Integrity measures quantifying the erosion of smooth and fractal basins of attraction

Soliman, Mohamed S.; Thompson, J. M. T.

doi:10.1016/0022-460x(89)90699-8

Cited by 154 publications

(91 citation statements)

References 8 publications

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“…This difference, which can be appreciated in Fig. 9, ensues from the different asymptotic behaviour of the saddles, which can be seen by comparing (10) with (15).…”

mentioning

confidence: 96%

“…It was Thompson that, around the 90's [14][15][16], discovered that (local) stability is not enough, and that the relevant results do not actually guarantee the load carrying capacity of systems. By considering a global approach, and by truly considering the dynamical behavior (even in the absence of an external excitation), Thompson introduced the notion of dynamical integrity, which in the authors' opinion is fundamental for properly pursuing the safety of structures in an evolutionary context, although to date it is not yet a commonly addressed concept.…”

mentioning

confidence: 99%

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Load carrying capacity of systems within a global safety perspective. Part I. Robustness of stable equilibria under imperfections

Lenci

Rega

2011

International Journal of Non-Linear Mechanics

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This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. 1 LOAD CARRYING CAPACITY OF SYSTEMS WITHIN Abstract:The problem of the practical stability of structures is addressed in a modern way by considering the effects of both static and dynamic perturbations. The major historical contributions, due to Euler, Koiter and Thompson, are reviewed and illustrated by an archetypal model permitting to highlight the main mechanical and dynamical points. It is found that a global approach is necessary for a reliable safety estimation, especially in the neighborhood of (static) critical loads. Considering that the admissible load threshold has to account for robustness to finite perturbations, the Koiter critical load must be lowered, obtaining the so called Thompson critical load. It is shown how these two thresholds share some properties (e.g. both depend in a sensitive way on imperfections, which must be known for practical calculations), while having a deep different meaning: the former is related to static imperfections, and requires only a local analysis, while the latter is related to dynamical imperfections, and requires a global analysis. It is shown that crit p Euler ≥ crit p Koiter ≥ crit p Thompson , i.e., that the advancement of knowledge leads to a lower estimation of the actual critical load.

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“…This difference, which can be appreciated in Fig. 9, ensues from the different asymptotic behaviour of the saddles, which can be seen by comparing (10) with (15).…”

mentioning

confidence: 96%

mentioning

confidence: 99%

Load carrying capacity of systems within a global safety perspective. Part I. Robustness of stable equilibria under imperfections

Lenci

Rega

2011

International Journal of Non-Linear Mechanics

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“…Special attention is dedicated to the presence of disturbances. As previously observed in Thompson and coworkers [19,20], disturbances are inevitably encountered under realistic conditions. They produce small, but finite perturbations, which may significantly affect and alter the system response.…”

Section: Introductionmentioning

confidence: 77%

Jump and pull-in dynamics of an electrically actuated bistable MEMS device

Ruzziconi

Lenci

Younis

2014

MATEC Web of Conferences

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Abstract. This study analyzes a theoretical bistable MEMS device, which exhibits a considerable versatility of behavior. After exploring the coexistence of attractors, we focus on each rest position, and investigate the final outcome, when the electrodynamic voltage is suddenly applied. Our aim is to describe the parameter range where each attractor may practically be observed under realistic conditions, when an electric load is suddenly applied. Since disturbances are inevitably encountered in experiments and practice, a dynamical integrity analysis is performed in order to take them into account. We build the integrity charts, which examine the practical vulnerability of each attractor. A small integrity enhances the sensitivity of the system to disturbances, leading in practice either to jump or to dynamic pull-in. Accordingly, the parameter range where the device, subjected to a suddenly applied load, can operate in safe conditions with a certain attractor is smaller, and sometimes considerably smaller, than in the theoretical predictions. While we refer to a particular case-study, the approach is very general.

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“…The perfect system, Figure 4a, displays the four symmetric saddles corresponding to the four unstable postbuckling descending branches shown in Figure 2a and a minimum corresponding to the stable prebuckling solution. The saddles and their invariant manifolds separate the initial conditions that lead to bounded solutions surrounding the prebuckling configuration -which identify the so-called safe region [Soliman and Thompson 1989;Rega and Lenci 2005] -from the unbounded escape solutions. For φ = 1 • and ψ = 0 • , Figure 4b shows that the safe region is bounded by the two saddles corresponding to the blue equilibrium path in Figure 2b.…”

Section: Formulation Of the Problemmentioning

confidence: 99%

Nonlinear dynamics and sensitivity to imperfections in Augusti’s model

Orlando¹,

Gonçalves²,

Rega³

et al. 2011

J. Mech. Mater. Struct.

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The influence of geometric imperfections on the nonlinear behavior and stability of Augusti's model under static and dynamic loads is analyzed. This 2-DOF lumped-parameter system is an archetypal model of modal interaction in stability theory representing a large class of structural problems. When the system displays coincident buckling loads, several postbuckling paths emerge from the bifurcation point (critical load) along the fundamental path, in particular coupled unstable postbuckling paths that control the nonlinear dynamics of the system for load levels lower than the critical load. Systems displaying unstable postbuckling behavior are particularly sensitive to initial imperfections. They decrease the static buckling load and distort the topology of the safe potential well. Herein, coupled/uncoupled dynamic responses, bifurcations, escape from the prebuckling potential well, stability, space-time-varying displacements, and attractor-manifold-basin phase portraits are numerically evaluated with the aim of enlightening the effect of system imperfection sensitivity. In particular, the investigation of the reduction of escape load for several varying system parameters highlights the remarkable loss of safety and dynamic integrity of the structure due to penetration of eroding fractal tongues into the safe basin.

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Integrity measures quantifying the erosion of smooth and fractal basins of attraction

Cited by 154 publications

References 8 publications

Load carrying capacity of systems within a global safety perspective. Part I. Robustness of stable equilibria under imperfections

Load carrying capacity of systems within a global safety perspective. Part I. Robustness of stable equilibria under imperfections

Jump and pull-in dynamics of an electrically actuated bistable MEMS device

Nonlinear dynamics and sensitivity to imperfections in Augusti’s model

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