Kinematic Synthesis of Adjustable Mechanisms—Part 2: Path Generation

1973 ◽  
Vol 95 (2) ◽  
pp. 423-429 ◽  
Author(s):  
Joseph F. McGovern ◽  
George N. Sandor
Keyword(s):  
Closed Form ◽  
Higher Order ◽  
Complex Numbers ◽  
Path Generation ◽  
Function Generator ◽  
Kinematic Synthesis ◽  
Closed Form Solutions ◽  
Straight Line ◽  
Adjustable Mechanisms

A method utilizing complex numbers similar to that used in Part 1 for adjustable function generator synthesis is applied to the synthesis of adjustable path generators. Finitely separated path points with prescribed timing as well as higher order approximations (infinitesimally separated path points) are treated, by way of analytical and closed form solutions. Adjustment of the path generator mechanism is accomplished by moving a fixed pivot. Mechanisms adjustable for different approximate straight line motions, for various path curvatures, and path tangents as well as several arbitrary paths can be synthesized. Four-bar and geared five-bar mechanisms are considered. Examples are included describing synthesized mechanisms.

Kinematic Synthesis of Adjustable Mechanisms—Part 1: Function Generation

1973 ◽  
Vol 95 (2) ◽  
pp. 417-422 ◽  
Author(s):  
Joseph F. McGovern ◽  
George N. Sandor
Keyword(s):  
Complex Number ◽  
Relative Orientation ◽  
Complex Numbers ◽  
Input Output ◽  
Kinematic Synthesis ◽  
Digital Computation ◽  
Closed Form Solutions ◽  
Function Generation ◽  
Planar Linkages ◽  
Adjustable Mechanisms

One of the major advantages of linkages over cams is the ease with which they can be adjusted to modify their output. Incorporating an adjustment into the design of a linkage makes it possible to make a selection from two, three, or more different outputs by simply making the adjustment. Adjustable mechanisms make it possible to use the same hardware for different input-output relationships. The adjustment considered in this paper is the changing of a fixed pivot location. This paper presents a method of synthesizing adjustable planar linkages for function generation with finitely separated precision points and higher order synthesis involving prescribed velocities, accelerations, and higher accelerations. The method of synthesis is analytical with closed form solutions and utilizes complex numbers. The method is programmed for automatic digital computation. The linkages considered are a four-bar, a geared five-bar, and a geared six-bar mechanism. Examples include adjustable mechanisms which have been successfully synthesized with the method developed here. Future extensions of the complex number method to include adjustment by changing the length of a link and by changing of the relative orientation of the gears in geared linkages are outlined.

Kinematic Synthesis of a Geared Five-Bar Function Generator

1971 ◽  
Vol 93 (1) ◽  
pp. 11-16 ◽  
Author(s):  
Arthur G. Erdman ◽  
George N. Sandor
Keyword(s):  
Computer Program ◽  
Closed Form ◽  
Complex Numbers ◽  
Function Generator ◽  
Kinematic Synthesis ◽  
Numerical Examples ◽  
Function Generation ◽  
Structural Error ◽  
Dependent Variables ◽  
Angular Displacements

A general closed form method of planar kinematic synthesis, using complex numbers to represent link vectors, is applied to the synthesis of a geared five-bar linkage for function generation. Equations are derived and a computer program is developed to yield several solutions. Angular displacements of the input, a cycloidal crank, and the output, a simple follower, are used as linear analogs of the independent and the dependent variables, respectively. A method is demonstrated for six precision conditions (three first, three second-order precision conditions). Numerical examples are included, and the structural error of these geared five-bars are compared to that of optimized four-bar linkages generating the same functions.

scholarly journals Definite Integral of Algebraic, Exponential and Hyperbolic Functions Expressed in Terms of Special Functions

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1425
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Special Function ◽  
Special Functions ◽  
Complex Numbers ◽  
Definite Integral ◽  
Hyperbolic Functions ◽  
Closed Form Solutions ◽  
Definite Integrals ◽  
Arbitrary Complex ◽  
Famous Book

