On Infinitesimal Cycloidal Kinematic Theory of Planar Motion

1966 ◽  
Vol 33 (4) ◽  
pp. 927-933 ◽  
Author(s):  
G. N. Sandor
Keyword(s):  
Closed Form ◽  
Computer Programs ◽  
Plane Motion ◽  
Higher Order ◽  
Planar Motion ◽  
Kinematic Theory ◽  
Special Cases ◽  
Moving Plane ◽  
Motion Generator

A kinematic theory is developed involving infinitesimally close coplanar positions of a moving plane, based on a finite cycloidal kinematic theory of plane motion, which was shown to include Burmester’s classical centerpoint-circlepoint theory as one of several special cases. The infinitesimal theory is developed for both the general cycloidal and for the special circular (Burmester) cases. Applications are derived for closed-form analytic synthesis of higher-order approximate path and motion generator linkages. Examples are given, and implementations of the theory in the form of computer programs are made available.

On the Existence of a Cycloidal Burmester Theory in Planar Kinematics

1964 ◽  
Vol 31 (4) ◽  
pp. 694-699 ◽  
Author(s):  
George N. Sandor
Keyword(s):  
Plane Curves ◽  
Motion Generation ◽  
Kinematic Synthesis ◽  
Multiple Generation ◽  
Special Cases ◽  
Moving Plane ◽  
Motion Generator ◽  
Linkage Synthesis

One of the basic theories of kinematic synthesis, namely, Burmester’s classical centerpoint-circlepoint theory, is shown to be one of several special cases of a broader, more general new theory, involving points of the moving plane whose several corresponding positions lie on cycloidal curves. These curves may be generated by “cycloidal cranks.” Such “cycloidpoints,” centers of their generating circles (“circlepoints”) and base circles (“centerpoints”) are proposed to be called “Burmester point trios” (BPT’s). In case of 6 prescribed arbitrary positions, such BPT’s appear to lie, respectively, on three higher plane curves proposed to be called “cycloidpoint,” “circlepoint” and “centerpoint curves,” or, collectively, “generalized Burmester curves.” In the case of hypocycloidal cranks with “Cardanic” proportions, the hypocycloids become ellipses. For 7 prescribed positions, the number of BPT’s is finite. Application to linkage synthesis for motion generation with prescribed order and timing is presented and cognate-motion generator linkages, based on multiple generation of cycloidal curves, are shown to exist. Analytical derivations are outlined for the equations of the “generalized Burmester curves,” and possible further specializations and generalizations are indicated.

scholarly journals Brownian Swarm Dynamics and Burgers’ Equation with Higher Order Dispersion

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 57
Author(s):  
Max-Olivier Hongler
Keyword(s):  
Burgers Equation ◽  
Higher Order ◽  
Underlying Structure ◽  
Swarm Dynamics ◽  
Special Cases ◽  
Kink Waves ◽  
Systematic Derivation ◽  
Fifth Order ◽  
Higher Order Dispersion ◽  
Order Dispersion

The concept of ranked order probability distribution unveils natural probabilistic interpretations for the kink waves (and hence the solitons) solving higher order dispersive Burgers’ type PDEs. Thanks to this underlying structure, it is possible to propose a systematic derivation of exact solutions for PDEs with a quadratic nonlinearity of the Burgers’ type but with arbitrary dispersive orders. As illustrations, we revisit the dissipative Kotrweg de Vries, Kuramoto-Sivashinski, and Kawahara equations (involving third, fourth, and fifth order dispersion dynamics), which in this context appear to be nothing but the simplest special cases of this infinitely rich class of nonlinear evolutions.

On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers

2020 ◽  
Vol 70 (3) ◽  
pp. 641-656
Author(s):  
Amira Khelifa ◽  
Yacine Halim ◽  
Abderrahmane Bouchair ◽  
Massaoud Berkal
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Difference Equations ◽  
Fibonacci Numbers ◽  
Higher Order ◽  
Real Numbers ◽  
Lucas Numbers ◽  
Rational Difference Equations ◽  
Initial Values ◽  
Fibonacci And Lucas Numbers

AbstractIn this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations$$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$where n, k∈ ℕ0, the initial values x−k, x−k+1, …, x0, y−k, y−k+1, …, y0, z−k, z−k+1, …, z1 and z0 are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.

