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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]

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 paper, we consider hemivariational–variational inequalities driven by uniformly monotone or d -monotone operators in Banach spaces. We establish related minimization principles leading to the existence and uniqueness of solutions to the inequality considered as well as we suggest the Ritz type numerical approximations. The theoretical results obtained are next applied to some problems inspired by models from contact mechanics. [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]

We investigate the existence of variational generalized solutions to coercive competitive equations driven by a competing p , q − Laplacian type operator with weigth depending on the gradient. An abstract existence result based on a Galerkin scheme is therefore obtained and subsequently applied. [ABSTRACT FROM AUTHOR]

The paper presents the theoretical background and four applications examples of the new method for the estimation of support stiffness coefficients of complex structures modelled discretely (e.g. with the use of the Finite Element Model (FEM) method based on the modified Particle Swarm Optimization (PSO) algorithm. In real-life cases, exact values of the supports' stiffness coefficients may change for various reasons (e.g. order of fastening, state of the contact surfaces, environment changes, etc.). Because of the unknown coefficients, reliable simulations of fixed structure (i.e. mounted, assembled, not in a free-free state) are difficult to perform. The method serves as a tool to obtain good correlation between the FEM of a structure and the experimental data. Simple modal tests are required to estimate the first few modes of the fixed system. The FEM of the structure is considered in a free-free state and the support stiffness coefficients of the FEM are estimated by the proposed method. Further simulations with test-tuned and correlated complete FEM could be performed in time or frequency domain. [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]

Using microcontroller systems becomes a routine in various measurement and control tasks. Their wide availability together with a huge potential of extending their functionality by additional modules allows developing advanced measuring and monitoring systems by non-specialists. However, using popular example codes often leads the user to pass over or not to be aware of the limitations of the system and drawing too farreaching conclusions on the basis of incorrectly performed measurements This paper deals with the problem of choosing the right method for performing measurements using an acquisition system based on the budget Arduino UNO solution. The main assumption was to use the standard, widely available Arduino libraries. The work focuses on the scenario when data should be subject to time and frequency analysis in the later processing. The operating limits of the device were also determined depending on the data transmission method used. [ABSTRACT FROM AUTHOR]

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]

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 note we consider Dirichlet boundary value problem on a half line. Using critical point theory we prove the existence of at least one nontrivial solution. [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 use the direct variational method, the Ekeland variational principle, the mountain pass geometry and Karush–Kuhn–Tucker theorem in order to investigate existence and multiplicity results for boundary value problems connected with the discrete p ( · ) − Laplacian on weighted finite graphs. Several auxiliary inequalities for the discrete p ( · ) − Laplacian on finite graphs are also derived. Positive solutions are considered. [ABSTRACT FROM AUTHOR]

The paper presents the new method of the natural frequencies and damping identification based on the Artificial Intelligence (AI) Particle Swarm Optimization (PSO) algorithm. The identification is performed in the frequency domain. The algorithm performs two PSO-based steps and introduces some modifications in order to achieve quick convergence and low estimation error of the identified parameters’ values for multi-mode systems. The first stage of the algorithm concentrates on the natural frequencies estimation. Using the information about the natural frequencies, measurement data are filtered and corrected dampings as well as amplitudes are calculated for each preliminary identified mode. This allows regrouping particles to the area around proper parameters values. Particle regrouping is based on the physical properties of modally tested structures. This differs the algorithm from other PSO based algorithms with particles regrouping. In the second stage of the algorithm parameters of all modes are tuned together in order to adjust estimates. The procedure of identification, as well as the appropriate algorithm, is presented and some SISO examples are provided. Results are compared with the results obtained for the selected, already developed modal identification methods. The paper presents practical application of AI method for mechanical systems identification. [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]

We investigate the existence of an optimal process for such an optimal control problem in which the dynamics is given by the Dirichlet problem driven by weak Laplacian on the Sierpinski gasket. To accomplish this task using a direct variational approach with no global growth conditions on the nonlinear term, we consider the existence of solutions, their uniqueness and their dependence on a functional parameter for mentioned Dirichlet problem. This allows us to prove that the optimal control problem admits at least one solution. [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]

An n -th order delayed differential equation y (n) (t) = f (t , y t , y t ′ , ... , y t (n − 1) ) is considered, where y t (θ) = y (t + θ) , θ ∈ [ − τ , 0 ] , τ > 0 , if t → ∞. A criterion is formulated guaranteeing the existence of a solution y = y (t) in a cone 0 < (− 1) i − 1 y (i − 1) (t) < (− 1) i − 1 φ (i − 1) (t) , i = 1 , ... , n where φ is an n -times continuously differentiable function such that 0 < (− 1) i φ (i) (t) , i = 0 , ... , n. The proof is based on a similar result proved first for a system of delayed differential equations equivalent in a sense. Particular linear cases are considered and an open problem is formulated as well. [ABSTRACT FROM AUTHOR]

