Spatial Modulation of Repeated Vibration Modes in Rotationally Periodic Structures

1997 ◽  
Author(s):  
Mintae Kim ◽  
Joonho Moon ◽  
Jonathan A. Wickert
Keyword(s):  
Mode Shape ◽  
Natural Frequencies ◽  
Periodic Structures ◽  
Spatial Modulation ◽  
Vibration Modes ◽  
Algebraic Relation ◽  
Natural Frequencies And Modes ◽  
Axisymmetric Structure ◽  
Technical Interest ◽  
Model Features

Abstract When a structure deviates from axisymmetry because of circumferentially varying model features, significant changes can occur to its natural frequencies and modes, particularly for the doublet modes that have non-zero nodal diameters and repeated natural frequencies in the limit of axisymmetry. Of technical interest are configurations in which inertia, dissipation, stiffness, or domain features are evenly distributed around the structure. Aside from the well-studied phenomenon of eigenvalue splitting, whereby the natural frequencies of certain doublets split into distinct values, modes of the axisymmetric structure that are precisely harmonic become contaminated by certain additional wavenumbers in the presence of periodically spaced model features. From analytical, numerical, and experimental perspectives, this paper investigates spatial modulation of the doublet modes, particularly those retaining repeated natural frequencies for which modulation is most acute. In some cases, modulation can be sufficiently severe that a mode shape will beat spatially as harmonics with commensurate wavenumbers combine, just as the superposition of time records having nearly equal frequencies leads to classic temporal beating. A straightforward algebraic relation and a graphical checkerboard diagram are discussed with a view towards predicting the wavenumbers present in modulated eigenfunctions given the number of nodal diameters in the base mode and the number of equally spaced model features.

Weakly Isotopic Planar Ternary Rings

1975 ◽  
Vol 27 (1) ◽  
pp. 32-36
Author(s):  
Frederick W. Stevenson
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Ternary Ring ◽  
Algebraic Relation ◽  
Planar Ternary Ring ◽  
Algebraic Properties

This paper introduces two relations both weaker than isotopism which may hold between planar ternary rings. We will concentrate on the geometric consequences rather than the algebraic properties of these relations. It is well-known that every projective plane can be coordinatized by a planar ternary ring and every planar ternary ring coordinatizes a projective plane. If two planar ternary rings are isomorphic then their associated projective planes are isomorphic; however, the converse is not true. In fact, an algebraic bond which necessarily holds between the coordinatizing planar ternary rings of isomorphic projective planes has not been found. Such a bond must, of course, be weaker than isomorphism; furthermore, it must be weaker than isotopism. Here we show that it is even weaker than the two new relations introduced.This is significant because the weaker of our relations is, in a sense, the weakest possible algebraic relation which can hold between planar ternary rings which coordinatize isomorphic projective planes.

On the connexion of algebraic functions with automorphic functions

1898 ◽  
Vol 63 (389-400) ◽  
pp. 267-268 ◽  
Keyword(s):  
Riemann Surface ◽  
Existence Theorem ◽  
Algebraic Equation ◽  
Rational Functions ◽  
Algebraic Relation ◽  
Periodic Functions ◽  
Algebraic Functions ◽  
Automorphic Functions

If u and z are variables connected by an algebraic equation, they are, in general, multiform functions of each other; the multiformity can be represented by a Riemann surface, to each point of which corresponds a pair of values of u and z . Poincaré and Klein have proved that a variable t exists, of which u and z are uniform automorphic functions; the existence-theorem, however, does not connect t analytically with u and z . When the genus ( genre, Geschlecht ) of the algebraic relation is zero oi unity, t can be found by known methods; the automorphic functions required are rational functions, and doubly periodic functions, in the two case respectively.

scholarly journals Karmarkar scalar condition

2020 ◽  
Vol 80 (2) ◽  
Author(s):  
J. Ospino ◽  
L. A. Núñez
Keyword(s):  
Differential Equation ◽  
Energy Density ◽  
Density Profile ◽  
Riemann Tensor ◽  
Orthogonal Decomposition ◽  
Algebraic Relation ◽  
Structure Scalars ◽  
Physical Variables ◽  
Adiabatic Case ◽  
Dissipative Dynamic

