A Parametric Study on the Baumgarte Stabilization Method for Forward Dynamics of Constrained Multibody Systems

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Paulo Flores ◽  
Margarida Machado ◽  
Eurico Seabra ◽  
Miguel Tavares da Silva
Keyword(s):  
Parametric Study ◽  
Multibody Systems ◽  
Initial Conditions ◽  
Differential Algebraic Equations ◽  
Forward Dynamics ◽  
Stabilization Method ◽  
Time Step ◽  
Constrained Multibody Systems ◽  
Laplace Transform Technique ◽  
Baumgarte Stabilization

This paper presents and discusses the results obtained from a parametric study on the Baumgarte stabilization method for forward dynamics of constrained multibody systems. The main purpose of this work is to analyze the influence of the variables that affect the violation of constraints, chiefly the values of the Baumgarte parameters, the integration method, the time step, and the quality of the initial conditions for the positions. In the sequel of this process, the formulation of the rigid multibody systems is reviewed. The generalized Cartesian coordinates are selected as the variables to describe the bodies’ degrees of freedom. The formulation of the equations of motion uses the Newton–Euler approach, augmented with the constraint equations that lead to a set of differential algebraic equations. Furthermore, the main issues related to the stabilization of the violation of constraints based on the Baumgarte approach are revised. Special attention is also given to some techniques that help in the selection process of the values of the Baumgarte parameters, namely, those based on the Taylor’s series and the Laplace transform technique. Finally, a slider-crank mechanism with eccentricity is considered as an example of application in order to illustrate how the violation of constraints can be affected by different factors.

A Parametric Study on the Baumgarte Stabilization Method for Forward Dynamics of Constrained Multibody Systems

2009 ◽  
Author(s):  
Paulo Flores ◽  
Margarida Machado ◽  
Eurico Seabra ◽  
Miguel Tavares da Silva
Keyword(s):  
Parametric Study ◽  
Multibody Systems ◽  
Initial Conditions ◽  
Differential Algebraic Equations ◽  
Forward Dynamics ◽  
Stabilization Method ◽  
Time Step ◽  
Constrained Multibody Systems ◽  
Laplace Transform Technique ◽  
Baumgarte Stabilization

This paper presents and discusses the results obtained from a parametric study on the Baumgarte stabilization method for forward dynamics of constrained multibody systems. The main purpose of this work is to analyze the influence of the variables that affect the violation of constraints, chiefly the values of the Baumgarte parameters, the integration method, the time step and the quality of the initial conditions for the positions. In the sequel of this process the formulation of the rigid multibody systems is reviewed. The generalized Cartesian coordinates are selected as the variables to describe the bodies’ degrees of freedom. The formulation of the equations of motion uses the Newton-Euler approach that is augmented with the constraint equations that lead to a set of differential algebraic equations. Furthermore, the main issues related to the stabilization of the violation of constraints based on the Baumgarte approach are revised. Special attention is also given to some techniques that help in the selection process of the values of the Baumgarte parameters, namely those based on the Taylor’s series and Laplace transform technique. Finally, a slider crank mechanism with eccentricity is considered as an example of application in order to illustrated how the violation of constraints can be affected by different factors such as the Baumgarte parameters, integrator, time step and initial guesses.

A Constraint Stabilization Method for Time Integration of Constrained Multibody Systems in Lie Group Setting

2014 ◽  
Author(s):  
Andreas Müller ◽  
Zdravko Terze
Keyword(s):  
Vector Space ◽  
Lie Group ◽  
Multibody Systems ◽  
Differential Algebraic Equations ◽  
Iteration Step ◽  
Stabilization Method ◽  
Constrained Multibody Systems ◽  
Integration Schemes ◽  
Local Coordinates ◽  
Classical Vector

