scholarly journals On the Effects of Circulation around a Circle on the Stability of a Thomson Vortex N-gon

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1033
Author(s):  
Leonid Kurakin ◽  
Irina Ostrovskaya
Keyword(s):  
Circular Cylinder ◽  
Hamiltonian Systems ◽  
Nonlinear Stability ◽  
Vortex Dynamics ◽  
Stability Problem ◽  
Orbital Stability ◽  
Stability And Instability ◽  
The Stability ◽  
Parameter Values ◽  
Linear Setting

The stability problem of the stationary rotation of N identical point vortices is considered. The vortices are located on a circle of radius R 0 at the vertices of a regular N-gon outside a circle of radius R. The circulation Γ around the circle is arbitrary. The problem has three parameters N, q, Γ , where q = R 2 / R 0 2 . This old problem of vortex dynamics is posed by Havelock (1931) and is a generalization of the Kelvin problem (1878) on the stability of a regular vortex polygon (Thomson N-gon) on the plane. In the case of Γ = 0 , the problem has already been solved: in the linear setting by Havelock, and in the nonlinear setting in the series of our papers. The contribution of this work to the solution of the problem consists in the analysis of the case of non-zero circulation Γ ≠ 0 . The linearization matrix and the quadratic part of the Hamiltonian are studied for all possible parameter values. Conditions for orbital stability and instability in the nonlinear setting are found. The parameter areas are specified where linear stability occurs and nonlinear analysis is required. The nonlinear stability theory of equilibria of Hamiltonian systems in resonant cases is applied. Two resonances that lead to instability in the nonlinear setting are found and investigated, although stability occurs in the linear approximation. All the results obtained are consistent with those known for Γ = 0 . This research is a necessary step in solving similar problems for the case of a moving circular cylinder, a model of vortices inside an annulus, and others.

scholarly journals ON THE STABILITY PROBLEM FOR THE $\mathfrak{so}(5)$ FREE RIGID BODY

2011 ◽  
Vol 08 (06) ◽  
pp. 1205-1223 ◽  
Author(s):  
IOAN CAŞU
Keyword(s):  
Rigid Body ◽  
Weyl Group ◽  
Nonlinear Stability ◽  
Stability Problem ◽  
Integrals Of Motion ◽  
Free Rigid Body ◽  
Stability And Instability ◽  
Cartan Subalgebras ◽  
The Stability ◽  
Adjoint Orbit

In the general case of the [Formula: see text] free rigid body, we will give a list of integrals of motion, which generate the set of Mishchenko's integrals. In the case of [Formula: see text], we prove that there are 15 coordinate-type Cartan subalgebras which on a regular adjoint orbit give 15 Weyl group orbits of equilibria. These coordinate-type Cartan subalgebras are the analogs of the three axes of equilibria for the classical rigid body on [Formula: see text]. The nonlinear stability and instability of these equilibria is analyzed. In addition to these equilibria there are 10 other continuous families of equilibria.

scholarly journals On the equilibrium and stability of a row of point vortices

1995 ◽  
Vol 290 ◽  
pp. 167-181 ◽  
Author(s):  
Hassan Aref
Keyword(s):  
Vortex Dynamics ◽  
Stability Problem ◽  
Regular Polygon ◽  
Symmetric Matrix ◽  
Classical Problem ◽  
Point Vortices ◽  
Instability Modes ◽  
Systematic Enumeration ◽  
The Stability ◽  
Accessible Form

The equilibrium and stability of a single row of equidistantly spaced identical point vortices is a classical problem in vortex dynamics, which has been addressed by several investigators in different ways for at least a century. Aspects of the history and the essence of these treatments are traced, stating some in more accessible form, and pointing out interesting and apparently new connections between them. For example, it is shown that the stability problem for vortices in an infinite row and the stability problem for vortices arranged in a regular polygon are solved by the same eigenvalue problem for a certain symmetric matrix. This result also provides a more systematic enumeration of the basic instability modes. The less familiar theory of equilibria of a finite number of vortices situated on a line is also recalled.

scholarly journals ENERGY METHODS IN THE STABILITY PROBLEM FOR THE 𝔰𝔬(4) FREE RIGID BODY

2013 ◽  
Vol 23 (02) ◽  
pp. 1350032 ◽  
Author(s):  
PETRE BIRTEA ◽  
IOAN CAŞU
Keyword(s):  
Lyapunov Function ◽  
Rigid Body ◽  
Nonlinear Stability ◽  
Lyapunov Functions ◽  
Stability Problem ◽  
Energy Methods ◽  
Linear Combinations ◽  
Free Rigid Body ◽  
Constants Of Motion ◽  
The Stability

For the 𝔰𝔬(4) free rigid body the stability problem for the isolated equilibria has been completely solved using Lie-theoretical and topological arguments. For each case of nonlinear stability previously found, we construct a Lyapunov function. These Lyapunov functions are linear combinations of Mishchenko's constants of motion.

scholarly journals A Note on the Stability and Instability of Travelling Wave of Korteweg-de Vries Type: The Periodic Case

2011 ◽  
Vol 14 ◽  
pp. 57-72
Author(s):  
José R Quintero
Keyword(s):  
Orbital Stability ◽  
Type Equation ◽  
Travelling Wave ◽  
Periodic Case ◽  
Travelling Wave Solutions ◽  
Stability And Instability ◽  
Periodic Travelling Wave ◽  
De Vries ◽  
The Mean ◽  
The Stability

In this paper we adapt the work of M. Grillakis, J. Shatah, and W. Strauss, or J. Bona, P. Souganidis and W. Strauss to the periodic case in spaces having the mean zero property in order to establish the orbital stability/instability of periodic travelling wave solutions of a generalized Korteweg-de Vries type equation.

