A New Sparse Solver for Kinematics Simulation of Multibody Systems in Natural Coordinates Based on Tearing Methods

2001 ◽  
Author(s):  
Francisco Javier Funes ◽  
José Manuel Jiménez ◽  
José Ignacio Rodríguez ◽  
Javier García de Jalón
Keyword(s):  
Multibody Systems ◽  
Linear Equations ◽  
New Method ◽  
Natural Coordinates ◽  
Open Chain ◽  
Triangular Form ◽  
Constraint Equations ◽  
Closed Loop Systems ◽  
Kinematics Simulation ◽  
The Matrix

Abstract This paper presents a new method for the factorization of the sparse system of linear equations arising from the kinematic simulation of multibody systems using natural coordinates. A special reordering of the jacobian matrix of the mechanism constraint equations will be described. This reordering depends only on the topology of the mechanism. In open-chain systems the matrix can be reordered to a block triangular form. For closed-loop systems this matrix can take a bordered block triangular form, with very few columns in the border. A modification of the Harwell’s implementation of the P5 algorithm from Erisman et al. [1] is used for reordering the rows and columns of the matrix. A new method of factorization is described. This method reduces the number of floating-point operations and the fill-ins. An efficient way for solving the least-squares problem arising from over-determined systems is explained.

Formulation of Three-Dimensional Rigid-Flexible Multibody Systems

2007 ◽  
Author(s):  
Daniel Garci´a-Vallejo ◽  
Jose´ L. Escalona ◽  
Juana M. Mayo ◽  
Jaime Domi´nguez
Keyword(s):  
Multibody Systems ◽  
Multibody System ◽  
Three Dimensional ◽  
Natural Coordinates ◽  
Flexible Multibody System ◽  
Absolute Nodal Coordinates ◽  
Constraint Equations ◽  
Flexible Multibody ◽  
The Matrix ◽  
Nodal Coordinates

Multibody systems generally contain solids the deformations of which are appreciable and which decisively influence the dynamics of the system. These solids have to be modeled by means of special formulations for flexible solids. At the same time, other solids are of such a high stiffness that they may be considered rigid, which simplifies their modeling. For these reasons, for a rigid-flexible multibody system, two types of formulations co-exist in the equations of the system. Among the different possibilities provided in bibliography on the material, the formulation in natural coordinates and the formulation in absolute nodal coordinates are utilized in this article to model the rigid and flexible solids, respectively. This article contains a mixed formulation based on the possibility of sharing coordinates between a rigid solid and a flexible solid. In addition, the fact that the matrix of the global mass of the system is shown to be constant and that many of the constraint equations obtained upon utilizing these formulations are linear and can be eliminated. In this work, the formulation presented is utilized to simulate a mechanism with both rigid and flexible components.

On a Numerical Method for Solution of the Mathieu-Hill Type Equations

1980 ◽  
Vol 102 (3) ◽  
pp. 619-626 ◽  
Author(s):  
A. Midha ◽  
M. L. Badlani
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Initial Conditions ◽  
Linear Equations ◽  
Solution Process ◽  
Constraint Equations ◽  
Step Functions ◽  
The Matrix ◽  
Constant Coefficients ◽  
Simultaneous Linear Equations

This paper presents a computer-programmable numerical method for the solution of a class of linear, second order differential equations with periodic coefficients of the Mathieu-Hill type. The method is applicable only when the initial conditions are prescribed and the solution is not requiried to be periodic. The solution is facilitated by representing the coefficient functions as a sum of step functions over corresponding sub-intervals of the fundamental interval. During each sub-interval, the solution form is assumed to be that of the differential equations with “constant” coefficients. Constraint equations are derived from imposing the conditions of “compatibility” of response at the end nodes of the intermediate sub-intervals. This set of simultaneous linear equations is expressed in matrix form. The matrix of coefficients may be represented as a triangular one. This form greatly simplifies the solution process for simultaneous equations. The method is illustrated by its application to some specific problems.

