Effect of grinding forces on the vibration of grinding machine spindle system

Transcription

1 International Journal of Machine Tools & Manufacture 40 (2000) Effect of grinding forces on the vibration of grinding machine spindle system Mohammed Alfares *, Abdallah Elsharkawy Department of Mechanical and Industrial Engineering, College of Engineering and Petroleum, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait Received 29 October 1999; accepted 12 May 2000 Abstract In this paper the effects of dynamic changes in the grinding force components due to changes in the grinding wheel wear flat area and the workpiece material on the vibration behavior of the grinding spindle are studied. The steady-state dynamics and vibration behavior of the grinding machine spindle is simulated by a five degree of freedom model. The results indicate that when grinding different materials using a grinding wheel with fixed wear flat area, different vibration behavior is noticed. As the grinding wheel flat area increases, the level of vibration increases for all the degrees of freedom, which indicates that there is an upper limit for the level of wear on the grinding wheel that can be tolerated, and after that level dressing operation must be conducted on the grinding wheel Elsevier Science Ltd. All rights reserved. 1. Introduction In today s industry, there are numerous applications where the performance of the spindle system (i.e. shaft bearing housing system) is critical. The grinding machine spindle is one such system where vibration causes major problems in the grinding process as the grinding wheel depth of cut is so small that even the slightest amplitude of vibration can have dramatically damaging effects on surface finish, wheel wear, and form-holding. As with other machine tools, grinding machine vibration are usually classified into two types [1]: forced vibration and self-excited (chatter) vibration. Forced vibrations are caused by periodic disturbance external to the grinding process. Such external sources of disturbances may be caused by factors affecting the spindle system such as out-of-balance shaft, bearing variable compliance vibration, manufacturing errors, * Corresponding author. Tel.: ; fax: address: alfares@kuc01.kuniv.edu.kw (M. Alfares) /00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S (00)

2 2004 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Nomenclature A distance between centers of curvature of inner and outer race grooves, m a 0 distance from the center of gravity to the position of the applied grinding forces, m a 1 distance between the left bearing and the center of gravity, m b 1 distance between the right bearing and the center of gravity, m D ball diameter, m E Young s modulus, N/m 2 d bearing pitch diameter, m d i inner race diameter, m d o outer race diameter, m F x (t) grinding force component in the x-direction, N F y (t) grinding force component in the y-direction, N F z (t) grinding force component in the z-direction, N F(r) curvature difference g acceleration of gravity, m/s 2 I moment of inertia, kg.m 2 K stiffness per ball, N/m m s mass of the shaft, kg m g mass of the grinding wheel, kg M total mass, M=m s +m g,kg n number of balls in a bearing set P r axial preload, N P d bearing diameteral clearance, m P d radial interference, m r radii of curvature of bearings rings, m t time, s W load per ball, N x, y, z Cartesian coordinate set, m X, Y, Z displacements along x, y, z, m a contact angle, rad a 0 unloaded contact angle due to interference fit of bearings, rad a p preload contact angle, rad q ball center angular displacement angular velocity of the spindle, rpm c cage set angular velocity, rpm d elastic deflection at each contact, m rocking motion of spindle about y-axis, rad. φ rocking motion of spindle about x-axis, rad. n Poisson s ratio

