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Featured Application: Prediction and selection of the clamping conditions of large-size workpieces for the purpose of milling, based on modal tests and milling simulations. Vibrations occurring during milling operations are one of the main issues disturbing the pursuit of better efficiency of milling operations and product quality. Even in the case of a stable cutting process, vibration reduction is still an important goal. One of the possible solutions to obtain it is selection of the favorable conditions for clamping the workpiece to the machine table. In this paper, a method for predicting and selecting the clamping condition of a large-size workpiece for the reduction in vibrations during milling is presented. A modal test of the workpiece is performed first for a selected set of tightening screw settings. Next, one milling pass is performed to obtain reference data which are then used to tune the hybrid computational model. In the subsequent step, milling simulations are performed for a set of tightening variants, and the best one is selected, providing the lowest vibrations, assessed as the root mean square (RMS) of vibration displacements. In this paper, the description of the clamping selection procedure, key elements of the simulation model, and simulation and experimental results obtained for the milling of the test workpiece performed for a set of different clamping conditions are provided. The proposed method accurately predicts not only the best but also the worst clamping conditions. [ABSTRACT FROM AUTHOR]

The paper presents results on the solvability and parameter dependence for problems driven by weakly continuous potential operators with continuously differentiable and coercive potential. We provide a parametric version on the existence result to nonlinear equations involving coercive and weakly continuous operators. Applications address a variant of elastic beam equation. [ABSTRACT FROM AUTHOR]

The article introduces a method for selecting the best clamping conditions to obtain vibration reduction during the milling of large-size workpieces. It is based on experimental modal analysis performed for a set of assumed, fixing conditions of a considered workpiece to identify frequency response functions (FRFs) for each tightening torque of the mounting screws. Simulated plots of periodically changing nominal cutting forces are then calculated. Subsequently, by multiplying FRF and spectra of cutting forces, a clamping selection function (CSF) is determined, and, thanks to this function, vibration root mean square (RMS) is calculated resulting in the clamping selection indicator (CSI) that indicates the best clamping of the workpiece. The effectiveness of the method was evidenced by assessing the RMS value of the vibration level observed in the time domain during the real-time face milling process of a large-sized exemplary item. The proposed approach may be useful for seeking the best conditions for fixing the workpiece on the table. [ABSTRACT FROM AUTHOR]

We investigate fourth‐order equations with Dirichlet‐type boundary conditions with perturbation unbounded from above making the problem non‐potential. We apply variational method to some auxiliary problem and conclude about the existence and uniqueness to the original one. Multiple solutions are also considered. We conclude our note with the result pertaining to the continuous dependence on parameters. [ABSTRACT FROM AUTHOR]

We introduce a type of mixed variational inequality driven by a uniformly monotone operator for which we investigate the existence and uniqueness problem by applying techniques suggested in [12]. We apply the abstract results obtained to a mixed boundary value problems inspired by contact mechanics but with the elasticity operator being not necessarily strongly monotone. We allow this operator to be a uniformly monotone acting from a reflexive Banach space into its dual. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of a Nash‐type equilibrium for a non‐potential nonlinear system by combining variational methods with the monotonicity approach. The advance over existing research is that we can consider systems of Dirichlet problems in which the operator is not necessarily linear. [ABSTRACT FROM AUTHOR]

This paper concerns the problem of vibration reduction during milling. For this purpose, it is proposed that the standard supports of the workpiece be replaced with adjustable stiffness supports. This affects the modal parameters of the whole system, i.e., object and its supports, which is essential from the point of view of the relative tool–workpiece vibrations. To reduce the vibration level during milling, it is necessary to appropriately set the support stiffness coefficients, which are obtained from numerous milling process simulations. The simulations utilize the model of the workpiece with adjustable supports in the convention of a Finite Element Model (FEM) and a dynamic model of the milling process. The FEM parameters are tuned based on modal tests of the actual workpiece. For assessing simulation results, the proper indicator of vibration level must be selected, which is also discussed in the paper. However, simulating the milling process is time consuming and the total number of simulations needed to search the entire available range of support stiffness coefficients is large. To overcome this issue, the artificial intelligence salp swarm algorithm is used. Finally, for the best combination of stiffness coefficients, the vibration reduction is obtained and a significant reduction in search time for determining the support settings makes the approach proposed in the paper attractive from the point of view of practical applications. [ABSTRACT FROM AUTHOR]

