Stabilization Method for Numerical Integration of Multibody Mechanical Systems

1998 ◽  
Vol 120 (4) ◽  
pp. 565-572 ◽  
Author(s):  
Shih-Tin Lin ◽  
Ming-Chong Hong
Keyword(s):  
Numerical Integration ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Constraint Equation ◽  
Differential Algebraic Equations ◽  
Integration Step ◽  
Algebraic Equations ◽  
Stabilization Method ◽  
Step Size ◽  
Integration Methods

The object of this study is to solve the stability problem for the numerical integration of constrained multibody mechanical systems. The dynamic equations of motion of the constrained multibody mechanical system are mixed differential-algebraic equations (DAE). In applying numerical integration methods to this equation, constrained equations and their first and second derivatives must be satisfied simultaneously. That is, the generalized coordinates and their derivatives are dependent. Direct integration methods do not consider this dependency and constraint violation occurs. To solve this problem, Baumgarte proposed a constraint stabilization method in which a position and velocity terms were added in the second derivative of the constraint equation. The disadvantage of this method is that there is no reliable method for selecting the coefficients of the position and velocity terms. Improper selection of these coefficients can lead to erroneous results. In this study, stability analysis methods in digital control theory are used to solve this problem. Correct choice of the coefficients for the Adams method are found for both fixed and variable integration step size.

A PID Type Constraint Stabilization Method for Numerical Integration of Multibody Systems

2011 ◽  
Vol 6 (4) ◽  
Author(s):  
Shih-Tin Lin ◽  
Ming-Wen Chen
Keyword(s):  
Numerical Integration ◽  
Pid Controller ◽  
Controller Design ◽  
Equations Of Motion ◽  
Constraint Equation ◽  
Correct Choice ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Stabilization Method ◽  
Constraint Stabilization

The dynamic equations of motion of the constrained multibody mechanical system are mixed differential-algebraic equations (DAEs). The numerical solution of the DAE systems solved using ordinary-differential equation (ODE) solvers may suffer from constraint drift phenomenon. To solve this problem, Baumgarte proposed a constraint stabilization method in which a position and velocity terms were added in the second derivative of the constraint equation. Baumgarte’s method is a proportional-derivative (PD) type controller design. In this paper, an Iintegrator controller is included to form a proportional-integral-derivative (PID) controller so that the steady state error of the numerical integration can be reduced. Stability analysis methods in the digital control theory are used to find out the correct choice of the coefficients for the PID controller.

A Geometric Derivation of the Equations of Motion for Mechanical Systems With Scleronomic Constraints

2015 ◽  
Author(s):  
Sotirios Natsiavas ◽  
Elias Paraskevopoulos
Keyword(s):  
Numerical Methods ◽  
Lagrange Multipliers ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Main Idea ◽  
Constraint Equation ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Motion Constraints ◽  
Stiffness Properties

A new set of equations of motion is presented for a class of mechanical systems subjected to equality motion constraints. Specifically, the systems examined satisfy a set of holonomic and/or nonholonomic scleronomic constraints. The main idea is to consider the equations describing the action of the constraints as an integral part of the overall process leading to the equations of motion. The constraints are incorporated one by one, in a process analogous to that used for setting up the equations of motion. This proves to be equivalent to assigning appropriate inertia, damping and stiffness properties to each constraint equation and leads to a system of second order ordinary differential equations for both the coordinates and the Lagrange multipliers associated to the motion constraints automatically. This brings considerable advantages, avoiding problems related to systems of differential-algebraic equations or penalty formulations. Apart from its theoretical value, this set of equations is well-suited for developing new robust and accurate numerical methods.

scholarly journals Numerical solution for consistent initial conditions of constrained mechanical systems

2003 ◽  
Vol 25 (3) ◽  
pp. 170-185
Author(s):  
Dinh Van Phong
Keyword(s):  
Numerical Solution ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Flexible Approach ◽  
Constrained Mechanical Systems ◽  
Consistent Initial Values

The article deals with the problem of consistent initial values of the system of equations of motion which has the form of the system of differential-algebraic equations. Direct treating the equations of mechanical systems with particular properties enables to study the system of DAE in a more flexible approach. Algorithms and examples are shown in order to illustrate the considered technique.

