A New Algorithm for Calculating the Degrees of Freedom of Complex Mechanisms

2011 ◽  
Vol 346 ◽  
pp. 324-331
Author(s):  
Wei Jiang ◽  
Xin Luo ◽  
Wen Chuan Jia ◽  
Yuan Tai Hu ◽  
Hong Ping Hu
Keyword(s):  
Degrees Of Freedom ◽  
Linear Equations ◽  
Linear Constraint ◽  
Coefficient Matrix ◽  
Constraint Matrix ◽  
Constraint Equations ◽  
Complex Mechanisms ◽  
Traditional Approaches ◽  
Homogeneous Linear

A new algorithm is presented to calculate the degrees of freedom (DOFs) of spatial complex mechanisms by using the coefficient matrix of the linear constraint equations. A joint constraint matrix is firstly put forward for each kind of joint to formulate linear constraint equations in terms of spatial fine displacements of joint acting point with respect to joint frame. Two kinds of transformation are then proposed to rewrite all the constraint equations in terms of a set of fine displacements of all bodies and it leads to a set of homogeneous linear equations. The rank of the resulting coefficient matrix stands for the number of effective constraints and therefore the DOFs of the mechanism can be easily figured out. The proposed method can be widely used to solve the problem of DOFs for many spatial complex mechanisms, which may not be correctly solved with traditional approaches. Besides, the proposed method is very easy for implementation.

The DOMAIN family of propagation operations for intervals on simultaneous linear equations

1995 ◽  
Vol 9 (3) ◽  
pp. 197-210 ◽  
Author(s):  
R. Chen ◽  
A.C. Ward
Keyword(s):  
Linear Equations ◽  
Constraint Equation ◽  
Linear Constraint ◽  
The Other ◽  
Interval Matrix ◽  
Domain Family ◽  
Output Vector ◽  
Constraint Equations ◽  
Simultaneous Linear Equations ◽  
Mathematical Operations

AbstractThis paper defines, develops algorithms for, and illustrates the utility in design of a class of mathematical operations. These accept as inputs a system of linear constraint equations, Ax = b, an interval matrix of values for the coefficients, A, and an interval vector of values for either x or b. They return a set of values for the “domain” of the other vector, in the sense that all combinations of the output vector values set and values for A, when inserted into the constraint equation, correspond to values for the input vector that lie within the input interval. These operations have been mostly overlooked by the interval matrix arithmetic community, but are mathematically interesting and useful in the design, for example, of structures.

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.

The SUFFICIENT-POINTS family of propagation operations for intervals on simultaneous linear equations

1995 ◽  
Vol 9 (3) ◽  
pp. 211-217 ◽  
Author(s):  
R. Chen ◽  
A.C. Ward
Keyword(s):  
Linear Equations ◽  
Linear Constraint ◽  
Input Vector ◽  
The Other ◽  
Interval Matrix ◽  
Interval Vector ◽  
Constraint Equations ◽  
Simultaneous Linear Equations ◽  
Mathematical Operations ◽  
Mathematics Community

AbstractThis paper defines, develops algorithms for, and illustrates the design use of a class of mathematical operations. These operations accept as inputs a system of linear constraint equations, Ax = b, an interval matrix of values for the coefficients A, and an interval vector of values for either x or b. They return a set of values for the other variable that is “sufficient” in this sense. Suppose that ◯ is an interval of input vectors, and  an interval matrix. Then, one Sufficient-Points operation returns a set of vectors ~ such that for each b in ~, the set of x values that can be produced by inserting all the values of  into Ax = b is a superset of the input vector x. These operations have been partly overlooked by the interval matrix mathematics community, but are mathematically interesting and useful in the design, for example, of circuits.

scholarly journals A Globally Convergent Parallel SSLE Algorithm for Inequality Constrained Optimization

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Zhijun Luo ◽  
Lirong Wang
Keyword(s):  
Constrained Optimization ◽  
Optimization Problems ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Descent Direction ◽  
Constrained Optimization Problems ◽  
Inequality Constrained Optimization ◽  
Globally Convergent ◽  
Variable Distribution ◽  
Systems Of Linear Equations

A new parallel variable distribution algorithm based on interior point SSLE algorithm is proposed for solving inequality constrained optimization problems under the condition that the constraints are block-separable by the technology of sequential system of linear equation. Each iteration of this algorithm only needs to solve three systems of linear equations with the same coefficient matrix to obtain the descent direction. Furthermore, under certain conditions, the global convergence is achieved.