3 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Subscripts 0 initial conditions or outer race ir inner race i left hand bearing counter j right hand bearing counter L left hand bearing R right hand bearing x, y, z coordinates, x, y, z r curvature sum, 1/m Superscripts. the first derivative with respect to time.. the second derivative with respect to time and/or the method of assembly [2]. On the other hand, self-excited vibrations are generally associated with natural vibration modes of the machine-tool structure. The causes of self-excited vibrations are much more complicated compared with forced vibrations, and a great deal of research has been devoted to this issue [3 5]. Sources of self-excited vibrations may be referenced to the application of cyclic forces, out-of-balance forces, and/or chatter that can be introduced by a host of undesired causes. One such important cause for chattering in grinding process is uneven in-process grinding wheel wear. Uneven in-process grinding wheel wear generation may generate irregularities in the grinding process causing variations in the grinding forces that can dynamically excite the shaft bearing structure. This may lead to variations in the local depth of cut during successive passes of the wheel, thereby generating undulations on the previously generated surface causing vibration to regenerate rather than die out. The vibration behavior of the grinding machine spindle system is conditioned by both the vibration characteristics of the assembly components, i.e. shaft bearing housing system and the external grinding process forces. While vibration characteristics of the shaft bearing housing components were approached by many reliable mathematical models [6 12] little attention was given to study the effect of the external grinding forces on the vibration behavior of the grinding machine. Various simulation studies [13 15] concentrated on the dynamic stability of other types of machines such as drilling and face milling machines affected by external factors such as cutting forces as a function of cutting velocity, rake angle variation, chip thickness variation, and friction between tool and workpiece, without considering the direct influence of these external factors on the vibrational characteristics and behavior of the machine tool spindle system. Both Aini et al. [2] and Akturk et al. [9] investigated and modeled the vibration of a grinding spindle supported by a pair of angular contact ball bearings. Both investigators used a three or five degrees of freedom model to simulate the steady-state dynamics of a grinding machine rigid shaft supported

4 2006 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) by a pair of angular ball bearings. In their models, the effects of different factors such as natural frequency of the shaft and bearing assembly, varying the number of balls, ball passage frequency, spindle speed variation, preload, misalignment, and manufacturing defects influencing ball bearing vibration were investigated. In both studies, no one investigated the dynamical changes in the external grinding forces due to changes in grinding conditions and its direct influence on the vibration performance of the grinding spindle system. One of the major factors affecting the grinding conditions is the grinding wheel wear rate as it changes with time. Many studies [15,16,18 20] showed that the wheel wear can have an important influence on the topography of the grinding wheel, which consequently affects grinding forces. These changes in grinding forces may have adverse consequences on the vibration of the grinding machine spindle. The objective of the present paper is to study and analyze the effects of the dynamic changes in the grinding force components due to the changes in the grinding wheel wear on the vibration behavior of the grinding spindle. The five degree of freedom model developed by Aini et al. [2] is utilized to simulate the steady-state dynamics and vibration behavior of the grinding machine spindle. 2. Analysis 2.1. Grinding wheel wear and force analysis Of all the machining processes in common use these days, grinding is undoubtedly the least understood and the most involved than the others machining used in practice. This is due to the complexity of the grinding process with its complex tooling structure, high cutting speeds, and small depth of cut that vary from grain to grain. Like other metal cutting processes, material removal by grinding involves a shearing process of chip formation. Yet, the theories of chip flow, cutting mechanics, cutting temperatures, and tool life, that are generally applicable to other single or multi-point machining operations, such as turning, milling, and drilling, are not entirely relevant to grinding [1]. In many studies [15 18] theoretical models have been developed for representing the grinding forces on the workpiece. These models are based on the fact that grinding is basically a chipremoval process in which the cutting tool is an individual abrasive grain and chip formation during grinding process consists of three stages, as shown in Fig. 1: sliding stage, ploughing stage, and cutting or chip formation stage. In the grinding process, cutting is not the only mechanism that produces forces on the grain. In addition to cutting, there is sliding of the cutting edge formed by the grain where both the cutting edge and the workpiece are elastically deformed. Then, as the stresses between the cutting edge formed by the grain and the workpiece are increased beyond elastic limit, plastic deformation occurs. At this stage the cutting edge penetrates into the plastic matrix and ploughing or plastic flow to the front and the side of the grain will form a groove. If the deformed material ahead of the cutting edge comes into contact with the cutting edge profile, then a transition from ploughing to chip formation occurs. All three stages increase the grinding force and the specific energy in grinding process. Therefore, the total force in grinding process can be expressed as

5 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Fig. 1. Grinding process stages. F total F sliding F ploughing F cutting (1) For plunge grinding operations, as illustrated in Fig. 2, the total grinding force vector F total exerted by the wheel on the workpiece can be resolved into a horizontal component F H and a vertical component F V. The main force components are those caused by sliding (i.e. friction) and ploughing, while shearing will slightly contribute to the resultant force [15]. The topography of the grinding wheel cutting surface is created initially by the dressing oper-