The stabilization of geometric constraints is vital for an accurate numerical solution of the differential-algebraic equations (DAE) governing the dynamics of constrained multibody systems (MBS). Although this has been a central topic in numerical MBS dynamics using classical vector space formulations, it has not yet been sufficiently addressed when using Lie group formulations. A straightforward approach is to impose constraints directly on the Lie group elements that represent the MBS motion, which requires additional constraints accounting for the invariants of the Lie group. On the other hand, most numerical Lie group integration schemes introduce local coordinates within the integration step, and it is natural to perform the stabilization in terms of these local coordinates. Such a formulation is presented in this paper for index 1 formulation. The stabilization method is applicable to general coordinate mappings (canonical coordinates, Cayley-Rodriguez, Study) on the MBS configuration space Lie group. The stabilization scheme resembles the well-known vectors space projection and pseudo-inverse method consisting in an iterative procedure. A numerical example is presented and it is shown that the Lie group stabilization scheme converges normally within one iteration step, like the scheme in the vector space formulation.

Combined DAE and Sliding Mode Control Methods for Simulation of Constrained Mechanical Systems

2000 ◽  
Vol 122 (4) ◽  
pp. 691-698 ◽  
Author(s):  
M. D. Compere ◽  
R. G. Longoria
Keyword(s):  
Sliding Mode Control ◽  
Numerical Method ◽  
Hybrid Method ◽  
Sliding Mode ◽  
Initial Conditions ◽  
Differential Algebraic Equations ◽  
Control Law ◽  
Stabilization Method ◽  
Mode Control ◽  
Hybrid Numerical Method

In dynamic analysis of constrained multibody systems (MBS), the computer simulation problem essentially reduces to finding a numerical solution to higher-index differential-algebraic equations (DAE). This paper presents a hybrid method composed of multi-input multi-output (MIMO), nonlinear, variable-structure control (VSC) theory and post-stabilization from DAE solution theory for the computer solution of constrained MBS equations. The primary contributions of this paper are: (1) explicit transformation of constrained MBS DAE into a general nonlinear MIMO control problem in canonical form; (2) development of a hybrid numerical method that incorporates benefits of both Sliding Mode Control (SMC) and DAE stabilization methods for the solution of index-2 or index-3 MBS DAE; (3) development of an acceleration-level stabilization method that draws from SMC’s boundary layer dynamics and the DAE literature’s post-stabilization; and (4) presentation of the hybrid numerical method as one way to eliminate chattering commonly found in simulation of SMC systems. The hybrid method presented can be used to simulate constrained MBS systems with either holonomic, nonholonomic, or both types of constraints. In addition, the initial conditions (ICs) may either be consistent or inconsistent. In this paper, MIMO SMC is used to find the control law that will provide two guarantees. First, if the constraints are initially not satisfied (i.e., for inconsistent ICs) the constraints will be driven to satisfaction within finite time using SMC’s stabilization method, urobust,i=−ηisgnsi. Second, once the constraints have been satisfied, the control law, ueq and hybrid stabilization techniques guarantee surface attractiveness and satisfaction for all time. For inconsistent ICs, Hermite-Birkhoff interpolants accurately locate when each surface reaches zero, indicating the transition time from SMC’s stabilization method to those in the DAE literature. [S0022-0434(00)02404-7]

Developing a Numerical Simulation Software for 3D Multibody Systems Based on a Unified Computational Modeling Technique

2009 ◽  
Author(s):  
Shahram Shokouhfar ◽  
Sayyid Mahdi Khorsandijou
Keyword(s):  
Numerical Simulation ◽  
Multibody Systems ◽  
Dynamic Models ◽  
Initial Conditions ◽  
Rigid Bodies ◽  
Simulation Software ◽  
Computational Technique ◽  
Main Task ◽  
Algebraic Equations ◽  
Time Step

