scholarly journals Analytical stability in the Caledonian Symmetric Five-Body Problem

2020 ◽  
Vol 132 (11-12) ◽  
Author(s):  
B. A. Steves ◽  
M. Shoaib ◽  
Winston L. Sweatman
Keyword(s):  
Stability Criterion ◽  
Body Problem ◽  
Dynamical Symmetry ◽  
Centre Of Mass ◽  
Real Motion ◽  
Analytical Stability ◽  
Hierarchical Stability ◽  
Dimensionless Function ◽  
The Stability ◽  
The Masses

AbstractIn this paper, we develop an analytical stability criterion for a five-body symmetrical system, called the Caledonian Symmetric Five-Body Problem (CS5BP), which has two pairs of equal masses and a fifth mass located at the centre of mass. The CS5BP is a planar problem that is configured to utilise past–future symmetry and dynamical symmetry. The introduction of symmetries greatly reduces the dimensions of the five-body problem. Sundman’s inequality is applied to derive boundary surfaces to the allowed real motion of the system. This enables the derivation of a stability criterion valid for all time for the hierarchical stability of the CS5BP. We show that the hierarchical stability depends solely on the Szebehely constant $$C_0$$ C 0 which is a dimensionless function involving the total energy and angular momentum. We then explore the effect on the stability of the whole system of varying the relative sizes of the masses. The CS5BP is hierarchically stable for $$C_0 > 0.065946$$ C 0 > 0.065946 . This criterion can be applied in the investigation of the stability of quintuple hierarchical stellar systems and symmetrical planetary systems.

Intrinsic Thermal Stability of Anisotropic Thin-Film Superconductors

1990 ◽  
Vol 112 (1) ◽  
pp. 10-15 ◽  
Author(s):  
M. I. Flik ◽  
C. L. Tien
Keyword(s):  
Thin Film ◽  
Thermal Stability ◽  
Stability Criterion ◽  
High Temperature Superconductors ◽  
Stability Criteria ◽  
Anisotropic Thermal Conductivity ◽  
Operating Current ◽  
Released Energy ◽  
The Stability ◽  
Thermal Stability Of

Intrinsic thermal stability denotes a situation where a superconductor can carry the operating current without resistance at all times after the occurrence of a localized release of thermal energy. This novel stability criterion is different from the cryogenic stability criteria for magnets and has particular relevance to thin-film superconductors. Crystals of ceramic high-temperature superconductors are likely to exhibit anisotropic thermal conductivity. The resultant anisotropy of highly oriented films of superconductors greatly influences their thermal stability. This work presents an analysis for the maximum operating current density that ensures intrinsic stability. The stability criterion depends on the amount of released energy, the Biot number, the aspect ratio, and the ratio of the thermal conductivities in the plane of the film and normal to it.

On the Stability of the Linearly Related Modes of Certain Nonlinear Two-Degree-of-Freedom Systems

1961 ◽  
Vol 28 (1) ◽  
pp. 71-77 ◽  
Author(s):  
C. P. Atkinson
Keyword(s):  
Nonlinear Differential Equations ◽  
Mass Ratio ◽  
Positive Mass ◽  
Fourth Degree ◽  
Real Roots ◽  
General Stability ◽  
The Stability ◽  
The Masses ◽  
Spring Constants ◽  
Two Degree Of Freedom

This paper presents a method for analyzing a pair of coupled nonlinear differential equations of the Duffing type in order to determine whether linearly related modal oscillations of the system are possible. The system has two masses, a coupling spring and two anchor springs. For the systems studied, the anchor springs are symmetric but the masses are not. The method requires the solution of a polynomial of fourth degree which reduces to a quadratic because of the symmetric springs. The roots are a function of the spring constants. When a particular set of spring constants is chosen, roots can be found which are then used to set the necessary mass ratio for linear modal oscillations. Limits on the ranges of spring-constant ratios for real roots and positive-mass ratios are given. A general stability analysis is presented with expressions for the stability in terms of the spring constants and masses. Two specific examples are given.