6 2008 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Fig. 2. Three stages of chip generation and grinding force components. ation (see Fig. 3). The grain cutting edge shape and the percentage of wheel surface consisting of flattened area (i.e. wear flat) can vary greatly depending on the dressing conditions employed [17,18]. During the grinding process, the sharp grains will start to develop wear flats as a result of the grinding operation (similar to flank wear in cutting tools) while the already flattened grains become enlarged due to the overall grinding wheel wear and adhesion of metal particles. The overall grinding wheel wear (Fig. 4), was considered as the sum of grain fracture, bond fracture, and attritious wear [19,21,22]. Most of the grinding wheel wear is due to grain and bond fracture, while attritious wear constitute almost insignificant portion of the total grinding wheel wear. However, the attritious wear is the most important form of wear since it directly related to the size of wear flat and hence the grinding force. As the grinding process progresses, the wear flat slides along the surface being ground, because of friction, requires energy for sliding. The larger the wear flat, the higher the grinding force is. The relationship between the grinding force and the wear flat area has been previously investigated theoretically and experimentally [15,16,19,21 23]. Based on these investigations and the above discussion, there is a direct relationship between the grinding forces and the magnitude of wear flat area x (percentage of wheel surface consisting of wear flats) of the grinding wheel. Experiments conducted by Malkin and Cook [21] showed a linear relationship between the grinding force and the wear flat area which indicates that the average contact pressure between the loaded or wear flat areas is constant depending on the hardness of the workpiece material. Also,

7 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Fig. 3. Grinding wheel dressing operation. Fig. 4. Types of wear in grinding wheel. (A) Attritious wear. (B) Grain fracture. (C) Bond fracture. the results of these experiments confirm that the friction coefficient between the workpiece material and the wear flat area of the grinding wheel is constant. It was therefore decided to utilize this linear relationship to show how the grinding forces change during grinding process which will affect the dynamical behavior of the grinding shaft bearing system. One such relation-

8 2010 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) ship between the grinding forces (horizontal and vertical components) and the wear of the grinding wheel for different materials was developed by Malkin [1] using experimental techniques. Table 1 shows the mathematical representation of such experimental results Modeling of grinding spindle system Fig. 5 shows a schematic representation for a grinding spindle supported by a pair of angular contact bearings. The bearings are identical and in a back-to-back configuration. The five degrees of freedom model of a rigid shaft supported by a pair of angular contact ball bearings developed by Aini et al. [2] is adopted and modified to study the effect of grinding force components on the vibration of the grinding spindle. The five degrees of freedom are along the radial directions x and y, in the vertical plane x y, the axial or longitudinal direction z, and the rocking motions about the y-axis and x-axis (i.e. and φ respectively). The mathematical formulation of the model is presented in Appendix A. Note that, for this model, the spindle is assumed to be rigid, the centrifugal effects and torsional vibrations of the spindle are neglected, the balls of the angular contact bearings are assumed to be massless, the angular velocity of the cage is assumed to be constant. Furthermore, sources of damping either from the elastohydrodynamic film at the contact, or from any friction present between the cage and the balls and between the various races to shaft and housing joints, are assumed to be negligible. The mathematical formulation presented in Appendix A is an initial value problem, whose Table 1 Grinding force components as a function of wear flat area x for different materials Material AISI Steel SAE 1018 Steel Molybdenum (Mo) Niobium (Nb) Grinding force components F H (x)= x x F V (x)= x x F H (x)= x x F V (x)= x x F H (x)= x F V (x)= x F H (x)= x 0 x 2.5 x =2.5 0 x = x x x 3.44 x 3.44 F V (x)= x 130 A Titanium F H (x)= x F V (x)= x

9 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Fig. 5. Five degrees of freedom for grinding spindle. solution is provided by step by step iteration procedure. A computer program is developed to solve this initial value problem numerically. The numerical scheme presented by Aini et al. [2] has been refined as shown in Appendix A. The remarks mentioned by Akturk et al. [24] about the stiffness coefficient (or load-deflection constant) K are considered. In order to demonstrate the validity of the solution algorithm as well as that of the computer program, all the results presented by Aini et al. [2] were recovered. 3. Results and discussion In the present study the following data are obtained from Aini et al. [2] and Malkin and Cook [21]: Mass of the shaft, m s =5.5 kg Diameter of grinding wheel d g = m Width of grinding wheel w g = m Mass of the grinding wheel, m g =1kg

