Analysis of Constraint Forces in Multibody Systems Based on the Differential Null Space Matrix Method

2019 ◽  
Author(s):  
Keisuke Kamiya
Keyword(s):  
Lagrange Multipliers ◽  
Multibody Systems ◽  
Null Space ◽  
Governing Equations ◽  
Constraint Forces ◽  
Numerical Examples ◽  
Space Matrix ◽  
Rigid Multibody Systems ◽  
Rigid Multibody

Abstract This paper treats a problem to determine constraint forces in rigid mutibody systems. One of the most often applied algorithms for determination of constraint forces is based on the use of recursive Newton-Euler formalism. Another algorithm often applied for determination of constraint forces is based on the use of Lagrange multipliers. This paper presents a new method to determine constraint forces in rigid multibody systems. First relative displacements which violate the constraints, called anti-constraint relative displacements, are introduced, and governing equations which involve the constraint forces explicitly are derived. In the derived equations, the constraint forces appear independently from the Lagrange multipliers. Then, a method is proposed to determine the constraint forces by eliminating the Lagrange multipliers based on the methods proposed in previous papers by the author. The method is extended to have ability to treat systems with redundant constraints. Finally, validity of the proposed method is confirmed via numerical examples.

Null Space Method of Differential Equation Type for Motion Analysis of Multibody Systems

2017 ◽  
Author(s):  
Keisuke Kamiya ◽  
Yusaku Yamashita
Keyword(s):  
Lagrange Multipliers ◽  
Multibody Systems ◽  
Null Space ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Governing Equations ◽  
Accurate Analysis ◽  
Dependent Variables ◽  
Space Matrix ◽  
Space Method

The governing equations of multibody systems are, in general, formulated in the form of differential algebraic equations (DAEs) involving the Lagrange multipliers. For efficient and accurate analysis, it is desirable to eliminate the Lagrange multipliers and dependent variables. Methods called null space method and Maggi’s method eliminate the Lagrange multipliers by using the null space matrix for the constraint Jacobian. In previous reports, one of the authors presented methods which use the null space matrix. In the procedure to obtain the null space matrix, the inverse of a matrix whose regularity may not be always guaranteed. In this report, a new method is proposed in which the null space matrix is obtained by solving differential equations that can be always defined by using the QR decomposition, even if the constraints are redundant. Examples of numerical analysis are shown to validate the proposed method.

Continuous Null Space Method for Constrained Mechanical Systems

2011 ◽  
Author(s):  
Keisuke Kamiya ◽  
Makoto Sawada ◽  
Yuji Furusawa
Keyword(s):  
Lagrange Multipliers ◽  
Null Space ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Governing Equations ◽  
Accurate Analysis ◽  
Constrained Mechanical Systems ◽  
Dependent Variables ◽  
Space Matrix ◽  
Space Method

The governing equations for multibody systems are, in general, formulated in the form of differential algebraic equations (DAEs) involving the Lagrange multipliers. It is desirable for efficient and accurate analysis to eliminate the Lagrange multipliers and dependent variables. As a method to solve the DAEs by eliminating the Lagrange multipliers, there is a method called the null space method. In this report, first, it is shown that using the null space matrix one can eliminate the Lagrange multipliers and reduce the number of velocities to that of the independent ones. Then, a new method to obtain the continuous null space matrix is presented. Finally, the presented method is applied to four-bar linkages.

Elimination of Redundant Cut Joint Constraints for Multibody System Models

2003 ◽  
Vol 126 (3) ◽  
pp. 488-494 ◽  
Author(s):  
A. Mu¨ller
Keyword(s):  
Multibody Systems ◽  
Computational Effort ◽  
Basis Matrix ◽  
Constraint Equations ◽  
Kinematic Loops ◽  
Joint Constraints ◽  
Rigid Multibody Systems ◽  
Kinematic Loop ◽  
Rigid Multibody

The problem of dependent cut joint constraints for kinematic loops in rigid multibody systems is addressed. The constraints are reduced taking into account the subalgebra generated by the screw system of the kinematic loop. The elimination of dependent constraint equations is based on constructing a basis matrix of the screw algebra generated by loop’s screw system. This matrix is configuration independent and thus always valid. The determination of the sufficient constraints is achieved with a SVD or QR decomposition of this matrix. Unlike all other proposed approaches the presented method is singularity consistent because it is not the Jacobian which is decomposed, but instead a basis matrix for the loop algebra. Since this basis is obtained after a finite number of cross products the computational effort is negligible. Furthermore, because the elimination process is only necessary once in advance of the integration/simulation process, it proved valuable even if it does not remove all dependent constraints, as for paradoxical mechanisms.

Multiple impacts with friction in rigid multibody systems

1995 ◽  
Vol 7 (4) ◽  
pp. 471-497 ◽  
Author(s):  
Ch. Glocker ◽  
F. Pfeiffer
Keyword(s):  
Multibody Systems ◽  
Multiple Impacts ◽  
Rigid Multibody Systems ◽  
Rigid Multibody ◽  
Impacts With Friction

Time-Differentiable Null Space Method for Constrained Mechanical Systems (Application to a Rheonomic System)

2013 ◽  
Author(s):  
Keisuke Kamiya
Keyword(s):  
Lagrange Multipliers ◽  
Null Space ◽  
Constraint Equation ◽  
Coefficient Matrix ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Accurate Analysis ◽  
Constrained Mechanical Systems ◽  
Space Matrix ◽  
Space Method

The governing equations of multibody systems are, in general, formulated in the form of differential algebraic equations (DAEs) involving the Lagrange multipliers. For efficient and accurate analysis, it is desirable to eliminate the Lagrange multipliers and dependent variables. Methods called null space method and Maggi’s method eliminate the Lagrange multipliers by using the null space matrix for the coefficient matrix which appears in the constraint equation in velocity level. In a previous report, the author presented a method to obtain a time differentiable null space matrix for scleronomic systems, whose constraint does not depend on time explicitly. In this report, the method is generalized to rheonomic systems, whose constraint depends on time explicitly. Finally, the presented method is applied to four-bar linkages.

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