This paper was contributed to the Journal of the American Society for Precision Engineering, Vol.17, No.1, 1995.
Study of cylindrical grinding using ball and cone centers
: static and dynamic characteristics of cone and ball center systems
Takaaki OIWA* and Akira KYUSOJIN**
*Faculty of Engineering, Shizuoka University, Hamamatsu, 432 Japan
**Faculty of Engineering, Nagaoka University of Technology, Nagaoka, 940-21 Japan
In cylindrical grinding, it is known, that balls used in place of conventional cone machine centers (CMCs), enables one to grind a workpiece without the influences of the misalignments and the geometrical deviations of the workpiece center holes and the machine centers. The purpose of this study is to determine the characteristics of these ball machine centers (BMCs) and CMCs. In this paper, the stiffness and the damping capacity of CMC and BMC systems were measured to investigate the potential of chatter in stability of BMCs compared to convention 60 degree CMCs. The main results obtained are as follows : (1) The stiffness difference between BMCs and CMCs was small or negligible, because the stiffness of the taper connection of our machine center shank was remarkably low. (2) The damping ratio of the BMCs was slightly larger than that of the CMCs. (3) The misalignments of the workpiece and CMC reduced the stiffness, while the stiffness of the BMCs was inherently independent of any misalignment. (4) Using BMCs, a lubricant with a center hole diameter appropriate to the workpiece size can be used because the balls run-in to the center holes rapidly. (5) Using CMCs, the stiffness depends on the center hole diameter. (6) Using BMCs, the waviness of the socket center hole containing the steel ball influences the stiffness. The ball diameter, moreover, affects the damping. In conclusion, there is no need to worry about a more frequent occurrence of chatter, when using BMCs, because the dynamic characteristics of BMCs are almost the same as that of CMCs.
Key words: cylindrical grinding, ball machine center, cone machine center, dynamic characteristics, natural frequency, damping ratio, chatter stability, run-in
In conventional cylindrical grinding, the cone-shaped dead machine centers have been used to support the workpiece. With this combination, misalignments and geometrical deviations of the machine centers and the workpiece center holes influence the roundness of the ground surface.[1-3] However, a ball used instead of the cone machine center (CMC) produces rotational accuracy of the workpiece without the above influences.[4,5] The purpose of this study is to determine the mechanical characteristics of these ball machine centers (BMCs). We already discussed axial and radial static contact stiffness at the support in our first report. The finding was that the stiffness of BMCs was slightly lower but more stable relating to chatter as compared with CMCs, because the BMC was independent of the misalignment of the centers. In the second report we showed that the wear of the steel ball had little effect on the rotational accuracy of the workpiece. Moreover, intermittent lubrication was effective in preventing the exfoliation and the wear of the ball.
In this article, we discuss the static and dynamic characteristics of CMC and BMC systems. Work regenerative chatter occurs frequently in cylindrical grinding. The damping capacity of the grinding machine usually represses the chatter. When it does not, an improvement in the damping capacity and the stiffness of center system comes up as an important consideration. We were concerned that chatter might occur easily when using BMCs because of its lower contact stiffness. In this study, we measured the natural frequency and the damping rate of the BMC system, and compared them with those of the CMC system.
Considering plunge grinding, we adopted a workpiece vibration model (Figure 1) having one degree of freedom. We will neglect the contact stiffness between the workpiece and the grinding wheel, because we will only discuss the radial contact stiffness between the machine center and the workpiece center. Consequently, the natural frequency and the damping rate between the workpiece center and the machine center were determined from the damped vibration wave form produced by hammering the workpiece.
Figure 2 shows the experimental device and the dimensions of the workpiece. The workpiece was supported on the grinding machine with dead machine centers in both the head and tailstock. To specifically investigate the stiffness and the damping characteristics between the workpiece and the machine center of the tail stock side, the center of gravity of the workpiece was shifted to near the tail stock. Thus, it can be considered that there is little effect of the stiffness between the workpiece and the machine center near the head stock, if the percussion point is situated near the center of gravity on the workpiece.