10 2012 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Total mass of the spindle, M=m s +m g =6.5 kg Moment of inertia of the grinding spindle around x-axis or y-axis, I x =I y = kg.m 2 Moment of inertia of the grinding spindle around z-axis, I z = kg.m 2 Distance from the center of gravity to the position of the applied grinding forces, a 0 = m Distance between the left bearing and the center of gravity, a 1 = m Distance between the right bearing and the center of gravity, b 1 = m Angular velocity of the spindle, =2984 rpm Bearing pitch diameter, d=0.054 m Ball diameter, D= m Inner race radius of curvature, r ir = m Outer race radius of curvature, r or = m Axial preload, P r =100 N Inner race diameter, d i =0.046 m Outer race diameter, d o =0.062 m Radial interference, P d =4.82 µm Modulus of elasticity, E=207 GPa Poisson s ratio, n=0.3 Maximum time, t max =0.03 s The external forces F x (x) and F y (x) in the model are taken from Table 1, where F x (x)=f V (x), and F y (x)=f H (x). Note that these forces are considered to be constant with respect to time in the simulation process and they are only a function of wear flat area x. Due to the bearing nonlinearity and to avoid numerical instability, the time step t has to be very small. This study was carried out with time step, t= s. Fig. 6 shows the time domain response of the five degrees of freedom for workpiece material AISI at wear flat area x=2%. Both x and y directions time response show beat type responses which indicate two dominant frequencies close to each other as shown in Fig. 7. The first frequency is close to 800 Hz which represents the natural frequency in the x-direction while the second frequency is around 1300 Hz which represents the natural frequency in the -direction. A third frequency is also noticed at 1200 Hz which may be related to the rocking motion in ϕ- direction. It is clear that the natural frequency in z-direction is less than the natural frequency of the system in the x and y directions. These results are in agreement with the results presented by Aini et al. [2] and Akturk [9]. The amplitude of oscillations in the z-direction (axial direction) is very small compared to the x and y responses because there is no external load in the z-direction and the direct effect of preload which minimizes the motion in the axial direction. Furthermore, the peak-to-peak values of the z-direction are varying as time progress (as shown in Fig. 6) which indicate the direct effect of the variation in the contact angle as time proceeds. The variation of the contact angle is due to the rotation of the balls of the bearing along the cage orbit which directly influences the amplitude variation in the z-direction. Comparing the time response of the angular displacement ϕ with shows that the peak-to-peak rocking amplitude of is higher than that of ϕ. Also, the frequency of the oscillations of is higher than that of ϕ. Fig. 8 displays the effect of grinding wheel wear flat area x on the time domain response of the spindle x-displacement. The workpiece material in this case is AISI steel. It is clear

11 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Fig. 6. Time domain response of the five degrees of freedom (x, y, z,, φ) for AISI and wear rate of 2%.

12 2014 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Fig. 7. 2%. Frequency domain response of the five degrees of freedom (x, y, z,, φ) for AISI and wear rate of

13 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Fig. 8. Time domain response of the x-displacement for AISI and different values wear rate (0%, 1%, 2%, 3%, and 4%).

