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Nonlinear Functional Analysis And Its Applications

The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one{dimensional second order elliptic operators with random coeecients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coeecient. Abstract. In the theory of homogenisation it is of particular interest to determine the classes of problems which are stable on taking the homogenisation limits. A notable situation where the limit enlarges the class of original problems is known as memory (nonlocal) eeects. A number of results in that direction has been obtained for linear problems. Tartar (1990) innitiated the study of the eeective equation corresponding to nonlinear equation: @ t u n + a n u 2 n = f: Signiicant progress has been hampered by the complexity of required computations needed in order to obtain the terms in power{series expansion. We propose a method which overcomes that diiculty by introducing graphs representing the domain of integration of the integrals in each term. The graphs are relatively simple, it is easy to calculate with them and they give us a clear image of the form of each term. The method allows us to discuss the form of the eeective equation and the convergence of power{series expansions. The feasibility of our method for other types of nonlinearities will be discussed as well.

Applied Mathematics and Computation

Chatter suppression techniques in metal cutting

Differential equations with non-standard growth have been a very active field of investigation in recent years. In this survey we present an overview of the field, as well as several of the most important results. We consider both existence and regularity questions. Finally, we provide a comprehensive list of papers published to date.

Overview of differential equations with non-standard growth

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Applied Functional Analysis

Chatter vibration surveillance by the optimal-linear spindle speed control

In this note, a critical point result for differentiable functionals is exploited in order to prove that a suitable class of one-dimensional fractional problems admits at least one non-trivial solution under an asymptotical behaviour of the nonlinear datum at zero. A concrete example of an application is then presented. Copyright © 2015 John Wiley & Sons, Ltd.

Existence results for one-dimensional fractional equations

In this note a critical point result for differentiable functionals is exploited in order to prove that a suitable class of one-dimensional fractional problems admits at least one non-trivial solution under an asymptotical behaviour of the nonlinear datum at zero. A concrete example of an application is then presented.

In the paper a method of optimal spindle speed determination for vibration reduction during ball-end milling of flexible details is proposed. In order to reduce vibration level, an original procedure of the spindle speed optimisation, based on the Liao–Young criterion [ 1 ], is suggested. As the result, an optimal, constant spindle speed value is determined. For this purpose, non-stationary computational model of machining process is defined. As a result of modelling, a hybrid system is described. This model consists of following subsystems, i.e. stationary model of one-side-supported flexible workpiece (modal subsystem), non-stationary discrete model of ball-end mill (structural subsystem) and conventional contact point between tool and workpiece (connective subsystem). The method requires identification of some natural frequencies of stationary modal subsystem. To determine them, appropriate modal experiments have to be performed on the machine tool, just before machining. Examples of vibration surveillance during cutting process on two high speed milling machines Mikron VCP 600 and Alcera Gambin 120CR are illustrated.

Optimal spindle speed determination for vibration reduction during ball-end milling of flexible details

Using critical point theory methods we undertake the existence and multiplicity of solutions for discrete anisotropic two-point boundary value problems.

Marek Galewski

Papers

Chatter vibration surveillance by the optimal-linear spindle speed control

Existence results for one-dimensional fractional equations

Existence results for one-dimensional fractional equations

Optimal spindle speed determination for vibration reduction during ball-end milling of flexible details

On the discrete boundary value problem for anisotropic equation