In this work, we shifted a recent multiplicity result by B. Ricceri from a Hilbert space to a Banach space setting by making use of a duality mapping relative to some increasing function. Using the min–max arguments, we provide conditions for an action functional to have at least two global minima. [ABSTRACT FROM AUTHOR]

We provide parameter‐dependent version of the Browder–Minty Theorem in case when the solution is unique utilizing different types of monotonicity and compactness assumptions related to condition (S)2. Potential equations and the convergence of their Euler action functionals are also investigated. Applications towards the dependence on parameters for both potential and nonpotenial nonlinear Dirichlet boundary problems are given. [ABSTRACT FROM AUTHOR]

In this note we are concerned with the proposal of case study approach in teaching mathematical analysis. By describing a simple student's project about the solvability of (nonlinear) equations we aim at indicating that many possible further directions are also possible. [ABSTRACT FROM AUTHOR]

The paper presents an innovative method of solving the problem of vibration suppression during milling of large-size details. It consists in searching for the best conditions for clamping the workpiece based on a rapid modal identification of the dominant natural frequencies only and requires repetitive changes in the tightening torque of the clamping screws. Then, by estimating the minimum work of the cutting forces acting in the direction of the width of the cutting layer, it is possible to predict the best fixing of the workpiece. Application of the method does not require the creation and identification of a computational model of the process or preliminary numerical simulations. The effectiveness of this method was confirmed by the evaluation of the Root Mean Square (RMS) of the vibration level in the time domain observed during the actual face milling process. The worst results were obtained for the configuration of supports tightened with a torque of 90–110 Nm, and the best—with a torque of 50 Nm. [ABSTRACT FROM AUTHOR]

The paper presents an original method concerning vibration suppression problem during milling of large-size and geometrically complicated workpieces with the use of novel way of selecting the spindle speed. This consists in repetitive simulations of the cutting process for subsequent values of the spindle speed, until the best vibration state of the workpiece is reached. An appropriate method of obtaining a computational model, called a modal approach, consists in identifying the parameters of the workpiece model created using the Finite Element Method (FEM). Thanks to the results of the identification of the modal subsystem obtained by the Experimental Modal Analysis (EMA) method, it can be stated that the parameters obtained from the experiment and delivered from the computational model have been correctly determined and constitute reliable process data for the simulation tests. The Root Mean Square (RMS) values of time domain displacements are evaluated. The efficiency of the proposed approach is evidenced by chosen technique of mechatronic design, called Experiment Aided Virtual Prototyping (EAVP). The proposed method is verified on the basis of the results of the experimental research of the relevant milling process. [ABSTRACT FROM AUTHOR]

Using monotonicity methods and some variational argument we consider nonlinear problems which involve monotone potential mappings satisfying condition (S) and their strongly continuous perturbations. We investigate when functional whose minimum is obtained by a direct method of the calculus of variations satisfies the Palais-Smale condition, relate minimizing sequence and Galerkin approximaitons when both exist, then provide structure conditions on the derivative of the action functional under which bounded Palais-Smale sequences are convergent. Finally, we make some comment concerning the convergence of Palais-Smale sequence obtained in the mountain pass theorem due to Rabier. [ABSTRACT FROM AUTHOR]

Using monotonicity methods, the Lagrange multiplier rule, and some variational arguments, we consider a type of localization results pertaining to the existence of critical points to action functionals on a closed ball. A variant of the Schechter critical point theorem on a ball in Hilbert and Banach spaces is obtained. Applications to nonlinear Dirichlet problem and to partial difference equations are given in the final part of this paper. [ABSTRACT FROM AUTHOR]

In this work we apply global invertibility result in order to examine the solvability of elliptic equations with both Neumann and Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

We show that the monotonicity methods can also be applied in the fractal setting to examine existence and also existence and uniqueness. Furthermore, we investigate the continuous dependence on parameters for the problem under consideration. [ABSTRACT FROM AUTHOR]

