Computer-Aided Displacement Analysis of Spatial Mechanisms

1994 ◽  
Author(s):  
Sean Thompson ◽  
Harry H. Cheng
Keyword(s):  
Programming Languages ◽  
Design Automation ◽  
Data Type ◽  
Complex Numbers ◽  
Dual Numbers ◽  
Displacement Analysis ◽  
Spatial Mechanism ◽  
Programming Paradigm ◽  
Spatial Mechanisms ◽  
Programming Techniques

Abstract Recently, Cheng (1993) introduced the CH programming language. CH is designed to be a superset of ANSI C with all programming features of FORTRAN. Many programming features in CH are specifically designed and implemented for design automation. Handling dual number as a basic built-in data type in the language is one example. Formulas with dual numbers can be translated into CH programming statements as easily as formulas with real and complex numbers. In this paper we will show that both formulation and programming with dual numbers are remarkably simple for analysis of complicated spatial mechanisms within the programming paradigm of CH. With computational capabilities for dual formulas in mind, formulas for analysis of spatial mechanisms are derived differently from those intended for implementation in computer programming languages without dual data type. We will demonstrate some formulation and programming techniques in the programming paradigm of CH through a displacement analysis of the RCRCR five-link spatial mechanism. A CH program that can obtain both numerical and graphical results for complete displacement analysis of the RCRCR mechanism will be presented.

Computations of Dual Numbers in the Extended Finite Dual Plane

1993 ◽  
Author(s):  
Harry H. Cheng
Keyword(s):  
Programming Language ◽  
Data Type ◽  
Complex Numbers ◽  
Dual Numbers ◽  
Linguistic Features ◽  
Mathematical Functions ◽  
Dual Plane ◽  
Spatial Mechanisms ◽  
Computational Aspects ◽  
Dual Functions

Abstract The numerical computational aspects of dual numbers in the CH programming language are presented in this paper. Dual is a built-in data type in CH. Dual numbers and dual metanumbers are described in the extended dual plane and extended finite dual plane. The arithmetic and relational operations, and built-in mathematical functions are defined for both dual numbers and dual metanumbers of DualZero, DualInf, and DualNaN. Due to polymorphism, the syntax of dual arithmetic and relational operations, and built-in dual functions are the same as those for real and complex numbers in the CH programming language. The linguistic features and handling of user’s dual functions and dual arrays, as well as applications of dual numbers in robotics and spatial mechanisms in the CH programming language can be found in (Cheng, 1993c).

Exact Displacement Analysis of Four-Link Spatial Mechanisms by the Direction Cosine Matrix Method

1984 ◽  
Vol 51 (4) ◽  
pp. 921-928 ◽  
Author(s):  
T. C. Huang ◽  
Y. Youm
Keyword(s):  
Local Coordinate System ◽  
Direction Cosine ◽  
Displacement Analysis ◽  
Spatial Mechanism ◽  
Spatial Mechanisms ◽  
Local Coordinates ◽  
Development Direction ◽  
Direction Cosine Matrix ◽  
Direction Cosines ◽  
Unit Vectors

A method of displacement analysis of the four-link spatial mechanism is developed. The results through this analysis will be exact solutions that can be obtained without resorting to numerical or iteration schemes. In the analysis, the position of a link in a mechanism can be fully defined if its direction and length are known. Therefore, this analysis involves the calculation of the unknown direction cosines and length of each link for a given configuration of the mechanism. In finding the direction cosines of the unknown unit vectors involved for each link and rotating axis, two types of coordinates, the global and the local, are generally used. Then, a direction cosine matrix between each local coordinate system and the global coordinates is established. Thus, the unknown direction cosines of the local coordinates, the links, and the rotating axes are obtained in global coordinates. In this development, direction cosine matrices are used throughout the analysis. As an illustration, the application of this method to the study of four-link spatial mechanisms, RGGR, RGCR, RRGG, and RRGC will be presented.

Dual Polynomials and Complex Dual Numbers for Analysis of Spatial Mechanisms

1996 ◽  
Author(s):  
Harry H. Cheng ◽  
Sean Thompson
Keyword(s):  
Numerical Solutions ◽  
Polynomial Equation ◽  
Mathematical Tool ◽  
Polynomial Equations ◽  
Dual Numbers ◽  
Output Displacement ◽  
Spatial Mechanism ◽  
Dual Number ◽  
Spatial Mechanisms ◽  
Language Environment

Abstract Complex dual numbers w̌1=x1+iy1+εu1+iεv1 which form a commutative ring are for the first time introduced in this paper. Arithmetic operations and functions of complex dual numbers are defined. Complex dual numbers are used to solve dual polynomial equations. It is shown that the singularities of a dual input-output displacement polynomial equation of a mechanism correspond to its singularity positions. This new method of identifying singularities provides clear physical insight into the geometry of the singular configurations of a mechanism, which is illustrated through analysis of special configurations of the RCCC spatial mechanism. Numerical solutions for dual polynomial equations and complex dual numbers are conveniently implemented in the CH language environment for analysis of the RCCC spatial mechanism. Like the dual number, the complex dual number is a useful mathematical tool for analytical and numerical treatment of spatial mechanisms.