The flow of a micropolar liquid layer down an inclined plane

1968 ◽  
Vol 64 (2) ◽  
pp. 513-526 ◽  
Author(s):  
A. J. Willson
Keyword(s):  
Thin Films ◽  
Steady State ◽  
Liquid Layer ◽  
Stability Criterion ◽  
Long Waves ◽  
Inclined Plane ◽  
The Stability ◽  
Micropolar Liquid

AbstractConsideration is given to the flow of a micropolar liquid down an inclined plane. The steady state is analysed and Yih's technique is employed in an investigation of the stability of this flow with respect to long waves. Detailed calculations are given for thin films and it is shown that the micropolar properties of the liquid play an important role in the stability criterion.

Parallel-Process (Simultaneous) Machining and its Stability

1999 ◽  
Author(s):  
O. Burak Ozdoganlar ◽  
William J. Endres
Keyword(s):  
Initial Conditions ◽  
General System ◽  
Parallel Process ◽  
Analytical Stability ◽  
Multiple Processes ◽  
Or Practice ◽  
Machining System ◽  
Eigenvector Analysis ◽  
The Stability ◽  
Machining Operations

Abstract This paper presents a mathematical perspective, to complement the intuitive or practice-oriented perspective, to classifying machining operations as parallel-process (simultaneous) or single-process in nature. Illustrative scenarios are provided to demonstrate how these two perspectives may lead in different situations to the same or different conclusions regarding process parallelism. A model representation of a general parallel-process machining system is presented, based on which the general parallel-process stability eigenvalue problem is formulated. For a special simplified case of the general system, analytical methods are employed to derive a fully analytical stability solution. Thorough study of this solution through eigenvector analysis sheds light on some fundamental phenomena of parallel-process machining stability, such as dependence of the stability solution on phasing of the initial conditions (disturbances). This establishes the importance, when employing numerical time-domain simulation for such analyses, of specifying initial conditions for the multiple processes to be arbitrarily phased so that correct results are achieved across all spindle speeds.

scholarly journals INTEGRAL TRANSFORM TECHNIQUE FOR MESON WAVE FUNCTIONS

1996 ◽  
Vol 11 (20) ◽  
pp. 1611-1626 ◽  
Author(s):  
A.P. BAKULEV ◽  
S.V. MIKHAILOV
Keyword(s):  
Wave Function ◽  
Sum Rules ◽  
Integral Transform ◽  
Wave Functions ◽  
Sum Rule ◽  
New Approach ◽  
Exactly Solvable ◽  
The Stability ◽  
The Masses ◽  
Integral Transform Technique

In a recent paper1 we have proposed a new approach for extracting the wave function of the π-meson φπ(x) and the masses and wave functions of its first resonances from the new QCD sum rules for nondiagonal correlators obtained in Ref. 2. Here, we test our approach using an exactly solvable toy model as illustration. We demonstrate the validity of the method and suggest a pure algebraic procedure for extracting the masses and wave functions relating to the case under investigation. We also explore the stability of the procedure under perturbations of the theoretical part of the sum rule. In application to the pion case, this results not only in the mass and wave function of the first resonance (π′), but also in the estimation of π″-mass.

scholarly journals Retrograde resonances in compact multi-planetary systems: a feasible stabilizing mechanism

2007 ◽  
Vol 3 (S249) ◽  
pp. 511-516 ◽  
Author(s):  
Julie Gayon ◽  
Eric Bois
Keyword(s):  
Body Problem ◽  
Extrasolar Planets ◽  
Planetary Systems ◽  
Central Star ◽  
Point Of View ◽  
Outer Planet ◽  
Mean Motion Resonances ◽  
Hot Jupiters ◽  
The Stability ◽  
Orbital Data

AbstractMulti-planet systems detected until now are in most cases characterized by hot-Jupiters close to their central star as well as high eccentricities. As a consequence, from a dynamical point of view, compact multi-planetary systems form a variety of the general N-body problem (with N ≥ 3), whose solutions are not necessarily known. Extrasolar planets are up to now found in prograde (i.e. direct) orbital motions about their host star and often in mean-motion resonances (MMR). In the present paper, we investigate a theoretical alternative suitable for the stability of compact multi-planetary systems. When the outer planet moves on a retrograde orbit in MMR with respect to the inner planet, we find that the so-called retrograde resonances present fine and characteristic structures particularly relevant for dynamical stability. We show that retrograde resonances and their resources open a family of stabilizing mechanisms involving specific behaviors of apsidal precessions. We also point up that for particular orbital data, retrograde MMRs may provide more robust stability compared to the corresponding prograde MMRs.

