scholarly journals The vibrations of an infinite system of vortex rings

1927 ◽  
Vol 116 (774) ◽  
pp. 352-379 ◽  
Keyword(s):  
Infinite System ◽  
Vortex Rings ◽  
Critical Ratio ◽  
Single Vortex ◽  
Natural Modes ◽  
Finite Section ◽  
Modes Of Vibration ◽  
Central Filament ◽  
Set Up ◽  
The Stability

The vibrations that may be set up and maintained in the central filament of a single vortex ring of small but finite section have been investi­gated by Thomson and others, but no corresponding investigation appears to have been undertaken for a system of parallel rings, although the matter is of some importance in connection with the state of motion behind a moving body. In a previous paper the authors have examined the stability of an infinite system of equal vortex rings situated in parallel planes with their centres evenly spaced along an infinite line and with their planes at right angles to that line. Instability was there found to occur for disturbances confined to displacements of the centre of each ring along the central axis, the filament of each ring still remaining circular. In the present paper the investigation is extended to deformation of the vortex filaments, and some interesting conclusions are drawn regarding natural modes of vibration of the infinite system of vortex rings, such as may occur without the longitudinal instability referred to in the previous paper becoming apparent. It is found, for example, that for any given ratio of radius of ring section to radius of ring there exists a critical ratio of ring spacing to radius, separating the region of stable oscillation from that of instability, a result in some respects closely analogous to that found by Kármán for the stability of two infinite parallel rows of rectilinear vortices.

scholarly journals XX. The potential of an anchor ring.-Part II

1893 ◽  
Vol 184 ◽  
pp. 1041-1106 ◽  
Keyword(s):  
Vortex Ring ◽  
Cross Section ◽  
Steady Motion ◽  
Vortex Rings ◽  
Laplace’S Equation ◽  
Single Vortex ◽  
Laplace's Equation ◽  
The Stability ◽  
Work Done

This paper is a continuation of that at pp. 43-95 suprd , on “The Potential of an Anchor Bing.” In that paper the potential of an anchor ring was found at all external points; in this/its value is determined at internal points. The annular form of rotating gravitating fluid was also discussed in that paper; here the stability of such a ring is considered. In addition, the potential of a ring whose cross-section is elliptic, being of interest in connection with Saturn, is obtained. The similarity of the methods employed, as well as of the analysis, has led me to give in this paper also a determination of the steady motion of a single vortex-ring in an infinite fluid, and of several fine vortex rings on the same axis. In Section I. solutions of Laplace’s equation applicable to space inside an anchor ring are obtained. These results are applied to obtain the potential of a solid ring at internal points, and also of a distribution of matter on the surface of the ring. The work done in collecting the ring from infinity is obtained.

scholarly journals The stability of an infinite system of circular vortices

1927 ◽  
Vol 114 (768) ◽  
pp. 594-604 ◽  
Keyword(s):  
Infinite Number ◽  
Degrees Of Freedom ◽  
Infinite System ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Vortex Rings ◽  
Infinite Cylinder ◽  
Vortex Filaments ◽  
Straight Line ◽  
The Stability

A system of equal circular vortex filaments have their centres evenly spaced along a straight line, and their planes at right angles to this line. The present investigation is concerned with the stability or instability of such an arrangement. The corresponding problem in two dimensions has been dealt with by Kármán who considered the case of two infinite trails of parallel rectilinear vortices, with the object of applying his results to the resistance of an infinite cylinder moving in a fluid and to the state of motion in the rear of the cylinder. The infinite system of circular vortex filaments, on the other hand, may be supposed, in certain circumstances to be discussed in a later paper, to be generated in the rear of a three-dimensional body in motion in a fluid. The present investigation may therefore be regarded as a first step towards an examination of the three-dimensional problem analogous to that treated by Kármán for two dimensions. A special difficulty arises in an investigation of this type from the fact that the system of vortex rings, possessing an infinite number of degrees of freedom, are capable of adopting an infinite number of possible configurations about the position of equilibrium.

scholarly journals On Kaufmann's theory of the impact of the pianoforte hammer

1920 ◽  
Vol 97 (682) ◽  
pp. 99-110 ◽  
Keyword(s):  
Initial Conditions ◽  
Practical Importance ◽  
Prima Facie ◽  
Point Of View ◽  
Infinite Length ◽  
Modes Of Vibration ◽  
Rigorous Treatment ◽  
The Subject ◽  
Set Up ◽  
The Impact