Stability and Accuracy Analysis of Baumgarte’s Constraint Violation Stabilization Method

1995 ◽  
Vol 117 (3) ◽  
pp. 446-453 ◽  
Author(s):  
S. Yoon ◽  
R. M. Howe ◽  
D. T. Greenwood
Keyword(s):  
Equations Of Motion ◽  
Integration Step ◽  
State Variables ◽  
Stabilization Method ◽  
Lagrange Equations ◽  
Step Size ◽  
Constraint Violation ◽  
Truncation Errors ◽  
Numerical Stability Analysis ◽  
Stability And Accuracy

When Baumgarte’s Constraint Violation Stabilization Method (CVSM) is used in the simulation of Lagrange equations of motion with holonomic constraints, it is shown that, with suitable assumptions on the integration step size h and the eigenvalues (λ’s) of the linearized system, the constraint variables are effectively integrated by the same algorithm as that used for the state variables. A numerical stability analysis of the constraint violations can be performed using this so-called pseudo-integration equation. A study is also made of truncation errors and their modeling in the continuous time domain. This model can be used to determine the effectiveness of various constraint controls and integration methods in reducing the errors in the solution due to truncation errors. Examples are presented to illustrate the use of a higher-order truncation error model which leads to an accurate quantitative steady-state analysis of the constraint violations.

Application of Scaling to Multibody Dynamics Simulations

2015 ◽  
Author(s):  
William Prescott
Keyword(s):  
Equations Of Motion ◽  
Industrial Applications ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Step Size ◽  
Virtual Lab ◽  
Long Run ◽  
Multibody Dynamic ◽  
Dynamics Simulations ◽  
Stability Issues

This paper will examine the importance of applying scaling to the equations of motion for multibody dynamic systems when applied to industrial applications. If a Cartesian formulation is used to formulate the equations of motion of a multibody dynamic system the resulting equations are a set of differential algebraic equations (DAEs). The algebraic components of the DAEs arise from appending the joint equations used to model revolute, cylindrical, translational and other joints to the Newton-Euler dynamic equations of motion. Stability issues can arise in an ill-conditioned Jacobian matrix of the integration method this will result in poor convergence of the implicit integrator’s Newton method. The repeated failures of the Newton’s method will require a small step size and therefore simulations that require long run times to complete. Recent advances in rescaling the equations of motion have been proposed to address this problem. This paper will see if these methods or a variant addresses not only stability concerns, but also efficiency. The scaling techniques are applied to the Gear-Gupta-Leimkuhler (GGL) formulation for multibody problems by embedding them into the commercial multibody code (MBS) Virtual. Lab Motion and then use them to solve an industrial sized automotive example to see if performance is improved.

A Computer-Aided Environment for Analysis and Design of Mechanical Systems

1991 ◽  
Author(s):  
Hamid M. Lankarani ◽  
Behnam Bahr ◽  
Saeid Motavalli
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
General Purpose ◽  
Design System ◽  
Integrated Analysis ◽  
Algebraic Equations ◽  
Efficient Design ◽  
Analysis And Design ◽  
Wide Range

Abstract This paper presents the description of an ideal tool for analysis and design of complex multibody mechanical systems. It is in the form of a general-purpose computer program, which can be used for simulation of many different systems. The generality of this computer-integrated environment allows a wide range of applications with significant engineering importance. No matter how complicated the mechanical system under consideration is, a numerical multibody model of the system is constructed. The governing mixed differential/algebraic equations of motion are automatically formulated and numerically generated. State-of-the-art numerical techniques and computational methods are employed and developed which produce in the response of the system at discrete time junctures. Postprocessing of the results in the form of graphical images or real-time animations provides an enormous aid in visualizing motion of the system. The analysis package may be merged with an efficient design optimization algorithm. The developed integrated analysis/design system is a valuable tool for researchers, design engineers, and analysts of mechanical systems. This computer-integrated tool provides an important bridge between the classical decision making process by an engineer and the emerging technology of computers.