The Meniscus Instability of a Thin Liquid Film

1988 ◽  
Vol 55 (4) ◽  
pp. 975-980 ◽  
Author(s):  
H. Koguchi ◽  
M. Okada ◽  
K. Tamura
Keyword(s):  
Thin Film ◽  
Analytical Solution ◽  
Liquid Film ◽  
Tilt Angle ◽  
Stability Criterion ◽  
Thin Liquid Film ◽  
Normal Direction ◽  
Wave Number ◽  
The Stability ◽  
Maximum Amplification

This paper reports on the instability for the meniscus of a thin film of a very viscous liquid between two tilted plates, which are separated at a constant speed with a tilt angle in the normal direction of the plates. The disturbances on the meniscus moving with movement of the plates are examined experimentally and theoretically. The disturbances are started when the velocity of movement of the plates exceeds a critical one. The wavelength of the disturbances is measured by using a VTR. The instability of the meniscus is studied theoretically using the linearized perturbation method. A simple and complete analytical solution yields both a stability criterion and the wave number for a linear thickness geometry. These results compared with experiments for the instability show the validity of the stability criterion and the best agreement is obtained with the wave number of maximum amplification.

scholarly journals On the Stability of Collinear Points of the RTBP with Triaxial and Oblate Primaries and Relativistic Effects

2021 ◽  
pp. 56-73
Author(s):  
S. E. Abd El-Bar
Keyword(s):  
Characteristic Equation ◽  
Body Problem ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Relativistic Effects ◽  
Three Body Problem ◽  
Total Potential ◽  
The Mean ◽  
The Stability ◽  
Three Body

Under the influence of some different perturbations, we study the stability of collinear equilibrium points of the Restricted Three Body Problem. More precisely, the perturbations due to the triaxiality of the bigger primary and the oblateness of the smaller primary, in addition to the relativistic effects, are considered. Moreover, the total potential and the mean motion of the problem are obtained. The equations of motion are derived and linearized around the collinear points. For studying the stability of these points, the characteristic equation and its partial derivatives are derived. Two real and two imaginary roots of the characteristic equation are deduced from the plotted figures throughout the manuscript. In addition, the instability of the collinear points is stressed. Finally, we compute some selected roots corresponding to the eigenvalues which are based on some selected values of the perturbing parameters in the Tables 1, 2.

scholarly journals THE QUESTION OF BOOM LIFTING EQUIPMENT OPERATION STABILITY ON A WHEELED CHASSIS

2021 ◽  
pp. 100-108
Author(s):  
В.И. Токарев ◽  
Н.В. Бабоченко
Keyword(s):  
Longitudinal Direction ◽  
Equipment Operation ◽  
Mathematical Expressions ◽  
Maximum Stability ◽  
Operation Stability ◽  
Quality Of Movement ◽  
The Stability ◽  
The Masses ◽  
Constituent Elements ◽  
Lifting Equipment