Parameter Optimization of Passive Dynamic Walking Biped with Knees

2012 ◽  
Vol 532-533 ◽  
pp. 379-384 ◽  
Author(s):  
Pei Jie Zhang ◽  
Ke Fei Song
Keyword(s):  
Stability Property ◽  
Mechanical Design ◽  
Slope Angle ◽  
Orbital Stability ◽  
Physical Parameters ◽  
Dynamic Walking ◽  
Passive Dynamic Walking ◽  
Eigen Values ◽  
The Stability ◽  
Parameter Values

A method to find the optimized parameter values of passive dynamic walking biped is presented. The effects of biped physical parameters on the stability property of passive gaits are studied by simulation experiments. The chosen parameters include the mass distribution, length of leg and slope angle. The stability property of passive walking limit cycles is used as criterion of optimization calculation, including the orbital stability described by eigen-values of linearized Poincaré map and the global property described by size of attraction region. The simulation results show how the stability of limit cycle varies when physical parameters of the passive biped change. The work is useful to explore the inherent property of passive dynamic walking and can be used as an important instruction in the mechanical design of biped robots based on principle of passive dynamic walking.

scholarly journals DETERMINATION OF THE STABILITY BOUNDARIES FOR THE HAMILTONIAN SYSTEMS WITH PERIODIC COEFFICIENTS

2005 ◽  
Vol 10 (2) ◽  
pp. 191-204 ◽  
Author(s):  
A. N. Prokopenya
Keyword(s):  
Hamiltonian Systems ◽  
Computer Algebra System ◽  
Many Body ◽  
Stability Boundaries ◽  
Periodic Coefficients ◽  
Stability And Instability ◽  
Existence Of Periodic Solutions ◽  
Linear Differential ◽  
The Stability

We consider the hamiltonian system of linear differential equations with periodic coefficients. Using the infinite determinant method based on the existence of periodic solutions on the boundaries between the domains of stability and instability in the parameter space we have developed the algorithm for analytical computation of the stability boundaries. The algorithm has been realized for the second and the fourth order hamiltonian systems arising in the restricted many-body problems. The stability boundaries have been found in the form of powers series, accurate to the sixth order in a small parameter. All the computations are done with the computer algebra system Mathematica. Nagrinejama Hamiltono tiesiniu diferencialiniu lygčiu su periodiniais koeficientais sistema. Remiantis tuo, kad parametru erdveje stabilumo ir nestabilumo sritis skiriančioje sienoje egzistuoja periodinis sprendinys, sukurtas analitinis minetos sienos apskaičiavimo algoritmas. Algoritmas realizuotas antros ir ketvirtos eiles Hamiltono sistemoms, kylančioms nagrinejant apribotu keleto kūnu uždavinius. Stabilumo srities siena randama laipsnines eilutes pavidalu mažojo parametro šešto laipsnio tikslumu. Skaičiavimai atlikti skaičiavimo algebros paketo Mathematica pagalba.

The identity policy and its effect in the stability of Middle East after 2010

2019 ◽  
pp. 434
Author(s):  
Ali Hussein Kadhim Alesammi
Keyword(s):  
Middle East ◽  
International Relations ◽  
Social Construct ◽  
Stability And Instability ◽  
The Social ◽  
The Stability ◽  
International Interactions ◽  
Regional Interests ◽  
Political Groups ◽  
Identity Policy

Since 2010 Middle East have many events or what they call "Arab spring events" which it result of overthrow governments and the rise of new political groups, all of this elements was resulting of many international and regional activities and making new regional and international axles, as well as the intersections of the different regional interests, therefore this research will try to study the stability and instability in the region as an independent variable not according to the neorealism or neoliberalism theories, but according to the constructivism theory which it base their assumptions on:  "In the international relations the non-physical structures of international interactions are determined by the identities of the players, which in turn determine the interests that determine the behavior of international players." So the research questions are: 1-What is the identity policy and haw affect in international relations? 2-How the social construct affect in international relations? 3-How the elite's identities for the main actors in the Middle East affect in the regional axles?  

scholarly journals The two-dimensional hydrodynamical theory of moving aerofoils. IV

1947 ◽  
Vol 188 (1015) ◽  
pp. 439-463
Keyword(s):  
Stability Problem ◽  
Two Dimensional ◽  
Centre Of Gravity ◽  
First Approximation ◽  
Von Karman ◽  
Moving Cylinder ◽  
Period Equation ◽  
The Stability ◽  
Von Kármán ◽  
Hydrodynamical Theory

In the first part of this paper opportunity has been taken to make some adjustments in certain general formulae of previous papers, the necessity for which appeared in discussions with other workers on this subject. The general results thus amended are then applied to a general discussion of the stability problem including the effect of the trailing wake which was deliberately excluded in the previous paper. The general conclusion is that to a first approximation the wake, as usually assumed, has little or no effect on the reality of the roots of the period equation, but that it may introduce instability of the oscillations, if the centre of gravity of the element is not sufficiently far forward. During the discussion contact is made with certain partial results recently obtained by von Karman and Sears, which are shown to be particular cases of the general formulae. An Appendix is also added containing certain results on the motion of a vortex behind a moving cylinder, which were obtained to justify certain of the assumptions underlying the trail theory.

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