While browsing through the famous book of Bierens de Haan, we came across a table with some very interesting integrals. These integrals also appeared in the book of Gradshteyn and Ryzhik. Derivation of these integrals are not listed in the current literature to best of our knowledge. The derivation of such integrals in the book of Gradshteyn and Ryzhik in terms of closed form solutions is pertinent. We evaluate several of these definite integrals of the form ∫0∞(a+y)k−(a−y)keby−1dy, ∫0∞(a+y)k−(a−y)keby+1dy, ∫0∞(a+y)k−(a−y)ksinh(by)dy and ∫0∞(a+y)k+(a−y)kcosh(by)dy in terms of a special function where k, a and b are arbitrary complex numbers.

scholarly journals Note on a Stieltjes Transform in terms of the Lerch Function

2021 ◽  
Vol 14 (3) ◽  
pp. 723-736
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Special Functions ◽  
Complex Numbers ◽  
Definite Integral ◽  
Fundamental Constants ◽  
Logarithmic Function ◽  
Stieltjes Transform ◽  
Closed Form Solutions ◽  
General Complex

In this work the authors derive the Stieltjes transform of the logarithmic function in terms of the Lerch function. This transform is used to derive closed form solutions involving fundamental constants and special functions. Specifically we derive the definite integral given by\[\int_{0}^{\infty} \frac{(1-b x)^m \log ^k(c (1-b x))+(b x+1)^m \log ^k(c (b x+1))}{a+x^2}dx\]where $a,b,c,m$ and $k$ are general complex numbers subject to the restrictions given in connection with the formulas.

An Algorithm for Reducing the Size of Finite Element Closed-Form Source Code Files

2009 ◽  
Author(s):  
Sara McCaslin ◽  
Kent Lawrence
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Closed Form ◽  
Source Code ◽  
Simple Algorithm ◽  
Higher Order ◽  
Error Estimator ◽  
Element Analysis ◽  
Closed Form Solutions ◽  
Large Source

Closed-form solutions, as opposed to numerically integrated solutions, can now be obtained for many problems in engineering. In the area of finite element analysis, researchers have been able to demonstrate the efficiency of closed-form solutions when compared to numerical integration for elements such as straight-sided triangular [1] and tetrahedral elements [2, 3]. With higher order elements, however, the length of the resulting expressions is excessive. When these expressions are to be implemented in finite element applications as source code files, large source code files can be generated, resulting in line length/ line continuation limit issues with the compiler. This paper discusses a simple algorithm for the reduction of large source code files in which duplicate terms are replaced through the use of an adaptive dictionary. The importance of this algorithm lies in its ability to produce manageable source code files that can be used to improve efficiency in the element generation step of higher order finite element analysis. The algorithm is applied to Fortran files developed for the implementation of closed-form element stiffness and error estimator expressions for straight-sided tetrahedral finite elements through the fourth order. Reductions in individual source code file size by as much as 83% are demonstrated.

Closed-Form Solutions of Higher-Order Duration Measures

1988 ◽  
Vol 44 (6) ◽  
pp. 82-84 ◽  
Author(s):  
Sanjay K. Nawalkha ◽  
Nelson J. Lacey
Keyword(s):  
Closed Form ◽  
Higher Order ◽  
Closed Form Solutions

Higher-Order Plane Motion Theories in Kinematic Synthesis

1967 ◽  
Vol 89 (2) ◽  
pp. 223-230 ◽  
Author(s):  
G. N. Sandor ◽  
F. Freudenstein
Keyword(s):  
Plane Motion ◽  
Higher Order ◽  
Conic Section ◽  
Path Generation ◽  
Kinematic Synthesis

Algebraic multiple-position theories in kinematic synthesis are classified in a manner which also suggests various extensions of classical circular theory. In the case of infinitesimally separated positions, parabolic, elliptic, hyperbolic, and general conic-section theories are developed. The results, which are generally applicable, are illustrated with reference to the synthesis of cycloidal motions involving higher-order path generation.

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