OPERATION OF METAL-CUTTING MACHINE TOOL UNITS UNDER CONDITIONS OF INTENSIVE TOOL WEAR

V. I. Livshits

(Presented by the scientific seminar of the departments of machine tools and metal cutting and of mechanical-engineering technology)

Cutting difficult-to-machine materials is, from the very beginning, accompanied by intensive tool wear, which substantially changes the character and the result of the working process even over a small machining length. The need for a quantitative and qualitative assessment of the influence of tool wear on the cutting process and of its effect on the operation of machine-tool units is beyond doubt.

The most important factor that changes as the tool wears is the change in the effectiveness of the cutting wedge, i.e., the ability of the tool to penetrate under the chip. For a sharp tool this ability differs for different tool geometries, different cutting conditions, and different pairs of materials “tool—workpiece.” As the ability of the tool to penetrate under the chip changes, the component of the cutting resistance force \(P_x\) increases. In experiments on comparatively non-rigid lathes, and also on transverse-planing machines, cases were observed in which the growth of \(P_x\) leads to a stoppage of the tool displacement in the direction of the \(X\)-axis, i.e., to cessation of its actual feed. Then, owing to the continuing motion of the feed drive, additional tensioning of the feed chain occurs, ending in a jump of the tool that removes, wholly or partially, the accumulated thickness of cut. Similar phenomena may be repeated several times in succession.

The periodic alternation of penetration and non-penetration of the tool under the chip may be represented as intermittent motion consisting of jumps and stops. Considering the cutting process as a process of intermittent motion appears to be a more general approach than the traditional kinematic scheme of continuous motion, for example in turning. Under real conditions machining is carried out with stops, after which there inevitably occur starting from rest of heavy machine units, their acceleration, penetration into the workpiece, tensioning of the elastic system of the feed drive, tool exit, etc.

An analysis of intermittent motion as applied to machine-tool units was carried out by V. E. Push [1], in whose works an expression was obtained for the critical sliding velocity determining the transition from continuous motion to intermittent motion. On the basis of the approach developed by V. E. Push, it appears possible to find the critical value of the velocity of tool entry under the chip, at which

for this amount of wear (for example, on the flank face of the tool), the cutting wedge still retains the ability to penetrate beneath the chip. It is also of practical importance to determine the limiting value of the force \(P_n\) that is permissible from the condition of continuous motion.

The elastic system of the machine-tool feed chain can be represented by the diagram shown in Fig. 1 [1]. It consists of a driving member (drive) moving uniformly at velocity \(V_0\), transmission elastic members of reduced stiffness \(C\), and a load of mass \(m\) (the machine-tool unit) sliding along the guides. The load is acted upon by the following set of forces: (a) inertia forces; (b) the elastic force due to the action of the drive; (c) the cutting-resistance force \(P_x\); (d) the friction force on the guides.

Fig. 1

On the basis of metalworking practice it may be concluded that the force \(P_x\) in the elastic system of the feed chain should be considered for the following cases:

I. \(P_x=\mathrm{const}\). This case is characteristic of the machining of ordinary structural materials possessing good machinability by cutting, when tool wear proceeds with low intensity and affects the process only at the very end of the tool-life period.

II. \(P_x=f(t)\), where \(f(t)\) is an explicit and continuous function of time. This case is characteristic of the machining of difficult-to-machine materials, as was mentioned above.

III. \(P_x=\Phi(t)\), where \(\Phi(t)\) is a periodic discontinuous function, for example, of the “rectangular sine” type.

Such cases arise in the machining of complex noncontinuous surfaces with tools of the type of form cutters, round self-rotating cutters, cutters for power cutting, etc.

In what follows, the action on the elastic system of the feed chain of a linearly increasing-in-time force \(P_x\) is considered. This case corresponds to the real machining conditions of difficult-to-machine steels [2].

Of substantial importance is the question of the nature of the friction forces on the guides. V. E. Push considers the operation of an elastic system under conditions of viscous friction [1].

According to data from ENIMS [3], on feed guides it is not possible to attain purely fluid friction over a considerable part of the speed range. Depending on the operating conditions, the type of lubricant, and the oil-distribution system, the guides operate in regimes of boundary friction, mixed friction, fluid hydrodynamic friction, and hydrostatic friction.