This article represents the features and capabilities of a newly developed application namely MASS (Mechanisms Analysis and Simulation Software) and the formulation and techniques therein. MASS is a general C++ application program whose main task is to construct and solve the governing algebraic differential motion equations of 3D multibody systems automatically in matrix forms complying with the computational algorithms required for numerical simulation. Newton-Raphson and SVD methods have been used for kinematical assembling and producing consistent initial conditions. Adaptive time-step Runge-Kutta-Fehlberg numerical integration methods might be used for forward dynamics problems. The governing equations perfectly describe the kinematics and dynamics of multibody systems within which 3D kinematical joints and collisions between rigid bodies might be taken into consideration. The unified computational technique for mathematical modeling of kinematical joints is the most important concept on top of which MASS has been implemented. It has occurred due to the existence of thirteen basic kinematical constraint equations. Each kinematical joint might be defined by a set of algebraic equations being selected from the mentioned basic equations. The unified dynamic models for collisions and impulsive loads have been also achieved using the mentioned technique. Simulation results obtained from MASS have been compared with that of the corresponding software of Working Model ver. 6 and a discussion about the coincidences and differences has been exposed.

Third-Order Differential-Algebraic Equations for Improved Integration of Multibody Dynamics

2017 ◽  
Author(s):  
H. J. Sommer
Keyword(s):  
Multibody Systems ◽  
Computation Time ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Planar Mechanisms ◽  
Third Order ◽  
Simulation Accuracy ◽  
Constrained Multibody Systems ◽  
Predictor Corrector ◽  
Reduce Computation Time

This paper introduces the concept of third-order differential-algebraic equations (DAE) for dynamics of constrained multibody systems. Third-order DAE provide jerk of components which can be integrated simultaneously with acceleration to provide improved simulation accuracy. A new Obreshkov predictor-corrector multistep integrator was developed to test this concept. Results from simulations of two planar mechanisms indicate that third-order DAE can reduce computation time by a factor of ten with equivalent accuracy compared to classical methods.

Two Lie Group Formulations for Dynamic Multibody Systems With Large Rotations

2011 ◽  
Author(s):  
Olivier Bru¨ls ◽  
Martin Arnold ◽  
Alberto Cardona
Keyword(s):  
Numerical Solution ◽  
Lie Group ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Time Integration ◽  
Computational Cost ◽  
Differential Algebraic Equations ◽  
Rigid Body Dynamics ◽  
Time Step ◽  
Iteration Matrix

This paper studies the formulation of the dynamics of multibody systems with large rotation variables and kinematic constraints as differential-algebraic equations on a matrix Lie group. Those equations can then be solved using a Lie group time integration method proposed in a previous work. The general structure of the equations of motion are derived from Hamilton principle in a general and unifying framework. Then, in the case of rigid body dynamics, two particular formulations are developed and compared from the viewpoint of the structure of the equations of motion, of the accuracy of the numerical solution obtained by time integration, and of the computational cost of the iteration matrix involved in the Newton iterations at each time step. In the first formulation, the equations of motion are described on a Lie group defined as the Cartesian product of the group of translations R3 (the Euclidean space) and the group of rotations SO(3) (the special group of 3 by 3 proper orthogonal transformations). In the second formulation, the equations of motion are described on the group of Euclidean transformations SE(3) (the group of 4 by 4 homogeneous transformations). Both formulations lead to a second-order accurate numerical solution. For an academic example, we show that the formulation on SE(3) offers the advantage of an almost constant iteration matrix.

Adaptive Time Integration Method for DAEs of Multibody Systems

2012 ◽  
Author(s):  
Jieyu Ding ◽  
Zhenkuan Pan
Keyword(s):  
Multibody Systems ◽  
Integration Method ◽  
Time Integration ◽  
Differential Algebraic Equations ◽  
Richardson Extrapolation ◽  
Time Interval ◽  
Time Step ◽  
Time Integration Method ◽  
Adaptive Time Integration ◽  
Adaptive Time

An adaptive time integration method is developed for the index-3 differential-algebraic equations (DAEs) of multibody systems to improve the computational efficiency as well as the accuracy of the results. Based on the modified general-α method, the adaptive time integration is presented. At each discrete time interval, the time step size is changed through Richardson extrapolation with definable computation accuracy. A rotary rod slider system is used to validate the presented adaptive time integration. The accuracy and efficiency are determined by the expected order of the accuracy in Richardson extrapolation.

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