Implicit Integration of the Equations of Multibody Dynamics in Descriptor Form

1997 ◽  
Author(s):  
Edward J. Haug ◽  
Mirela Iancu ◽  
Dan Negrut
Keyword(s):  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Time Derivative ◽  
Kinetic Equations ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Integration Formulas ◽  
Multibody Dynamic ◽  
Constrained Equations

Abstract An implicit numerical integration approach, based on generalized coordinate partitioning of the descriptor form of the differential-algebraic equations of motion of multibody dynamics, is presented. This approach is illustrated for simulation of stiff mechanical systems using the well known Newmark integration method from structural dynamics. Second order Newmark integration formulas are used to define independent generalized coordinates and their first time derivative as functions of independent accelerations. The latter are determined as the solution of discretized equations obtained using the descriptor form of the equations of motion. Dependent variables in the formulation, including Lagrange multipliers, are determined to satisfy all the kinematic and kinetic equations of multibody dynamics. The approach is illustrated by solving the constrained equations of motion for mechanical systems that exhibit stiff behavior. Results show that the approach is robust and has the capability to integrate differential-algebraic equations of motion for stiff multibody dynamic systems.

Scaling of Constraints and Augmented Lagrangian Formulations in Multibody Dynamics Simulations

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Olivier A. Bauchau ◽  
Alexander Epple ◽  
Carlo L. Bottasso
Keyword(s):  
Physical Properties ◽  
Augmented Lagrangian ◽  
Equations Of Motion ◽  
Time Integration ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Integration Schemes

This paper addresses practical issues associated with the numerical enforcement of constraints in flexible multibody systems, which are characterized by index-3 differential algebraic equations (DAEs). The need to scale the equations of motion is emphasized; in the proposed approach, they are scaled based on simple physical arguments, and an augmented Lagrangian term is added to the formulation. Time discretization followed by a linearization of the resulting equations leads to a Jacobian matrix that is independent of the time step size, h; hence, the condition number of the Jacobian and error propagation are both O(h0): the numerical solution of index-3 DAEs behaves as in the case of regular ordinary differential equations (ODEs). Since the scaling factor depends on the physical properties of the system, the proposed scaling decreases the dependency of this Jacobian on physical properties, further improving the numerical conditioning of the resulting linearized equations. Because the scaling of the equations is performed before the time and space discretizations, its benefits are reaped for all time integration schemes. The augmented Lagrangian term is shown to be indispensable if the solution of the linearized system of equations is to be performed without pivoting, a requirement for the efficient solution of the sparse system of linear equations. Finally, a number of numerical examples demonstrate the efficiency of the proposed approach to scaling.

On an Implementation of the Hilber-Hughes-Taylor Method in the Context of Index 3 Differential-Algebraic Equations of Multibody Dynamics (DETC2005-85096)

2006 ◽  
Vol 2 (1) ◽  
pp. 73-85 ◽  
Author(s):  
Dan Negrut ◽  
Rajiv Rampalli ◽  
Gisli Ottarsson ◽  
Anthony Sajdak
Keyword(s):  
Size Control ◽  
Differential Algebraic Equations ◽  
Integration Step ◽  
Algebraic Equations ◽  
Step Size ◽  
Stopping Criteria ◽  
Flexible Bodies ◽  
Engine Model ◽  
Numerical Integrator ◽  
Speedup Factor

The paper presents theoretical and implementation aspects related to a numerical integrator used for the simulation of large mechanical systems with flexible bodies and contact/impact. The proposed algorithm is based on the Hilber-Hughes-Taylor (HHT) implicit method and is tailored to answer the challenges posed by the numerical solution of index 3 differential-algebraic equations that govern the time evolution of a multibody system. One of the salient attributes of the algorithm is the good conditioning of the Jacobian matrix associated with the implicit integrator. Error estimation, integration step-size control, and nonlinear system stopping criteria are discussed in detail. Simulations using the proposed algorithm of an engine model, a model with contacts, and a model with flexible bodies indicate a 2 to 3 speedup factor when compared against benchmark MSC.ADAMS runs. The proposed HHT-based algorithm has been released in the 2005 version of the MSC.ADAMS/Solver.

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