scholarly journals The Stability of the Equilibrium of a Conservative System

1996 ◽  
Vol 202 (1) ◽  
pp. 133-149 ◽  
Author(s):  
Bin Liu
Keyword(s):  
Conservative System ◽  
The Stability

The Loading-Frequency Relationship in Multiple Eigenvalue Problems

1971 ◽  
Vol 38 (4) ◽  
pp. 1007-1011 ◽  
Author(s):  
K. Huseyin ◽  
J. Roorda
Keyword(s):  
Eigenvalue Problems ◽  
Stability Boundary ◽  
Conservative System ◽  
Free Vibrations ◽  
Degree Of Freedom ◽  
External Loading ◽  
Loading Frequency ◽  
Multiple Loading ◽  
The Stability ◽  
Frequency Relationship

The free vibrations of a linear conservative system with multiple loading parameters are studied, attention being restricted to pure eigenvalue problems. It is shown that the smallest frequency and external loading parameters of such a system constitute a strictly convex (synclastic) surface which cannot have convexity toward the origin of the “parameter space.” It is further proved that in the case of systems with one degree of freedom only, the surface takes the form of a plane. The practical implications of these results regarding the estimation of the frequencies and/or the stability boundary of the system are discussed. Thus it is observed that, on the basis of the established theorems, lower bounds to the frequencies at any stage of external loading and/or upper bounds to the stability boundary are readily obtainable. A two-degree-of-freedom illustrative example is discussed.

The Real-Valued Maxwell–Bloch Equations with Controls: From a Hamilton–Poisson System to a Chaotic One

2017 ◽  
Vol 27 (09) ◽  
pp. 1750143 ◽  
Author(s):  
Cristian Lăzureanu
Keyword(s):  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Dissipative System ◽  
Homoclinic Orbits ◽  
Conservative System ◽  
Poisson System ◽  
Bloch Equations ◽  
The Stability ◽  
Transitional System ◽  
Maxwell Bloch Equations

Applying parametric controls to the 3D real-valued Maxwell–Bloch equations, we obtain a Hamilton–Poisson system, a dissipative system with chaotic behavior, and a transitional system between the aforementioned states, which is a conservative system that has only one constant of motion. In the Hamiltonian case, we present some connections of the energy-Casimir mapping with the equilibrium states and the existence of the homoclinic orbits. We study the stability of the equilibrium points of the transitional system and the dissipative system. Furthermore, we point out the chaotic behavior of the introduced system.

Sensitivity Analysis of Dissipative Reversible and Hamiltonian Systems: A Survey

2009 ◽  
Author(s):  
Oleg N. Kirillov ◽  
Ferdinand Verhulst
Keyword(s):  
Sensitivity Analysis ◽  
Hopf Bifurcation ◽  
Structural Stability ◽  
Hamiltonian Systems ◽  
Perturbation Analysis ◽  
Stability Boundary ◽  
Conservative System ◽  
Small Dissipation ◽  
Whitney Umbrella ◽  
The Stability

The paradox of destabilization of a conservative or non-conservative system by small dissipation, or Ziegler’s paradox (1952), has stimulated an ever growing interest in the sensitivity of reversible and Hamiltonian systems with respect to dissipative perturbations. Since the last decade it has been widely accepted that dissipation-induced instabilities are closely related to singularities arising on the stability boundary. What is less known is that the first complete explanation of Ziegler’s paradox by means of the Whitney umbrella singularity dates back to 1956. We revisit this undeservedly forgotten pioneering result by Oene Bottema that outstripped later findings for about half a century. We discuss subsequent developments of the perturbation analysis of dissipation-induced instabilities and applications over this period, involving structural stability of matrices, Krein collision, Hamilton-Hopf bifurcation and related bifurcations.

