Different techniques are used to obtain approximate solutions for delayed functional differential equations (RFDEs). All these models used the so-called stability lobe diagrams, to choose the maximum axial depth of cut for a given spindle speed associated with a free chatter in machining. In this research paper, the ZOA (Zeroth Order Approximation) and SD (Semi Discretization) methods are explained, developed and used to obtain the stability lobe diagrams for a milling cutting system with two degree of freedom, in high speed machining case.

Recently, the investigation of periodic motion of the delay differential (DDEs) and the associated variation systems become into the focus of many studies. One of the most important motivations is the milling process analysis. In this work, the semi discretization method is briefly explained and have been applied for 1-DOF (degree of freedom) and 2-DOF milling system in order to build the stability lobes charts.

The machining of the complex forms is a characteristic of moulds and matrices manufacturing. Owing to the fact that these forms became increasingly complicated, the production of moulds and matrices require tolerances which are particularly severe. One of the principal objectives to be reached is the precision of machining and the improvement of the micro geometrical state of machined surfaces, for the minimization of polishing operations after machining which still necessary to obtain a good finished part, which is required by the car industry and the aeronautic industry. To reach these goals, it is important to choose a tool which fulfils the requirements of a machining which could be carried out in various orientations. This will make possible the machining of complex forms by using a milling machine with several axes. Among the tools allowing the realization of these complex forms, there is the ball-end mill. The goal of this work is to study the geometry of the mill's spherical part, in order to determine the contact zones between the part of complex form and the tool which is following a trajectory in space.

We have determined in which conditions we have a regenerative vibration cases (chatter). We have used the simulation of a machining system dynamic behaviour. A cutting force law was obtained by an indirect method, by comparing between simulation results and experimental ones.

During manufacturing process of mechanical parts, the method of material remove by cutting tools have a significant importance, according to the matter of the part manufacturing process and to its raw elaboration method. On one hand, the characteristics of machining operations depend on the “studies office” choices during the part conception (the shape, the material, ...). On the other hand, they also depend on the “methods office” choices (machining gamut, choice of: machines, tools, cutting parameters, and tools paths, ...). For the “methods office”, some bad choices of cutting forces could change the final quality of the machined part. In this order, we have studied the dynamic cutting law evaluation problem. This consists in developing a simplified phenomenological cutting force model, in a case of turning under orthogonal machining conditions in presence of vibrations.

The simulation of the global Part-Tool-Machine-System (PTMS) dynamic behaviour during cutting is one of the validation procedures used to prepare a machining process. Various objectives could be reached when using simulation and this depends on the user who could be the machine-tool designer, the tool designer or the machinist. Two kinds of mathematical models, in order to develop simulation programs, are suggested for the (PTMS) modelling: the first one describes the (PTMS) as a mechanical structure and the second one describes the interaction between a tool and a worked part. In our study, we are interested in the second mathematical model and we must firstly establish a dynamic cutting force law, including different parameters and a damping term. The aim of this research paper is to suggest an identification method for the dynamic cutting force law parameters, calling (CDFC), by using a dynamometer in the case of an orthogonal cutting process.