Expansions of the mildly relativistic dispersion function for nearly perpendicular electron cyclotron ray tracing

1999 ◽  
Vol 61 (1) ◽  
pp. 121-128 ◽  
Author(s):  
I. P. SHKAROFSKY
Keyword(s):  
Ray Tracing ◽  
Computer Programs ◽  
Higher Order ◽  
Electron Cyclotron ◽  
Dispersion Function ◽  
Relativistic Dispersion ◽  
Plasma Dispersion ◽  
Derivatives Of ◽  
Cyclotron Harmonic ◽  
Experienced Difficulty

To trace rays very close to the nth electron cyclotron harmonic, we need the mildly relativistic plasma dispersion function and its higher-order derivatives. Expressions for these functions have been obtained as an expansion for nearly perpendicular propagation in a region where computer programs have previously experienced difficulty in accuracy, namely when the magnitude of (c/vt)2 (ω−nωc)/ω is between 1 and 10. In this region, the large-argument expansions are not yet valid, but partial cancellations of terms occur. The expansion is expressed as a sum over derivatives of the ordinary dispersion function Z. New expressions are derived to relate higher-order derivatives of Z to Z itself in this region of concern in terms of a finite series.

scholarly journals Inversion of higher order isotropic tensors with minor symmetries and solution of higher order heterogeneity problems

2010 ◽  
Vol 467 (2126) ◽  
pp. 314-332 ◽  
Author(s):  
Vincent Monchiet ◽  
Guy Bonnet
Keyword(s):  
Closed Form ◽  
Strain Field ◽  
Spherical Inclusion ◽  
Higher Order ◽  
Quadratic Polynomial ◽  
Elastic Matrix ◽  
Closed Form Expression ◽  
Form Expression ◽  
Order Tensor ◽  
Strain Fields

In this paper, the derivation of irreducible bases for a class of isotropic 2 n th-order tensors having particular ‘minor symmetries’ is presented. The methodology used for obtaining these bases consists of extending the concept of deviatoric and spherical parts, commonly used for second-order tensors, to the case of an n th-order tensor. It is shown that these bases are useful for effecting the classical tensorial operations and especially the inversion of a 2 n th-order tensor. Finally, the formalism introduced in this study is applied for obtaining the closed-form expression of the strain field within a spherical inclusion embedded in an infinite elastic matrix and subjected to linear or quadratic polynomial remote strain fields.

Thermal Stresses Due to Disturbance of Uniform Heat Flow by an Insulated Ovaloid Hole

1960 ◽  
Vol 27 (4) ◽  
pp. 635-639 ◽  
Author(s):  
A. L. Florence ◽  
J. N. Goodier
Keyword(s):  
Heat Flow ◽  
Closed Form ◽  
Thermal Stresses ◽  
Thermoelastic Problem ◽  
Uniform Heat ◽  
Special Cases

The linear thermoelastic problem is solved for a uniform heat flow disturbed by a hole of ovaloid form, which includes the ellipse and circle as special cases. Results for stress and displacement are found in closed form, by reducing the problem to one of boundary loading solvable by a method of Muskhelishvili.

scholarly journals A Note on a Triple Integral

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2056
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Closed Form Expression ◽  
Polynomial Functions ◽  
Fundamental Constants ◽  
Form Expression ◽  
Zero Distribution ◽  
Special Cases ◽  
Almost All

A closed form expression for a triple integral not previously considered is derived, in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. The kernel of the integral involves the product of the logarithmic, exponential, quotient radical, and polynomial functions. Special cases are derived in terms of fundamental constants; results are summarized in a table. All results in this work are new.

scholarly journals On the Signatures of Ordered System Lifetimes

2014 ◽  
Vol 51 (1) ◽  
pp. 82-91 ◽  
Author(s):  
N. Balakrishnan ◽  
William Volterman
Keyword(s):  
Closed Form ◽  
Coherent Systems ◽  
Special Cases

The idea of the system signature is extended here to the case of ordered system lifetimes arising from a test of coherent systems with a signature. An expression is given for the computation of the ordered system signatures in terms of the usual system signature for system lifetimes. Some properties of the ordered system signatures are then established. Closed-form expressions for the ordered system signatures are obtained in some special cases, and some illustrative examples are presented.

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