The Synthesis of Planar RR and Spatial CC Chains and the Equation of a Triangle

1995 ◽  
Vol 117 (B) ◽  
pp. 101-106 ◽  
Author(s):  
J. M. McCarthy
Keyword(s):  
Rotation Angle ◽  
Standard Form ◽  
Slight Modification ◽  
Complex Numbers ◽  
Simultaneous Solution ◽  
Form Equation ◽  
Equivalent Equation ◽  
Fixed Axis ◽  
Spatial Linkages ◽  
Triangle Equation

This paper formulates the planar and spatial versions of an equation that determines one vertex of a triangle in terms of the other two vertices and their interior angles. The fact that a slight modification of Sandor and Erdman’s standard form equation for the design of RR chains yields this planar triangle equation is the basis for identifying the equivalent equation for a spatial triangle as the standard form equation for CC chains. The simultaneous solution of two of the planar equations yields an analytical expression of Burmester’s relationship between the fixed pivot of an RR chain and the relative position poles of its floating link. A similar solution of simultaneous spatial triangle equations yields Roth’s generalization of this insight, specifically, the fixed axis of a CC chain views two relative screw axes in one-half the dual crank rotation angle. These results provide the foundation for generalizing planar linkage synthesis techniques based on complex numbers to the synthesis of spatial linkages.

The Synthesis of Planar RR and Spatial CC Chains and the Equation of a Triangle

1995 ◽  
Vol 117 (B) ◽  
pp. 101-106 ◽  
Author(s):  
J. M. McCarthy
Keyword(s):  
Rotation Angle ◽  
Standard Form ◽  
Slight Modification ◽  
Complex Numbers ◽  
Simultaneous Solution ◽  
Form Equation ◽  
Equivalent Equation ◽  
Fixed Axis ◽  
Spatial Linkages ◽  
Triangle Equation

This paper formulates the planar and spatial versions of an equation that determines one vertex of a triangle in terms of the other two vertices and their interior angles. The fact that a slight modification of Sandor and Erdman’s standard form equation for the design of RR chains yields this planar triangle equation is the basis for identifying the equivalent equation for a spatial triangle as the standard form equation for CC chains. The simultaneous solution of two of the planar equations yields an analytical expression of Burmester’s relationship between the fixed pivot of an RR chain and the relative position poles of its floating link. A similar solution of simultaneous spatial triangle equations yields Roth’s generalization of this insight, specifically, the fixed axis of a CC chain views two relative screw axes in one-half the dual crank rotation angle. These results provide the foundation for generalizing planar linkage synthesis techniques based on complex numbers to the synthesis of spatial linkages.

A reduction theory of second order meromorphic differential equations, I

1987 ◽  
Vol 101 (2) ◽  
pp. 323-342
Author(s):  
W. B. Jurkat ◽  
H. J. Zwiesler
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Standard Form ◽  
Reduction Theory ◽  
Fundamental Solution Matrix ◽  
Equivalent Equation ◽  
The Difference ◽  
Solution Matrix ◽  
Meromorphic Differential Equation ◽  
Meromorphic Differential

In this article we investigate the meromorphic differential equation X′(z) = A(z) X(z), often abbreviated by [A], where A(z) is a matrix (all matrices we consider have dimensions 2 × 2) meromorphic at infinity, i.e. holomorphic in a punctured neighbourhood of infinity with at most a pole there. Moreover, X(z) denotes a fundamental solution matrix. Given a matrix T(z) which together with its inverse is meromorphic at infinity (a meromorphic transformation), then the function Y(z) = T−1(z) X(z) solves the differential equation [B] with B = T−1AT − T−1T [1,5]. This introduces an equivalence relation among meromorphic differential equations and leads to the question of finding a simple representative for each equivalence class, which, for example, is of importance for further function-theoretic examinations of the solutions. The first major achievement in this direction is marked by Birkhoff's reduction which shows that it is always possible to obtain an equivalent equation [B] where B(z) is holomorphic in ℂ ¬ {0} (throughout this article A ¬ B denotes the difference of these sets) with at most a singularity of the first kind at 0 [1, 2, 5, 6]. We call this the standard form. The question of how many further simplifications can be made will be answered in the framework of our reduction theory. For this purpose we introduce the notion of a normalized standard equation [A] (NSE) which is defined by the following conditions:(i) , where r ∈ ℕ and Ak are constant matrices, (notation: )(ii) A(z) has trace tr for some c ∈ ℂ,(iii) Ar−1 has different eigenvalues,(iv) the eigenvalues of A−1 are either incongruent modulo 1 or equal,(v) if A−1 = μI, then Ar−1 is diagonal,(vi) Ar−1 and A−1 are triangular in opposite ways,(vii) a12(z) is monic (leading coefficient equals 1) unless a12 ≡ 0; furthermore a21(z) is monic in case that a12 ≡ 0 but a21 ≢ 0.

