Countermeasures against polygonal deformation of borehole in reaming process

Reaming is a processing method for widening pilot holes such as drill holes and press holes to obtain a highly accurate finish. However, in machining using a regular-pitch reamer, polygonal deformation of the borehole occurs. This deformation is also called a spiral mark because the polygonal shape twists in the feed direction. Although several papers have dealt with vibration phenomena during reaming, countermeasures to date have not been sufficient. In previous studies, the authors considered polygonal deformation of a machined hole during reaming as a self-excited vibration caused by a time delay and clarified its mechanism. In the present study, we theoretically and experimentally investigated the suppression of polygonal deformation by optimizing the angular arrangement of the cutting edges of irregular-pitch reamers for an 8-flute reamer. In addition, we suggested a new evaluation standard to reduce the calculation load and to evaluate the optimum angular arrangement of the irregular-pitch cutting edges of reamers with other numbers of flutes.


Introduction
Reaming is a finishing process to improve hole accuracy after drilling.However, the tool vibrates during machining, which creates the problem of reduced hole accuracy.When vibration occurs, the machined hole deforms into a polygonal shape.This phenomenon is called polygonal deformation or pattern formation, and it occurs also in drilling and BTA deep hole drilling.The polygonal deformation of a hole is characterized by a helical twist in the processing direction.Although several papers have dealt with vibration phenomena during reaming [1]- [6], countermeasures against polygonal deformation of reamed boreholes have not been sufficient.
The authors previously clarified the mechanism behind polygonal deformation in reaming with 6flute reamers and BTA deep hole drilling as self-excited vibration due to time delay [7], [8], and also discussed the suppressive effect of reaming with an irregular pitch, as well as the countermeasure of an additional guide pad in BTA tooling.However, the optimum angular arrangement of the reamer with irregular-pitch cutting edges (referred to hereafter as an irregular-pitch reamer) to prevent polygonal deformation has not yet been determined.
In the present study, we theoretically and experimentally investigate the optimum angular arrangement of an irregular-pitch reamer with 8 flutes.In addition, we suggest a new evaluation standard to reduce the calculation load and to evaluate the optimum angular arrangement of the irregular-pitch cutting edges of reamers with other numbers of flutes.

Analytical model
Figure 1 shows the analysis model for 8-flute reaming.The same theoretical analysis as reported in a previous paper [7] was used in this study, except the number of cutting edges was different.In general reaming, the workpiece is fixed in space and the reamer rotates.To consider the fundamental mechanism of polygonal deformation in reaming, it is assumed that the reamer is fixed in the space and the workpiece rotates in the analytical model.The workpiece has a pilot hole and rotates clockwise at angular velocity .Defining the x and y axes from the center of the prepared hole, the cutting edges are numbered 1, 2, ..., n (n=8) from the x-axis.Let  i be the angle of cutting edge i from the x-axis.Primary and radial cutting forces are represented by P i and Q i , respectively.The normal force and the frictional force acting on the cutting edge are represented by N i and F i =N i ( is the coefficient of dynamic friction), respectively.Let M be the equivalent mass of the reamer and spindle, and K and C be the equivalent spring constant and equivalent viscous damping coefficient in the x and y directions, the equations of motion for the reamer and spindle are given by equations ( 1) and ( 2 The radial cutting force acting on each cutting edge is assumed to be proportional to the primary cutting force.The radial displacement r i of the i-th cutting edge is expressed as follows: The primary cutting force is assumed to be proportional to the radial displacement r i of the i-th cutting edge.The contact force is assumed to be proportional to the difference between the radial displacement of the cutting edge and the hole surface profile in contact.The hole profile is equal to the radial displacement of the cutting edge at the time the hole profile was formed.Therefore, the equations of motion (1) include a time delay related to the angular difference between the cutting edges.The details of the theoretical analysis are the same as in the previous paper [7], so they are omitted in this paper.The solution to equation (1) written as: ( ) , ( ) The characteristic root s= +iN (i= 1  ) is obtained from the characteristic equation, and the stability is determined.The real part  represents the instability of the system, and if there exists more than one characteristic root with a positive , the system becomes unstable and polygonal deformation occurs.The imaginary part N of the unstable characteristic root represents the number of sides of polygon.