On the Sufficiency of the Energy Criterion for the Stability of Certain Nonconservative Systems of the Follower-Load Type

1972 ◽  
Vol 39 (3) ◽  
pp. 717-722 ◽  
Author(s):  
H. H. E. Leipholz
Keyword(s):  
Lower Bound ◽  
Critical Load ◽  
Energy Criterion ◽  
Conservative System ◽  
Buckling Load ◽  
Follower Load ◽  
Nonconservative System ◽  
Nonconservative Systems ◽  
Divergence Type ◽  
The Stability

Using Galerkin’s method it is shown that in the domain of divergence, the nonconservative system of the follower-load type is always more stable than the corresponding conservative system. Hence, for nonconservative systems of the divergence type, the critical load of the corresponding conservative system becomes a lower bound for the buckling load, and the energy criterion remains sufficient for predicting stability. Moreover, it is proven that even for more general nonconservative systems, the energy criterion is sufficient under certain restrictions.

scholarly journals On the instability of equilibrium of a mechanical system with nonconservative forces

2004 ◽  
Vol 31 (3-4) ◽  
pp. 411-424
Author(s):  
Miroslav Veskovic ◽  
Vukman Covic
Keyword(s):  
Equilibrium Position ◽  
Nonholonomic Constraints ◽  
Nonholonomic Systems ◽  
Conservative System ◽  
Dissipation Function ◽  
Applied Mechanics ◽  
Differentiable Functions ◽  
Infinitely Differentiable Functions ◽  
Positional Force ◽  
The Stability

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]. .

An energy criterion for the linear stability of conservative flows

2000 ◽  
Vol 402 ◽  
pp. 329-348 ◽  
Author(s):  
P. A. DAVIDSON
Keyword(s):  
Variational Principle ◽  
Linear Stability ◽  
Energy Criterion ◽  
Conservative System ◽  
Base Flow ◽  
Recirculating Flow ◽  
Ideal Mhd ◽  
Mhd Equilibria ◽  
Euler Flows ◽  
The Stability

We investigate the linear stability of inviscid flows which are subject to a conservative body force. This includes a broad range of familiar conservative systems, such as ideal MHD, natural convection, flows driven by electrostatic forces and axisymmetric, swirling, recirculating flow. We provide a simple, unified, linear stability criterion valid for any conservative system. In particular, we establish a principle of maximum action of the formformula herewhere η is the Lagrangian displacement,e is a measure of the disturbance energy, T and V are the kinetic and potential energies, and L is the Lagrangian. Here d represents a variation of the type normally associated with Hamilton's principle, in which the particle trajectories are perturbed in such a way that the time of flight for each particle remains the same. (In practice this may be achieved by advecting the streamlines of the base flow in a frozen-in manner.) A simple test for stability is that e is positive definite and this is achieved if L(η) is a maximum at equilibrium. This captures many familiar criteria, such as Rayleigh's circulation criterion, the Rayleigh–Taylor criterion for stratified fluids, Bernstein's principle for magnetostatics, Frieman & Rotenberg's stability test for ideal MHD equilibria, and Arnold's variational principle applied to Euler flows and to ideal MHD. There are three advantages to our test: (i) d2T(η) has a particularly simple quadratic form so the test is easy to apply; (ii) the test is universal and applies to any conservative system; and (iii) unlike other energy principles, such as the energy-Casimir method or the Kelvin–Arnold variational principle, there is no need to identify all of the integral invariants of the flow as a precursor to performing the stability analysis. We end by looking at the particular case of MHD equilibria. Here we note that when u and B are co-linear there exists a broad range of stable steady flows. Moreover, their stability may be assessed by examining the stability of an equivalent magnetostatic equilibrium. When u and B are non-parallel, however, the flow invariably violates the energy criterion and so could, but need not, be unstable. In such cases we identify one mode in which the Lagrangian displacement grows linearly in time. This is reminiscent of the short-wavelength instability of non-Beltrami Euler flows.

