General Stability Theory Research Articles

Stable generalised predictive control: an algorithm with guaranteed stability

Generalised predictive control (GPC) has many attributes which make it an effective practical tool for the control of scalar systems. The method is easy to understand/implement and has won popularity among many practicising engineers. Despite its success it does not have a general stability and robust stability theory. The paper proposes a new algorithm which deploys most of the ideas of GPC but yields control configurations with guaranteed stability and provides a systematic means of optimising robust stability margins. It achieves this through the use of a particular stabilising feedback controller which replaces the relevant transfer function operators by simple polynomial operators. This simplification also reduces the computational burden of a computer implementation of the method. The superiority of the results of the paper are illustrated by means of numerical examples.

Post-buckling of elastic cone-cylinder and sphere-cylinder complex shells

A study is presented of the elastic buckling of pressurized complex shells of revolution which are formed by the intersection of cylindrical with spherical or conical shells. The studies are based on the general theory of elastic stability using an asymptotic approximation to the secondary path. An axisymmetric finite element is used for the discretization of the shell. First, the behaviour of the individual components is presented for simply supported and clamped boundary conditions. Next, the complex shells are investigated in terms of the buckling load, the meridional and circumferential shape of the buckling mode, and the initial characteristics of the post-buckling path. The results show that the bifurcation loads of complex shells are lower than those of the individual components; but the post-critical path does not decrease as severely as in the isolated shells.

An asymptotic modal approach to nonlinear structural elastic instability

An asymptotic modal formulation for nonlinear structural instability analysis is presented. The approach to this problem is based on the classical concepts and criteria of the general theory of elastic stability which are applied together with the standard procedures of the finite element method. An automatic and consistent method, extracted from the general perturbation scheme, is proposed for the choice of a reduced number of fundamental, critical and post-critical modes to be retained in the analysis. The potentiality is emphasized of the proposed approach in detecting not only limit loads but also bifurcations, in cases where a conventional nonlinear finite element model, combined with usual incremental/iterative procedures, fails to give any predictions. These main features are demonstrated through numerical application to elastic stability problems of perfect or initially imperfect space frame structures, which display stable or unstable bifurcations along a moderately nonlinear fundamental equilibrium path.

Multiple interaction and localization phenomena in the postbuckling of compressed thin-walled members

In this paper, the effect of boundary conditions on multiple interactive buckling in thin-walled members under compression is analyzed by means of the general theory of stability. The total potential energy is expanded to up to third-order terms to investigate structures with asymmetric postbuckling behavior; initial geometric imperfections are taken into account. Under simplifying assumptions, the nonlinear equations governing the postcritical behavior reduce to a linear eigenvalue problem that is independent of the geometric and mechanical properties of the column and depends on the boundary conditions only. Localization of the buckling patterns is verified when a large number of local buckling modes occur simultaneously. A continuum representation, based on the substitution of the algebraic eigenvalue problem with a differential equation, is proposed.

Postbuckling behavior of infinite beams on elastic foundations using Koiter's improved theory

This study deals with the postbuckling behavior of infinite beams on non-linear elastic foundations subjected to axial compression. The analysis utilizes an improved Koiter's postbuckling theory such that the postbuckling coefficients are evaluated at the actual applied load rather than at the classical buckling load. The improved postbuckling paths are found to agree well with the non-linear large deflection solution using the Ritz procedure. This paper substantiates Koiter's conjecture that the general theory of elastic stability may be improved. The implications of various lower-bound buckling loads are examined.

Asymmetric interactive buckling of thin-walled columns with initial imperfections

In this paper the effect of the interaction between two or more simultaneous buckling modes on the postbuckling behaviour of uniformly compressed thin-walled members (TWM) is analysed by means of the general theory of elastic stability. The analysis is restricted to third-order terms of the energy expansion and therefore can be fruitfully applied to the investigation of structures with asymmetric postbuckling behaviour only. Initial imperfection effect is taken into account. A simplified procedure is suggested for solving the nonlinear equations relative to the evaluation of the bifurcated paths. By using the finite strip method an extensive parametric analysis is performed. It is found that when the flexural-torsional (FT) buckling interacts with a local symmetric and antisymmetric mode, sensitivity to initial inperfections is remarkable and is comparable to the one arising from the interaction between the Euler (E) and any local buckling.

