Dynamic model of the three-dimensional cut.

Bisu, Claudiu Florinel ; Darnis, Philippe ; K'nevez, Jean yves 等

Abstract: The determination of a dynamic law of cut is complex and often very difficult to develop. Several formulations were developed, in very complex ways being given that 3D crosses from there, the number of variables is much higher than out of orthogonal cut. The existence of the plan of displacements and the correlations with the elastic characteristics of the machining system thus make it possible to simplify the dynamic model 3D. A dynamic model is proposed on the basis of experimental approach. Simulation is in concord with the experimental results.

Keywords: self-excited vibrations, dynamic model 3D in turning, dynamic model 2D equivalent in turning,

1. INTRODUCTION

In order to reduce the costs and the times of adjustment of the manufacturing processes, the model approach seems an ideal solution. To represent as well, as possible, the process of cut, these models must integrate the physical, thermal and dynamic phenomena related to the formation of the chip and the generation of surfaces. Industrial configurations of machining as well as preceding research tasks (Darnis et al., 2000), (Cahuc et al., 2001) show the need for a three-dimensional modeling of the process, which increases the complexity of the resolution of the problem of the dynamics of the cut.

The dynamic model is based on semi-analytical thermo-mechanical model of three-dimensional cutting (Laheurte, 2004), which describes finely the contact tool/part/chip. The model takes into account various areas of stresses such as plastic deformation, in the primary area of shearing, or of the stresses of contact, during the process of formation of the chip. Modeling is carried out under the conditions of self-excited vibrations.

[FIGURE 1 OMITTED]

2. CONSTRUCTION OF MODEL

2.1 Formulation of the problem

The model of cut is based on a 3D configuration of machining, with a system with 3 dof (degrees of freedom). The movement of the tool is expressed in a reference coordinate system related to the tool, which is then projected in the coordinate system related to the machining system.

[FIGURE 2 OMITTED]

2.2 Dynamic description of the contact tool/part/chip

By assumption, the dynamic law of cut defined in established mode. The transient states are in general too dependent on the initial conditions so that the results of simulation can be compared quantitatively with those of the experiment. With this method, we determine the relation between the cutting forces and the instantaneous parameters of machining. The principle consists in determining the cutting forces starting from the optimization of the angle of shear integrated in the dynamic model. These variations of the contact tool/part/chip are due to relative displacements tool/part, tool/chip, and generate variations of the section of chip, speed of the chip, rake angle, clearance angle and shearing angle. The variations of the contact tool/part/chip are then examined, on each direction. The relative displacement of the tool causes certain physical variations of the tool section, (Bisu, 2007, a). The dynamic model is based on semi-analytical thermo-mechanical model of three-dimensional cutting (Laheurte, 2004), which describes finely the contact tool/part/chip. The model takes into account various areas of stresses such as plastic deformation, in the primary area of shearing, or of the stresses of contact, during the process of formation of the chip. Modeling is carried out under the conditions of self-excited vibrations.

[FIGURE 3 OMITTED]

3 MODELING OF THE CUTTING FORCES

3.1 Three dimensional model

The determination of the parameters of contact and the dynamic geometrical parameters constitute the first stage, which leads to the evaluation of the cutting forces. Dynamic modeling is based on the semi-analytical model developed by (Dargnat, 2006), which provide us the nominal values of the results for an iteration. In this model it is possible to know the forces in each zone of contact tool/chip/workpiece (Fig.3):

* in the secondary shearing zone along (OB)

* the area of skin is divided into two parts: the area of the radius of acuity (area OJ) and the rectilinear area (area JK), Projection in the coordinate system x2, y2, z2 of the cutting forces (Fig.3) and then in the global coordinate system makes it possible to write the differential system of equation in the matrix form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

with [[M.sub.3]] the mass matrix, [[C.sub.3]] the damping matrix and [[K.sub.3]] the stiffness matrix. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are respectively the displacement vector and the force vector in the three-dimensional reference coordinates (subscript 3). The dote is the time derivative.

