight manipulators. There are several approaches to the numerical computation of the exact inversion of a flexible multibody system. In this paper, the singularly perturbed form of such mechanical systems is used to give a semianalytic solution to the tracking control design. The control makes the end-effector to even though not exactly, but approximately track a certain trajectory. Thereby, the control signal is computed as a series expansion in terms of an overall flexibility of the bodies of the multibody system. Due to the use of symbolic computations, the main calculations are independent of given parameters (e.g., the desired trajectories), such that the feedforward control can be calculated online. The effectiveness of this approach is shown by the simulation of a two-link flexible manipulator.

scholarly journals Evolution of the DeNOC-based dynamic modelling for multibody systems

2013 ◽  
Vol 4 (1) ◽  
pp. 1-20 ◽  
Author(s):  
S. K. Saha ◽  
S. V. Shah ◽  
P. V. Nandihal
Keyword(s):  
Multibody Systems ◽  
Essential Role ◽  
Closed Loop ◽  
Multibody System ◽  
Dynamic Modelling ◽  
Efficient Manner ◽  
Numerical Examples ◽  
Cam Follower ◽  
Dynamics Modelling ◽  
Denoc Matrices

Abstract. Dynamic modelling of a multibody system plays very essential role in its analyses. As a result, several methods for dynamic modelling have evolved over the years that allow one to analyse multibody systems in a very efficient manner. One such method of dynamic modelling is based on the concept of the Decoupled Natural Orthogonal Complement (DeNOC) matrices. The DeNOC-based methodology for dynamics modelling, since its introduction in 1995, has been applied to a variety of multibody systems such as serial, parallel, general closed-loop, flexible, legged, cam-follower, and space robots. The methodology has also proven useful for modelling of proteins and hyper-degree-of-freedom systems like ropes, chains, etc. This paper captures the evolution of the DeNOC-based dynamic modelling applied to different type of systems, and its benefits over other existing methodologies. It is shown that the DeNOC-based modelling provides deeper understanding of the dynamics of a multibody system. The power of the DeNOC-based modelling has been illustrated using several numerical examples.

scholarly journals A new version of the Riccati transfer matrix method for multibody systems consisting of chain and branch bodies

2019 ◽  
Vol 49 (3) ◽  
pp. 337-354 ◽  
Author(s):  
Xue Rui ◽  
Dieter Bestle ◽  
Guoping Wang ◽  
Jiangshu Zhang ◽  
Xiaoting Rui ◽  
...  
Keyword(s):  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Multibody System ◽  
Branch System ◽  
Computational Speed ◽  
Speed Accuracy

Abstract Computational speed and stability are two important aspects in the dynamics analysis of large-scale complex multibody systems. In order to improve both in the context of the multibody system transfer matrix method, a new version of the Riccati transfer matrix method is presented. Based on the new version of the general transfer matrix method for multibody system simulation, recursive formulae are developed which not only retain all advantages of the transfer matrix method, but also reduce the truncation error. As a result, the computational speed, accuracy and efficiency are improved. Numerical computation results obtained by the proposed method and an ordinary multibody system formulation show good agreement. The successful computation for a spatial branch system with more than 100000 degrees of freedom validates that the proposed method is also working for huge systems.

Formulation of Three-Dimensional Rigid-Flexible Multibody Systems

2007 ◽  
Author(s):  
Daniel Garci´a-Vallejo ◽  
Jose´ L. Escalona ◽  
Juana M. Mayo ◽  
Jaime Domi´nguez
Keyword(s):  
Multibody Systems ◽  
Multibody System ◽  
Three Dimensional ◽  
Natural Coordinates ◽  
Flexible Multibody System ◽  
Absolute Nodal Coordinates ◽  
Constraint Equations ◽  
Flexible Multibody ◽  
The Matrix ◽  
Nodal Coordinates