EQUATIONS OF MOTION OF COMPLEX MULTIBODY SYSTEMS USING KINEMATICAL DIFFERENTIALS

1989 ◽  
Vol 13 (4) ◽  
pp. 113-121 ◽  
Author(s):  
M. HILLER ◽  
A. KECSKEMETHY
Keyword(s):  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Multibody System ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Nonholonomic Systems ◽  
Automatic Generation ◽  
Minimal Order ◽  
Closed Form Solutions ◽  
Constraint Equations

In complex multibody systems the motion of the bodies may depend on only a few degrees of freedom. For these systems, the equations of motion of minimal order, although more difficult to obtain, give a very efficient formulation. The present paper describes an approach for the automatic generation of these equations, which avoids the use of LAGRANGE-multipliers. By a particular concept, designated “kinematical differentials”, the problem of determining the partial derivatives required to state the equations of motion is reduced to a simple re-evaluation of the kinematics. These cover the solution of the global position, velocity and acceleration problems, i.e. the motion of all bodies is determined for given generalized (independent) coordinates. For their formulation and solution, the multibody system is mapped to a network of nonlinear transformation elements which are connected by linear equations. Each transformation element, designated “kinematical transformer”, corresponds to an independent multibody loop. This mapping of the constraint equations makes it possible to find closed-form solutions to the kinematics for a wide variety of technical applications, and (via kinematical differentials) leads also to an efficient formulation of the dynamics. The equations are derived for holonomic, scleronomic systems, but can also be extended to general nonholonomic systems.

On Some Properties of Semi-rings

2018 ◽  
pp. 35-50
Author(s):  
A. I. Belousov
Keyword(s):  
Linear Equations ◽  
Oriented Graph ◽  
Theoretical Computer Science ◽  
Alternative Methods ◽  
Main Role ◽  
Cost Matrix ◽  
Sequential Elimination ◽  
The Matrix ◽  
Systems Of Linear Equations ◽  
The Cost

The main objective of this paper is to prove a theorem according to which a method of successive elimination of unknowns in the solution of systems of linear equations in the semi-rings with iteration gives the really smallest solution of the system. The proof is based on the graph interpretation of the system and establishes a relationship between the method of sequential elimination of unknowns and the method for calculating a cost matrix of a labeled oriented graph using the method of sequential calculation of cost matrices following the paths of increasing ranks. Along with that, and in terms of preparing for the proof of the main theorem, we consider the following important properties of the closed semi-rings and semi-rings with iteration.We prove the properties of an infinite sum (a supremum of the sequence in natural ordering of an idempotent semi-ring). In particular, the proof of the continuity of the addition operation is much simpler than in the known issues, which is the basis for the well-known algorithm for solving a linear equation in a semi-ring with iteration.Next, we prove a theorem on the closeness of semi-rings with iteration with respect to solutions of the systems of linear equations. We also give a detailed proof of the theorem of the cost matrix of an oriented graph labeled above a semi-ring as an iteration of the matrix of arc labels.The concept of an automaton over a semi-ring is introduced, which, unlike the usual labeled oriented graph, has a distinguished "final" vertex with a zero out-degree.All of the foregoing provides a basis for the proof of the main theorem, in which the concept of an automaton over a semi-ring plays the main role.The article's results are scientifically and methodologically valuable. The proposed proof of the main theorem allows us to relate two alternative methods for calculating the cost matrix of a labeled oriented graph, and the proposed proofs of already known statements can be useful in presenting the elements of the theory of semi-rings that plays an important role in mathematical studies of students majoring in software technologies and theoretical computer science.

scholarly journals A new method for identifying the role of marital preferences at shaping marriage patterns

2021 ◽  
pp. 1-27
Author(s):  
Anna Naszodi ◽  
Francisco Mendonca
Keyword(s):  
Contingency Table ◽  
Transformation Method ◽  
New Method ◽  
Marriage Patterns ◽  
The Us ◽  
Analytical Properties ◽  
The Matrix ◽  
Transformation Methods

Abstract We develop a method which assumes that marital preferences are characterized either by the scalar-valued measure proposed by Liu and Lu, or by the matrix-valued generalized Liu–Lu measure. The new method transforms an observed contingency table into a counterfactual table while preserving its (generalized) Liu–Lu value. After exploring some analytical properties of the new method, we illustrate its application by decomposing changes in the prevalence of homogamy in the US between 1980 and 2010. We perform this decomposition with two alternative transformation methods as well where both methods capture preferences differently from Liu and Lu. Finally, we use survey evidence to support our claim that out of the three considered methods, the new transformation method is the most suitable for identifying the role of marital preferences at shaping marriage patterns. These data are also in favor of measuring assortativity in preferences à la Liu and Lu.