On the effect of the local overall interaction on the postbuckling of uniformly compressed channels

On the basis of the general theory of elastic stability due to Koiter, postbuckling analysis of simply supported channels under uniform compression is performed. Attention is essentially focused on local/eularian and local/flexural-torsional simultaneous bucklin modes interaction. The column is treated as a plate assemblage. Linearized expressions for the displacement field are employed while assuming strain-displacements relationships that are linear for the curvatures and up to second order terms for the in-plane strains. The total potential energy is hence written up to third order terms in order to investigate asymmetric buckling phenomena. A discrete model is developed through an automatic procedure of algebraic manipulation, and an extensive parametric analysis is performed. After determining the range of geometric parameters which characterize different types of interaction, it is found that in the postbuckling range the local/eulerian interaction is more dangerous than the local/flexural-torsional one, due to the column higher imperfection sensitivity.

The political economy of international shipping: Europe versus America

The political challenge to the post-World War II order in shipping has been issued in the context of the North-South debate, but American power and interest are central to current developments. In the bulk and tanker sector the United States retains a strong interest in stability and successfully defends the existing order. In the liner sector, on the other hand, the United States has participated in recent assaults on the postwar order, producing great tension between Europe and America. There is a strong correlation between this growing maritime conflict and the political processes anticipated by the general theory of hegemonic stability. But “hegemony” and “power” are distinct concepts. Instability in international shipping arises neither from America's loss of power in shipping nor from challenges from Europe and the Third World. Rather, instability reflects American attempts to establish a closer identity between the existing regime and short-term national interest.

Stability Theory of Synchronized Motion in Coupled-Oscillator Systems

The general stability theory of the synchronized motions of the coupled· oscillator systems is developed with the use of the extended Lyapunov matrix approach. We give the explicit formula for a stability parameter of the synchronized state W unlfWhen the coupling strength is weakened, the coupled system may exhibit several types of non· synchronized motion. In particular, if W Unlf is chaotic, we always get a transition from chaotic Wunlf to a certain non· uniform state and finally the non·uniform chaos. Details associated with such transition are investigated for the coupled Lorenz model. As an application of the theory, we propose a new experimental method to directly measure the positive Lyapunov exponent of intrinsic chaos in reaction systems.

Dynamik und Stabilität verfahrenstechnischer Prozesse

AbstractDynamics and stability of chemical engineering processes. Peculiarities of dynamic behaviour as well as stability phenomena in the field of chemical engineering are reviewed with emphasis on the stability of single evaporator tubes and on the stability and dynamics of chemical reactors with strongly exothermic reactions. It is shown that the basic behaviour can be explained in both cases by fairly simple considerations. Some new aspects of general stability theory are also discussed briefly.

Sul criterio dell'energia per la stabilitá di un sistema termomeccanico

The energy criterion for mechanical stability asserts that the stable configurations are those that minimize the potential energy. Recent studies have shown that the energy criterion can be extended to stability of thermomechanical systems under suitable environment conditions, provided that the «stored energy» is interpreted as the equilibrium free-energy at the environmental temperature θe. The aim of this paper is to provide a contribution to a general theory of thermomechanical stability. Essentially we have restated the theory for general materials introduced by Gurtin with a new framework in the light of recent theories of Noll and Coleman-Owen on simple materials and on thermodynamical potentials. We define a «thermomechanical system» which posseses two main features: i) state space has a «natural topology» depending on the thermodynamical behaviour of system; ii) internal energy E and entropy S are not supposed to exist but are expressely obtained with their smoothness properties.

Stochastic metastability and Hamiltonian dynamics

Stochastic metastability is stability weakened by chance. In a definition of stability certain conditions must be satisfied for sufficiently small displacements. Corresponding to a definition of stability we introduce a definition of (stochastic) metastability, in which the same conditions must be satisfied with probability approaching unity as the maximum displacement approaches zero. Measurable sets are essential to metastability as open sets are to stability. Because of Arnol’d diffusion, the invariant tori of conservative Hamiltonian systems of n ≽ 3 degrees of freedom are not usually dynamically stable with respect to changes in initial conditions. However, we prove using the Lebesgue density theorem that if the proper invariant tori of phase space fill a bounded domain of positive measure, then they are almost all dynamically metastable. Metastability is difficult to distinguish from stability in physical systems or numerical experiments, so the result is consistent with the difficulty of observing Arnol’d diffusion in the neighbourhood of an invariant torus in practice. Probability and measure are essential to the general theory of stability of nonlinear systems.