В статье представлены на рассмотрение характеристики стабильности работы стреловых грузоподъемных средств на колесном шасси в форме математических выражений. Математические выражения представлены в виде не громоздких зависимостей от конкретно заданных параметров. Качество движения зависит от возрастания линейных размеров, масс, моментов инерции, а также скоростей и других механических параметров грузоподъемных средств. Достижение стабильности работы выносных стреловых грузоподъемных средств достигается путем распределения нагрузки между утлегарью (выносной стрелой грузоподъемного средства) и опорными колесами колесного шасси. Считаем, что при существовании ряда концепцией со своими теориями. возможно определение стабильности работы стрелового грузоподъемного средства на колесном шасси. Нами установлено, что возможно обеспечить стабильность работы путем выбора целесообразных значений механических составляющих всех звеньев рабочего механизма для спланировано составленных рабочих ситуаций. В зависимости от возможного размещения грузоподъемного устройства показатели стабильности работы меняются и это подтверждают составленные нами математические выражения, которые приводятся в статье. Установлено, что путем варьирования различными вариантами положений и массой составляющих элементов конструкции грузоподъемного средства, а также графически определяя возможные варианты перемещения груза в зависимости от заданной длины утлегарьи, имеет место выражение, позволяющее определить ряд значений масс, безопасно поднимаемых грузоподъемным средством. Нами получены значения необходимых для графических построений грузовых характеристик грузоподъемного средства, выражающие зависимость между массой груза и вылетом утлегарьи с весом ее элементов. Реакции в шарнирах утлегарьи и усилия в ее составляющих звеньях возможно установить из данных грузовой характеристики. Стремление обеспечить максимальную стабильность работы грузоподъемного средства накладывает ограничения на контроль за несколькими подвижными операциями одновременно, что неблагоприятно сказывается на эффективности рабочего процесса. Установили, что обеспечение стабильности работы в поперечной и продольной плоскостях грузоподъемного средства является необходимым компонентом безопасной эксплуатации. По зависимостям для определения показателя грузового равновесия возможно определение предварительного места установки выносных опор грузоподъемного средства. Как подтверждают полученные результаты, стабильность работы грузоподъемного средства в продольном направлении определяется аналогично стабильности работы в поперечном направлении и для номинальной массы груза при наибольшем вылете утлегарьи и выставленных выносных опорах. В итоге отметим, что показателем грузового равновесия служит отношение удерживающего момента относительно ребра опрокидывания, создаваемого весом грузоподъемного средства на колесном шасси с учетом уменьшающих его дополнительных внешних нагрузок и влияния уклона площадки к опрокидывающему моменту, создаваемому рабочим грузом. The article presents for consideration the characteristics of the stability of the boom lifting equipment on a wheeled chassis in the form of mathematical expressions. Mathematical expressions are presented in the form of not cumbersome dependencies on specified parameters. The quality of movement depends on the increase in linear dimensions, masses, moments of inertia, as well as speeds, and other mechanical parameters of the lifting equipment. Achievement of the stability of the outboard boom lifting device is achieved by distributing the load between the jib boom (outboard boom of the lifting device) and the support wheels of the wheeled chassis. We believe that with the existence of a number of concepts with their theories, it is possible to determine the stability of the boom lifting device on a wheeled chassis. It has been found that it is possible to ensure the stability of work by choosing the appropriate values of the mechanical components of all links of the working mechanism for planned working situations. Depending on the possible placement of the lifting device, the stability indicators are changed, and this is confirmed by the mathematical expressions we compiled, which are given in the article. It has been established that by varying the positions and the mass of the constituent elements of the structure of the lifting device, as well as graphically defining the possible options of the load moving, depending on the given length of the jib boom, an expression takes place that makes it possible to determine a number of values of the masses safely lifted by the lifting device. There have been obtained the values of the cargo characteristics of the lifting device necessary for graphic constructions, expressing the relationship between the weight of the cargo and the overhanging of the jib boom with the weight of its elements. The reactions in the joints of the jig boom and the forces in its constituent links can be established from the data of the load characteristics. The desire to ensure maximum stability in the operation of the lifting device imposes restrictions on the control of several mobile operations at the same time, which adversely affects the efficiency of the work process. It has been established that ensuring the stability of operation in the transverse and longitudinal planes of the lifting device is a necessary component of safe operation. According to the dependencies for determining the indicator of cargo balance, it is possible to determine the preliminary installation site of the outriggers of the lifting device. As the results obtained confirm, the stability of the operation of the lifting device in the longitudinal direction is determined similarly to the stability of the operation in the transverse direction and for the nominal weight of the load with the greatest overhanging of the jib boom and the set outriggers. As a result, we note that the ratio of the holding moment relative to the overturning rib created by the weight of the lifting device on the wheeled chassis, taking into account the additional external loads that reduce it and the influence of the platform slope to the overturning moment created by the working load, serves as an indicator of the cargo balance.

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