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Internal spherical surface grinding
281
Wear, 133 (1989) 281 - 294
INTERNAL SPHERICAL SURFACE GRADING KAZIMIERZ
E. OCZOS and TADEUSZ S. DZIOCH
Mechanical Faculty, Department of Machining and Machine Tools, Technical University, P.O. Box 85, Rzeszitw 35-959 (Poland) (Received August 8,1988;
revised February 9,1989;
accepted March 23,198Q)
Summary Roth internal spherical surface grinding and external spherical surface grinding require the application of specific kinematics of the grinding process. In this paper a study is made of the dimensional interrelations between the ground surface diameter and the crossing angle of the tool and workpiece axes as well as the grinding wheel dimensions, A definition is also proposed for the effect on the grinding wheel axial wear of the size of the slot between the grinding wheel and the ground socket. The effect is shown of the rigidity of the wheelhead on the quality of the ground spherical surface. In test results presented here particular attention has been given to defining the dependence of the grinding wheel axial wear rate and the material removal rate (linear) on the radial feed rate and the grinding wheel peripheral speed as well as the effect on the quality of the spherical socket ground surface.
1. Introduction Constructional elements, with both external spherical surfaces (ESSs) and internal spherical surfaces (ISSs) are becoming more and more widely used in various fields of technology. Spherical surfaces mated as kinematic pairs ensure - besides a maximum number of degrees of freedom - great durability and reliability in such motor couplings. These features are dependent not only on the material properties of the mating elements but also on the workmanship of their spherical surface macro- and micromorphology. The quality of the latter is mainly limited by the conditions during machining. Different methods of machining [l] are applied depending on the spherical surface diameter, the properties of the material being ground and the required accuracy and quality of production. The higher the required quality and usability and the lower the degree of mach~abili~ inherent in the material, the more complex is the process of machining a spherical surface. The final shape and usability properties of a spherical surface are produced primarily by abrasive machining in which grinding predominates 0043-1648/$9/$3.50
@ Elsevier Sequoia/Printed in The Netherlands
282
and particularly generation grinding (“along the chord”) with the grinding wheel face. This paper presents the problems connected with ISS shaping conditions as well as some experimental results from the grinding process, giving particular attention to the problem of grinding wheel wear, and using spherical socket production for hydraulic pump shafts as an illustration. 2. Internal spherical surface (ISS) shaping Let an arbitrary point G (see Fig. 1) on the grinding wheel cutting edge S turn around the y-y axis; then on contact with the wurkpieee it will describe a circular arc with the diameter of the grinding wheel d,, By coordinating the rotational speeds of the grinding wheel n, and the workpiece n, one obtains a circular annulus P of diameter &. The width of this armulus depends on the grinding wheel diameter d, but its position is dependent on the angle (x. The nature of ISS shaping kinematics is illustrated in Fig. 1. Since the position of the sphere centre Ok is invariant it is possible to shape ISSs of different diameters &, changing only the distance E and the diameter d,, as expressed by the dependence cd, = (d,2 + 412)“2
(11
An incrementation in t of Al produces an increment in dk of Adk. A necessary condition for proper ISS shaping is the intersection of the X-X and y-y axes at the point Ok as well as that the direction of the radial
Fig. 1, Kinematic-geometrical
dependence in internal sphericat surface shaping.
283
feed f, (in-feed) should be parallel to the plane intersecting the x-x and y-y axes. Failure to meet this condition results in “out-of-roundness” in the plane intersecting the x-x axis. 3. Dimensional interrelations between the ground surface diameter dk, the angle (Yand the grinding wheel dimensions ISS grinding according to the kinematics followed here - “along the chord” [l, 21 imposes geometrical dependence between the spherical (socket) surface diameter dk, the grinding wheel diameter d, and its width b, at the specified angle (Y. The chosen variants in ISS shaping with a space at the bottom of the spherical socket are presented in Fig. 2. (Fig. 2(a)) is theoretically the smallest grinding The diameter d,,i, wheel diameter which does not ensure the presence of a gap W for conveying the cutting fluid to the contact zone and carrying it away, together with the cutting products, from the contact zone of a tool and a workpiece. The presence, and any change in the size of the gap W will require adjustment of the width b, of the grinding wheel, the mandrel diameter d, and the angle CY. Because of its function the presence of W is of great importance; therefore it is essential to treat it as one of the output quantities determined by the choice of the grinding wheel. Having assumed a value for the size of the gap W it is possible to set the grinding wheel diameter d, in the range d,, & d, < ds2
(2)
To establish the necessary grinding wheel dimensions together with those of the holder used during specialized machine tool designing it it possible to assume a nominal grinding wheel setting as in Fig. 3.