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In this work we consider second order Dirichelet boundary value problem with nonlinearity depending on the derivative. Using a global diffeomorphism theorem we propose a new variational approach leading to the existence and uniqueness result for such problems. [ABSTRACT FROM AUTHOR]

We prove a non-smooth generalization of the global implicit function theorem. More precisely we use the non-smooth local implicit function theorem and the non-smooth critical point theory in order to prove a non-smooth global implicit function theorem for locally Lipschitz functions. A comparison between several global inversion theorems is discussed. Applications to algebraic equations are given. [ABSTRACT FROM AUTHOR]

Using the global diffeomorphism theorem we consider the existence of solutions to the following Dirichlet problem,, whereis a jointly continuous function subject to some further growth conditions. Together with a given problem we consider the family of its discretizations and as a result we prove the existence of a non-spurious solution. [ABSTRACT FROM PUBLISHER]

In this article we continue the study of discrete anisotropic equations and we will provide a new multiplicity results of the solutions for a discrete anisotropic equation. The procedure viewed here is according to variational methods and critical point theory. In fact, using a consequence of the local minimum theorem due Bonanno and mountain pass theorem we look into the existence results for our problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Furthermore, by mingling two algebraic conditions on the nonlinear term employing two consequences of the local minimum theorem due Bonanno we guarantee the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of third solution for our problem. [ABSTRACT FROM AUTHOR]

Modern industry expectations in terms of milling operations often demand the milling of the flexible details by using slender ball-end tools. This is a difficult task because of possible vibration occurrence. Due to existence of certain conditions (small depths of cutting, regeneration phenomena), cutting process may become unstable and self-excited chatter vibration may appear. Frequency of the chatter vibration is close to dominant natural frequency of the workpiece or the tool. One of the methods of chatter vibration avoidance is matching the spindle speed to the optimum phase shift between subsequent cutting edges passes (i.e. the Liao-Young condition). However, the set of optimum spindle speeds from the point of view of vibration reduction may be not optimum one from other points of view. For example milling efficiency or machine tool capabilities cannot be assured. This article presents the idea of a workpiece holder with adjustable stiffness and discusses a new variant of its realization. In the holder, milling process is performed at constant spindle speed and feed speed. In order to avoid vibration the holder stiffness is modified. Stiffness changes modify natural frequencies of the workpiece and thus it is possible to modify dynamic properties of the workpiece in such a way that arbitrary chosen, constant spindle speed will be optimum, due to the Liao-Young condition performance. Calculation of the optimum stiffness is performed before milling, based on the workpieces modal identification results and the finite element model simulations. [ABSTRACT FROM AUTHOR]

We provide sufficient conditions for a mappingf:Rn → Rnto be a global diffeomorphism in casefneed not be continuously differentiable. Instead it is assumed to be strictly (Hadamard-like) and Fréchet differentiable. We use classical local invertibility conditions together with the non-smooth critical point theory. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Using direct variational method we consider the existence of non-spurious solutions to the following Dirichlet problem,whereis a jointly continuous function convex inwhich does not need to satisfy any further growth conditions. [ABSTRACT FROM PUBLISHER]

In this work we investigate multiple solutions to discrete inclusions with the-Laplacian problem. We focus on the existence of three solutions with the aid of linking arguments and a new non-smooth three critical points tool, which we provide. [ABSTRACT FROM AUTHOR]

Using the Fenchel-Young duality we derive a new critical point theorem. We illustrate our results with solvability for certain discrete BVP. Multiple solutions are also considered. [ABSTRACT FROM AUTHOR]

In this note we derive a type of a three critical point theorem which we further apply to investigate the multiplicity of solutions to discrete anisotropic problems with two parameters. [ABSTRACT FROM AUTHOR]

We investigate the existence of multiple periodic solutions to the anisotropic discrete system. We apply the linking method and a new three critical point theorem which we provide. [ABSTRACT FROM AUTHOR]

We discuss the eigenvalue problem for n-th order equations with multipoint boundary conditions. We present the variational method in spite of the fact that the nonlinearity is dependent on the derivative of n/2 order. The results can be applied applied to both sublinear and superlinear cases. [ABSTRACT FROM AUTHOR]

