The Emergence of Multiple Equilibria in Electromechanical Systems

1994 ◽  
Vol 116 (4) ◽  
pp. 735-744 ◽  
Author(s):  
Neyram Hemati
Keyword(s):  
Multiple Equilibria ◽  
Chaotic Behavior ◽  
Multiple Equilibrium ◽  
Equilibrium Solutions ◽  
Multiple Steady State ◽  
Brushless Dc ◽  
Variable Reluctance ◽  
Stable Operating ◽  
The Stability ◽  
Steady State Solutions

The dynamic characteristics of brushless dc machines are considered. It is demonstrated that due to the inherent nonlinear dynamics of these systems, multiple steady-state solutions can exist. The presence of multiple equilibrium solutions is in turn used to provide an explanation for the loss of stability associated with a locally stable operating state. Numerical simulations are used to help verify the presence of multiple equilibria and their effect on the stability of the systems under investigation. Finally, an example is presented which demonstrates that the presence of multiple equilibria can lead to chaotic behavior in bldcm systems with variable reluctance.

Clathrate hydrate equilibrium modeling: Do self-consistent cell models provide unique equilibrium solutions?

2015 ◽  
Vol 93 (8) ◽  
pp. 826-830
Author(s):  
Patrick G. Lafond ◽  
R. Gary Grim ◽  
Amadeu K. Sum
Keyword(s):  
Solid Phase ◽  
Mean Field ◽  
Melting Behavior ◽  
Multiple Equilibrium ◽  
Stability Field ◽  
Equilibrium Solutions ◽  
Scanning Calorimetry ◽  
Cell Potential ◽  
Hydrate Phase ◽  
The Stability

When clathrate hydrates of xenon gas are formed deep within the stability field, anomalous melting behavior is readily observed in differential scanning calorimetry (DSC). In the DSC thermograms, multiple dissociation events may be observed, suggesting the presence of more than one solid phase. Following a suite of diffraction and NMR measurements, we are only able to detect the presence of simple structure I hydrate. Recognizing that hydrates are nonstoichiometric compounds, we look back to how the molar composition of a hydrate phase is determined. Making a mean-field improvement to current equilibrium models, we find that some conditions yield multiple solutions to the cage filling of the hydrate phase. Though the solutions are not truly stable, they would result in a kinetically trapped system. If such a case existed experimentally, this could explain the dissociation behavior observed for xenon hydrates. More importantly, this raises the question of how well defined the equilibrium condition is for a cell potential model, and whether or not multiple equilibrium solutions could exist.

Stability of Thermoelastic Contact for the Aldo Model

1981 ◽  
Vol 48 (3) ◽  
pp. 555-558 ◽  
Author(s):  
J. R. Barber
Keyword(s):  
Thermal Contact Resistance ◽  
Thermal Contact ◽  
Multiple Steady State ◽  
Imperfect Contact ◽  
One Dimensional ◽  
Thermoelastic Contact ◽  
The Stability ◽  
Contact Pressures ◽  
Steady State Solutions ◽  
Small Separation

A perturbation method is used to investigate the stability of a simple one-dimensional rod model of thermoelastic contact which exhibits multiple steady-state solutions. A thermal contact resistance is postulated which is a continuous function of the contact pressure or separation. It is found that solutions involving substantial separation and/or contact pressures are always stable, but these are separated by unstable “imperfect contact” solutions in which one of the rods is very lightly loaded or has a very small separation. The results can be expressed in terms of the minimization of a certain energy function.

On the Dynamics and Multiple Equilibria of an Inverted Flexible Pendulum With Tip Mass on a Cart

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Ojas Patil ◽  
Prasanna Gandhi
Keyword(s):  
Dynamic Model ◽  
Multiple Equilibria ◽  
Chaotic Behavior ◽  
Elastic Beam ◽  
Superior Performance ◽  
Flexible Link ◽  
Equilibrium Solutions ◽  
Lagrange Formulation ◽  
Compact Design ◽  
Tip Mass

Flexible link systems are increasingly becoming popular for advantages like superior performance in micro/nanopositioning, less weight, compact design, lower power requirements, and so on. The dynamics of distributed and lumped parameter flexible link systems, especially those in vertical planes are difficult to capture with ordinary differential equations (ODEs) and pose a challenge to control. A representative case, an inverted flexible pendulum with tip mass on a cart system, is considered in this paper. A dynamic model for this system from a control perspective is developed using an Euler Lagrange formulation. The major difference between the proposed method and several previous attempts is the use of length constraint, large deformations, and tip mass considered together. The proposed dynamic equations are demonstrated to display an odd number of multiple equilibria based on nondimensional quantity dependent on tip mass. Furthermore, the equilibrium solutions thus obtained are shown to compare fairly with static solutions obtained using elastica theory. The system is demonstrated to exhibit chaotic behavior similar to that previously observed for vibrating elastic beam without tip mass. Finally, the dynamic model is validated with experimental data for a couple of cases of beam excitation.

scholarly journals Multiple equilibria in a simple elastocapillary system

2012 ◽  
Vol 712 ◽  
pp. 273-294 ◽  
Author(s):  
Michele Taroni ◽  
Dominic Vella
Keyword(s):  
Multiple Equilibria ◽  
Microelectromechanical Systems ◽  
Numerical Solutions ◽  
Multiple Equilibrium ◽  
Type Model ◽  
Initial State ◽  
Micro Channels ◽  
Simple Model System ◽  
Stable Equilibria ◽  
The Stability

AbstractWe consider the elastocapillary interaction of a liquid drop placed between two elastic beams, which are both clamped at one end to a rigid substrate. This is a simple model system relevant to the problem of surface-tension-induced collapse of flexible micro-channels that has been observed in the manufacture of microelectromechanical systems (MEMS). We determine the conditions under which the beams remain separated, touch at a point, or stick along a portion of their length. Surprisingly, we show that in many circumstances multiple equilibrium states are possible. We develop a lubrication-type model for the flow of liquid out of equilibrium and thereby investigate the stability of the multiple equilibria. We demonstrate that for given material properties two stable equilibria may exist, and show via numerical solutions of the dynamic model that it is the initial state of the system that determines which stable equilibrium is ultimately reached.

