Pub. online:1 Jan 2001Type:Research ArticleOpen Access
Volume 12, Issue 2 (2001), pp. 303–314
In the practice of metal treatment by cutting it is frequently necessary to deal with self-excited oscillations of the cutting tool, treated detail and units of the machine tool. In this paper are presented differential equations with the delay of self-excited oscillations. The linear analysis is performed by the method of D-expansion. There is chosen an area of asymptotically stability and area D2. It is prove that, in the area D2 the stable periodical solution appears. The non-linear analysis is performed by the theory of bifurcation. The computational experiment of metal cutting process and results of these experiments are presented.
Pub. online:1 Jan 1994Type:Research ArticleOpen Access
Volume 5, Issues 3-4 (1994), pp. 385–413
This paper establishes sufficient conditions for stability of linear and time-invariant delay differential systems including their various usual subclasses (i.e., point, distributed and mixed point-distributed delay systems). Sufficient conditions for stability are obtained in terms of the Schur's complement of operators and the frequency domain Lyapunov equation. The basic idea in the analysis consists in the use of modified Laplace operators which split the characteristic equation into two separate multiplicative factors whose roots characterize the system stability. The method allows a simple derivation of stabilizing control laws.
Pub. online:1 Jan 1991Type:Research ArticleOpen Access
Volume 2, Issue 2 (1991), pp. 155–170
This paper is devoted to the construction and investigation of difference schemes for the solution of one-dimensional parabolic problems with non-classical boundary conditions. The stability of schemes and the convergence of a numerical solution is proved in the norms L1 and C.