512 Analysis of Multibody Systems Based on The Modified Hamilton's Principle : Elimination of the Lagrange Multipliers Based on the Null 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.

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

Volume 7A: 9th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

10.1115/detc2013-12963 ◽

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

Volume 6: 15th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

10.1115/detc2019-97730 ◽

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.

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Continuous Null Space Method for Constrained Mechanical Systems

Volume 4: 8th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A and B ◽

10.1115/detc2011-48150 ◽

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.

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Derivation of Equations for Flexible Multibody Systems in Terms of Quasi-Coordinates from the Extended Hamilton’s Principle

Shock and Vibration ◽

10.1155/1993/915264 ◽

1993 ◽

Vol 1 (2) ◽

pp. 107-119 ◽

Cited By ~ 5

Author(s):

L. Meirovitch

Keyword(s):

Multibody Systems ◽

Equations Of Motion ◽

Rigid Bodies ◽

Flexible Multibody Systems ◽

Hamilton’S Principle ◽

Hamilton's Principle ◽

Lagrange’S Equations ◽

Extended Hamilton’S Principle ◽

Flexible Multibody ◽

Lagrange's Equations

Early derivations of the equations of motion for single rigid bodies, single flexible bodies, and flexible multibody systems in terms of quasi-coordinates have been carried out in two stages. The first consists of the use of the extended Hamilton’s principle to derive standard Lagrange’s equations in terms of generalized coordinates and the second represents a transformation of the Lagrange’s equations to equations in terms of quasi-coordinates. In this article, hybrid (ordinary and partial) differential equations for flexible multibody systems are derived in terms of quasi-coordinates directly from the extended Hamilton's principle. The approach has beneficial implications in an eventual spatial discretization of the problem.

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