The mathematical model of the cutting threads kinematic axis without reversing the main movement.

Ciupan, Cornel

Abstract: The following paper presents the achievement of cutting thread kinematic axis without reversing the main axis. This axis was conceived for lathe and other cutting thread machines. The novelty of the solution consists of introducing reversing gear with timing within the axis. This paper shows research of the behaviour of this axis.

Key words: thread, lathe, kinematic, axis.

1. INTRODUCTION

For cutting thread on lathe it is used the reversing of the advance movement method through the main movement. The disadvantage of the method consists of reversing the main movement (the rotation movement of the piece) because of the high inertia of the main kinematic axis (Galis, et al., 1994). This disadvantage leads to the usage of low revolution for the cutting thread. The cutting thread that consists of more passes assumes the existence of an active stroke followed by a retreat race. In the case of known solutions, the retreat stroke has higher speed of about 15-20% than the active stroke. The statements above lead to low productivity (Nzumbe-Mesape, 2005) (Popescu, et al., 1995). Although this field was less researched in the past years (Vanin, 2004), the author of this paper has patented a new solution that allows the cutting of the thread without reversing the main movement (Ciupan, 1993). The return of the cutting tool at the beginning of the cutting thread is made with fast advancing.

2. DESCRIPTION OF THE AXIS

The author has conceived and developed a new method of threading lathes. The solution, patented by the author--RO 105781, is shown in figure 1. The novelty of this solution consists of a supplementary reverser Is in the advancing axis Z. The return of the cutting tool in achieved only by reversing the advance axis from the reverser Is. The reverser Is is made out of a differential mechanism D, a coupling Cd and a break B. The necessary stages for the cutting thread are presented in table 1.

[FIGURE 1 OMITTED]

A feed reversing system FR commands the stages of the cutting thread, manually or automatically.

3. THE MATHEMATICAL MODEL

In the case of classical cutting thread kinematic axis, the reversing of the movement is realized by lamellar coupling witch attenuates the shocks. The kinematic axis without reversing the main movement uses, for self-synchronizing, a shifting-jay clutch coupling. This coupling produces a shock at the beginning of every cutting pass. The shock occurs because of the dynamic load.

Considering the big inertia of the main kinematic axis of the mechanisms and the fact that the reversing of the advancing movement takes place only in the kinematic axis of advance, the influence of the dynamic load on the modification of the angle speed U0i of the engine M is negligible. Even if U0i is modified, it does not influence the thread pitch.

A big influence on the process of cutting thread and its precision can be made by the elastically deformation of the transmitting elements. Considering constant the speed of the part, due to the elastic deformation of the transmission elements it will produce an oscillation of the pitch thread around the fixed value.

The simplifying diagram from figure 2 was used for studying the dynamic of the coupling process of the advance kinematic axis.

Taking into account the forces that act upon the mass M and based on figure 2, the dynamic equation of movement could be obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[FIGURE 2 OMITTED]

In relation (1), we have:

[[beta].sub.m]--the tilt angle of the medium helix of the leader screw; [phi]--the friction angle; [k.sub.S]--the constant of the spring; d--the medium diameter of the screw; [i.sub.1], [i.sub.2]--kinematic transfer rate; z--displacement of tool.

Applying to the differential equation (1) the Laplace transformed formula, we obtain:

M[s.sup.2]z(s) + f s z(s) + [C.sub.1][k.sub.S] [theta](s) (2)

[C.sub.1] = 4[pi] / [P.sub.S][d.sub.S][t.sub.g]([[beta].sub.m]] + [phi]) (3)

From relation (2) we get the transfer function of the system:

H(s) = [C.sub.2][k.sub.S] / M[s.sup.2] + f s + [C.sub.1][k.sub.S]. (4)

A = f / 2M; [omega] = 1 / 2M [square root of (4M[C.sub.1][k.sub.S]-[f.sup.2]] (5)

Using the notations (5), the next form of the transfer function could be obtained:

H(s) = [C.sub.2][k.sub.S] / [(s - a).sup.2] + [[omega].sup.2] (6)

The beginning of the pitch of the thread screw suffers a declination, damped around to the fixed value PE:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

A programme was developed which helped determine the answer of the system for step input signal following relations (6) and (7). The program contains specific data for the lathe SN 560x1000, because on this machine took place the experimentation of the invention and the experimental research.

The data studied were the stability of the system depending on the transmission's mechanical characteristics and the influence of the coupling shock on the pitch of the thread.

The elasticity of the transmission ks has a direct effect on the behaviour of the system. The different values of the constants were introduced in the program as a j parameter for obtaining the diagrams from only one pass. Thus, the output signal is dependent of two parameters t and j.

The system behaviour for a unit step input signal is shown in the next two figures. Figure 3 presents the evolution of the tool movement along the advance axis Z, considering the amortization factor a=12,5 x j. The influence of the ks upon the axis behaviour is shown in figure 4.

The transitory regime produced by coupling the cutting thread faze, produces a declination of the tool from the helicoidally trajectory, phenomenon that modifies the pitch of the tread.

For studying this phenomenon the program was run with four values of the pitch thread, pj=2xj mm; j=1 ... 4, as shown in figure 5.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

4. CONCLUSION

Analysing the graphics presented in figures 3, 4 and 5, the following have been established:

- the system is stable and the output signal is delayed related to the input signal for all the studied cases;

- the beginning of the helicoidally trajectory is strongly affected, but, due to the low transitory regime, (0,2 seconds), the tool enters again on the helicoidally trajectory before beginning to cut;

- the performance of the system can be increased by redesigning the transmission elements.

5. REFERENCES

Ciupan. C. (1993) Kinematic chain of thread. Patent RO 105781

Galis M. & all. (1994). Machine tools design. Transilvania Press, ISBN 973-95-635-4-3, Cluj-Napoca

Nzumbe-Mesape N. (2005). Design and manufacture of screw threads. The International Journal of Mechanical Engineering Education. Vol. 33, No. 3, (July 2005) page numbers (208-214), ISSN 0306-4190

Popescu S. & all. (1995). Special machine tools. Casa Cartii de Stiinta, ISBN 973-9204-15-5, Cluj-Napoca

Vanin, V. A. (2004). Kinematic structure of machines for cutting screw with a variable pitch. Chemical and Petroleum Engineering, Vol. 39, No. 3-4, (March 2003) page numbers (244-248), ISSN 1573-8329

Table 1. The phases of the cutting thread

Phase Cd B Transfer rate

Cutting 1 0 i = 1
Return 0 1 i = [z.sub.1] / [z.sub.3]
Stop 0 0 i = 0