Several drilled attachments were attached to the workpiece to investigate the influences of the geometrical deviations and the diameters of the workpiece center hole. Thus, when we refer the workpiece center hole, we mean actually the attachment center hole. Using BMC system, the steel ball was fixed in the center hole of the attachment. The steel balls used were made from high carbon chromium bearing steels (JIS SUJ2).
The cemented carbide machine center having the shape of male cone, and the BMCs having the shape of female cones, were employed as the machine center of the tail stock (see Figure 3). The dimensions of the BMC (type II) are similar to the CMC(type I). The sockets, having the same dimensions as the attachment, were attached to the taper shank (type III) in order to investigate the effect of the geometrical deviations of the center hole of the socket supporting the steel ball. These three type taper shanks were ground with the contact area of more than 80%; the Morse taper gauge and red lead were used for checking.
The axial force was varied from 100 N to 500 N. We rotated the workpiece at a speed of 100 rpm with an axial force of 300N for one minute to establish a good fit of the contact surfaces between the machine centers and the workpiece center holes. A spindle oil (#22) was used for lubrication between the machine center and the workpiece center hole. With the workpiece not rotating, the hammer vibrated the workpiece in the horizontal infeed direction (see Figure 2). A noncontacting eddy-current probe (frequency response: DC-20 kHz) was used to detect the horizontal displacement of the workpiece.
Supporting stiffness of the centers
Before taking measurements of the workpiece/machine center combination, we needed to measure the stiffness of the machine center shanks shown in Figure 3. This is cause the stiffness of the structure supporting the workpiece includes the contact stiffness between the machine center and workpiece center and also the stiffness of the machine center shank friction locked into the tailstock. We installed just the center shanks into the tail stock, and measured the natural frequencies and damping rates of the tips of the machine centers without the workpiece. It was expected that the natural frequency and the damping capacity of the machine center shank would change owing to the increase of the axial force when the workpiece is mounted on the machine centers. However, the axial force has little effect on the stiffness and the damping rate because the Morse taper, having a very small taper angle, is friction locked into the taper sleeve (see Figure 9). Therefore, the presence of the workpiece has little influence on the stiffness of the taper joint.
We prepared two new CMCs (type I shown in Figure 3), two used CMCs having worn taper surfaces(type I), two BMCs (type II) and two BMCs(type III). In this paper, we used the centers of type I and type II for comparison of CMC and BMC systems, and use the center of type III to investigate the effects of the geometrical deviations of the socket center hole containing the steel ball.
Figure 4 shows the natural frequencies and the logarithmic decrements of the three types of the machine center shanks. The symbols show the average of six measurements. The natural frequencies of the new CMCs (type I)(open triangle) and the BMC (type II)(open circle) were almost equal. The natural frequency of the BMC (type III)(closed circle) was slightly higher. The logarithmic decrements of the machine centers of type I and type II were also almost equal. Static stiffness measurements also showed similar tendencies. The worn machine center (type I)(closed triangle) shows lower natural frequencies. Thus, the contact condition of the taper shank has stronger effect on the stiffness than the profile and the material of the machine centers. We used these machine centers for comparison of CMC and BMC systems because the machine centers of type I and type II shown in Table 1 have a similar stiffness and damping rate.
Wave form of damped vibration
Figure 5 shows an example of the damped transient wave at which the workpiece is supported by the BMCs. The amplitude of all damped linear systems decay exponentially.
The results of the frequency analysis by Fast Fourier Transform(FFT) are shown in Figure 6. We can consider the vibration system to be of one-degree of freedom, because the wave form consists of almost one frequency component. Higher amplitudes of 240 Hz and 360 Hz relate to the natural frequencies of the magnet stand supporting the eddy current probe.
Comparison of natural frequency
Figure 7 shows a relationship between the axial force and the natural frequencies of the workpiece supported by both types of the machine center systems. Each natural frequencies increase gradually as the axial force increases. The natural frequency of the BMC (type II) was lower than that of the CMC (type I). This agrees with the result of static stiffness measurement in our first report.
Figure 8 shows that the resultant stiffness coefficient, k, of the machine center system, consists of the contact stiffness, kc, and the taper shank stiffness, kt.