3.2 Two-dimensional dynamic model after projection

The experimental analysis showed the existence of a specific plan of displacement of the tool, in which the point of the tool describes an ellipse (Bisu, 2007, b). Displacements are generated during the variation of the cutting forces, which are located on a level equivalent to the plan of displacements. The determination of this plan enables us to adopt a real configuration of the cut, in this coordinate system related to the axes of the ellipse in the plan described. This new two-dimensional model is equivalent to characterize the three-dimensional cutting. A plan different from the orthogonal plan of cutting machining is defined but it characterizes the dynamic behavior of the machining system.

The cutting forces are defined in this two-dimensional coordinate system ([n.sub.fa], [n.sub.fb]) (Fig. 4). Displacements are also defined in this coordinate system.

[FIGURE 4 OMITTED]

Then the system can be written in the matrix form :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

with [[M.sub.2]] the mass matrix, [[C.sub.2]] the damping matrix and [[K.sub.2]] the stiffness matrix; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are respectively the projection of displacement vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and force vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from 3D rectangular reference coordinates to 2D reference coordinates.

4. RESULTS AND ANALYZES

The two systems of equations are solved separately using the Runge Kutta order 5 method. The results obtained by simulation in the two-dimensional case are coherent with the experimental data (Table 1), (Bisu, 2007, c):

where [a.sub.u] and [b.sub.u] represent the large axis and the small axis of the ellipse displacement, in the theoretical case and the experimental case.

5. CONCLUSIONS

A model of dynamic cut original was developed. It integrates the stationary model of cut developed by the laboratories of Bordeaux: LMP and [LGM.sup.2] B (University Bordeaux 1). While reviewing some existing dynamic phenomena during the vibratory cut, and by assumptions for this modeling, a dynamic model is designed. A first resolution of the system is carried out and the results obtained are coherent. The existence and the analysis of the plan of displacements make it possible to transform the 3D problem to a 2D problem, faster and more easy to solve. It represents a first evolution of a three-dimensional dynamic model of the cut. The model makes it possible to obtain results according to the experimental values.

The prospects offered by this model it allows predicting the dynamic forces of cutting but also the morphology of the chip.

6. REFERENCES

Darnis, P.; Cahuc, O. & Couetard Y. (2000). Energy balance with mechanical actions measurement during turning process, International Seminar on Improving Machine Tool Performance, La baule, 3-5 July.

Cahuc, O.; Darnis, P.; Gerard, A. & Bataglia J. (2001). Experimental and analytical balance sheet in turning applications, International Journal of Advanced Manufacturing Technologies, Vol. 18, No. 9, pp. 648-656

Laheurte R. (2004). Second gradient theory applied to cutting of materials, PhD Thesis (in French), University Bordeaux 1- France.

Dargnat, F. (2006) Semi-analytical modelling by energy approach of monolithic materials drilling process, PhD Thesis (in French), University Bordeaux 1- France.

Bisu, C. (2007, a). Self-excited vibrations study in three-dimensional cut : new modelling applied to turning, PhD Thesis (in French), University Bordeaux 1--University POLITEHNICA of Bucharest.

Bisu, C., Darnis, P. K'nevez, J-Y., Cahuc, O., Laheurte,R., Gerard, A., Ispas, C., (2007, b)). New vibrations phenomena analysis to turning. Mecanique et Industries.

Bisu, C., Laheurte, R., Gerard, A., K'nevez, J-Y. (2006, c) The regenerative vibration influence on the mechanical actions turning , 15th Int. Conf. on Manufact. Syst., Bucharest, Roumanie, 26-27 October.

Table 1 Comparison experiment/simulation of displacements.

[f.sub.(mm/tr)] [a.sub.u(theorique)] [a.sub.u(experimental)]

0.1 8.5 x [10.sup.-5] m 7.71 x [10.sup.-5]
0.075 6.2 x [10.sup.-5] m 5.6 x [10.sup.-5]
0.0625 4.1 x [10.sup.-5] m 3.87 x [10.sup.-5]
0.05 2.9 x [10.sup.-5] m 2.83 x [10.sup.-5]

[f.sub.(mm/tr)] [b.sub.u(theorique)] [b.sub.u(experimental)]

0.1 1.5 x [10.sup.-5] m 1.72 x [10.sup.-5]
0.075 1.28 x [10.sup.-5] m 1.23 x [10.sup.-5]
0.0625 1.1 x [10.sup.-5] m 0.83 x [10.sup.-5]
0.05 0.7 x [10.sup.-5] m 0.51 x [10.sup.-5]