Radial Forces in Hole Burnishing with a Ball Mandrel

V. D. Varlakov

(Presented by Prof. A. N. Eremin, Doctor of Technical Sciences)

The question of the radial forces arising during the burnishing of internal surfaces has not been covered at all in the literature. Knowledge of the magnitude of these forces is necessary in order to make a correct calculation of the design of tools for processing holes by burnishing. The data of all studies on the pressures necessary for surface work-hardening, or accompanying this work-hardening, relate exclusively to external surfaces, considering the question of obtaining work-hardening on external surfaces. Researchers usually set quite definite pressures and then determined how these pressures affect the depth of work-hardening. This was the case in the studies of Föppl [1], Herger [2], and a number of others. All of them place the depth of work-hardening in dependence on the pressure created or arising during processing. In our experiments [10], complete agreement was obtained between the depth of penetration of work-hardening and the data of other researchers. However, we examined the question of the change in the depth of work-hardening as a function of the thickness of the layer being deformed.

In the present article the author discusses questions of the method for determining the magnitude of radial pressures and their dependence on a number of factors.

Method for Conducting Experiments and Apparatus for Measuring Radial Forces

In our investigations, holes were processed with a ball mandrel. The principal pressure arising on the balls is the radial pressure. This pressure is caused by the elasticity and plasticity of the material being processed. In setting up experiments to determine the magnitude of radial pressures during the burnishing of holes with a ball mandrel, the author designed a hydraulic dynamometer, manufactured in the metal-cutting laboratory of the Tomsk Polytechnic Institute. A drawing of the dynamometer is given in Fig. 1, and the scheme of its operation is presented in Fig. 2. Here 1 is the specimen being burnished, 2 and 4 are cones of hardened steel, between which the balls 3 are placed. The angle of inclination of the cones is equal to 45°, as a result of which the magnitude of the reaction along the rod of the dynamometer is equal to the total pressure on the balls. Cone 2, through rod 9, is rigidly connected with the body of the dynamometer, while cone 4 sits freely on rod 9 and can move along this rod together with piston 10. When moving to the left, the piston acts on rubber membrane 8, which hermetically closes cavity 6. Cavity 6 is filled with a mixture of glycerin and water and is connected by a tube to pressure gauge 5.

Fig. 1. Hydraulic dynamometer for measuring radial forces during hole burnishing.

Visible drawing annotations: “to the pressure gauge”; “M24×2”; “Morse taper No. 2.”
Visible dimensions include: 120; ⌀20; ⌀35; ⌀95; ⌀45; ⌀130; 30; 30.

The pressure arising on the balls during the process of burnishing the hole is transmitted through the piston to the liquid and to the pressure gauge. The harmful resistances of the dynamometer pressure gauge were eliminated by us during its calibration.

The dynamometer was calibrated in the following manner. A sleeve 1 (Fig. 3) was clamped in the chuck of the headstock of a powerful lathe,

Fig. 2
Schematic of the hydraulic dynamometer

against which the body of the dynamometer rested; in this way its accurate centering was ensured. During calibration it was necessary for us to create pressures analogous to the working pressures. These pressures were applied to the piston through the cup 3 by the center of the tailstock. The magnitude of the pressure of the tailstock center 4 on the dynamometer piston was checked by means of the control dynamometer 5, placed between center 4 and cup 3.

Fig. 3
Schematic of dynamometer calibration

At the same time, the readings of the control pressure gauge and of our pressure gauge were recorded. On the basis of the data obtained, a calibration graph was constructed. Simultaneously with calibration of the load, calibration of the piston displacement was carried out; this was needed by us in order to determine the true size of the dynamometer mandrel under load, since here, as the load increases, the piston displacement also increases. The method for determining the true size of the hydraulic mandrel will be described below.

The calibration graph for piston displacement was constructed in the following manner. We used two indicators (Fig. 3). The spindle of one of them (6) rested on the piston, and the spindle of the second (7) on the body of the dynamometer. The difference between the readings of the first and second indicators gives the true value of the piston displacement under the corresponding load relative to the dynamometer body.

Experiments to determine the radial forces were carried out on a lathe of the Kuibyshev Plant. The dynamometer was fastened by its shank in the lathe carriage, exactly along the line of its centers.

The experiments were conducted on various carbon steels, with different feeds, speeds, and numbers of passes. As test specimens, rings 23 mm high with holes previously machined by two reaming operations were used.

Dependence of radial forces on the burnishing conditions

Before considering the radial forces that arise during burnishing, let us dwell on the features of our dynamometer.

Under the action of the radial forces arising during burnishing of holes, the movable cone of the dynamometer acts on its piston. Displacement of the movable cone can be obtained only through the mutual action on it of several balls acting simultaneously on the machined surface. If the balls of the dynamometer mandrel are regarded as analogous to the balls of a thrust bearing, then in practice it turns out that the position of the movable cone is determined by the mutual arrangement of three balls. These three balls take up the action of the material being machined and transmit it to the dynamometer piston. All the remaining balls merely align their position relative to the first three. The pressure arising on all the balls except the three does not cause any additional displacement of the piston. The proposition we have stated is fully confirmed by experiment (Table 1).