AbstractIn this work we present the Karmarkar condition in terms of the structure scalars obtained from the orthogonal decomposition of the Riemann tensor. This new expression becomes an algebraic relation among the physical variables, and not a differential equation between the metric coefficients. By using the Karmarkar scalar condition we implement a method to obtain all possible embedding class I static spherical solutions, provided the energy density profile is given. We also analyse the dynamic adiabatic case and show the incompatibility of the Karmarkar condition with several commonly assumed simplifications to the study of gravitational collapse. Finally, we consider the dissipative dynamic Karmarkar collapse and find a new solution family.

The unbounded two-dimensional guiding-centre plasma

1995 ◽  
Vol 54 (1) ◽  
pp. 11-29 ◽  
Author(s):  
Michael K. -H. Kiessling
Keyword(s):  
Angular Momentum ◽  
Variational Principle ◽  
Mean Field ◽  
Analytical Techniques ◽  
Larmor Frequency ◽  
Two Dimensional ◽  
Algebraic Relation ◽  
Density Profiles ◽  
Translation Invariant ◽  
Computer Based

The thermal mean-field equilibrium of a translation-invariant, unbounded one- component guiding-centre plasma is studied by analytical techniques. A variational principle is constructed. It is shown that only radial symmetric, decreasing density profiles occur. Prescribing the total number of gyro centres N ∈ (0, ∞), the energy E ∈ (E0, ∞) and the canonical angular momentum M ∈ (0, ∞]) uniquely determines a profile. Metastable or unstable profiles do not exist. A simple algebraic relation between N, M, the guiding-centre temperature β−1 and the characteristic Larmor frequency ω is derived. This explains Williamson's computer-based observations.

On Euclidean invariance of algebraic Reynolds stress models in turbulence

2003 ◽  
Vol 476 ◽  
pp. 63-68 ◽  
Author(s):  
J. WEIS ◽  
K. HUTTER
Keyword(s):  
Reynolds Stress ◽  
Reynolds Stress Model ◽  
Simple Extension ◽  
Stress Model ◽  
Algebraic Relation ◽  
Reynolds Stresses ◽  
Coordinate Systems ◽  
Stress Models ◽  
Algebraic Reynolds Stress Model ◽  
Euclidean Invariance

This article shows how Euclidean invariance can be preserved in the so-called algebraic Reynolds stress model (ARSM) approximation. This approximation is used to reduce the transport equation for the Reynolds stresses to an explicit algebraic relation. A number of known models, which make use of this approximation, are not form-invariant under transformations to rotating coordinate systems. A simple extension is presented to show how this artifact can be removed.

scholarly journals Function Theories in Cayley-Dickson Algebras and Number Theory

2021 ◽  
Author(s):  
Rolf Sören Kraußhar
Keyword(s):  
Clifford Analysis ◽  
Clifford Algebras ◽  
Function Theory ◽  
Elliptic Functions ◽  
Number Fields ◽  
Algebraic Relation ◽  
Algebraic Number Fields ◽  
Explicit Formulas ◽  
Hyperholomorphic Functions ◽  
Set Up

AbstractIn the recent years a lot of effort has been made to extend the theory of hyperholomorphic functions from the setting of associative Clifford algebras to non-associative Cayley-Dickson algebras, starting with the octonions.An important question is whether there appear really essentially different features in the treatment with Cayley-Dickson algebras that cannot be handled in the Clifford analysis setting. Here we give one concrete example: Cayley-Dickson algebras admit the construction of direct analogues of so-called CM-lattices, in particular, lattices that are closed under multiplication.Canonical examples are lattices with components from the algebraic number fields $$\mathbb{Q}{[\sqrt{m1}, \ldots \sqrt{mk}]}$$ Q [ m 1 , … mk ] . Note that the multiplication of two non-integer lattice paravectors does not give anymore a lattice paravector in the Clifford algebra. In this paper we exploit the tools of octonionic function theory to set up an algebraic relation between different octonionic generalized elliptic functions which give rise to octonionic elliptic curves. We present explicit formulas for the trace of the octonionic CM-division values.

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