The theory of the vibrations of the pianoforte string put forward by Kaufmann in a well-known paper has figured prominently in recent discussions on the acoustics of this instrument. It proceeds on lines radically different from those adopted by Helmholtz in his classical treatment of the subject. While recognising that the elasticity of the pianoforte hammer is not a negligible factor, Kaufmann set out to simplify the mathematical analysis by ignoring its effect altogether, and treating the hammer as a particle possessing only inertia without spring. The motion of the string following the impact of the hammer is found from the initial conditions and from the functional solutions of the equation of wave-propagation on the string. On this basis he gave a rigorous treatment of two cases: (1) a particle impinging on a stretched string of infinite length, and (2) a particle impinging on the centre of a finite string, neither of which cases is of much interest from an acoustical point of view. The case of practical importance treated by him is that in which a particle impinges on the string near one end. For this case, he gave only an approximate theory from which the duration of contact, the motion of the point struck, and the form of the vibration-curves for various points of the string could be found. There can be no doubt of the importance of Kaufmann’s work, and it naturally becomes necessary to extend and revise his theory in various directions. In several respects, the theory awaits fuller development, especially as regards the harmonic analysis of the modes of vibration set up by impact, and the detailed discussion of the influence of the elasticity of the hammer and of varying velocities of impact. Apart from these points, the question arises whether the approximate method used by Kaufmann is sufficiently accurate for practical purposes, and whether it may be regarded as applicable when, as in the pianoforte, the point struck is distant one-eighth or one-ninth of the length of the string from one end. Kaufmann’s treatment is practically based on the assumption that the part of the string between the end and the point struck remains straight as long as the hammer and string remain in contact. Primâ facie , it is clear that this assumption would introduce error when the part of the string under reference is an appreciable fraction of the whole. For the effect of the impact would obviously be to excite the vibrations of this portion of the string, which continue so long as the hammer is in contact, and would also influence the mode of vibration of the string as a whole when the hammer loses contact. A mathematical theory which is not subject to this error, and which is applicable for any position of the striking point, thus seems called for.

scholarly journals Galloping Stability and Wind Tunnel Test of Iced Quad Bundled Conductors considering Wake Effect

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Xiaohui Liu ◽  
Ming Zou ◽  
Chuan Wu ◽  
Mengqi Cai ◽  
Guangyun Min ◽  
...  
Keyword(s):  
Wind Tunnel ◽  
Transmission Lines ◽  
Degrees Of Freedom ◽  
Wind Tunnel Test ◽  
Aerodynamic Coefficients ◽  
Wake Effect ◽  
Aerodynamic Coefficient ◽  
Set Up ◽  
The Stability ◽  
Three Degrees Of Freedom

A new quad bundle conductor galloping model considering wake effect is proposed to solve the problem of different aerodynamic coefficients of each subconductor of iced quad bundle conductor. Based on the quasistatic theory, a new 3-DOF (three degrees of freedom) galloping model of iced quad bundle conductors is established, which can accurately reflect the energy transfer and galloping of quad bundle conductor in three directions. After a series of formula derivations, the conductor stability judgment formula is obtained. In the wind tunnel test, according to the actual engineering situation, different variables are set up to accurately simulate the galloping of iced quad bundle conductor under the wind, and the aerodynamic coefficient is obtained. Finally, according to the stability judgment formula of this paper, calculate the critical wind speed of conductor galloping through programming. The dates of wind tunnel test and calculation in this paper can be used in the antigalloping design of transmission lines.

The Mitigation of the French Nuclear Option: New Industrial Realism and Technical Democracy

2002 ◽  
Vol 13 (2) ◽  
pp. 263-279 ◽  
Author(s):  
Dominique Finon
Keyword(s):  
Waste Management ◽  
Electricity Markets ◽  
Social Preferences ◽  
Public Inquiry ◽  
The Future ◽  
Parliamentary Committee ◽  
Set Up ◽  
The Stability ◽  
The Cost ◽  
Phase Out

Nuclear phase-out policies and the European obligation to liberalise electricity markets could put the French nuclear option dramatically at risk by influencing social preferences or by constraining power producers' investment choices in the future. So far, the particular institutional set-up which has allowed the efficient build-up and operation of several series of standardised reactors preserves the stability of the main elements of the option. However, important adaptations to the evolving industrial and political environment occur and contribute to changing the option. Some institutional changes (such as local public inquiry, creation of a Parliamentary committee, independence of safety authorities) and divergence between industrial interests already allow debates on internal options such as reprocessing, type of waste management deposits, ordering of an advanced PWR. These changes improve the cost transparency, even if internalisation of nuclear externalities (cost of insurance, provisions for waste management) is still incomplete. However, when effective, this internalisation would not affect definitively the competitive position of the nuclear production because of the parallel internalisation of CO2 externalities from fossil fuel power generation in the official rationale. Consequently the real issue for the future of the nuclear option in France remains the preservation of social acceptability in the perception of nuclear risks.