Table 1

Influence of the number of balls on the magnitude of radial pressure

As can be seen from the table, the number of balls varied from 8 to 3; in this case the manometer readings remained stable, which indicates the absence of any influence of the number of balls on the magnitude of the total radial force. We did not use fewer than three balls, since in that case additional supports would have been required to hold two balls in a strictly diametral position, or else a special device creating pressure on one ball (when the ball diameter is less than the hole diameter). Our aim, however, was to determine the radial forces arising during hole burnishing and caused by the elasticity of the material being machined.

When working with a rigid mandrel, the question of the mandrel size was solved very simply—by measuring it with a precision caliper or micrometer; the size thus obtained is the working size. When working with a hydraulic mandrel, however, the situation is different. The magnitude of the pressures is recorded by us from the manometer readings, and for this purpose we must act on the rubber membrane with a piston. Hence, during operation the size of the mandrel will be different from that when it is installed in the unloaded state. Setting the size with a micrometer or caliper is impossible in this case, since the presence of a massive piston precludes any guarantee of its tight contact with the dynamometer membrane without creating a preliminary interference on the tapers. Obtaining such an interference with the aid of a micrometer is excluded. These circumstances compelled us to use the following method for setting the mandrel size. We made a hardened, ground ring with a size of 38.1 mm. Placing the ring on the mandrel, we created a certain preliminary-interference pressure. Suppose this pressure is equal to 1.5 atm; the displacement of the piston from 0 to 1.5 atm is known to us from the calibration curve and is equal to 0.03 mm. During operation a pressure equal to 5.7 atm arose, and the piston was displaced by 0.23 mm. Then the actual size of the mandrel will be:

\[ d_y = 38.1 + 0.03 - 0.23 = 37.9\ \text{mm}. \]

By changing the preliminary-interference pressure, we can obtain different setting diameters with one and the same ring. Thus, at \(P_{\text{set}} = 2.7\) atm, for a ring 38.1 mm in diameter, the piston displacement during setting will be 0.1 mm.

We used this method to vary the setting diameter of the mandrel and to determine the influence of the difference between the setting diameter and the hole diameter on the magnitude of the radial forces.

The correctness of our reasoning was fully confirmed by the verification carried out for the depth of penetration of work hardening. The verification consisted in drawing on the graph in Fig. 4 a straight line corresponding to the depth of penetration of work hardening at different diameter differences, obtained from formula [10] for the given speed and feed. After this, points were plotted corresponding to the penetration depths at different differences between the diameters of the mandrel and the hole in this series of experiments. The graph shows complete agreement between the calculated straight line and the experimental points.

The dependence of the radial forces arising during hole burnishing on the difference in diameters is presented in the graph in Fig. 5.

It is seen from the graph that the magnitude of the radial pressure depends on the thickness of the deformed layer and can be expressed by the formula

\[ P = C_p \delta d y, \tag{1} \]

where \(P\) is the radial pressure,
\(\delta d\) is the difference between the diameters of the “burnisher” and of the hole before machining,

\(C_p\)—a constant coefficient depending on the material and the experimental conditions,

\(y\)—an exponent depending on the material.

For steel with hardness \(H_b = 197\) and balls of diameter \(7\) mm, equation (1) is written as

\[ P = 6790\,\delta d^y . \tag{2} \]

Fig. 4. Dependence of the depth of work-hardening penetration on the difference in diameters

Fig. 5. Dependence of the radial force on the difference in diameters

Next we carried out a series of experiments to determine the influence of speed and of the number of layer passes on the magnitude of the radial force. The speed was varied within the range from \(8.85\) to \(68.2\) m/min, and the number of layer passes from 1 to 5 (Figs. 6 and 7).

From both graphs it is evident that there is a complete absence of influence of the burnishing speed on the magnitude of the radial pressure arising during burnishing of holes. This is explained by the fact that in the experiments we maintained a constant difference between the diameters of the mandrel and the hole, i.e., the thickness of the deformed layer.

These graphs show a decrease in radial pressure as the number of layer passes increases. This phenomenon is explained by the fact that after each pass the hole of the specimen or part increases its diameter by a certain, though small, amount, while the size of the mandrel does not change. Hence the layer of metal subjected to deformation gradually decreases—

Fig. 6. Dependence of the radial force on the number of passes and speed

and, consequently, a natural decrease in the force required for its deformation. In the limiting case, the increase in the diameter of the hole may be such that it becomes equal to the initial difference between the diameters of the mandrel and the hole, and the hole will become equal to the mandrel diameter. In this case the pressure will become equal to zero.

Fig. 7. Dependence of the radial force on the number of passes and speed

In determining the influence of feed on the magnitude of the radial pressure, we varied the feed within the range from 0.018 to 1.760 mm/rev (Fig. 8).