For the case of dry friction of the load on the guides, let \(R_1\) denote the limiting static-friction force, and \(R_2\) the force of friction of motion. A motion of the system is possible in which the velocity of the load is equal to the velocity \(V_0\) of the driving member. In this case the compressed spring creates a force that overcomes the forces resisting the motion of the load, \(R_2\) and \(P_x\). If the velocity \(V_0\) is small, then some random obstacle may be sufficient to stop the load. Such an obstacle may be a stepwise increase in the force \(P_x\) as a result of a sharp change in the geometry of the cutting edge of the tool owing to its wear. After this instant the driving member, continuing its motion, will compress the spring until the compressive force overcomes the forces \(R_1\) and

the increased value of \(P_x\). Here the load will break away, and the friction force will sharply decrease to the value \(R_2\). The equilibrium of the forces acting on the load will be disturbed.

If the origin of time \(t=0\) is taken to coincide with the moment of breakaway, and displacements are measured from the place where the load stopped, then at the instant

\[ t=0,\ x=0,\ \dot{x}=0. \tag{1} \]

Expressions (1) may be taken as the initial conditions for the system in Fig. 1.

The differential equation of motion of the system is written in the following form [4]:

\[ m\ddot{x}=R_1-c(x-V_0t)-R_2-P_x. \tag{2} \]

Introducing the notation

\[ \kappa^2=\frac{c}{m}\quad \text{and}\quad P_x=\beta t, \]

we obtain

\[ \ddot{x}+\kappa^2 x=\left(\kappa^2 V_0-\frac{\beta}{m}\right)t+\frac{R_1-R_2}{m}. \tag{3} \]

The solution of the equation satisfying the initial conditions (1) is obtained in the following form:

\[ x=\left(V_0-\frac{\beta}{c}\right)t-\frac{1}{\kappa}\left(V_0-\frac{\beta}{c}\right)\sin\kappa t+\frac{R_1-R_2}{c}(1-\cos\kappa t). \tag{4} \]

In the right-hand side, the first term expresses uniform motion, while the remaining terms express additional oscillations of the load. Ya. G. Panovko shows that after breakaway, after some time, stopping again occurs, followed by a new breakaway, and the next cycle begins, completely coinciding with the preceding one [4]. The process under consideration is stationary self-oscillation about the state of uniform motion of the load, which has proved to be unstable. The lower the speed of the driving link, the more sharply the self-oscillation process is expressed. Under the conditions of a complex multi-joint elastic feed-chain system, these self-oscillations may be damped to a considerable extent and therefore may manifest themselves weakly.

It follows from solution (4) that the uniform motion of the load occurs with velocity \(V_0-\dfrac{\beta}{c}\). This makes it possible to draw an important conclusion for the operation of machine tools: when the cutting resistance force \(P_x\) increases linearly with time, the feed during the steady-state process is less than the nominal feed. The additional compressive force of the elastic feed-chain system, arising due to the difference in the velocities of the load and the driving link, is expended in overcoming the increment in the cutting resistance force \(P_x\). This is a substantial difference between the process and operation at \(P_x=\mathrm{const}\), when the feed during the steady-state process does not differ from the nominal one [5]. The increase in feed at tool exit, known in practice when operating under conditions of a linearly increasing force \(P_x\), is aggravated, which leads to a greater increase in the cross section of the cut and to a jump in force at exit.

On the basis of solution (4), one may also conclude that, in the case of a large value of \(\beta\) with small \(C\), the increase in force \(P_x\) will lead to stoppage of the feed motion.

Obviously, there exists a limiting value of the elastic deformation of the feed chain during cutting with a linearly increasing force \(P_x\). V. E. Push

[5] proposes the following expression for determining the total deformation of the feed chain:

\[ x=t\cdot \frac{1}{2c_x \tg \varphi}\left(2c_x \tg \varphi\, S_0 + c - \sqrt{c^2+4c_x \tg \varphi\, S_0 c}\right), \tag{5} \]

where

\(C_x\) is the coefficient in the empirical formula for the cutting force, characterizing the properties of the material being machined;

\(\varphi\) is the angle in the tool plane;

\(S_0\) is the nominal feed value.

In our view, this expression does not reflect the essence of the operation of the feed chain in machine tools, since here, as \(t\) increases, the deformation of the feed chain \(x\) increases without bound, which cannot occur in real designs.

REFERENCES

1. V. E. Push. Small Displacements in Machine Tools. Moscow, Mashgiz, 1961.

2. V. I. Livshits. Investigation of the Machinability by Cutting of High-Manganese Steels, dissertation. Tomsk, 1966.

3. Parts and Mechanisms of Metal-Cutting Machine Tools. Vol. 1, edited by D. N. Reshetov. Moscow, Mashinostroenie, 1972.

4. Ya. G. Panovko. Introduction to the Theory of Mechanical Vibrations. Moscow, Nauka, 1971.

5. V. E. Push. On the Jump in Force When the Tool Exits. Stanki i instrument, 1948, No. 6.