Numerical calculations
Table 1 shows the angular arrangement of the cutting edges for the selected reamers.Consecutive numbers are the angles between adjacent cutting edges, shown in counterclockwise order in figure 1.
Tool A is a commonly used reamer with even-pitch cutting edges (referred to hereafter as the regular-pitch reamer), and Tools B, C, and D are reamers with irregular-pitch cutting edges.Numerical calculations were performed up to 17 polygons and at a rotation speed lower than 100 Hz.All characteristic roots were calculated near 0 rad/s, and the loci of the roots were calculated continuously up to 100 Hz.The tool was evaluated using the maximum value of the real part of all characteristic roots  max .
Figure 2 shows the relationship between the maximum value of the real part for each locus of the characteristic root and its imaginary part.Figure 2(a) shows the results for Tool A. Among all characteristic roots, the maximum real part  max is 3.19  10 -2 .The tool was evaluated using the maximum value of the real part of the characteristic roots  max .Polygons with 7, 9, 15, and 17 sides are unstable.When these tools are used, polygonal deformation is highly likely to occur.In reaming, it is empirically known that polygons with an integer multiple of the number of teeth 1  are likely to occur [9], and the analytical results are consistent with this.Tool B and Tool C are two types of tools selected by changing the angle of the cutting edge in 5-deg increments.Figure 2 2(a), it can be seen that the characteristic roots near N=7, 15, and 17, which were unstable, are stabilized.However, the characteristic root near N=9 is still unstable although the value of the real part has decreased, and the characteristic root near N=11, which was stable with the regular-pitch reamer, has become unstable.The maximum real part is 2.05  10 -2 .
Figure 2(c) shows the results when using Tool C with angles of 30, 40, 30, 40, 65, 55, 50, and 50 deg.All the characteristic roots are stable in the numerical calculation.The maximum real part  max is -9.12  10 -2 .The system can be stabilized by appropriately choosing the angular arrangement of the irregular pitch.Figure 2(d) shows the results when using Tool D with angles of 30, 57, 46, 47, 37, 64, 48, and 31 deg.These angles were given by a calculation using a genetic algorithm.In general, more than one pair of opposing cutting edges is often present, so the analysis was performed under this assumption.The initial population was set to 100 with 10 bits, and the fitness function was defined as  max .Roulette wheel selection and two-point crossover were used to produce new generations.Mutations were applied every five generations.In figure 2(d), the values of  are all negative, indicating that the system is stable.The maximum real part  max is -9.42  10 -2 .Because both figures 2(c) and 2(d) show stable results, it seems that polygonal deformation does not occur when using Tools C and D.

Cutting conditions
Experiments were conducted using reamers with the same angular arrangement as that for which the numerical calculations were performed in Section 2. Table 2 shows the cutting conditions.Two holes were machined under each experimental condition.After machining, the roundness was measured.

Experimental results
In previous studies [7], [9], roundness and harmonic analyses were performed to evaluate the accuracy of machined holes.From the experimental results, in the case of an 8-flute reamer, the roundness was found to be smaller than that for a 4-flute or 6-flute reamer, and the diameter of the machined hole was larger than that for a 4-flute or 6-flute reamer.In this case, a strong influence of the oval shape due to the inclination of the workpiece was found.Therefore, in the present study, the contour shape of the machined hole determined by the roundness measurement was Fourier transformed, and the contour shape was separated into the component of each number of polygon sides.The amplitude for each number of polygon sides was evaluated.Twelve layers were measured every 3 mm in depth, and the contour shapes for the bottom-most six layers were evaluated.
Figures 3(a) to 3(d) show the amplitude for each number of polygon sides for the contour shapes associated with Tools A to D, respectively.In the experiment, the number of sides of polygon from 5 to 17 is evaluated.The red line in the figure is the average value of the amplitude for each polygon.From the analytical results shown in figure 2(a) using Tool A, the 7-sided and 9-sided polygons were unstable.This is consistent with the experimental results shown in figure 3(a), where the amplitudes for the 7-sided and 9-sided polygons are large.From the analytical results shown in figure 2(b) using Tool B, the 9-sided and 11-sided polygons were unstable, which is again consistent with figure 3(b) where the amplitudes for these polygons are large.The theoretical analysis results in figures 2(c) and 2(d) show that all characteristic roots are stable for both Tool C and Tool D. From the experimental results in figures 3(c) and 3(d), the amplitudes of the profile shape for each polygon are small.It can be seen that an optimum arrangement of the irregular pitch suppresses polygonal deformation.