A Dual Iterative Method for Displacement Analysis of Spatial Mechanisms

1994 ◽  
Author(s):  
Sean Thompson ◽  
Harry H. Cheng
Keyword(s):  
Iterative Method ◽  
Programming Language ◽  
Displacement Analysis ◽  
Spatial Mechanism ◽  
Dual Transformation ◽  
Transformation Matrices ◽  
Spatial Mechanisms

Abstract A dual iterative method for displacement analysis of a spatial mechanism is presented in this paper. The algorithm and formulation based upon 3 × 3 dual transformation matrices are succinct. They can be easily implemented in the CH programming language. The algorithm has been numerically verified by dual iterative displacement analysis of an RCCC four-link spatial mechanism.

Singularity Analysis of Spatial Mechanisms Using Dual Polynomials and Complex Dual Numbers

1999 ◽  
Vol 121 (2) ◽  
pp. 200-205 ◽  
Author(s):  
H. H. Cheng ◽  
S. Thompson
Keyword(s):  
Numerical Solutions ◽  
Polynomial Equation ◽  
Singularity Analysis ◽  
Polynomial Equations ◽  
Dual Numbers ◽  
Output Displacement ◽  
Spatial Mechanism ◽  
Spatial Mechanisms ◽  
Physical Insight ◽  
Language Environment

Complex dual numbers wˇ = x + iy + εu + iεv which form a commutative ring are introduced in this paper to solve dual polynomial equations numerically. It is shown that the singularities of a dual input-output displacement polynomial equation of a mechanism correspond to its singularity positions. This new method of identifying singularities provides clear physical insight into the geometry of the singular configurations of a mechanism, which is illustrated through analysis of special configurations of the RCCC spatial mechanism. Numerical solutions for dual polynomial equations and complex dual numbers are conveniently implemented in the CH language environment for analysis of the RCCC spatial mechanism.

On the shaft locations of two contra-rotating counterweights for balancing spatial mechanisms

1997 ◽  
Vol 211 (8) ◽  
pp. 567-578 ◽  
Author(s):  
S-T Chiou ◽  
J-C Tzou
Keyword(s):  
Root Mean Square ◽  
Mean Square ◽  
Spatial Mechanism ◽  
Spatial Mechanisms ◽  
Crank Mechanism ◽  
Frequency Term ◽  
Shaking Moment

It has been shown in a previous work that a frequency term of the shaking force of spatial mechanisms, whose hodograph is proved to be an ellipse, can be eliminated by a pair of contrarotating counterweights. In this work, it is found that the relevant frequency term of the shaking moment is minimized if the balancing shafts are coaxial at the centre of a family of ellipsoids, called isomomental ellipsoids, with respect to (w.r.t.) any point on an ellipsoid, as is also the root mean square (r.m.s.) of the relevant frequency term of the shaking moment. It can also be minimized even though the location of either shaft, but not both, is chosen arbitrarily on a plane. The location of the second shaft is then determinate. In order to locate the centre, a derivation for the theory of isomomental ellipsoids of a frequency term of the shaking moment of spatial mechanisms is given. It is shown that the r.m.s. of a frequency term shaking moment of a spatial mechanism w.r.t. the concentric centre of the isomomental ellipsoids is the minimum. Examples of a seven-link 7-R spatial linkage and a spatial slider-crank mechanism are included.

Kinematic Analysis of Spatial Mechanisms by Means of Screw Coordinates. Part 2—Analysis of Spatial Mechanisms

1971 ◽  
Vol 93 (1) ◽  
pp. 67-73 ◽  
Author(s):  
M. S. C. Yuan ◽  
F. Freudenstein ◽  
L. S. Woo
Keyword(s):  
Computer Program ◽  
Kinematic Analysis ◽  
Static Force ◽  
Single Degree Of Freedom ◽  
Numerical Examples ◽  
Spatial Mechanism ◽  
Single Degree ◽  
Spatial Mechanisms ◽  
Torque Analysis ◽  
Basic Concepts

The basic concepts of screw coordinates described in Part I are applied to the numerical kinematic analysis of spatial mechanisms. The techniques are illustrated with reference to the displacement, velocity, and static-force-and-torque analysis of a general, single-degree-of-freedom spatial mechanism: a seven-link mechanism with screw pairs (H)7. By specialization the associated computer program is capable of analyzing many other single-loop spatial mechanisms. Numerical examples illustrate the results.

Integrated Kinematic and Dynamic Optimal Synthesis of Spatial Mechanisms

1987 ◽  
Author(s):  
A. J. Kakatsios ◽  
S. J. Tricamo
Keyword(s):  
Nonlinear Programming ◽  
Simultaneous Optimization ◽  
Programming Formulation ◽  
Joint Torques ◽  
Spatial Mechanism ◽  
Spatial Mechanisms ◽  
Structural Error ◽  
Integrated Technique ◽  
Driving Torque ◽  
Shaking Moment

Abstract A novel integrated technique permitting the simultaneous optimization of kinematic and dynamic characteristics in the synthesis of spatial mechanisms is shown. The nonlinear programming formulation determines mechanism variables which simultaneously minimize the maximum values of bearing reactions, joint torques, driving torque, shaking moment, and shaking force while constraining the maximum kinematic structural error to a prescribed bound. The method is applied to the design of a path generating RRSS spatial mechanism with prescribed input link timing. Dynamic reactions in the mechanisms synthesized using the integrated technique were substantially reduced when compared to those of a mechanism synthesized to satisfy only the specified kinematic conditions.

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