On the Stability of Equilibrium Sets of Discrete Conservative Mechanical Systems

AIAA Journal ◽  
1974 ◽  
Vol 12 (8) ◽  
pp. 1057-1059 ◽  
Author(s):  
PETER HAGEDORN
Keyword(s):  
Mechanical Systems ◽  
Stability Of Equilibrium ◽  
The Stability

scholarly journals Numerical Exploration of Kaldorian Interregional Macrodynamics: Enhanced Stability and Predominance of Period Doubling under Flexible Exchange Rates

2010 ◽  
Vol 2010 ◽  
pp. 1-29 ◽  
Author(s):  
Toichiro Asada ◽  
Christos Douskos ◽  
Vassilis Kalantonis ◽  
Panagiotis Markellos
Keyword(s):  
Exchange Rates ◽  
Government Expenditure ◽  
Period Doubling ◽  
Numerical Approach ◽  
Capital Movement ◽  
Dimensional Parameter ◽  
Stability Of Equilibrium ◽  
The Stability ◽  
Flexible Exchange Rates ◽  
Enhanced Stability

We present a discrete two-regional Kaldorian macrodynamic model with flexible exchange rates and explore numerically the stability of equilibrium and the possibility of generation of business cycles. We use a grid search method in two-dimensional parameter subspaces, and coefficient criteria for the flip and Hopf bifurcation curves, to determine the stability region and its boundary curves in several parameter ranges. The model is characterized by enhanced stability of equilibrium, while its predominant asymptotic behavior when equilibrium is unstable is period doubling. Cycles are scarce and short-lived in parameter space, occurring at large values of the degree of capital movementβ. By contrast to the corresponding fixed exchange rates system, for cycles to occur sufficient amount of trade is requiredtogetherwith high levels of capital movement. Rapid changes in exchange rate expectations and decreased government expenditure are factors contributing to the creation of interregional cycles. Examples of bifurcation and Lyapunov exponent diagrams illustrating period doubling or cycles, and their development into chaotic attractors, are given. The paper illustrates the feasibility and effectiveness of the numerical approach for dynamical systems of moderately high dimensionality and several parameters.

scholarly journals Non Smooth Contact Dynamics Approach for Mechanical Systems Subjected to Friction-Induced Vibration

Lubricants ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 59 ◽  
Author(s):  
Jean-Jacques Sinou ◽  
Olivier Chiello ◽  
Lucien Charroyer
Keyword(s):  
Mechanical Systems ◽  
Mathematical Community ◽  
Contact Dynamics ◽  
Strong Impact ◽  
Brake Squeal ◽  
Excited Vibration ◽  
Smooth Contact ◽  
Frictional Interface ◽  
The Stability ◽  
Friction Induced Vibration

The modeling of contact is one of the main features of contact dynamics in the context of friction-induced vibrations. It can have a strong impact on the numerical results and consequently on the design choices during the optimization or specification of industrial mechanical systems. This is particularly the case for scientific studies interested in brake squeal. The objective of the paper is to recall and to promote developments concerning the use of non smooth contact dynamics approach for numerical simulations based on finite element method. The specific problem of the prediction of self-excited vibration in the context of brake squeal is discussed. In order to illustrate the potential benefit for the mechanical community of using formulations and theoretical developments from the mathematical community, the stability analysis and the estimation of nonlinear vibrations of a brake system with multiple frictional interface is investigated.

Thermodynamic conditions for stability in materials with rate-independent dissipation

2005 ◽  
Vol 363 (1836) ◽  
pp. 2479-2515 ◽  
Author(s):  
Henryk Petryk
Keyword(s):  
Standard Condition ◽  
Martensitic Transformations ◽  
Internal Variable ◽  
Irreversible Processes ◽  
Deformation Bands ◽  
Stability Of Equilibrium ◽  
Variable Approach ◽  
Thermodynamic Conditions ◽  
The Stability ◽  
Time Specific

A distinctive feature of the examined class of solids is that a part of the entropy production is due to rate-independent dissipation, as in models of plasticity, damage or martensitic transformations. The standard condition for thermodynamic stability is shown to be too restrictive for such solids and, therefore, an extended condition for stability of equilibrium is developed. The classical thermodynamic theory of irreversible processes is used along with the internal variable approach, with the emphasis on the macroscopic effects of micro-scale instabilities in the presence of two different scales of time. Specific conditions for material stability against internal structural rearrangements under deformation-sensitive loading are derived within the incremental constitutive framework of multi-mode inelasticity. Application to spontaneous formation of deformation bands in a continuum is presented. Conditions for stability or instability of a quasi-static process induced by varying loading are given under additional constitutive postulates of normality and symmetry. As illustration of the theory, the stability of equilibrium or a deformation path under uniaxial tension is analysed for a class of inelastic constitutive laws for a metal crystal deformed plastically by multi-slip or undergoing stress-induced martensitic transformation.

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