In this paper we consider the Dirichlet problem for a discrete anisotropic equation with some function α a nonlinear term f, and a numerical parameter λ : Δ (α (k) |Δu(k - 1)|p(k - 1)-2 Δu(k - 1)) + λf(k, u(k)) = 0; k ∈ [1, T]. We derive the intervals of a numerical parameter λ for which the considered BVP has at least 1, exactly 1, or at least 2 positive solutions. Some useful discrete inequalities are also derived. [ABSTRACT FROM AUTHOR]

We investigate the existence of solutions of the system of anisotropic discrete boundary value problems using critical point theory. We apply direct variational approach, the Ky Fan min–max inequality and the mountain pass lemma. [ABSTRACT FROM AUTHOR]

In this paper we investigate via variational methods and critical point theory the existence of solutions, uniqueness and continuous dependence on parameters to impulsive problems with ap(t)-Laplacian and Dirichlet boundary value conditions. [ABSTRACT FROM PUBLISHER]

The main goal of modern machining operations is to achieve increasingly better performance. High Speed Machining and/or High Performance Cutting, despite a lot of advantages, have also some drawbacks, for example, a possibility of losing stability and development of self-excited chatter vibration. This paper presents an approach of vibration surveillance during high speed milling with a use of active optimal control. Non-stationary model of the milling process is described with the use of convention of the discrete hybrid system, and the problem of optimal control at energy performance index is defined and resolved. Numerous computer simulations disclosed efficiency of the surveillance by the active optimal control of flexible workpiece vibration. The performed real experiments of the milling process support the meaning of the method of vibration surveillance. [ABSTRACT FROM AUTHOR]

Using the variational approach, we investigate the existence of solutions and their dependence on functional parameters for classical solutions to the second order impulsive boundary value Dirichlet problems with L1 right hand side. [ABSTRACT FROM AUTHOR]

In this paper we investigate the existence of solutions to impulsive problems with a p( t)-Laplacian and Dirichlet boundary value conditions. We introduce two types of solutions, namely a weak and a classical one which coincide because of the fundamental lemma of the calculus of variations. Firstly we investigate the existence of solution to the linear problem, i.e. a problem with a fixed rigth hand side. Then we use a direct variational method and next a mountain pass approach in order to get the existence of at least one weak solution to the nonlinear problem. [ABSTRACT FROM AUTHOR]

Using the direct method of the calculus of variations, we investigate the existence, uniqueness and continuous dependence on parameters for solutions of second-order discrete anisotropic equations with Dirichlet boundary conditions. [ABSTRACT FROM PUBLISHER]

Using monotonicity theory, we investigate the continuous dependence on parameters for the discrete BVPs, which can be written in a form of a nonlinear system. [ABSTRACT FROM AUTHOR]

Using Fan's Min-Max Theorem we investigate existence of solutions and their dependence on parameters for some second order discrete boundary value problem. The approach is based on variational methods and solutions are obtained as saddle points to the relevant Euler action functional. [ABSTRACT FROM AUTHOR]

We investigate the dependence on parameters for second-order difference equations with two-point boundary value conditions using a variational method in case when the corresponding Euler action functional is coercive. Some applications for discrete Emden-Fowler equation are also given. [ABSTRACT FROM AUTHOR]

We improve the results of Li by using less restrictive assumptions. [ABSTRACT FROM AUTHOR]

Using critical point theory and some monotonicity results we consider the existence of solutions to a boundary value problem connected with the discrete elliptic equation with a positive parameter. [ABSTRACT FROM AUTHOR]

We apply the dual variational method in order to prove that a certain system of difference equations with a parameter has at least one nontrivial bounded solution. [ABSTRACT FROM AUTHOR]

We provide existence and stability results for semilinear Dirichlet problems with nonlinearity satisfying general growth conditions.We consider the case when both the coefficients of the differential operator and the nonlinear term depend on the numerical parameter. We show applications for the fourth order semilinear Dirichlet problem. [ABSTRACT FROM AUTHOR]