14 2016 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) that the grinding wheel wear flat rate has a significant effect on the vibrations of the spindle system. As the wear flat rate increases from 0 to 2.5% the peak-to-peak value of the oscillations decreases while above 2.5% the peak-to-peak value of the oscillations increases. This is due to the variation of the grinding forces with the flat wear rate. Fig. 9 shows the effect of grinding wheel wear flat area x on the frequency domain response of the spindle x-displacement. The results presented in Fig. 9 confirms those of time domain shown in Fig. 8 with respect to decrease and increase in the x-displacement amplitude. It is noticed that as the wear flat rate increases, there is no change in the dominant frequencies of the system. Fig. 10 shows that the grinding wheel wear flat area has no significant effect on the time domain z-displacement of the spindle when its value is less than 2.5%. On the other hand, it has significant influence on the vibrations of the system when its value is higher than 2.5%. At wear flat area equals 4%, it is noticed that the oscillations are controlled by multiple frequencies which indicate that the oscillations noticed at 4% are triggered by more than one dominant frequency. Fig. 11 shows the frequency domain for the results presented in Fig. 10. Figs. 12 and 13 show the time and frequency domain responses in x-direction for different workpiece material. The grinding wheel wear flat area was held fixed at 2%. The results indicate that the workpiece material has a significant effect on the vibrations of the spindle system. From the results we still notice the beat type response for all the materials. Two dominant frequencies around 800 and 1300 Hz are shown for all the materials. For harder materials such as SAE 1018 and Molybdenum (Mo) the first natural frequency in the x-direction dominates the response. As the material gets harder, the stiffness of the system in this direction increases which results in domination of the first natural frequency. The same conclusions can be drawn from the results presented in Figs. 14 and 15 for the vibration response in z-direction. Fig. 16 shows the peak-to-peak values of the amplitude of the oscillations for the five degrees of freedom versus the grinding wheel wear flat area for different workpiece materials. As the grinding wheel wear flat area increases, the level of vibration increases for all the degrees of freedom. These results are similar to the relationship between the grinding force components and wear flat area [1]. For Molybdenum (Mo) workpiece material, the level of vibration for all the degrees of freedom is relatively high. This reflects the level of hardness of Mo relative to other materials. 4. Conclusions A numerical algorithm based on a five degrees of freedom model was presented to simulate the dynamic response of the spindle of a grinding machine. A detailed study of the effects of the dynamic changes in the grinding force components due to the changes in the grinding wheel wear flat area and the workpiece material on the vibration behavior of the grinding spindle has been introduced. Results were presented in both time and frequency domains. From the results presented in this study the following conclusions can be drawn: 1. The system response along the x and y directions (radial directions) always shows two dominant frequencies. The first one represents the system natural frequency along the direction of that

15 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Fig. 9. Frequency domain response of the x-displacement for AISI and different values wear rate (0%, 1%, 2%, 3%, and 4%).

16 2018 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Fig. 10. Time domain response of the z-displacement for AISI and different values wear rate (0%, 1%, 2%, 3%, and 4%).

17 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Fig. 11. Frequency domain response of the z-displacement for AISI and different values wear rate (0%, 1%, 2%, 3%, and 4%).

18 2020 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Fig. 12. Time domain response of the x-displacement for different workpiece materials and wear rate of 2%. degree of freedom and the other represents the rocking motion around the other degree of freedom. 2. The amplitude of oscillations in the z-direction (axial direction) is very small compared to the x and y responses because there is no external load in the z-direction and the direct effect of

19 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Fig. 13. Frequency domain response of the x-displacement for different workpiece materials and wear rate of 2%.

20 2022 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Fig. 14. Time domain response of the z-displacement for different workpiece materials and wear rate of 2%.

21 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Fig. 15. Frequency domain response of the z-displacement for different workpiece materials and wear rate of 2%.

22 2024 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Fig. 16. Peak-to-peak values of the five degrees of freedom (x, y, z,, φ) as a function of the wear rate for different workpiece materials.

23 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) preload which minimizes the motion in the axial direction. Furthermore, the natural frequency of the system in the z-direction is less than the natural frequency in the radial directions. 3. As the grinding wheel wear flat area increases, the level of vibration increases for all the degrees of freedom while the natural frequencies of the system do not change. 4. When the wear rate exceeded 3% for all the workpiece materials used in the present study, the amplitude of vibration increased significantly. Therefore, the grinding process will degrade which influences the quality of the surface finish. Appendix A Following Aini et al. [2], Akturk et al. [24] and Harris [25], the equations of motion for the five degrees of freedom model can be written as: where n Mẍ i 1 n Mÿ n Mz i 1 i 1 n W i(l) cos a i cos q i j 1 n W i(l) cos a i sin q i n W i(l) sin a i n I y F I z φ a 1 i 1 d n 2 i 1 j 1 j 1 W j(r) cos a j cos q j Mg F x (t) (2) W j(r) cos a j sin q j F y (t) (3) W j(r) sin a j F z (t) (4) n W i(l) cos a i cos q i b 1 W i(l) sin a i cos q i d n 2 n I x φ I z Ḟ a 1 i 1 j 1 j 1 W j(r) sin a j cos q j n W i(l) cos a i sin q i b 1 d n W 2 i(l) sin a i sin q i d n 2 i 1 j 1 j 1 W j(r) sin a j sin q j W j(r) cos a j cos q j (a 0 a 1 )F x (t) (5) W j(r) cos a j sin q j (a 0 a 1 )F y (t) (6)