Determination of Bearing Reactions of Spatial Linkages and Manipulators

1990 ◽  
Vol 112 (2) ◽  
pp. 168-174 ◽  
Author(s):  
F. L. Litvin ◽  
J. Tan
Keyword(s):  
Virtual Work ◽  
Parallel Manipulators ◽  
System Of Equations ◽  
Principle Of Virtual Work ◽  
Simultaneous Solution ◽  
Combined Application ◽  
Spatial Linkages ◽  
Virtual Velocity ◽  
D'alembert's Principle

Application of D’Alembert’s principle for determination of dynamic bearing reactions in joints of spatial linkages and parallel manipulators needs the simultaneous solution of a large system of equations. The authors of this paper propose an approach that is a combined application of principle of virtual work and D’Alembert’s principle. The main advantages of the proposed approach are: (1) reduction of the number of equations that have to be solved simultaneously, and (2) simplification of the expressions for the relative virtual velocity. The proposed approach is illustrated with the example of a 7-bar linkage and its application is explained with the crank-slider linkage.

scholarly journals A slight modification of the first phase of the simplex algorithm

2012 ◽  
Vol 22 (1) ◽  
pp. 107-114
Author(s):  
Tomica Divnic ◽  
Ljiljana Pavlovic
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Canonical Form ◽  
Linear Programming Problem ◽  
Standard Form ◽  
Main Idea ◽  
Slight Modification ◽  
Simplex Algorithm ◽  
Feasible Set ◽  
Standard Simplex

In this paper we give a modification of the first phase procedure for transforming the linear programming problem, given in the standard form min{cTx Ax=b, x?0}, to the canonical form, i.e., to the form with one feasible primal basis where standard simplex algorithm can be applied directly. The main idea of the paper is to avoid adding m artificial variables in the first phase. Instead, Step 2 of the proposed algorithm transforms the problem into the form with m?1 basic columns. Step 3 is then iterated until the m?th basic column is obtained, or it is concluded that the feasible set of LP problem is empty.

Complex Modular Numbers: Complex Numbers Need not be Complex

1976 ◽  
Vol 69 (1) ◽  
pp. 53-54
Author(s):  
Susan J. Grant ◽  
Ward R. Stewart
Keyword(s):  
Real Number ◽  
Complex Number ◽  
Number System ◽  
Complex Numbers ◽  
Negative Number ◽  
Real Numbers ◽  
The Real ◽  
Equivalent Equation ◽  
Imaginary Number

Most students are faced with the task of solving the equation x2 + 1 = 0 over the real numbers at some time in their algebra classes. After they substitute values for x unsuccessfully, they usually attempt to solve the equivalent equation x2 = -1. They soon realize that it is impossible to square a real number and obtain a negative number. At this point their teacher may define the imaginary number i to be and then proceed to develop the complex number system.

A Tabular Method of Calculating Helicopter Blade Deflections and Moments

1948 ◽  
Vol 15 (2) ◽  
pp. 97-106
Author(s):  
N. O. Myklestad
Keyword(s):  
Cross Section ◽  
Constant Term ◽  
Constant Load ◽  
Complex Numbers ◽  
Radial Line ◽  
Helicopter Blade ◽  
Axis Of Rotation ◽  
Principal Axes ◽  
Fixed Axis ◽  
Phase Angles

Abstract It is assumed that the blade rotates about a fixed axis with a constant angular velocity and that it is subjected to a known load parallel to the axis of rotation. The axis of the blade, which is a line through the centroid of each cross section, is not necessarily a straight radial line, but its deviation from such a line must be small. For each cross section, one of the principal axes of moment of inertia is also considered to be parallel to the axis of rotation and bending is assumed to take place in a plane through this axis. The load as well as the mass of the blade are concentrated at distinct points on the blade axis. Under these assumptions the proposed method of analysis will give the shear forces, bending moments, slopes, and deflections by performing a series of tabular calculations. The load, which is always periodic, must first be put in the form of a constant term and a series of harmonic terms, each of which must be analyzed separately. The effects of the constant load and the eccentricity and slope at the base of the blade axis are found together and are easily disposed of in two simple tabular calculations involving only real quantities. The effect of each harmonic involves three tabular calculations with complex numbers. The complex numbers take care of the phase angles which vary along the axis of the blade.