Figure 9 shows the static stiffness of the BMC system (type II). Lines A and B show the radial displacements of the workpiece and the tip of the machine center, respectively. Line A-B shows the difference between line A and line B, or the approach between the workpiece and the machine center. We used lever type electrical comparators to measure.
The stiffness, kt, was 25-28 N/micrometer under the axial forces of 100 N and 300 N. Nevertheless, the stiffness, kc, was 200 N/micrometer under the axial force of 100 N from figure 9. In general, the stiffnesses of the BMC and CMC systems, kc, were 130-660 N/micrometer. Therefore, the stiffness, kc, is five to 25 times higher than the stiffness, kt. Consequently, the decrease of the stiffness when using the BMC system can be ignored, because the contact stiffness, kc, has less effect on the total stiffness, k, than the stiffness, kt.
Moreover, Figure 4 suggests that the contact condition between the machine center taper shank and the tail stock socket depends on the stiffness, k. The natural frequency, which is approximately 630 Hz, is computed from a stiffness of 25 N/micrometer and a workpiece mass of 1.6 kg. This agrees favorably with the experimental result.
Figure 10 shows the relationship between the axial force and the logarithmic decrements of the workpiece supported by the two types of the machine center systems. The solid lines in the figure refer to the mean values. The logarithmic decrement of the BMC system was larger than that of the CMC system contrary to the natural frequency.
The vibration energy loss, which causes the damping, is primarily a surface phenomenon e.g. a micro slip at the contacts, and does not arise from internal friction or material damping. Thus, the contact surface decreases the stiffness of the structure, however increases the damping capacity. Consequently, higher damping capacity of a structure having a lower stiffness, as described above, is often observed in bolted joints of machine tools. The logarithmic decrements of bolted joints and machine center system are generally 0.1 to 0.3, and larger than that of steels or high tensile cast irons (generally 0.005 to 0.01).
The energy dissipated at the contact will increase with the relative curvature of the contact region9. The relative curvature of a BMC is large, and that of a CMC is zero, if the machine center fits the workpiece center hole well. Thus, the damping rate of a BMC system becomes larger than that of a CMC system. The effect of the relative curvature will be described later.
Effect of misalignment
Figure 11 represents the effect of the misalignment between two center systems on the natural frequency. The misalignment was provided by inserting a spacer into the dovetail between the tail stock and the bed. Using CMC system, the misalignment of 27 min. of arc caused a decrease of natural frequency of 30-40 Hz (see open triangle and closed square in figure). This occurs because nonconformity between the machine center and the workpiece center hole decreases the contact stiffness. We observed little decrease of the natural frequency, even if the misalignment was 10 min. of arc. Using the CMC system. Thus, we must reduce the eccentricity and angular misalignment of the workpiece center hole drilled carelessly before rough turning.
Using the BMC system, there was no decrease of the stiffness even if the misalignment was 27 min. of arc(see open circle and closed circle). Therefore, because the sphere will always fit the cone inside snugly, we have no need to care about the misalignments of the workpiece center hole.
Effect of run-in
Generally speaking, the degree of the run-in between the machine center and the workpiece center hole has a strong effect on the roundness of the ground surface. The lubricant used for preventing wear, moreover, extends the run-in time period. Therefore, the CMC system has a contradicting content between the lubrication requirement and the run-in period. If we use a lubricant on the centers to prevent the wear, it takes a long time to complete the run-in. Thus, it would appear that the rotational accuracy and the stiffness would be improved gradually during the rotation of the workpiece.
Using the BMCs, a socket containing the steel ball is used over again and again through many workpieces. In addition, out-of-roundness, roughness, and diameter variation of the steel balls fixed to the workpiece center hole are small. Consequently, we expected that the ball rapidly runs in the center hole of the machine center socket used over and over again, even if a new ball is used.
Figure 12 shows the natural frequency variation in each type of centers over a period of one hour. First, open triangles in the figure refer to the CMCs. The natural frequency is low before rotation, and approaches a fixed value during rotation. Second, open circles represent the BMCs using a new socket. The stiffness is low until the new machine center socket runs-in to fit the ball. Last, closed circles refer the natural frequencies using the same machine center socket and new steel ball in the workpiece center hole, and preserves the same high values throughout. Therefore, if using the same BMC socket and the same ball diameter, there is no need to consider the run-in time period between the machine center and the workpiece center hole. Moreover, we can use high performance lubricants to prevent the wear of the ball.