From the graph in Fig. 8, a certain increase in radial pressure with increasing feed is evident. This increase in pressure is natural, since in the present case there is an increase in the volume of the metal being deformed. An analogous phenomenon is observed in cutting, where, with increasing feed, an increase in cutting force is also observed [3], although somewhat more intensive than in our case.

Fig. 8. Dependence of the radial force on feed

Next we carried out experiments to determine the influence of steel hardness on the magnitude of the radial pressure (Fig. 9). With increasing steel hardness, the magnitude of the radial pressure increases. The experiments showed that the pressure increases with increasing \(H_{\mathrm{B}}\). The somewhat smaller increase in the radial force of resistance of the metal to penetration of the mandrel (6.45%), compared with the increase in hardness (50.45%), may be explained by a change in the plasticity of the steel.

In conclusion, we set ourselves the task of determining the dependence of the radial force on the diameter of the hole being machined and of the ball. From the available data it could be assumed that this dependence should obey the law of similarity. The law of similarity was first formulated by Thompson and then developed by Barr [7]. Barr’s principle of similarity states: “Similar bodies of the same material, when subjected to loads that are identical in magnitude, stress, and distribution, are stressed and deformed to the same degree both within and beyond the elastic range, in proportion to the square of their limiting dimensions.” As [8] also establishes, the equivalence of deformation in geometrically similar bodies of the same material is determined by the magnitude of the work, which is proportional to the weight or volume of these bodies. The same regularity is confirmed in the studies of Meyer [9] and a number of other investigators.

Fig. 9. Dependence of the radial force on the initial hardness of the steel
(graph labels visible: \(P_{r}\), kg; \(HB\); \(n = 350\) rpm; \(S = 0.398\) mm/rev.)

As a consequence of the law of similarity one may write:

\[ \frac{P_1}{D_1^2}=\frac{P_2}{D_2^2}=\ldots=\frac{P_n}{D_n^2}, \tag{3} \]

where \(P\) is the applied load,
\(D\) is the diameter of the ball.

This regularity is valid for the same degree of deformation.

In our experiments, a constant difference of diameters, or thickness of the deformed layer, a constant burnishing speed, and feed per revolution of the part were used. The material was also constant in all cases. The results of the experiments are given in Table 2.

Table 2

Effect of ball diameter on the magnitude of the radial force

It is clear from the table presented that the law of similarity is entirely valid in our case; therefore, from equation (3),

\[ P_n=\frac{P_1D_n^2}{D_1^2}. \tag{4} \]

Taking \(P_1=C_p\delta d^y\) and substituting into equation (4), we obtain:

\[ P_n=\frac{C_pSd^yD_n^2}{D_1^2}. \tag{5} \]

If the specific values of \(C_p\), \(y\), and \(D_1=7\) mm are substituted into formula (5), then

\[ P_n=\frac{6790\delta d^{1.75}D_n^2}{49}=13.85d^{1.75}D_n^2. \tag{6} \]

Using equation (6), one can determine the value of the radial force arising in the mandrel when burnishing steel with \(H_B=197\) for any difference between the diameters of the mandrel and the hole, and for any ball diameter.

Knowledge of the total pressure in hole burnishing is necessary for calculating mandrels for strength. As shown above, increasing the number of passes over a layer leads to a decrease in the radial force, while the initial hardness of the material and the feed have little effect on its magnitude (the burnishing speed has no effect). Therefore, there is no need to introduce into equation (5) additional terms relating the radial pressure to the number of passes, feed, and initial hardness of the material. The maximum pressure is obtained during the first pass, at the greatest feed and material hardness. Introducing into the formula a dependence on the number of passes gives a reduction in the value of the radial pressure, while the mandrel will experience the greatest pressure during the first pass; hence equation (5) may be adopted for design calculations. Feed affects this pressure very little, and this effect may be neglected. Equation (6) gives values of radial pressure for steel with \(H_B=197\), that is, for a fairly hard structural steel. When calculating a mandrel for processing softer steels by this equation, a certain (very small) increase in its strength will result. Processing steels with a higher initial hardness is unlikely to occur in practice, since in that case there will be only a slight increase in the hardness of the surface layer due to work hardening. If such steels contain a high amount of carbon, an increase in hardness can be obtained by quenching.

In view of this, equation (6) may be proposed for calculating radial pressure on ball mandrels intended for machining all structural steels.

REFERENCES

1. Feppel O., Bekker E., Heidekamp G. Long-Term Testing of Metals, ONTI, 1935.
2. Herger, Die Werkzeugmaschine, No. 19, 1937.
3. Standardization of Machine-Tool Operations, Mashgiz, 1949.

1. Kuznetsov V. D. Physics of the Solid State, vols. 1, 2, 3, Tomsk, Krasnoye Znamya Publishing House.
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5. Kick, F. Das Gesetz d. proportionalen Widerstände. Leipzig, 1885.
6. Meyer, E. ZdvDJ, vol. 52, 1908.
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