A NEW EXPLICIT TIME INTEGRATION METHOD FOR STRUCTURAL DYNAMICS

2013 ◽  
Vol 13 (03) ◽  
pp. 1250068 ◽  
Author(s):  
SHIH-HSUN YIN
Keyword(s):  
Stability Condition ◽  
Integration Method ◽  
Mechanical Energy ◽  
Time Integration ◽  
Time History ◽  
Conservative System ◽  
Computational Accuracy ◽  
Explicit Time Integration ◽  
Time Integration Method ◽  
The Stability

A family of new explicit time-integration method is proposed herein, which inherits the numerical characteristics of any existing implicit Runge–Kutta algorithms for a linear conservative system. Based on an exact derivation of the increment of mechanical energy, the method proposed is demonstrated to be unconditionally stable. Also, the stability condition of the proposed method is derived when applied to solving a nonlinear system. The characteristics of the proposed method are investigated by observing the mechanical-energy time history of a nonlinear conservative system. The numerical results can be explained by the stability condition derived in the nonlinear regime. Finally, the computational accuracy and efficiency between the Newmark time integration method and the proposed explicit method are compared in solving the dynamic response of a couple of linear oscillators.

Dynamic Stability of an Elastic Beam Subjected to Follower Forces

2012 ◽  
Author(s):  
Katsuhisa Fujita ◽  
Akihide Gotou
Keyword(s):  
Normal Modes ◽  
Elastic Beam ◽  
Conservative System ◽  
Critical Force ◽  
Concentrated Force ◽  
Nonconservative System ◽  
Damped System ◽  
Follower Forces ◽  
The Stability ◽  
The Relationship

The stability of nonconservative system of a beam is investigated when an elastic beam is subjected to follower forces. The mathematical formulations for a conservative system and a nonconservative system are established regarding to a uniform cantilever subjected to a concentrated force and a uniform distributed force axially. The displacement of a uniform cantilever is assumed to be obtained by superposing the modal functions which are normal modes in a vacuum, and is estimated by applying the Galerkin’s method. Changing the forces, the eigenvalue analysis is performed, and the root locus is calculated for the stability analysis. And, the relationship between forces and frequencies for the undamped system and the damped system of the uniform cantilever subjected to a concentrated force and a uniform distributed force is investigated. When the system is considered to be conservative, the divergence phenomenon is confirmed to appear first. On the other hand, when the system is considered to be nonconservative, the flutter phenomenon is confirmed to appear first although the critical force becomes high. And, by changing the structural damping, the destabilized effect due to the damping is confirmed when an elastic beam is subjected to follower forces.

Morphological stability of two-dimensional nucleus

1969 ◽  
Vol 310 (1501) ◽  
pp. 277-296 ◽  
Keyword(s):  
Conservative System ◽  
Infinite Matrix ◽  
Two Dimensional ◽  
Morphological Stability ◽  
Nucleus Size ◽  
Shape Preserving ◽  
Adsorbed Atoms ◽  
The Mean ◽  
The Stability ◽  
Limiting Value

Morphological stability of a disk-shaped nucleus growing on a substrate by surface diffusion of adsorbed atoms is analysed for two physically distinct cases: ( a ) a conservative system, in which a nucleus grows without addition or removal of mass from the system; ( b ) a non-conservative system: nucleus growing on a substrate in presence of a vapour source. The system is idealized by considering such a nucleus surrounded by a supersaturated infinite matrix of diffusing adsorbed atoms. Following the method of Mullins & Sekerka (1963), to examine the stability of the shape of the nucleus a periodic small fluctuation represented by circular harmonics is introduced into the disk shape. The flux of adsorbed atoms at any point of the nucleus interface is determined by solving the Laplace equation (conservative system) and the Helmholtz equation (non-conservative system) together with appropriate boundary conditions. For the conservative system, it is found that a perturbation will grow monotonically as soon as the nucleus size is bigger than a certain limiting value. For the non-conservative system, the result is totally different. As long as the ratio R*/x s (where R * is the critical nucleus size for nucleation and x s the mean diffusion distance of an adsorbed atom on the substrate) is smaller than a limiting value ( R*/x s ) c , a selective amplification of a perturbation is possible only when the nucleus size R lies between two values R c1 and R c2 . At R c1 the perturbation begins to grow and at R c2 it begins to attenuate. The two systems are compared and discussed; it is seen that a two-dimensional nucleus growing in a non-conservative system is inherently more stable and shape-preserving than it is in a conservative system.

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