Ferenc Hartung

Approximate stability charts for milling processes using semi-discretization

Unwanted relative vibrations between the tool and the workpiece represent significant challenges in high-speed machining. In order to avoid this problem, one needs to specify ranges for system parameters (spindle speed, depth of cut) for which the process is stable, i.e., to obtain a so-called stability chart. Such stability charts usually can only be given by numerical means which is illustrated in the paper for a single degree of freedom model of milling. In this paper, we establish the convergence of the semi-discretization approximation method for a class of delay equations modeling the milling process. Moreover, we show that semi-discretization preserves asymptotic stability of the original equation, thus it can be used to obtain good approximations for the stability charts.

Numerical approximations for a class of differential equations with time- and state-dependent delays

Abstract We establish limiting relations between solutions for a large class of functional differential equations with time- and state-dependent delays and solutions of appropriately selected sequences of approximating delay differential equations with piecewise constant arguments. The approximating equations, generated in the above process, lead naturally to discrete difference equations, well suited for computational purposes, and thus provide an approximation framework for simulation studies.

On existence, uniqueness and numerical approximation for neutral equations with state-dependent delays

Abstract We consider a class of neutral functional differential equations with state-dependent delays, and discuss existence, uniqueness, and numerical approximation of solutions of corresponding initial value problems.

State dependent regenerative delay in milling processes

Traditional models of regenerative machine tool chatter use constant time delays assuming that the period between two subsequent cuts is a constant determined definitely by the spindle speed. These models result in delay-differential equations with constant time delay. If the vibrations of the tool relative to the workpiece are also includedin thesurface regenerationmodel, thentheresulted time delay is not constant, but it depends on the actual and a delayed position of the tool. In this case, the governing equation is a delay-differential equation with state dependent time delay. Equations with state dependent delays can not be linearized in the traditional sense, but there exists linear equations that can be associated to them. This way, the local behavior of the system with state dependent delays can be investigated. In this study, a two degree of freedom model is presented for milling process. A thorough modeling of the regeneration effect results in the governingdelay-differentialequationwith state dependent time delay. It is shown through the linearization of the nonlinear equation that an additional term arises in the linearized equation of motion due to the state-dependency of the time delay.

On numerical approximation using differential equations with piecewise-constant arguments

In this paper we give a brief overview of the application of delay differential equations with piecewise constant arguments (EPCAs) for obtaining numerical approximation of delay differential equations, and we show that this method can be used for numerical approximation in a class of age-dependent population models. We also formulate an open problem for stability and oscillation of a class of linear delay equations with continuous and piecewise constant arguments.

Stability analysis of a single neuron model with delay

In this paper we study the asymptotic behavior and numerical approximation of the single neuron model equation x(t)=-dx(t)+af(x(t))+bf(x(t-τ))+I, t ≥ 0 (1), where d > 0 and f(x)=0.5(|x+ 1| - |x-1|). We obtain new sufficient conditions for global asymptotic stability of constant equilibriums of (1), give several numerical examples to illustrate our results, and formulate conjectures on the asymptotic behavior of the solutions based on our numerical experiments.

Parameter identification in classes of hereditary systems of neutral type

In this paper we prove theoretical convergence for a variety of parameter identification schemes, based on approximations by equations with piecewise constant arguments, for classes of neutral differential equations.

BIBO stabilization of feedback control systems with time dependent delays

Abstract This paper investigates the bounded input bounded output (BIBO) stability in a class of control system of nonlinear differential equations with time-delay. The proofs are based on our studies on the boundedness of the solutions of a general class of nonlinear Volterra integral equations.

Numerical approximation of neutral differential equations on infinite interval

In this paper we study numerical approximation of linear neutral differential equations on infinite interval using equations with piecewise constant arguments. As an application of our approximation results, we obtain stability theorems for some classes of linear delay and neutral difference equations.

On a nonlinear delay population model

We investigate uniform permanence of positive solutions of nonlinear delay equations.In several examples we give explicit estimates for lower and upper limit of solutions.We give conditions which imply that all solutions are asymptotically equivalent. The nonlinear delay differential equation x ? ( t ) = r ( t ) g ( t , x t ) - h ( x ( t ) ) , t ? 0 is considered. Sufficient conditions are established for the uniform permanence of the positive solutions of the equation. In several particular cases, explicit formulas are given for the upper and lower limit of the solutions. In some special cases, we give conditions which imply that all solutions have the same asymptotic behavior, in particular, when they converge to a periodic or constant steady-state.