An Analytical Method for Locating Velocity Instantaneous Centers

1992 ◽  
Author(s):  
Hong-Sen Yan ◽  
Meng-Hui Hsu
Keyword(s):  
Analytical Method ◽  
Degrees Of Freedom ◽  
Coefficient Matrix ◽  
Multiple Degrees Of Freedom ◽  
Singular Configurations ◽  
Special Cases

Abstract An analytical method is presented for locating all velocity instantaneous centers of linkage mechanisms with single or multiple degrees of freedom. The method is based on the fact that the coefficient matrix of the derived velocity equations for vector loops, independent inputs, and instantaneous centers is singular. This approach also works for special cases with kinematic indeterminacy or singular configurations.

scholarly journals Modelling Unsaturated Flow in Porous Media Using an Improved Picard Iteration Scheme

2021 ◽  
Author(s):  
S.R. Zhu ◽  
L.Z. Wu ◽  
T. Ma ◽  
S.H. Li
Keyword(s):  
Porous Media ◽  
Convergence Rate ◽  
Unsaturated Flow ◽  
Control Volume ◽  
Linear Equations ◽  
Solution Process ◽  
Coefficient Matrix ◽  
Picard Iteration ◽  
Flow In Porous Media ◽  
Richards Equation

Abstract The numerical solution of various systems of linear equations describing fluid infiltration uses the Picard iteration (PI). However, because many such systems are ill-conditioned, the solution process often has a poor convergence rate, making it very time-consuming. In this study, a control volume method based on non-uniform nodes is used to discretize the Richards equation, and adaptive relaxation is combined with a multistep preconditioner to improve the convergence rate of PI. The resulting adaptive relaxed PI with multistep preconditioner (MP(m)-ARPI) is used to simulate unsaturated flow in porous media. Three examples are used to verify the proposed schemes. The results show that MP(m)-ARPI can effectively reduce the condition number of the coefficient matrix for the system of linear equations. Compared with conventional PI, MP(m)-ARPI achieves faster convergence, higher computational efficiency, and enhanced robustness. These results demonstrate that improved scheme is an excellent prospect for simulating unsaturated flow in porous media.

An Iterative Approach to Dynamic Elastic-Plastic Analysis

1998 ◽  
Vol 65 (4) ◽  
pp. 811-819 ◽  
Author(s):  
F. Giambanco ◽  
L. Palizzolo ◽  
L. Cirone
Keyword(s):  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Degrees Of Freedom ◽  
Linear Complementarity ◽  
Iterative Approach ◽  
Loading History ◽  
Constraint Matrix ◽  
Elastic Plastic ◽  
Recursive Solution ◽  
The Matrix

The step-by-step analysis of structures constituted by elastic-plastic finite elements, subjected to an assigned loading history, is here considered. The structure may possess dynamic and/or not dynamic degrees-of-freedom. As it is well-known, at each step of analysis the solution of a linear complementarity problem is required. An iterative method devoted to solving the relevant linear complementarity problem is presented. It is based on the recursive solution of a linear complementarity, problem in which the constraint matrix is block-diagonal and deduced from the matrix of the original linear complementarity problem. The convergence of the procedure is also proved. Some particular cases are examined. Several numerical applications conclude the paper.

Kinematic Analysis and Synthesis of Three-Input, Eight-Bar Mechanisms Driven by Relatively Small Cranks

1992 ◽  
Author(s):  
Kambiz Farhang ◽  
Partha Sarathi Basu
Keyword(s):  
Nonlinear Equations ◽  
Kinematic Analysis ◽  
Linear Equations ◽  
Output Link ◽  
Analysis And Design ◽  
Constraint Equations ◽  
Analysis And Synthesis ◽  
The Mean ◽  
Approximate Equations

Abstract Approximate kinematic equations are developed for the analysis and design of three-input, eight-bar mechanisms driven by relatively small cranks. Application of a method in which an output link is presumed to be comprised of a mean and a perturbational motions, along with the vector loop approach facilitates the derivation of the approximate kinematic equations. The resulting constraint equations are, (i) in the form of a set of four nonlinear equations relating the mean link orientations, and (ii) a set of four linear equations in the unknown perturbations (output link motions). The latter set of equations is solved, symbolically, to obtain the output link motions. The approximate equations are shown to be effective in the synthesis of three-input, small-crank mechanisms.

Architecture Optmization of a 3-DOF Parallel Mechanism

1998 ◽  
Author(s):  
J. A. Carretero ◽  
R. P. Podhorodeski ◽  
M. Nahon
Keyword(s):  
Parallel Mechanism ◽  
Architectural Design ◽  
Degrees Of Freedom ◽  
Jacobian Matrix ◽  
Design Parameters ◽  
Objective Functions ◽  
Constraint Equations ◽  
Three Degree Of Freedom ◽  
Architecture Optimization ◽  
Three Degrees Of Freedom

Abstract This paper presents a study of the architecture optimization of a three-degree-of-freedom parallel mechanism intended for use as a telescope mirror focussing device. The construction of the mechanism is first described. Since the mechanism has only three degrees of freedom, constraint equations describing the inter-relationship between the six Cartesian coordinates are given. These constraints allow us to define the parasitic motions and, if incorporated into the kinematics model, a constrained Jacobian matrix can be obtained. This Jacobian matrix is then used to define a dexterity measure. The parasitic motions and dexterity are then used as objective functions for the optimizations routines and from which the optimal architectural design parameters are obtained.

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