Multibody systems generally contain solids the deformations of which are appreciable 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 co-exist in the equations of the system. Among the different possibilities provided in bibliography on the material, the formulation in natural coordinates and the formulation in absolute nodal coordinates are utilized in this article to model the rigid and flexible solids, respectively. This article contains a mixed formulation based on the possibility of sharing coordinates between a rigid solid and a flexible solid. In addition, the fact that the matrix of the global mass of the system is shown to be constant and that many of the constraint equations obtained upon utilizing these formulations are linear and can be eliminated. In this work, the formulation presented is utilized to simulate a mechanism with both rigid and flexible components.

scholarly journals On the cosimulation of multibody systems and hydraulic dynamics

2020 ◽  
Vol 50 (2) ◽  
pp. 143-167
Author(s):  
Jarkko Rahikainen ◽  
Francisco González ◽  
Miguel Ángel Naya ◽  
Jussi Sopanen ◽  
Aki Mikkola
Keyword(s):  
Numerical Integration ◽  
Multibody Systems ◽  
Multibody System ◽  
Physical Nature ◽  
The Other ◽  
Numerical Examples ◽  
Complex Engineering ◽  
Coupling Variables ◽  
Mechanical Devices ◽  
Accuracy And Stability

Abstract The simulation of mechanical devices using multibody system dynamics (MBS) algorithms frequently requires the consideration of their interaction with components of a different physical nature, such as electronics, hydraulics, or thermodynamics. An increasingly popular way to perform this task is through co-simulation, that is, assigning a tailored formulation and solver to each subsystem in the application under study and then coupling their integration processes via the discrete-time exchange of coupling variables during runtime. Co-simulation makes it possible to deal with complex engineering applications in a modular and effective way. On the other hand, subsystem coupling can be carried out in a wide variety of ways, which brings about the need to select appropriate coupling schemes and simulation options to ensure that the numerical integration remains stable and accurate. In this work, the co-simulation of hydraulically actuated mechanical systems via noniterative, Jacobi-scheme co-simulation is addressed. The effect of selecting different co-simulation configuration options and parameters on the accuracy and stability of the numerical integration was assessed by means of representative numerical examples.

Numerical Computation of Differential-Algebraic Equations for Non-Linear Dynamics of Multibody Systems Involving Contact Forces

1999 ◽  
Vol 123 (2) ◽  
pp. 272-281 ◽  
Author(s):  
B. Fox ◽  
L. S. Jennings ◽  
A. Y. Zomaya
Keyword(s):  
Multibody Systems ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Variational Problems ◽  
Differential Algebraic Equations ◽  
Contact Forces ◽  
Algebraic Equations ◽  
Differential Algebraic Equation ◽  
Lagrange Equations ◽  
Constraint Equations

The well known Euler-Lagrange equations of motion for constrained variational problems are derived using the principle of virtual work. These equations are used in the modelling of multibody systems and result in differential-algebraic equations of high index. Here they concern an N-link pendulum, a heavy aircraft towing truck and a heavy off-highway track vehicle. The differential-algebraic equation is cast as an ordinary differential equation through differentiation of the constraint equations. The resulting system is computed using the integration routine LSODAR, the Euler and fourth order Runge-Kutta methods. The difficulty to integrate this system is revealed to be the result of many highly oscillatory forces of large magnitude acting on many bodies simultaneously. Constraint compliance is analyzed for the three different integration methods and the drift of the constraint equations for the three different systems is shown to be influenced by nonlinear contact forces.

Formulating Dynamic Equations of Elastic Multibody System via the Method of Typical Substructures

1992 ◽  
Author(s):  
Liu Hongzhao ◽  
Wei-qing Cao
Keyword(s):  
Multibody Systems ◽  
Present Method ◽  
Multibody System ◽  
Dynamic Equations ◽  
Closed Loops ◽  
Constraint Equations ◽  
Elastic Multibody System ◽  
Elastic Multibody Systems

Abstract In this paper, a method of using multibodies as substructures to establish the dynamic equations of elastic multibody systems involving closed loops is put forward. Plane elastic linkage is divided into four typical substructures, and equations of free two-link dyad — the most general one among the four substructures — are derived. Also some examples of substuctures’ partitioning of mechanisms are given to show the present method’s advantages. By means of the present method, the number of system’s constraint equations can be greatly reduced, and it will facilitate solving dynamic equations on a microcomputer.

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