Stability Considerations in Thermoelastic Contact

1980 ◽  
Vol 47 (4) ◽  
pp. 871-874 ◽  
Author(s):  
J. R. Barber ◽  
J. Dundurs ◽  
M. Comninou
Keyword(s):  
Steady State ◽  
Perturbation Method ◽  
Energy Function ◽  
First Principles ◽  
Dimensional Model ◽  
Contact Conditions ◽  
One Dimensional ◽  
Thermoelastic Contact ◽  
The Stability ◽  
Steady State Solutions

A simple one-dimensional model is described in which thermoelastic contact conditions give rise to nonuniqueness of solution. The stability of the various steady-state solutions discovered is investigated using a perturbation method. The results can be expressed in terms of the minimization of a certain energy function, but the authors have so far been unable to justify the use of such a function from first principles in view of the nonconservative nature of the system.

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.

scholarly journals Finite Element Analysis of Thermoelastic Contact Stability

1994 ◽  
Vol 61 (4) ◽  
pp. 919-922 ◽  
Author(s):  
Taein Yeo ◽  
J. R. Barber
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Steady State ◽  
Eigenvalue Problem ◽  
Linear Perturbation ◽  
Dimensional System ◽  
The Finite Element Method ◽  
One Dimensional ◽  
Thermoelastic Contact ◽  
Element Method

When heat is conducted across an interface between two dissimilar materials, theimoelastic distortion affects the contact pressure distribution. The existence of a pressure-sensitive thermal contact resistance at the interface can cause such systems to be unstable in the steady-state. Stability analysis for thermoelastic contact has been conducted by linear perturbation methods for one-dimensional and simple two-dimensional geometries, but analytical solutions become very complicated for finite geometries. A method is therefore proposed in which the finite element method is used to reduce the stability problem to an eigenvalue problem. The linearity of the underlying perturbation problem enables us to conclude that solutions can be obtained in separated-variable form with exponential variation in time. This factor can therefore be removed from the governing equations and the finite element method is used to obtain a time-independent set of homogeneous equations in which the exponential growth rate appears as a linear parameter. We therefore obtain a linear eigenvalue problem and stability of the system requires that all the resulting eigenvalues should have negative real part. The method is discussed in application to the simple one-dimensional system of two contacting rods. The results show good agreement with previous analytical investigations and give additional information about the migration of eigenvalues in the complex plane as the steady-state heat flux is varied.

One-dimensional model for the Rijke tube

1989 ◽  
Vol 202 ◽  
pp. 83-96 ◽  
Author(s):  
C. Nicoli ◽  
P. Pelcé
Keyword(s):  
Heat Transfer ◽  
Transfer Function ◽  
Dimensional Model ◽  
Unsteady Heat Transfer ◽  
Order Unity ◽  
Rijke Tube ◽  
Thermoacoustic Instability ◽  
One Dimensional ◽  
The Stability ◽  
Stability Limits

We develop a simple model in which longitudinal, compressible, unsteady heat transfer between heater and gas is computed in the small-Mach-number limit. This calculation is used to determine the transfer function of the heater, which plays an important role in the stability limits of the thermoacoustic instability of the Rijke tube. The transfer function is determined analytically in the limit of small expansion parameter γ, and numerically for γ of order unity. In the case ρμ/cp = constant, an analytical solution can be found.

scholarly journals A one-dimensional tearing mode equation for pedestal stability studies in tokamaks

2018 ◽  
Vol 84 (3) ◽  
Author(s):  
J. W. Connor ◽  
R. J. Hastie ◽  
C. Marchetto ◽  
C. M. Roach
Keyword(s):  
Cross Section ◽  
Approximate Equation ◽  
Stability Studies ◽  
Dimensional Model ◽  
Tearing Mode ◽  
One Dimensional ◽  
Tearing Modes ◽  
The Stability ◽  
Poloidal Flux

Starting from expressions in Connor et al. (Phys. Fluids, vol. 31, 1988, p. 577), we derive a one-dimensional tearing equation similar to the approximate equation obtained by Hegna & Callen (Phys. Plasmas, vol. 1, 1994, p. 2308) and Nishimura et al. (Phys. Plasmas, vol. 5, 1998, p. 4292), but for more realistic toroidal equilibria. The intention is to use this approximation to explore the role of steep profiles, bootstrap currents and strong shaping in the vicinity of a separatrix, on the stability of tearing modes which are resonant in the H-mode pedestal region of finite aspect ratio, shaped cross-section tokamaks, e.g. the Joint European Torus (JET). We discuss how this one-dimensional model for tearing modes, which assumes a single poloidal harmonic for the perturbed poloidal flux, compares with a model that includes poloidal coupling Fitzpatrick et al. (Nucl. Fusion, vol. 33, 1993, p. 1533).

