Aliki D. Muradova

Fuzzy Vibration Control of a Smart Plate

Vibration suppression of a smart thin elastic rectangular plate is considered. The plate is subjected to external disturbances and generalized control forces, produced, for instance, by electromechanical feedback. A nonlinear controller is designed, based on fuzzy inference. The initial-boundary value problem is spatially discretized by means of the time spectral method. The implicit Newmark-beta method is employed for time integration. Two numerical algorithms are proposed. The techniques have been implemented within MATLAB with the use of the Fuzzy Logic Toolbox. Representative numerical results are given.

Buckling Simulation of a Plate Embedded in a Unilaterally Supported Environment

Abstract Buckling of a von Kármán plate unilaterally connected with an elastic, Winkler-type, tensionless foundation on both sides is considered in this article. The plate is simply supported and subjected to compressive and tensile loads along its edges. A variational principle is formulated for the mechanical model. The variational problem is numerically solved by means of the spectral method and computational algorithm. The algorithm is based on Newtons scheme and a numerical continuation procedure which considers the contact conditions. The bifurcation phenomenon has been analyzed for different values of compressive and tensile loading parameters. The proposed algorithm is demonstrated with the help of numerical results. Finally, an application of the unilateral plate-bending model to layered tissues in biomechanics is considered.

Kinetic model of mass exchange with dynamic Arrhenius transition rates

We study a nonlinear kinetic model of mass exchange between interacting grains. The transition rates follow the Arrhenius equation with an activation energy that depends dynamically on the grain mass. We show that the activation parameter can be absorbed in the initial conditions for the grain masses, and that the total mass is conserved. We obtain numerical solutions of the coupled, nonlinear, ordinary differential equations of mass exchange for the two-grain system, and we compare them with approximate theoretical solutions in specific neighborhoods of the phase space. Using phase plane methods, we determine that the system exhibits regimes of diffusive and growth–decay (reverse diffusion) kinetics. The equilibrium states are determined by the mass equipartition and separation nullcline curves. If the transfer rates are perturbed by white noise, numerical simulations show that the system maintains the diffusive and growth–decay regimes; however, the noise can reverse the sign of equilibrium mass difference. Finally, we present theoretical analysis and numerical simulations of a system with many interacting grains. Diffusive and growth–decay regimes are established as well, but the approach to equilibrium is considerably slower. Potential applications of the mass exchange model involve coarse-graining during sintering and wealth exchange in econophysics.

Numerical simulation of a coupled nonlinear model for grain coarsening and coalescence

Abstract A kinetic nonlinear model of mass transfer, grain coarsening and coalescence with potential applications in sintering processes is studied. The model involves nonlinear ordinary differential equations that determine the transport of mass between grains. The rate of mass transfer is controlled by an Arrhenius factor leading to a nonlinear model of mass transfer and grain coarsening. The resulting dynamical system of coupled nonlinear differential equations with random initial conditions (i.e., initial grain mass configuration) is solved by means of the fourth order Runge–Kutta method. We conduct an analysis of the two-grain system and identify three dynamic regimes (diffusive, growth-decay and trapping). The same regimes are shown to persist in the multigrain system. We confirm the numerical performance of the Runge–Kutta method by means of a suitable convergence measure. The influence of the activation energy parameter on the dynamic regimes is investigated. It is shown that as the parameter grows the diffusive regime is progressively restricted to smaller values of the initial grain distribution. We introduce grain coalescence in the mass transfer equations, and we show that it accelerates the growth of the larger grains. Finally, we compare the dynamic evolution of the grain size distribution with the Ostwald ripening expression.

Numerical Investigation of Grain Coarsening and Coalescence Model

A kinetic nonlinear model of mass transfer, grain coarsening and coalescence with potential applications in sintering processes is studied. The model involves nonlinear differential equations that determine the transport of mass between grains. The rate of mass transfer is controlled by the activation energy (an Arrhenius factor) leading to a nonlinear model of mass transfer and grain coarsening. The resulting dynamical system of coupled nonlinear differential equations with random initial conditions (i.e., initial grain mass configuration) is solved by means of the Runge-Kutta method. An analysis of the fixed points of the two-grain system is carried out, and the solution of the multi-grain system is studied. We incorporate coalescence of smaller grains with larger neighbors using a cellular automaton step in the evolution of the system.

Nonlinear Time Spectral Analysis for a Dynamic Contact Model with Buckling for an Elastic Plate

In the present paper a dynamic nonlinear model with contact and buckling for an elasticplate is considered. The model consists of two coupled nonlinear hyperbolic type partial differentialequations. The plate is subjected to compressive and/or tensile moving loads on its edges. The foundationsare nonlinear elastic Winkler and Pasternak models. The initial-boundary value problems forthe model are solved with the use of the time spectral method for spatial discretization and after thediscretization the Newmark- time-stepping iterative scheme for the obtained system of nonlinear ordinarydifferential equations. The model is tested for the Winkler-type and shear Pasternak-type andas well for several values of the physical constants of the foundations.

Postbuckling Behaviour of a Rectangular Plate Surrounded by Nonlinear Elastic Supports

The postbuckling behaviour of a von Karman plate on a nonlinear elastic tensionless foundation is investigated. The foundation is modeled as linearly elastic, Winkler-type medium with softening cubic nonlinearity and shear deformable medium of Pasternak-type. The cases of compressive and tensile loadings along the edges of the plate are considered. The postbuckling behaviour of these plates is described by a system of nonlinear PDEs which takes into account in-plane compression and tension, and reaction forces of the foundation. A bifurcation analysis of the solution for simply supported boundary conditions is presented. The spectral method is applied for the discretization of the boundary value problem. A path-following numerical algorithm is introduced to trace branches of the solution. The Newton iterative scheme with the numerical continuation is employed to solve the resulting system of nonlinear relations. Numerical results are presented.

To problem of approximate solution of BVPs for ODEs with boundary layers by multipoint methods

Boundary value problems for ordinary differential equations with a small parameter in the leading derivative are solved by a high accuracy order multipoint method. The numerical scheme for a two-point boundary value problem for the second order ordinary differential equation is presented. The estimates for the derivatives of the solution are obtained with the use of the asymptotic expansions. The remainders of the difference scheme are evaluated. Numerical examples illustrate the efficiency of the proposed techniques.