Postbuckling Behavior of Thin-Walled Open Cross-Section Compression Members

ABSTRACT The postbuckling behavior of simply supported columns with thin-walled open cross section is investigated by means of the general nonlinear theory of elastic stability. Fourth-order terms in the series expansion of the total potential energy are disregarded. It is shown that interaction between linearly independent simultaneous buckling modes is responsible for neutral equilibrium at bifurcation if the column cross section has two axes of symmetry, and unstable if it has only one. If the buckling modes are not simultaneous, the equilibrium is neutral in both cases. Finally, the equilibrium at bifurcation is usually unstable if the cross section has no axis of symmetry.

Imperfection-sensitivity of semi-symmetric branching

The general theory of elastic stability is extended to include the imperfection-sensitivity of twofold compound branching points with symmetry of the potential function in one of the critical modes (semi-symmetric points of bifurcation). Three very different forms of imperfection-sensitivity can result, so a subclassification into monoclinal, anticlinal and homeoclinal semi-symmetric branching is introduced. Relating this bifurcation theory to René Thom’s catastrophe theory, it is found that the anticlinal point of bifurcation generates an elliptic umbilic catastrophe, while the monoclinal and homeoclinal points of bifurcation lead to differing forms of the hyperbolic umbilic catastrophe. Practical structural systems which can exhibit this form of branching include an optimum stiffened plate with free edges loaded longitudinally, and an analysis of this problem is presented leading to a complete description of the imperfection-sensitivity. The paper concludes with some general remarks concerning the nature of the optimization process in design as a generator of symmetries, instabilities and possible compound bifurcations.

Effects of diffusion on the stability of the equilibrium in multi-species ecological systems.

Continuous population distributions that undergo self-diffusion, migrational cross-diffusion and interaction in a region of (1-, 2- or 3-dimensional) space are described dynamically by a governing system of nonlinear reaction-diffusion equations. It is shown that the constants associated with migrational cross-diffusion are ordinarily nonnegative or nonpositive, contingent on the type of species interaction. A simple sign relationship obtains between the latter diffusivity constants and the rate constants for species interaction in the neighborhood of a spatially uniform equilibrium state, and this relationship of signs serves to simplify the general stability theory for the growth or decay of arbitrary perturbations on a spatially uniform equilibrium state. The stability of the equilibrium state is analyzed and discussed in detail for the case of a generic two-species model, where the self-diffusion and migrational cross-diffusion of species act to either stabilize or destabilize the equilibrium, depending essentially on the character of the species interaction and also on the magnitude of the Helmholtz eigenvalues associated with the region and boundary conditions. In particular, for a prey-predator or host-parasite model, self-diffusion usually helps to stabilize the equilibrium state and migrational cross-diffusion can only act as an additional stabilizing influence, as evidenced generally by the experiments on such two-species systems. Sufficient conditions are derived for stability of the equilibrium state in the general case for an arbitrarily large number of interacting species. It is shown that the equilibrium state is always stable if all species undergo significant self-diffusion and the Helmholtz eigenvalues are suitably large.

Classification of critical conditions in the general theory of stability

Classification of critical conditions in the general theory of stability

Local stability of the nerve impulse

Some general theory of nerve impulse stability is reviewed and is applied to the Hodgkin-Huxley equations. It is seen that the transition behavior of certain uniformly propagated waves with a velocity close to the velocity of an impulse has important implications with regard to stability. Numerical estimates of some of the principal objects in the stability theory for the Hodgkin-Huxley equations are given.

SOME ADMISSIBILITY AND DICHOTOMY QUESTIONS. THE AVERAGING PRINCIPLE

In this paper some questions in the general theory of stability are studied, with a subsequent application to the problem of averaging on the whole axis.Bibliography: 14 titles.

Lee-Wick states as an example in Thom's catstrophe theory

It is shown that the abnormal state of Lee and Wick results as a cusp catastrophe in Thom's general theory of stability.

A General Theory of Elastic Stability. J. M. T. Thompson and G. W. Hunt. John Wiley, London. 1973. 322 pp. Illustrated. £7.50.

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