Steady-state solutions of one-dimensional competition models in an unstirred chemostat via the fixed point index theory

2020 ◽  
pp. 1-25
Author(s):  
Kunquan Lan ◽  
Wei Lin
Keyword(s):  
Steady State ◽  
Integral Equations ◽  
Fixed Point Index ◽  
Index Theory ◽  
One Dimensional ◽  
Hammerstein Integral Equations ◽  
Competition Models ◽  
Fixed Point Index Theory ◽  
Two Parameters ◽  
Steady State Solutions

Abstract The existence and nonexistence of semi-trivial or coexistence steady-state solutions of one-dimensional competition models in an unstirred chemostat are studied by establishing new results on systems of Hammerstein integral equations via the classical fixed point index theory. We provide three ranges for the two parameters involved in the competition models under which the models have no semi-trivial and coexistence steady-state solutions or have semi-trivial steady-state solutions but no coexistence steady-state solutions or have semi-trivial or coexistence steady-state solutions. It remains open to find the largest range for the two parameters under which the models have only coexistence steady-state solutions. We apply the new results on systems of Hammerstein integral equations to obtain results on steady-state solutions of systems of reaction-diffusion equations with general separated boundary conditions. Such type of results have not been studied in the literature. However, these results are very useful for studying the competition models in an unstirred chemostat. Our results on Hammerstein integral equations and differential equations generalize and improve some previous results.

Experimental studies of the propagation of combustion in solids

1992 ◽  
Vol 339 (1654) ◽  
pp. 387-394 ◽  
Keyword(s):  
Heat Transfer ◽  
Thermal Analysis ◽  
Steady State ◽  
Chemical Kinetics ◽  
Profile Analysis ◽  
Numerical Models ◽  
Experimental Studies ◽  
Dimensional Model ◽  
Thermal Methods ◽  
One Dimensional

The application of thermal methods to the study of steady-state combustion is described. Such methods provide a route to information on heat transfer and chemical kinetics which forms a basis for the implementation of numerical models. The experimental results from thermal analysis and temperature profile analysis have been examined within the context of a simple pseudo one-dimensional model of propagation offering some confirmation of the validity of the approach.

SPATIAL DISORDER OF CNN — WITH ASYMMETRIC OUTPUT FUNCTION

2001 ◽  
Vol 11 (08) ◽  
pp. 2085-2095 ◽  
Author(s):  
JUNG-CHAO BAN ◽  
KAI-PING CHIEN ◽  
SONG-SUN LIN ◽  
CHENG-HSIUNG HSU
Keyword(s):  
Neural Networks ◽  
Steady State ◽  
Three Dimensional ◽  
Cellular Neural Networks ◽  
Output Function ◽  
One Dimensional ◽  
Spatial Entropy ◽  
The One ◽  
Steady State Solutions

This investigation will describe the spatial disorder of one-dimensional Cellular Neural Networks (CNN). The steady state solutions of the one-dimensional CNN can be replaced as an iteration map which is one dimensional under certain parameters. Then, the maps are chaotic and the spatial entropy of the steady state solutions is a three-dimensional devil-staircase like function.

Steady-State Bifurcation for a Biological Depletion Model

2016 ◽  
Vol 26 (04) ◽  
pp. 1650066 ◽  
Author(s):  
Yan’e Wang ◽  
Jianhua Wu ◽  
Yunfeng Jia
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Steady States ◽  
Two Dimensional ◽  
Implicit Function ◽  
Flux Boundary Condition ◽  
One Dimensional ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Theoretical Results

A two-species biological depletion model in a bounded domain is investigated in which one species is a substrate and the other is an activator. Firstly, under the no-flux boundary condition, the asymptotic stability of constant steady-states is discussed. Secondly, by viewing the feed rate of the substrate as a parameter, the steady-state bifurcations from constant steady-states are analyzed both in one-dimensional kernel case and in two-dimensional kernel case. Finally, numerical simulations are presented to illustrate our theoretical results. The main tools adopted here include the stability theory, the bifurcation theory, the techniques of space decomposition and the implicit function theorem.

Dynamics and pattern formation in a modified Leslie–Gower model with Allee effect and Bazykin functional response

2017 ◽  
Vol 10 (05) ◽  
pp. 1750073 ◽  
Author(s):  
Peng Feng
Keyword(s):  
Steady State ◽  
Pattern Formation ◽  
Spatial Pattern ◽  
Numerical Simulations ◽  
Functional Response ◽  
Allee Effect ◽  
Turing Instability ◽  
Spatial Pattern Formation ◽  
The Stability ◽  
Steady State Solutions

In this paper, we study the dynamics of a diffusive modified Leslie–Gower model with the multiplicative Allee effect and Bazykin functional response. We give detailed study on the stability of equilibria. Non-existence of non-constant positive steady state solutions are shown to identify the rage of parameters of spatial pattern formation. We also give the conditions of Turing instability and perform a series of numerical simulations and find that the model exhibits complex patterns.

Sign in / Sign up

Export Citation Format

Share Document