Workpiece center holes, having a relatively small diameter and a short generating line, have been used in conventional cylindrical grinding in general, because their small contact areas run-in immediately. However, BMC system can adapt to a center hole diameter appropriate to the shape and weight of the workpiece without the above considerations.
Effect of diameter of center hole and steel ball
Figure 13 shows the effects of the center hole diameter of the CMC system and the steel ball diameter of the BMC system on the natural frequency. We had to use the machine center shank (type III, shown in Figure 3) in order to investigate effects of the various specific characteristics of the center hole containing the steel ball. The center hole of the machine center shank (type III) was interchangeable with the socket as shown in Figure 3. The natural frequency using the machine center (type III) should not be compared directly with that using the machine center (type I). Because the stiffness of the taper shank (type III) is larger than that of the taper shank (type I) as shown in Table 1.
Closed triangles refer to the relationship between the natural frequency of the CMC system and the workpiece center hole diameter measured at the end face of the workpiece. The stiffness increases in proportion to the diameter of the workpiece center hole because the contact area is in proportion to the diameter. Therefore, decreasing the workpiece center hole diameter in order to shorten the run-in time period, substantially decreases the stiffness.
Open circles show the effect of the steel ball diameter on the stiffness using BMCs. There is little correlation between the natural frequency and the ball diameter, contrary to the CMCs. In our first report, the waviness of the center hole had a larger effect on the contact stiffness than the ball diameter did. Thus, it is assumed that the waviness also has a strong effect on the natural frequency.
Figure 14 shows the influence of the diameter, of the workpiece center hole in CMC system and the steel ball in BMC system, on the logarithmic decrement. The logarithmic decrement, using BMCs, is in inverse proportion to the ball diameter because the damping rate varies with the relative curvature of the contact region, as mentioned above. Although, there is little correlation between the logarithmic decrement and the workpiece center hole diameter using the CMCs, because the change of the workpiece center hole diameter has little effect on the relative curvature.
Effect of waviness of center hole
Figure 15 shows the relationship between the natural frequency and the waviness of the center hole for CMC and BMC systems. The waviness is represented by the standard deviation of the cross-sectional profile of the center hole. This profile was measured near the contact traces in the center hole, and passed through a polar filter of 1-15 undulations per revolution (Cut off: 1.8 mm). Using the CMCs shown as triangles, there is little correlation between the natural frequency and the roundness of the workpiece center hole. However, using the BMCs, an improvement of the roundness of the socket center hole results in a stiffness increase. This agrees with the measurement results of static contact stiffness in our first report. In addition, the logarithmic decrements of both CMC and BMC systems were independent of the waviness of the center hole.
In the cylindrical workpiece supporting systems using CMC and BMC systems, the following conclusions were obtained from the measurements of the natural frequency and the logarithmic decrement.
(1) In comparison of the natural frequency, the contact stiffness of the BMC is slightly lower than that of the CMC. However, the difference between the contact stiffnesses is negligible, because the stiffness of the taper shank is lower.
(2) The logarithmic decrement of the BMC is slightly higher than that of the CMC. Because the relative curvature radius of the contact part of the BMC is small.
(3) The misalignment of the CMC causes a decrease of the stiffness. The misalignment of the BMC has no effect.
(4) The stiffness of CMCs is low, until the machine center and the workpiece center hole fit each other. The steel ball of the BMC, however, rapidly runs-in the center hole. Therefore, you can choose an appropriate lubricant and workpiece center hole diameter appropriate to the workpiece size.
(5) The center hole diameter of the workpiece has a strong effect on the stiffness of CMCs, while the center hole waviness and the axial force have little effect on it. The damping rate is independent of the diameter and waviness of the center hole.
(6) The stiffness of the BMC varies inversely as the waviness of the machine center hole. The damping rate varies inversely as the ball diameter, and is independent of the waviness of the center hole.
As described above, the static and dynamic performance of BMC system is almost the same as that of CMC system. Therefore, there is no need to worry about a more frequent occurrence of chatter when adopting BMC system.
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