Frictional heating and the stability of rate and state dependent frictional sliding

Using an event-based switching law, we address the stability issue for continuous-time switched affine systems in the network environment. The state-dependent switching law in terms of the region function is firstly developed. We combine the region function with the event-triggering mechanism to construct the switching law. This can provide more candidates for the selection of the next activated subsystem at each switching instant. As a result, it is possible for us to activate the appropriate subsystem to avoid the sliding motion. The Zeno behavior for the switched affine system can be naturally ruled out by guaranteeing a positive minimum inter-event time between two consecutive executions of the event-triggering threshold. Finally, two numerical examples are given to demonstrate the effectiveness of the proposed method.

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Glacial lake drainage: a stability analysis

Annals of Glaciology ◽

10.3189/s0260305500012131 ◽

1997 ◽

Vol 24 ◽

pp. 175-180

Author(s):

Krzysztof Szilder ◽

Edward P. Lozowski ◽

Martin J. Sharp

Keyword(s):

Differential Equations ◽

Water Flow ◽

Frictional Heating ◽

Initial Perturbation ◽

Tunnel Wall ◽

Cross Sectional ◽

Linear Ordinary Differential Equations ◽

Simplifying Assumptions ◽

Lake Drainage ◽

The Stability

A model has been formulated to determine the stability regimes for water flow in a Subglacial conduit draining from a reservoir. The physics of the water flow is described with a set of differential equations expressing conservation of mass, momentum and energy. Non-steady flow of water in the conduit is considered, the conduit being simultaneously enlarged by frictional heating and compressed by plastic deformation in response to the pressure difference across the tunnel wall. With the aid of simplifying assumptions, a mathematical model has been constructed from two time-dependent, non-linear, ordinary differential equations, which describe the time evolution of the conduit cross-sectional area and the water depth in the reservoir. The model has been used to study the influence of conduit area and reservoir levels on the stability of the water flow for various glacier and ice-sheet configurations. The region of the parameter space where the system can achieve equilibrium has been identified. However, in the majority of cases the equilibrium is unstable, and an initial perturbation from equilibrium may lead to a catastrophic outburst of water which empties the reservoir.

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A Sufficient Condition for the Stability of Discrete-Time Systems With State-Dependent Coefficient Matrices

IEEE Transactions on Automatic Control ◽

10.1109/tac.2013.2270318 ◽

2014 ◽

Vol 59 (1) ◽

pp. 243-248 ◽

Cited By ~ 4

Author(s):

Tsuyoshi Yuno ◽

Toshiyuki Ohtsuka

Keyword(s):

Discrete Time ◽

Sufficient Condition ◽

Discrete Time Systems ◽

State Dependent ◽

The Stability ◽

Time Systems

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The Optimal Regulator Problem for a Stationary Linear System With State-Dependent Noise

Journal of Basic Engineering ◽

10.1115/1.3425003 ◽

1970 ◽

Vol 92 (2) ◽

pp. 363-368 ◽

Cited By ~ 3

Author(s):

P. J. McLane

Keyword(s):

Linear System ◽

Sufficient Conditions ◽

Quadratic Functional ◽

Optimal Controls ◽

Final Time ◽

State Dependent ◽

Continuous Dynamic ◽

The Stability ◽

And Control ◽

Stationary Linear System

The problem of minimizing a quadratic functional of the system outputs and control for a stationary linear system with state-dependent noise is solved in this paper. Both the finite final time and infinite final time versions of the problem are treated. For the latter case existence conditions are obtained using the second method of Lyapunov. The optimal controls for both problems are obtained using Bellman’s continuous dynamic programming. In light of this, the system dynamics are assumed to determine a diffusion process. For the infinite final time version of the problem noted above, sufficient conditions are obtained for the stability of the optimal system and uniqueness of the optimal control law. In addition, for this problem, an example is treated. The computational results for this example illustrate some of the qualitative features of regulators for linear, stationary systems with state-dependent disturbances.

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Dynamics Analysis and Biomass Productivity Optimisation of a Microbial Cultivation Process through Substrate Regulation

Discrete Dynamics in Nature and Society ◽

10.1155/2016/3685941 ◽

2016 ◽

Vol 2016 ◽

pp. 1-13

Author(s):

Kaibiao Sun ◽

Shan Liu ◽

Andrzej Kasperski ◽

Yuan Tian

Keyword(s):

Periodic Solution ◽

Substrate Concentration ◽

Process Model ◽

Biomass Productivity ◽

High Yield ◽

Dynamics Analysis ◽

State Dependent ◽

Cultivation Process ◽

Microbial Cultivation ◽

The Stability

A microbial cultivation process model with variable biomass yield, control of substrate concentration, and biomass recycle is formulated, where the biochemical kinetics follows an extension of the Monod and Contois models. Control of substrate concentration allows for indirect monitoring of biomass and dissolved oxygen concentrations and consequently obtaining high yield and productivity of biomass. Dynamics analysis of the proposed model is carried out and the existence of order-1 periodic solution is deduced with a formulation of the period, which provides a theoretical possibility to convert the state-dependent control to a periodic one while keeping the dynamics unchanged. Moreover, the stability of the order-1 periodic solution is verified by a geometric method. The stability ensures a certain robustness of the adopted control; that is, even with an inaccurately detected substrate concentration or a deviation, the system will be always stable at the order-1 periodic solution under the control. The simulations are carried out to complement the theoretical results and optimisation of the biomass productivity is presented.