24 2026 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) q i c t 2pi/n (7) q j c t 2pj/n (8) c 2 1 D d cos a p (the outer race is assumed to be fixed) (9) W i(l) Kd 3/2 i (10) W j(r) Kd 3/2 j (11) d i {[A sin a 0 Z L d/2( cos q i φ sin q i )] 2 [A cos a p d 0 cos a p X L cos q i (12) Y L sin q i ] 2 } 1/2 A d j {[A sin a 0 Z R d/2( cos q j φ sin q j )] 2 [A cos a p d 0 cos a p X R cos q j (13) Y R sin q j ] 2 } 1/2 A X L X a 1 sin (14) X R X b 1 sin (15) Y L Y a 1 sin φ (16) Y R Y b 1 sin φ (17) Z L Z 0 Z (18) Z R Z 0 Z (19)

25 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) Z 0 A sin(a p a 0 )/cos a p (20) A r or r ir D (21) a i tan 1 A sin a 0+Z L +d/2( cos q i +φ sin q i ) A cos a p +d 0 cos a p +X L cos q i +Y L sin q i (22) a j tan 1 A sin a 0+Z R d/2( cos q j +φ sin q j ) A cos a p +d 0 cos a p +X R cos q j +Y R sin q j (23) K 1 K ir 1 2/3+ 1 2/3 K or 3/2 (24) 2 2 E 3 1 n 2 2 K ir 1 ( r ir ) dir 3/2 (25) K or 2 2 E 3 1 n ( r or ) 3/2 dor (26) F(r) ir 1 + 2g f ir 1 g g f ir 1 g (27)

26 2028 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) F(r) or 1 2g f or 1+g 4 1 2g f or 1+g r ir D g f ir 1 g (28) (29) r or D g f or 1 g (30) f ir r ir D (31) f or r or D (32) g D cos a d where d* can be determined from the following equations If 0 F(r) (33) d (F(r)) (F(r)) (F(r)) (F(r)) 4 If F(r) d (F(r)) (F(r)) (F(r)) (F(r)) 4 If F(r) d (F(r)) (F(r)) (F(r)) (F(r)) 4 The preload contact angle a p, and the initial contact deflection d 0 due to the elasticity of the bearings for a given preload P r can be determined by solving the following equations simultaneously using iterative Newton Raphson scheme nkd 3/2 0 sin a p P r (35)

27 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) d 0 A cos a 0 cos a p 1 (36) where a 0 is the unloaded contact angle due to interference fitting of bearings and calculated from the following equation a 0 cos 1 1 P d P d (37) 2A where P d d o d i 2D (38) Eq. (35) is derived from the force balance in the axial direction and Eq. (36) from the initial contact deflection (see Harris [25] and Akturk et al. [24]). The applied preload will cause initial displacement Z 0 in the axial direction. The initial displacement can be determined from the following equation Z 0 d 0 sin a p (39) The initial contact load W 0 per ball is obtained from W 0 =Kd m 0. n W x0 W 0 cos q 0i cos a p (40) i 1 n W y0 W 0 sin q 0i cos a p (41) i 1 n W z0 W 0 sin a p (42) i 1 where q 0i =2πi/n. Note that, the same equations hold true for the ball component denoted by j. The initial accelerations Ẍ 0, Ÿ 0, Z 0, F 0 and φ 0 are calculated using Eqs. (2) (6) at t=0+, where a i =a j =a p for all values of i and j. References [1] S. Malkin, Grinding of metals: theory and applications, Journal of Applied Metalworking (1984) [2] R. Aini, H. Rahnejat, R. Gohar, A five degrees of freedom analysis of vibrations in precision spindles, International Journal of Machine Tools Manufacture 30 (1) (1990) [3] R.S. Hahn, Vibration problems and solutions in grinding, ASTME paper no. MR69-246, [4] R.A. Thompson, The character of regenerative chatter in cylindrical grinding, Transactions of ASME Journal of Engineering Industry August (1993)