Vibration of Rectangular Plates by the Ritz Method

1950 ◽  
Vol 17 (4) ◽  
pp. 448-453 ◽  
Author(s):  
Dana Young
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Modes Of Vibration ◽  
Ritz’S Method ◽  
Square Plate ◽  
Set Up

Abstract Ritz’s method is one of several possible procedures for obtaining approximate solutions for the frequencies and modes of vibration of thin elastic plates. The accuracy of the results and the practicability of the computations depend to a great extent upon the set of functions that is chosen to represent the plate deflection. In this investigation, use is made of the functions which define the normal modes of vibration of a uniform beam. Tables of values of these functions have been computed as well as values of different integrals of the functions and their derivatives. With the aid of these data, the necessary equations can be set up and solved with reasonable effort. Solutions are obtained for three specific plate problems, namely, (a) square plate clamped at all four edges, (b) square plate clamped along two adjacent edges and free along the other two edges, and (c) square plate clamped along one edge and free along the other three edges.

scholarly journals A Study of the Stability of a Plate-Like Load Towed beneath a Helicopter

1971 ◽  
Vol 13 (5) ◽  
pp. 330-343 ◽  
Author(s):  
D. F. Sheldon
Keyword(s):  
Equations Of Motion ◽  
Physical Parameters ◽  
Recent Experience ◽  
Wind Tunnel Testing ◽  
Stability Derivatives ◽  
Direct Indication ◽  
Metric Dimensions ◽  
Set Up ◽  
The Stability ◽  
Derivative Information

Recent experience has shown that a plate-like load suspended beneath a helicopter moving in horizontal forward flight has unstable characteristics at both low and high forward speeds. These findings have prompted a theoretical analysis to determine the longitudinal and lateral dynamic stability of a suspended pallet. Only the longitudinal stability is considered here. Although it is strictly a non-linear problem, the usual assumptions have been made to obtain linearized equations of motion. The aerodynamic derivative data required for these equations have been obtained, where possible, for the appropriate ranges of Reynolds and Strouhal number by means of static and dynamic wind tunnel testing. The resulting stability equations (with full aerodynamic derivative information) have been set up and solved, on a digital computer, to give direct indication of a stable or unstable system for a combination of physical parameters. These results have indicated a longitudinal unstable mode for all practical forward speeds. Simultaneously the important stability derivatives were found for this instability and modifications were made subsequently in the suspension system to eliminate the instabilities in the longitudinal sense. Throughout this paper, all metric dimensions are given approximately.

scholarly journals Enhancement the stability of power system using optimal location of FACTS devices

2020 ◽  
Vol 18 (2) ◽  
pp. 648
Author(s):  
Ali Najim Abdullah ◽  
Ahmed Majeed Ghadhban ◽  
Hayder Salim Hameed ◽  
Husham Idan Hussein
Keyword(s):  
Power Network ◽  
Performance Factors ◽  
Facts Devices ◽  
Clearing Time ◽  
Thyristor Controlled Series Capacitor ◽  
Overall Performance ◽  
Set Up ◽  
The Stability ◽  
And Control ◽  
Better Than

<p><span>This paper proposes a steady-state of the Static Var Compensator (SVC) &amp; Thyristor Controlled Series Capacitor (TCSC) set up for enhancing the damping overall performance and growing the integral clearing time (CCT) of a power network. The indispensable clearing time is carried out through increasing the time fault interval until the gadget loses stability. Increasing the CCT can be contribute to reliability of the safety gadget, decrease the protection machine ranking and cost. In order to attain most enhancement of machine stability via optimizing location, sizing and control modes of SVC and TCSC. Models and methodology for putting and designing shunt FACT’s units SVC (injected reactive strength Q) and series FACT’s devices TCSC (chose capacitive region) are examined in a 6-bus system. Performance factors are described to show validation of SVC and TCSC on extraordinary conditions. It is proven that the SVC is better than TCSC. </span></p>

The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem

2013 ◽  
Vol 18 (1-2) ◽  
pp. 33-62 ◽  
Author(s):  
Alexey V. Borisov ◽  
Alexander A. Kilin ◽  
Ivan S. Mamaev
Keyword(s):  
Stability Problem ◽  
Vortex Rings ◽  
The Stability
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