A method of vibration surveillance by using the optimal control at energy performance index has been creatively modified. The suggested original modification depends on consideration of direct relationship between the measured acceleration signal and the optimal control command. The paper presents the results of experiments and Hardware-in-the-loop simulations of a new active vibration reduction algorithm based on the energy performance index idea modified in such a way, that it directly utilises the acceleration feedback signal. Promising prospects towards real application of the modified method in case of the high speed milling are predicted as well. [ABSTRACT FROM AUTHOR]

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. [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]

Using the techniques connected with the measure of noncompactness we investigate the neutral difference equation of the following form Δ (rn (Δ (xn + pnxn-k))γ) + qnxna + anf(xn+1) = 0, where x:Ǖk→ ℝ, a,p,q: Ǖ0→ &#8477, r; Ǖ → ͭ \ {0}, f : ℝ → ℝ is continuous and k is a given positive integer, a≥ 1 is a ratio of positive integers with odd denominator, and γ ≤ 1 is ratio of odd positive integers; Ǖk : = {k,k + 1,...}. Sufficient conditions for the existence of a bounded solution are obtained. Also a special type of stability is studied. [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 provide sufficient conditions for a mapping acting between two Banach spaces to be a diffeomorphism. Standart arguments allow us to obtain a local diffeomorphism. It is proved to be global using mountain pass geometry. [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]

Abstract: We investigate the continuous dependence on parameters for the mountain pass solutions of second order discrete equations with -Laplacian and Dirichlet type boundary conditions. We show that assumptions leading to the existence of nontrivial solutions lead also to its continuous dependence on parameters. We investigate also the uniqueness of solutions. [Copyright &y& Elsevier]

Using mountain pass arguments and the Karsuh-Kuhn-Tucker Theorem, we prove the existence of at least two positive solution for anisotropic discrete Dirichlet boundary-value problems. Our results generalized and improve those in [16]. [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]

Abstract: We consider the existence and multiplicity of solutions to discrete boundary value problems with the use of critical point theory combined with some monotonicity argument. We consider both sub- and super-linear problems. [Copyright &y& Elsevier]

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]

Abstract: In this paper we consider the discrete anisotropic boundary value problem using critical point theory. Firstly we apply the direct method of the calculus of variations and the mountain pass technique in order to reach the existence of at least one nontrivial solution. Secondly we derive some version of a discrete three critical point theorem which we apply in order to get the existence of at least two nontrivial solutions. [Copyright &y& Elsevier]

Abstract: We derive a dual variational method in order to obtain the existence of a bounded solution to a certain nonlinear system with non-monotone nonlinearities. We consider such a system representing a difference equation arising from evaluating some Dirichlet boundary value problems. [Copyright &y& Elsevier]

Abstract: We investigate the dependence on parameters for the discrete boundary value problem connected with the Emden–Fowler equation. A variational method is used in order to obtain a general scheme allowing for investigating the dependence on parameters of discrete boundary value problems. [Copyright &y& Elsevier]

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]

We use direct variational method in order to investigate the dependence on parameter for the solution for a Duffing type equation with Dirichlet boundary value conditions. [ABSTRACT FROM AUTHOR]

Abstract: The paper concerns self-excited chatter vibration during high speed slender ball-end milling. Non-stationary cutting process, with inclusion of various approaches towards dynamic characteristics of the process, is described. Dynamic analysis of the milling process is performed and dynamics of controlled closed loop system with time-delay is presented. In order to reduce vibration level, instantaneous change in the spindle speed appears as a control command, and thus—the method of vibration surveillance by the spindle speed optimal-linear control is developed. Presented cutting models have been applied for the proposed method and procedure of the chatter vibration surveillance with a use of variable spindle speed has been developed. Computer simulations are performed for selected cases of ball-end milling at constant and variable spindle speed. The results of them are successfully confirmed by experimental investigations on the Alcera Gambin 120CR milling machine equipped with the S2M high speed electrospindle. [Copyright &y& Elsevier]

Abstract: Using a direct variational approach with no global growth conditions on the nonlinear term, we consider the existence of solutions and their dependence on a functional parameter for the fourth order Dirichlet problem connected with the elastic beam equation. We investigate also the existence of an optimal process for such an optimal control problem in which the dynamics is described by the beam equation. [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]

Abstract: Using critical point theory and monotonicity results, we consider the existence of solutions to certain types of difference equations depending on a parameter. [Copyright &y& Elsevier]