Production of New and Revised A.A.V.S.O. Charts

1979 ◽  
Vol 46 ◽  
pp. 368
Author(s):  
Clinton B. Ford
Keyword(s):  
Variable Star ◽  
Standard Form ◽  
Original Data ◽  
University Of Pennsylvania ◽  
Standard Format ◽  
Additional Information ◽  
The Gift ◽  
The University

A “new charts program” for the Americal Association of Variable Star Observers was instigated in 1966 via the gift to the Association of the complete variable star observing records, charts, photographs, etc. of the late Prof. Charles P. Olivier of the University of Pennsylvania (USA). Adequate material covering about 60 variables, not previously charted by the AAVSO, was included in this original data, and was suitably charted in reproducible standard format.Since 1966, much additional information has been assembled from other sources, three Catalogs have been issued which list the new or revised charts produced, and which specify how copies of same may be obtained. The latest such Catalog is dated June 1978, and lists 670 different charts covering a total of 611 variables none of which was charted in reproducible standard form previous to 1966.

Plasmalemma localisation of inorganic phosphate in B cells pancreas

1978 ◽  
Vol 36 (2) ◽  
pp. 528-529
Author(s):  
F. B. P. Wooding ◽  
K. Pedley ◽  
N. Freinkel ◽  
R. M. C. Dawson
Keyword(s):  
Electron Microscope ◽  
B Cells ◽  
B Cell ◽  
Pancreatic Islets ◽  
High Glucose ◽  
Lead Acetate ◽  
Inorganic Phosphate ◽  
Slight Modification ◽  
Uranyl Acetate ◽  
Lead Phosphate

Freinkel et al (1974) demonstrated that isolated perifused rat pancreatic islets reproduceably release up to 50% of their total inorganic phosphate when the concentration of glucose in the perifusion medium is raised.Using a slight modification of the Libanati and Tandler (1969) method for localising inorganic phosphate by fixation-precipitation with glutaraldehyde-lead acetate we can demonstrate there is a significant deposition of lead phosphate (identified by energy dispersive electron microscope microanalysis) at or on the plasmalemma of the B cell of the islets (Fig 1, 3). Islets after incubation in high glucose show very little precipitate at this or any other site (Fig 2). At higher magnification the precipitate seems to be intracellular (Fig 4) but since any use of osmium or uranyl acetate to increase membrane contrast removes the precipitate of lead phosphate it has not been possible to verify this as yet.

Role of spectrin in membrane fusion and in shape change of human erythrocyte ghost

1978 ◽  
Vol 36 (2) ◽  
pp. 328-329
Author(s):  
Hideo Hayashi ◽  
Yoshikazu Hirai ◽  
John T. Penniston
Keyword(s):  
Conformational Changes ◽  
Human Erythrocyte ◽  
Shape Change ◽  
Membrane Surface ◽  
Slight Modification ◽  
Active Role ◽  
Main Role ◽  
Bank Blood ◽  
Human Erythrocyte Ghost

Spectrin is a membrane associated protein most of which properties have been tentatively elucidated. A main role of the protein has been assumed to give a supporting structure to inside of the membrane. As reported previously, however, the isolated spectrin molecule underwent self assemble to form such as fibrous, meshwork, dispersed or aggregated arrangements depending upon the buffer suspended and was suggested to play an active role in the membrane conformational changes. In this study, the role of spectrin and actin was examined in terms of the molecular arrangements on the erythrocyte membrane surface with correlation to the functional states of the ghosts.Human erythrocyte ghosts were prepared from either freshly drawn or stocked bank blood by the method of Dodge et al with a slight modification as described before. Anti-spectrin antibody was raised against rabbit by injection of purified spectrin and partially purified.

Structure and migration of planar defects in crystals observed in atomic scale

1978 ◽  
Vol 36 (1) ◽  
pp. 284-285 ◽  
Author(s):  
H. Hashimoto ◽  
Y. Sugimoto ◽  
Y. Takai ◽  
H. Endoh
Keyword(s):  
Electron Microscope ◽  
Rotation Angle ◽  
Crystal Surface ◽  
Electron Gun ◽  
Atomic Scale ◽  
Present Observation ◽  
Twist Boundary ◽  
Optical Magnification ◽  
Zinc Crystal ◽  
And Migration

As was demonstrated by the present authors that atomic structure of simple crystal can be photographed by the conventional 100 kV electron microscope adjusted at “aberration free focus (AFF)” condition. In order to operate the microscope at AFF condition effectively, highly stabilized electron beams with small energy spread and small beam divergence are necessary. In the present observation, a 120 kV electron microscope with LaB6 electron gun was used. The most of the images were taken with the direct electron optical magnification of 1.3 million times and then magnified photographically.1. Twist boundary of ZnSFig. 1 is the image of wurtzite single crystal with twist boundary grown on the surface of zinc crystal by the reaction of sulphur vapour of 1540 Torr at 500°C. Crystal surface is parallel to (00.1) plane and electron beam is incident along the axis normal to the crystal surface. In the twist boundary there is a dislocation net work between two perfect crystals with a certain rotation angle.

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