Small Oscillations and the Stability of Nonholonomic Systems

1968 ◽  
Vol 90 (1) ◽  
pp. 97-102
Author(s):  
R. K. Duggins

An equation is derived to describe small oscillations in a simple surge tank for a wider range of operating conditions than hitherto considered. It is presented in a new non-dimensional form which facilitates an immediate prediction of surge phenomena when small disturbances are imposed on any given set of steady-state conditions. A characteristic of general form is assumed for conduit friction, and the effect is considered of deviation from the condition of constant hydraulic power delivered to the turbine. Some experiments have been carried out with a laboratory model simulating a variety of steady-state operating conditions, and surge measurements are presented for comparison with the calculated behavior.


1980 ◽  
Vol 1 ◽  
pp. 49-54 ◽  
Author(s):  
J. F. Nye ◽  
J. R. Potter

As an iceberg melts, the resulting change of shape can cause it to list gradually or to become unstable and topple over suddenly. Similarly, when an iceberg breaks up some of the individual pieces may capsize. We have used Zeeman’s analysis of the stability of ships, which is based on catastrophe theory, to examine this problem. We deal only with statical equilibrium; dynamical effects induced by water motion are important for ships, but very large icebergs have correspondingly small oscillations and therefore dynamical aspects are ignored in this first study. The advantage of the catastrophe-theory approach over the conventional stability theory used by naval architects lies in the conceptual clarity that it provides. In particular, it gives a three-dimensional geometrical picture that enables one to see all the possible equilibrium attitudes of a given iceberg, whether they are stable or unstable, whether a stable attitude is dangerously close to an unstable one, and how positions of stable equilibrium can be destroyed as the shape of the iceberg evolves with time.By making two-dimensional computations we examine the stability of two different shapes of cross-section, rectangles and trapezia, with realistic density distributions. These shapes may list gradually or topple suddenly as a single parameter is changed. For example, we find that a conversion of the vertical sides of a rectangular section into the slightly inward-sloping sides of a trapezium has a comparatively large adverse effect on stability. The main purpose of this work is to suggest how the stability characteristics of any selected iceberg may be investigated systematically.


1.1. The gliding of warm air over cold air, and of fresh water over salt water, without appreciable turbulent mixing, is a known phenomenon. Theoretically we know that, with a sudden transition in velocity and density, the laminar motion of superposed streams of fluid is unstable for any relative velocity, no matter how small, when viscosity is neglected. In reality, how­ ever, such sudden transitions cannot occur, or at any rate cannot be permanent, and when the width of the layer of transition is comparable with the wave­ length of the disturbance assumed, the analysis for an abrupt transition has no physical significance. The stability of such a system with a finite width of the layer of transition is therefore investigated below by the method of small oscillations. The investigation is a generalisation to heterogeneous stratified fluids of Rayleigh’s discussion of the homogeneous case. 1.2. The effect of a stratification of density in tending to prevent turbulence has lately been investigated by Prandtl, who, by considerations of energy, has arrived at the following criterion for the continuance of turbulence.


1983 ◽  
Vol 29 (3) ◽  
pp. 361-381 ◽  
Author(s):  
B. E. Meierovich

Small oscillations of a dense, equilibrium plasma in a strong current channel aro considered in the approximation of two-fluid hydrodynamics of ideal charged liquids. It is shown that chute-type radial oscillations are the ones most likely to affect the system's stability. The instability induced by spontaneous excitation of these oscillations results in a spallation of the diffuse equilibrium state into separate densely compressed current channels. The present investigation of the oscillations near the stability threshold enables one to formulate the stability criterion for the pinch system. It is shown that Bennett-type equilibrium distributions rapidly become unstable with a rise in the current, while the equilibrium distributions for the plasma compressed up to the electron degeneration, are stable, by a considerable margin. The theoretical conclusions on the pinch decay into separate current channels are in agreement with the available experimental data.


The stability of a cylindrical plasma with an axial magnetic field and confined between conducting walls is investigated by solving, for small oscillations about equilibrium, the linearized Boltzmann and Maxwell equations. A criterion for marginal stability is derived; this differs slightly from the one derived by Rosenbluth from an analysis of the particle orbits. However, Rosenbluth’s principal results on the possibility of stabilizing the pinch under suitable external conditions are confirmed. In the appendix a dispersion relation appropriate for plane hydromagnetic waves in an infinite medium is obtained; this relation discloses under the simplest conditions certain types of instabilities which may occur in plasma physics.


2004 ◽  
Vol 31 (3-4) ◽  
pp. 411-424
Author(s):  
Miroslav Veskovic ◽  
Vukman Covic

In this paper the stability of equilibrium of nonholonomic systems, on which dissipative and nonconservative positional forces act, is considered. We have proved the theorems on the instability of equilibrium under the assumptions that: the kinetic energy, the Rayleigh?s dissipation function and the positional forces are infinitely differentiable functions; the projection of the positional force component which represents the first nontrivial form of Maclaurin?s series of that positional force to the plane, which is normal to the vectors of nonholonomic constraints in the equilibrium position, is central and repulsive (with its centre of action in the equilibrium position). The suggested theorems are generalization of the results from [V.V. Kozlov, Prikl. Math. Mekh. (PMM), T58, V5, (1994), 31-36] and [M.M. Veskovic, Theoretical and Applied Mechanics, 24, (1998), 139-154]. The result obtained is analogous to the result from [D.R. Merkin, Introduction to theory of the stability of motion, Nauka, Moscow (1987)], which refers to the impossibility of equilibrium stabilization in a holonomic conservative system by dissipative and nonconservative positional forces in case when the potential energy in the equilibrium position has the maximum. The proving technique will be similar to that used in the paper [V.V. Kozlov, Prikl. Math. Mekh. (PMM), T58, V5, (1994), 31-36]. .


2020 ◽  
Vol 216 ◽  
pp. 01097
Author(s):  
K.R. Allaev ◽  
T.F. Makhmudov

The article discusses the results of the analysis of the static stability of complex electrical systems. The efficiency of the combined application of the equations of nodal voltages (ENV) and the Lyapunov function in quadratic form for the analysis of small oscillations of the electrical system is shown in the literature, which is called the Allaev method. A joint solution of the equations of nodal voltages and the matrix Lyapunov equation is given, which makes it possible to determine the stability conditions for the electrical system and identify the generator first approaching the stability limit. A study of small oscillations of complex electrical systems, which can be performed in full on the basis of matrix methods, successfully developed in recent decades, is carried out, which is associated with a sharp increase in the speed of computation and the amount of memory of modern computers.


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