Spectral analysis of M/G/1 and G/M/1 type Markov chains

1996 ◽  
Vol 28 (01) ◽  
pp. 114-165 ◽  
Author(s):  
H. R. Gail ◽  
S. L. Hantler ◽  
B. A. Taylor
Keyword(s):  
Markov Chains ◽  
Generating Functions ◽  
Complex Analysis ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Equilibrium Analysis ◽  
Transform Method ◽  
Technical Problems ◽  
The Matrix ◽  
Transient Case

When analyzing the equilibrium behavior of M/G/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid technical problems that arise in applying results from complex analysis and linear algebra. It is shown that such restrictive assumptions are unnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or second moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounded solutions of the system of homogeneous vector-valued Wiener–Hopf equations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are shown to correspond to a spectral basis of the shift operator on this subspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of properties of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary is identified and shown to be associated with certain eigenvalues and vectors of one of these generating functions. An equilibrium analysis of the class of G/M/1 type Markov chains by similar methods is also presented.

Efficient Dynamic Analysis Method for Multibody Systems With Constraint Violation Stabilization Method

1999 ◽  
Author(s):  
Apiwat Reungwetwattana ◽  
Shigeki Toyama
Keyword(s):  
Dynamic Analysis ◽  
Multibody Systems ◽  
Analysis Method ◽  
Stabilization Method ◽  
Kinematic Constraint ◽  
Constraint Violation ◽  
Closed Loops ◽  
Constraint Stabilization ◽  
Constraint Equations ◽  
Crank Mechanism

Abstract This paper presents an efficient extension of Rosenthal’s order-n algorithm for multibody systems containing closed loops. Closed topological loops are handled by cut joint technique. Violation of the kinematic constraint equations of cut joints is corrected by Baumgarte’s constraint violation stabilization method. A reliable approach for selecting the parameters used in the constraint stabilization method is proposed. Dynamic analysis of a slider crank mechanism is carried out to demonstrate efficiency of the proposed method.

scholarly journals Matrix and vector sequence transformations revisited

1995 ◽  
Vol 38 (3) ◽  
pp. 495-510 ◽  
Author(s):  
C. Brezinski ◽  
A. Salam
Keyword(s):  
Linear Equations ◽  
Convergence Acceleration ◽  
Difference Operators ◽  
Scalar Case ◽  
Extrapolation Methods ◽  
System Of Linear Equations ◽  
Vector Sequence ◽  
The Matrix ◽  
Sequence Transformations

Sequence transformations are extrapolation methods. They are used for the purpose of convergence acceleration. In the scalar case, such algorithms can be obtained by two different approaches which are equivalent. The first one is an elimination approach based on the solution of a system of linear equations and it makes use of determinants. The second approach is based on the notion of annihilation difference operators. In this paper, these two approaches are generalized to the matrix and the vector cases.

scholarly journals Matrix-Based RSA Encryption of Streaming Data

2021 ◽  
Author(s):  
Michael Prendergast
Keyword(s):  
Streaming Data ◽  
New Method ◽  
Encryption Algorithm ◽  
Public Key ◽  
Invertible Matrix ◽  
Integer Coefficients ◽  
Private Key ◽  
Matrix Generalization ◽  
Euler Totient Function ◽  
The Matrix

This paper describes a new method for performing secure encryption of blocks of streaming data. This algorithm is an extension of the RSA encryption algorithm. Instead of using a public key (e,n) where n is the product of two large primes and e is relatively prime to the Euler Totient function, φ(n), one uses a public key (n,m,E), where m is the rank of the matrix E and E is an invertible matrix in GL(m,φ(n)). When m is 1, this last condition is equivalent to saying that E is relatively prime to φ(n), which is a requirement for standard RSA encryption. Rather than a secret private key (d,φ(n)) where d is the inverse of e (mod φ(n)), the private key is (D,φ(n)), where D is the inverse of E (mod (φ(n)). The key to making this generalization work is a matrix generalization of the scalar exponentiation operator that maps the set of m-dimensional vectors with integer coefficients modulo n, onto itself.

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