SOLUTION UNIQUENESS IN A CLASS OF CURRENCY CRISIS GAMES

2005 ◽  
Vol 07 (04) ◽  
pp. 531-543 ◽  
Author(s):  
CHRISTIAN BAUER
Keyword(s):  
Incomplete Information ◽  
Expected Utility ◽  
Multiple Equilibria ◽  
Multiple Equilibrium ◽  
Equilibrium Strategy ◽  
Knightian Uncertainty ◽  
Equilibrium Solutions ◽  
Speculative Attack ◽  
Arbitrary Noise ◽  
Attack Models

A common feature of many speculative attack models on currencies is the existence of multiple equilibrium solutions. When choosing the equilibrium strategy, a trader faces Knightian uncertainty about the rational choice of the other traders. We show that the concept of Choquet expected utility maximization under Knightian uncertainty leads to unique equilibria. In games of incomplete information the optimal strategy maximizes the expected utility with respect to a two-dimensional information: environment and rationality. We define a new concept of equilibria, the Choquet-expected-Nash-equilibria, which allows the analysis of decisions under uncertainty, which result in multiple equilibria in standard analysis. We provide uniqueness theorems for a wide class of incomplete information games including global games and apply them to fairly general currency attack models. The uniqueness of the equilibrium remains valid for arbitrary noise distributions, positively correlated signals, the existence of large traders, individual payoff functions, and for the case that non attacking traders suffer a loss in case of a successful attack, as is the case for investors in the attacked country.

The Tangled Tale of Phase Space

2018 ◽  
Author(s):  
David D. Nolte
Keyword(s):  
Phase Space ◽  
Chaotic Behavior ◽  
Three Body Problem ◽  
Henri Poincaré ◽  
History Of ◽  
The Stability ◽  
Three Body ◽  
Space Phase ◽  
Central Paradigm ◽  
System Phase

This chapter presents the history of the development of the concept of phase space. Phase space is the central visualization tool used today to study complex systems. The chapter describes the origins of phase space with the work of Joseph Liouville and Carl Jacobi that was later refined by Ludwig Boltzmann and Rudolf Clausius in their attempts to define and explain the subtle concept of entropy. The turning point in the history of phase space was when Henri Poincaré used phase space to solve the three-body problem, uncovering chaotic behavior in his quest to answer questions on the stability of the solar system. Phase space was established as the central paradigm of statistical mechanics by JW Gibbs and Paul Ehrenfest.

Stability of slender inverted flags and rods in uniform steady flow

2016 ◽  
Vol 809 ◽  
pp. 873-894 ◽  
Author(s):  
John E. Sader ◽  
Cecilia Huertas-Cerdeira ◽  
Morteza Gharib
Keyword(s):  
Multiple Equilibria ◽  
Complex Dynamics ◽  
Flow Direction ◽  
Flow Speed ◽  
Uniform Flow ◽  
Biological Phenomena ◽  
Cantilever Length ◽  
The Stability ◽  
Elastic Sheets ◽  
Clamped Edge

Cantilevered elastic sheets and rods immersed in a steady uniform flow are known to undergo instabilities that give rise to complex dynamics, including limit cycle behaviour and chaotic motion. Recent work has examined their stability in an inverted configuration where the flow impinges on the free end of the cantilever with its clamped edge downstream: this is commonly referred to as an ‘inverted flag’. Theory has thus far accurately captured the stability of wide inverted flags only, i.e. where the dimension of the clamped edge exceeds the cantilever length; the latter is aligned in the flow direction. Here, we theoretically examine the stability of slender inverted flags and rods under steady uniform flow. In contrast to wide inverted flags, we show that slender inverted flags are never globally unstable. Instead, they exhibit bifurcation from a state that is globally stable to multiple equilibria of varying stability, as flow speed increases. This theory is compared with new and existing measurements on slender inverted flags and rods, where excellent agreement is observed. The findings of this study have significant implications to investigations of biological phenomena such as the motion of leaves and hairs, which can naturally exhibit a slender geometry with an inverted configuration.

scholarly journals Sequential eigenfunction expansion for a problem in combustion theory

2003 ◽  
Vol 45 (1) ◽  
pp. 35-48 ◽  
Author(s):  
M. Al-Refai ◽  
K. K. Tam
Keyword(s):  
Differential Equation ◽  
Eigenfunction Expansion ◽  
Time Dependent ◽  
Linear Parabolic Equation ◽  
System Of Equations ◽  
Multiple Steady State ◽  
Definitive Answer ◽  
Combustion Theory ◽  
The Galerkin Method ◽  
Steady State Solutions

AbstractA method of sequential eigenfunction expansion is developed for a semi-linear parabolic equation. It allows the time-dependent coefficients of the eigenfunctions to be determined sequentially and iterated to reach convergence. At any stage, only a single ordinary differential equation needs to be considered, in contrast to the Galerkin method which requires the consideration of a system of equations. The method is applied to a central problemin combustion theory to provide a definitive answer to the question of the influence of the initial data in determining whether the solution is sub- or super-critical, in the case of multiple steady-state solutions. It is expected this method will prove useful in dealing with similar problems.

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