Examination of new evaluation methods
Many calculations are required to theoretically evaluate polygonal deformation of a reamer.To solve this problem, we considered an evaluation method using quasi-static characteristic roots.The quasi-static characteristic roots were the characteristic roots at a rotational speed   0 rad/s, and the maximum value of the quasi-static characteristic roots max i  was used to evaluate the optimum arrangement for an irregular-pitch reamer.
However, it is difficult to simply evaluate by max i  because the value of max i  changes as the number of flutes changes.Therefore, system instability is evaluated using the  q defined below, and this value is used for optimization: where max r  is the maximum value of the quasi-static characteristic root for a regular-pitch reamer, and we call  q QCR value.
When the QCR value is used, it is not necessary to calculate all the characteristic roots for each rotation speed, so the calculation speed can be considerably increased.In addition, by using the difference from the maximum value of the quasi-static characteristic root for a regular-pitch reamer, it is possible to evaluate the degree of instability relative to the regular-pitch reamer.
It is necessary to confirm the correlation between  q and  max .Figure 6 shows the relationship between all QCR values  q and the maximum value of the real part of characteristic roots  max when the angle of each cutting edge is changed in 5-deg steps with the minimum angular interval of the cutting edges set to 30 deg for an 8-flute reamer.In the calculation of the QCR, to make the numerical calculation more general, K was set to the same value as the contact stiffness previously reported [7], and the time delay effect was considered for only one rotation.In figure 6, the cutting edge arrangement at the lower left side can suppress polygonal deformation.The correlation coefficient is 0.917, indicating a fairly strong positive correlation.The values of  q and  max for the tools used in the experiment are also indicated by red circles.To obtain an arrangement that can suppress polygonal deformation from among the infinite number of irregular-pitch reamer configurations, an angular arrangement with a smaller QCR value should be selected.

Conclusions
The polygonal deformation of a borehole during reaming was assumed to be a self-excited vibration due to time delay, and the suppression of polygonal deformation with an 8-flute reamer with an irregular pitch was verified by numerical calculations and experiments.The results are summarized as follows: (1) In the numerical calculation, three types of irregular-pitch reamers with different angular arrangements were selected.It was found that the regular-pitch reamer had the highest instability, and the system could be stabilized by appropriately arranging the angles of the irregular pitch.
(2) Experiments were conducted on the regular-pitch reamer and three different irregular-pitch reamers.It was found that amplitude of the polygon profile and its dispersion were smallest when the reamers that were stable in the numerical calculations were used.
(3) A QCR value was proposed to speed up the numerical calculations and to evaluate the optimum angular arrangement of the irregular-pitch cutting edges of reamers with other numbers of flutes.Because the correlation between the proposed QCR value and the maximum value of the real part of characteristic roots was strongly positive, the QCR value can be used to evaluate the angular arrangement of the irregular-pitch reamers.

Figure 2 .
Figure 2. Relationship between maximum real part and imaginary part of characteristic root.
shape (mm) Number of sides of polygon

Figure 3 .
Figure 3. Amplitude for each number of polygon sides.

Figures 4 (
Figures 4(a) and 4(b) show contour curves and profiles for points P and Q in figure 3(a), respectively.Figure 4(c) shows the contour curve and profile for a data in figure 3(d).A circular profile is also shown in the figure.The figure shows that the optimally designed Tool D forms an almost perfect circle.(a) (b)

Figure 5 Figure 5 .
Figure 5 shows the dispersion calculated for the amplitudes in each experiment in figure 3. Tool A, which is a regular-pitch reamer, has the largest dispersion, followed by Tool B. Tools C and D have a similar small dispersion to each other.The optimally designed Tools C and D have good accuracy in terms of dispersion.
x yMx t Cx t Kx t F t My t Cy t Ky t F t

Table 1 .
Angular arrangement of cutting edges.

.
Angular arrangement of cutting edges.