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.

The Automatic Stabilization Algorithm for Euler-Lagrange Equations With Holonomic/Nonholonomic Constraints in Multibody System Dynamics

2001 ◽  
Author(s):  
Zhenkuan Pan ◽  
Weijia Zhao
Keyword(s):  
System Dynamics ◽  
Multibody System ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Multibody System Dynamics ◽  
Integration Step ◽  
Lagrange Equations ◽  
Constraint Violation ◽  
Automatic Stabilization ◽  
The One

Abstract A new automatic constraint violation stabilization algorithm for numerical integration of Euler-Lagrange equations of motion of multibody system dynamics based on Baumgarte constraint violation stabilization method and Taylor expansion of constraint functions is presented. The constraint equations may be holonomic or nonholonomic. The parameters α, β, σ used in Baumgarte’s method are determined automatically according to integration step. Some numerical examples compare the accuracy between the traditional method and the one suggested in this paper are presented finally.

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 mechanistic model for simulating methane emissions from unstirred liquid manure storages

2010 ◽  
Vol 90 (3) ◽  
pp. 507-516 ◽  
Author(s):  
Q. Huang ◽  
O. Wohlgemut ◽  
N. Cicek ◽  
J. France ◽  
E. Kebreab
Keyword(s):  
Mechanistic Model ◽  
Correlation Coefficients ◽  
Methanogenic Bacteria ◽  
Integration Step ◽  
State Variables ◽  
Liquid Manure ◽  
Step Size ◽  
Intergovernmental Panel ◽  
Manure Storage ◽  
Time Courses

The objective of the study was to develop a mechanistic model of methane (CH4) producing processes in unstirred conditions with potential application for estimating CH4 emissions from anaerobic manure storage facilities. Although models for describing anaerobic digestion processes are available, they largely relate to anaerobic digesters, and do not directly apply to the prediction of CH4 emissions from liquid manure storage. Based on extant models, six biochemical steps were described: hydrolysis, acetogenesis, hydrogenogenesis, homoacetogenesis, hydrogenous methanogenesis and acetic methanogenesis, performed by five bacterial groups. The model contains six state variables, and mass flow is mostly generated and quantified using bacterial kinetics. The model was coded in acslX and a fourth-order Runge-Kutta method with an integration step size of 0.05 d was used for numerical integration. The time courses of CH4 production and volatile fatty acid (VFA) concentration of two laboratory-scale liquid manure storage tanks, both filled with liquid sow manures and running in unstirred and constant 25°C conditions, were well predicted, with correlation coefficients over 0.90. Discrepancies between predicted and measured CH4 production and VFA concentration were mainly due to random variation of observed data. The model was sensitive to parameters describing hydrolysis and the kinetics of acetogenic and acetate methanogenic bacteria. Simulations based on the Intergovernmental Panel on Climate Change model (Tier II) predicted 260 g CH4 kg-1 volatile solids (VS, assuming maximum CH4 producing capacity of 0.48 and methane conversion factor of 80%), whereas the measured value was 78.3 g CH4 kg-1 VS after 146 d and the mechanistic model predicted 74.8 g CH4 kg-1 VS. The model developed in this study appears to be better suited to batch manure storage than the IPCC model.

An Adaptive Constraint Violation Stabilization Method for Dynamic Analysis of Mechanical Systems

1985 ◽  
Vol 107 (4) ◽  
pp. 488-492 ◽  
Author(s):  
C. O. Chang ◽  
P. E. Nikravesh
Keyword(s):  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
General Purpose ◽  
Adaptive Mechanism ◽  
Stabilization Method ◽  
Constraint Violation ◽  
Transient Dynamic Analysis ◽  
Constrained Mechanical Systems ◽  
Feedback Control Theory

The transient dynamic analysis of equations of motion for constrained mechanical systems requires the solution of a mixed set of algebraic and differential equations. A constraint violation stabilization method, based on feedback control theory of linear systems, has been suggested by some researchers for solving these equations. However, since the value of damping parameters for this method are uncertain, the method is to some extent unattractive for general-purpose use. This paper presents an adaptive mechanism for determining the damping parameters. The results of the simulation for two examples illustrate the improvement in reducing the constraint violations when using this method.

A Compliant Double Pin Track Link Model for Multibody Tracked Vehicles

1999 ◽  
Author(s):  
J. H. Choi ◽  
D. S. Bae ◽  
H. S. Ryu
Keyword(s):  
High Speed ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
High Mobility ◽  
Contact Forces ◽  
Integration Step ◽  
Step Size ◽  
Tracked Vehicles ◽  
Accelerated Motion ◽  
Pin Track

