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.

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.

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.

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.

Improved Differential Nullspace Matrix Method for the Dynamic Analysis of Multibody Systems

2018 ◽  
Author(s):  
Keisuke Kamiya
Keyword(s):  
Lagrange Multipliers ◽  
Multibody Systems ◽  
Differential Algebraic Equations ◽  
Computational Time ◽  
Algebraic Equations ◽  
Accurate Analysis ◽  
Dynamics Of Multibody Systems ◽  
General Dynamics ◽  
Novel Method ◽  
Rigid Multibody Systems

This paper presents a novel method for motion analysis of rigid multibody systems. In general, dynamics of multibody systems is described by differential algebraic equations with Lagrange multipliers. For efficient and accurate analysis, it is desirable to eliminate the Lagrange multipliers and dependent variables. Methods called nullspace method and Maggi’s method eliminate the Lagrange multipliers by using the nullspace matrix for the constraint Jacobian. In a previous report, the author presented a method in which the nullspace matrix is obtained by solving a differential equation together with the equation of motion of the system. In that method QR decomposition is used. In this report, reduction in computational time with the LU decomposition is attempted. In addition, treatment of singular configurations for accurate analysis is presented. Validity of the presented method is confirmed via numerical examples.

A Practical Approach for the Linearization of the Constrained Multibody Dynamics Equations

2006 ◽  
Vol 1 (3) ◽  
pp. 230-239 ◽  
Author(s):  
Dan Negrut ◽  
Jose L. Ortiz
Keyword(s):  
Heterogeneous Systems ◽  
Reference Method ◽  
Nonholonomic Constraints ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Governing Equations ◽  
Constrained Mechanical Systems ◽  
Friction Control ◽  
Constrained Multibody Dynamics ◽  
Space Formulation

The paper presents an approach to linearize the set of index 3 nonlinear differential algebraic equations that govern the dynamics of constrained mechanical systems. The proposed method handles heterogeneous systems that might contain flexible bodies, friction, control elements (user-defined differential equations), and nonholonomic constraints. Analytically equivalent to a state-space formulation of the system dynamics in Lagrangian coordinates, the proposed method augments the governing equations and then computes a set of sensitivities that provide the linearization of interest. The attributes associated with the method are the ability to handle large heterogeneous systems, ability to linearize the system in terms of arbitrary user-defined coordinates, and straightforward implementation. The proposed approach has been released in the 2005 version of the MSC.ADAMS/Solver(C++) and compares favorably with a reference method previously available. The approach was also validated against MSC.NASTRAN and experimental results.

An Accurate Spatial Discretization and Substructural Method With Application to Moving Elevator Cable-Car Systems: Part I—Methodology

2011 ◽  
Author(s):  
H. Ren ◽  
W. D. Zhu
Keyword(s):  
Differential Algebraic Equations ◽  
Distributed Parameter ◽  
Algebraic Equations ◽  
Spatial Discretization ◽  
Matching Conditions ◽  
Generalized Coordinates ◽  
Axial Forces ◽  
Linear Algebraic Equations ◽  
Dependent Variables ◽  
Elevator Cable

A spatial discretization and substructure method is developed to calculate the dynamic responses of one-dimensional systems, which consist of length-variant distributed-parameter components such as strings, rods, and beams, and lumped-parameter components such as point masses and rigid bodies. The dependent variable, such as the displacement, of a distributed-parameter component is decomposed into boundary-induced terms and internal terms. The boundary-induced terms are interpolated from the boundary motions, and the internal terms are approximated by an expansion of trial functions that satisfy the corresponding homogeneous boundary conditions. All the matching conditions at the interfaces of the components are satisfied, and the expansions of the dependent variables of the distributed-parameter components absolutely and uniformly converge. The spatial derivatives of the dependent variables, which are related to the internal forces/moments, such as the axial forces, bending moments, and shear forces, can be accurately calculated. Assembling the component equations and the geometric matching conditions that arise from the continuity relations leads to a system of differential algebraic equations (DAEs). When some matching conditions are linear algebraic equations, some generalized coordinates can be represented by others so that the number of the generalized coordinates can be reduced. The methodology is applied to moving elevator cable-car systems in Part II of this work.

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.

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.

Simulation of Constrained Mechanical Systems—Part II: Explicit Numerical Integration

2012 ◽  
Vol 79 (4) ◽  
Author(s):  
David J. Braun ◽  
Michael Goldfarb
Keyword(s):  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Direct Consequence ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Constrained Motion ◽  
Constrained Mechanical Systems ◽  
Holonomic And Nonholonomic Constraints ◽  
Long Time ◽  
Constraint Correction

This paper presents an explicit to integrate differential algebraic equations (DAEs) method for simulations of constrained mechanical systems modeled with holonomic and nonholonomic constraints. The proposed DAE integrator is based on the equation of constrained motion developed in Part I of this work, which is discretized here using explicit ordinary differential equation schemes and applied to solve two nontrivial examples. The obtained results show that this integrator allows one to precisely solve constrained mechanical systems through long time periods. Unlike many other implicit DAE solvers which utilize iterative constraint correction, the presented DAE integrator is explicit, and it does not use any iteration. As a direct consequence, the present formulation is simple to implement, and is also well suited for real-time applications.

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