28 2030 M. Alfares, A. Elsharkawy / International Journal of Machine Tools & Manufacture 40 (2000) [5] S. Ohno, Self-excited vibration in cylindrical grinding, part 1 and 2, Bulletin of the JSME 13 (1970) [6] M. Matsubara, H. Rahnejat, R. Gohar, Computational modeling of precision spindles supported by ball bearings, International Journal of Machine Tools Manufacture 28 (4) (1988) [7] J.K. Choi, D.G. Lee, Characteristics of a spindle bearing system with a gear located on the bearing span, International Journal of Machine Tools Manufacture 37 (2) (1997) [8] G.D. Hagiu, M.D. Gafitanu, Dynamic characteristics of high speed angular contact ball bearings, Wear 211 (1997) [9] N. Akturk, Dynamics of a rigid shaft supported by angular contact ball bearings, Ph.D. thesis, Imperial College of Science, Technology and Medicine, London, [10] K.J. Al-Shareef, J.A. Brandon, On the effect of variations in the design parameters on the dynamics performance of machine tool spindle bearing systems, International Journal of Machine Tools Manufacture 30 (3) (1990) [11] F.P. Wardle, S.J. Lacey, Dynamic and static characteristics of a wide speed range machine tool spindle, Precision Engineering 15 (4) (1983) [12] A. Choudhury, N. Tandon, A theoretical model to predict vibration response of rolling bearings to distributed defects under radial load, Journal of Vibration and Acoustics 120 (1998) [13] S.A. Tobias, W. Fishwick, The vibration of radial drilling machines under test and working conditions, Proceedings of the Institute of Mechanical Engineers 170 (1956) [14] S.A. Tobias, The vibration of vertical milling machines under test and working conditions, Proceedings of the Insitute of Mechanical Engineers 173 (1959) [15] M. Younis, M.M. Sadek, T. El-Wardani, A new approach to development of a grinding force model, Transactions of ASME Journal of Engineering for Industry 109 (1987) [16] S. Malkin, Grinding technology: theory and application of machining with abrasive, in: Ellis Horwood Series in Mechanical Engineering, Ellis Horwood, UK, [17] X. Chen, W.B. Rowe, Analysis and simulation of the grinding process, part I II, International Journal of Machine Tools Manufacture 36 (8) (1996) [18] X. Chen, W.B. Rowe, B. Mills, D.R. Allanson, Analysis and simulation of the grinding process, part III: comparison with experiment, International Journal of Machine Tools Manufacture 36 (8) (1996) [19] X. Chen, W.B. Rowe, B. Mills, D.R. Allanson, Analysis and simulation of the grinding process, part IV: effect of wheel wear, International Journal of Machine Tools Manufacture 38 (1 2) (1998) [20] W.R. DeVries, Analysis of Material Removal Processes, Springer-Verlag, New York, [21] S. Malkin, N.H. Cook, The wear of grinding wheels, part 1: attritious wear, Transactions of ASME Journal of Engineering for Industry 93 (1971) [22] S. Malkin, N.H. Cook, The wear of grinding wheels, part 2: fracture wear, Transactions of ASME Journal of Engineering for Industry 93 (1971) [23] R.S. Hahn, R.P. Lindsay, Principles of grinding, Machinery, July November (1971). [24] N. Akturk, M. Uneeb, R. Gohar, The effects of number of balls and preload on vibrations associated with ball bearings, Transactions of ASME Journal of Tribology 199 (1997) [25] T.A. Harris, Rolling Bearing Analysis (3rd ed), John Wiley and Sons, New York, 1990.

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