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Thermal Instability of Sliding and Oscillations Due to Frictional Heating Effect

Journal of Tribology ◽

10.1115/1.3261577 ◽

1988 ◽

Vol 110 (1) ◽

pp. 69-72 ◽

Cited By ~ 7

Author(s):

I. L. Maksimov

Keyword(s):

Thermal Instability ◽

Initial Conditions ◽

Frictional Heating ◽

Interface Temperature ◽

Heating Effect ◽

Stick Slip ◽

Instability Development ◽

The Stability ◽

Sliding Dynamics ◽

Sliding Instability

The stability of sliding has been studied, taking into account frictional heating effect and friction coefficient dependence upon the interface temperature and sliding velocity. The collective—thermal and mechanical—sliding instability has been found to exist; instability emergence conditions and dynamics (both in linear and nonlinear stages) have been determined. It is shown that both the threshold and the dynamics of thermofrictional instability differ qualitatively from the analogous characteristics of “stick-slip” phenomenon. Namely, the oscillational instability behavior due to the energy exchange between thermal and mechanical modes has been found to occur under certain initial conditions; the velocities range has been determined for which collective sliding instability may occur whereas the stick-slips would be not possible. The nonlinear analysis of instability evolution has been carried out for pairs with the negative thermal-frictional sliding characteristics, the final stage of sliding dynamics has been described. It is found that stable thermofrictional oscillations can occur on the nonlinear stage of sliding instability development; the oscillations frequency and amplitude have been determined. The possibility has been discussed of the experimental observation of new dynamical sliding phenomena at low temperatures.

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Bifurcation and stability of periodic solutions of differential equations with state-dependent delays

European Journal of Applied Mathematics ◽

10.1017/s0956792502005053 ◽

2003 ◽

Vol 14 (1) ◽

pp. 3-14 ◽

Cited By ~ 1

Author(s):

D. SCHLEY

Keyword(s):

Time Delay ◽

Differential Equations ◽

Periodic Solutions ◽

Perturbation Methods ◽

Population Models ◽

Delay Function ◽

State Dependent ◽

The Stability ◽

Bifurcation And Stability

We consider periodic solutions which bifurcate from equilibria in simple population models which incorporate a state-dependent time delay of the discrete kind. The delay is a function of the current size of the population. Solutions near equilibria are constructed using perturbation methods to determine the sub/supercriticality of the bifurcation and hence their stability. The stability of the bifurcating solutions depends on the qualitative form of the delay function. This is in contrast to the stability of an equilibrium, which is determined purely by the actual value of this function at the equilibrium.

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Simulation of Frictional Sliding Under Impact Shear Loading

ASME/STLE 2004 International Joint Tribology Conference, Parts A and B ◽

10.1115/trib2004-64355 ◽

2004 ◽

Author(s):

Demir Coker ◽

Alan Needleman ◽

George Lykotrafitis ◽

Ares J. Rosakis

Keyword(s):

Constitutive Relation ◽

Wave Speed ◽

Constitutive Relations ◽

Shear Loading ◽

Stress And Strain ◽

Sliding Modes ◽

Elastic Plates ◽

Frictional Sliding ◽

State Dependent ◽

Dynamic Loading Conditions

Results from recent and ongoing investigations of frictional sliding under dynamic loading conditions are discussed. The configuration analyzed consists of two identical elastic plates with an interface characterized by a rate- and state-dependent frictional law. The calculations are carried out within a framework where two constitutive relations are used: a volumetric constitutive relation between stress and strain and a surface constitutive relation that characterizes the frictional behavior of the interface. The simulations discussed predict a variety of sliding modes including a crack-like mode and several pulse-like modes as well as circumstances where the sliding tip speed can exceed the longitudinal wave speed.

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On Some Sufficiency-Type Global Stability Results for Time-Varying Dynamic Systems with State-Dependent Parameterizations

International Journal of Differential Equations ◽

10.1155/2019/5097974 ◽

2019 ◽

Vol 2019 ◽

pp. 1-15

Author(s):

M. De la Sen

Keyword(s):

Global Stability ◽

Dynamic Systems ◽

Taylor Series Expansion ◽

Time Varying ◽

State Dependent ◽

The Matrix ◽

The Stability ◽

Interval Type ◽

Truncation Order ◽

Stability Results

This paper formulates sufficiency-type global stability and asymptotic stability results for, in general, nonlinear time-varying dynamic systems with state-trajectory solution-dependent parameterizations. The stability proofs are based on obtaining sufficiency-type conditions which guarantee that either the norms of the solution trajectory or alternative interval-type integrals of the matrix of dynamics of the higher-order than linear terms do not grow faster than their available supremum on the preceding time intervals. Some extensions are also given based on the use of a truncated Taylor series expansion of chosen truncation order with multiargument integral remainder for the dynamics of the differential system.

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