Abstract It is the objective of this investigation to develop compliant double pin track link models and investigate the use of these models in the dynamic analysis of high mobility tracked vehicles. There are two major difficulties encountered in developing the compliant track models discussed in this paper. The first is due to the fact that the integration step size must be kept small in order to maintain the numerical stability of the solution. This solution includes high oscillatory signals resulting from the impulsive contact forces and the use of stiff compliant elements to represent the joints between the track links. The characteristics of the compliant, elements used in this investigation to describe the track joints are measured experimentally. The second difficulty encountered in this investigation is due to the large number of the system equations of motion of the three dimensional multibody tracked vehicle model. The dimensionality problem is solved by decoupling the equations of motion of the chassis subsystem and the track subsystems. Recursive methods are used to obtain a minimum set of equations for the chassis subsystem. Several simulation scenarios including an accelerated motion, high speed motion, braking, and turning motion of the high mobility vehicle are tested in order to demonstrate the effectiveness and validity of the methods proposed in this investigation.

scholarly journals Holonomicity analysis of electromechanical systems

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 942-947 ◽  
Author(s):  
Miroslaw Wcislik ◽  
Karol Suchenia
Keyword(s):  
Equations Of Motion ◽  
Electromechanical System ◽  
State Variables ◽  
Lagrange Equations ◽  
Electromechanical Systems ◽  
Motion Equations ◽  
D’Alembert Equation ◽  
Reluctance Motor ◽  
Electric Parameters ◽  
Pendulum Oscillation

Abstract Electromechanical systems are described using state variables that contain electrical and mechanical components. The equations of motion, both electrical and mechanical, describe the relationships between these components. These equations are obtained using Lagrange functions. On the basis of the function and Lagrange - d’Alembert equation the methodology of obtaining equations for electromechanical systems was presented, together with a discussion of the nonholonomicity of these systems. The electromechanical system in the form of a single-phase reluctance motor was used to verify the presented method. Mechanical system was built as a system, which can oscillate as the element of physical pendulum. On the base of the pendulum oscillation, parameters of the electromechanical system were defined. The identification of the motor electric parameters as a function of the rotation angle was carried out. In this paper the characteristics and motion equations parameters of the motor are presented. The parameters of the motion equations obtained from the experiment and from the second order Lagrange equations are compared.

Optimal Design of Mechanical Systems With Constraint Violation Stabilization Method

1985 ◽  
Vol 107 (4) ◽  
pp. 493-498 ◽  
Author(s):  
C. O. Chang ◽  
P. E. Nikravesh
Keyword(s):  
Optimal Design ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Design Procedure ◽  
Optimization Procedure ◽  
Design Sensitivity ◽  
System Response ◽  
Stabilization Method ◽  
Sensitivity Matrix ◽  
Constraint Violation

This paper presents a comprehensive optimal design procedure for constrained dynamic systems. The constraint violation stabilization method for dynamic analysis of mechanical systems is briefly reviewed. A direct differentiation method is used to form the equations of design sensitivity analysis based on a constraint violation stabilization method. The sensitivity equations and the equations of motion are integrated simultaneously to obtain the system response, as well as the state sensitivity matrices. All integrations are performed using a multistep predictor-corrector method. The first order design sensitivity matrix is used to calculate the gradient of cost function and the performance constraint during the optimization procedure. An optimization routine is linked to the analysis/sensitivity algorithm. Two examples are given which illustrate the effectiveness of this method for determining the optimal design of a system.

Geometric Elimination of Constraint Violations in Numerical Simulation of Lagrangian Equations

1994 ◽  
Vol 116 (4) ◽  
pp. 1058-1064 ◽  
Author(s):  
S. Yoon ◽  
R. M. Howe ◽  
D. T. Greenwood
Keyword(s):  
Numerical Simulation ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Geometric Constraints ◽  
State Variables ◽  
Step Size ◽  
Constraint Equations ◽  
Lagrangian Equations ◽  
Small Step Size ◽  
Gradient Based

Conventional holonomic or nonholonomic constraints are defined as geometric constraints. The total enregy of a dynamic system can be treated as a constrained quantity for the purpose of accurate numerical simulation. In the simulation of Lagrangian equations of motion with constraint equations, the Geometric Elimination Method turns out to be more effective in controlling constraint violations than any conventional methods, including Baumgarte’s Constraint Violation Stabilization Method (CVSM). At each step, this method first goes through the numerical integration process without correction to obtain updated values of the state variables. These values are then used in a gradient-based procedure to eliminate the geometric and energy errors simultaneously before processing to the next step. For small step size, this procedure is stable and very accurate.

scholarly journals A least action principle for interceptive walking

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Soon Ho Kim ◽  
Jong Won Kim ◽  
Hyun Chae Chung ◽  
MooYoung Choi
Keyword(s):  
Social Systems ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Action Principle ◽  
Lagrange Equations ◽  
Least Action Principle ◽  
Least Action ◽  
The Least Action Principle ◽  
Principle Of Least Action ◽  
Human Participants

AbstractThe principle of least effort has been widely used to explain phenomena related to human behavior ranging from topics in language to those in social systems. It has precedence in the principle of least action from the Lagrangian formulation of classical mechanics. In this study, we present a model for interceptive human walking based on the least action principle. Taking inspiration from Lagrangian mechanics, a Lagrangian is defined as effort minus security, with two different specific mathematical forms. The resulting Euler–Lagrange equations are then solved to obtain the equations of motion. The model is validated using experimental data from a virtual reality crossing simulation with human participants. We thus conclude that the least action principle provides a useful tool in the study of interceptive walking.

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