Fig. 2. Geometrical interdependence between a grinding wheel and a ground workpiece: (a) minimum grinding wheel diameter dsdn with possible absence of gap W; (b) maximum mandrel diameter d,, - the influence of angle a! on d, and W values.
284
Fig. 3. Geometrical interpretation of the choice of grinding wheel dimensions us. dk, grinding-off allowance q and W: (a) initial size of gap We; (b) final size of gap Wk; Ic, length of grinding wheel travel before making contact; Zr, length of radial displacement of the grinding wheel; I,, length of grinding wheel displacement inside the allowance; I,, assumed distance of rear grinding wheel face from the ground surface.
Assuming a spherical surface diameter dk, grinding-off allowance value IJand initial gap size Ws, it is possible to define the following quantities: (a) grinding wheel diameter d,
dk2
ds= (dk2 +(dk -
(3)
2&)2}1’2
(b) the angle of inclination (x of the tool axis to the socket axis d 0. = arcsin 2
(4)
dk
(c) the height allowance along the generating grinding wheel axis I, 1, = $ [(dk’ - d,2)“2
-
{(dk-
2q)’
- d,2)l”]
(5)
(d) maximum grinding wheel with b, max(Fig. 3(a)) b Smax =dkCOSQ!-21q--ld-lt
(6)
the effect of the value of Wk on the grinding wheel axial wear Ab, (Fig. 3(b)) (e)
Ab, = It is essential to assume a permissible grinding wheel axial wear Ab, such that Wk > Wkperm. The amount of this wear significantly influences the macro- and micromorphology of the ISS being ground. An increase in the value of Ab, causes an increase in the contact area between a grinding wheel and a workpiece which hinders the access of the cutting fluid to the working zone. An excessive increase in Ab,, depending on the characteristics
285
of the grinding wheel, may result in burning of the ISS or crushing of the grinding wheel and may even cause it to break in its holder.
4. The influence of the rigidity of the wheelhead on the quality of the ground spherical surface ISS grinding is also constrained by machine rigidity. As with any internal grinding the rigidity of the various constructional elements of a machine tool on the side of the wheelhead si~ific~tly influences the quality of the shaped spherical surface. In~prop~a~ly chosen rigidity values in those elements may result in (i) an increase in “out-of-roundness”; (ii) the grinding wheel rapidly cutting into the workpiece material; (iii) violent self-dressing of the grinding wheel; (iv) loss of coherence in cutting-track crossings; (v) excessive grinding wheel wear. A tabulation of constructional element deflections of a wheelhead (Fig. 4) enables the total value of the grinding wheel deflection to be defined from the dependence Y=Yb+Ys+Ym-Ymb
(8)
If it is assumed that the spindle between the bearings in section B and the mandrel with the grinding wheel in section A (Fig. 4(a)) have stable and relatively large moments of inertia around their common axis of rotation, then it is possible to express the particular deflections (flexibility elements) in eqn. (8) by the following [ 31:
(11) YC
Ymb = cos(F.JEmIm)1’2a- yc
(12)
where F,, is the normal component of the grinding force, F, is the axial component of the grinding force, a is the length of reach of the grinding wheel mandrel, b is the distance between the spindle bearings, kA (kB) is the rigidity of the bearing A (B), Es (E,) is the modulus of elasticity of the spindel (mandrel) material, Is (I,) is the moment of inertia of the spindle (mandrel) and yc is the eccentricity of the force F,.
286
a)
YC
b
Fig. 4. Deflections of wheelhead constructional elements at the point of application of the force: (a) load model; (b) deflection yb caused by flexibility of the bearings; (c)deflection ys caused by spindle bending; (d) deflection ym caused by grinding-wheel mandrel bending; (e) deflection yu,r, caused by grinding-wheel mandrel buckling; (f) total deflection y.
The total deflection (flexibility) y (Fig. 4(f)) is dependent not only on the kind of material used for the mandrel and the spindle of the grinding wheel, on their method of connection and on the variable parameters kA, k,, I,, Inn,a and b, but also, to a small degree, on the angle cy. These considerations might suggest that a direct increase in the rigidity of the wheelhead could be acquired by envy the spindle and bearing diameters. However, the greater the diameter of the bearings the lower is the
287
limiting rotational speed and, indirectly, the permissible grinding speed. In consequence, for any particular constant radial feed rate ufr there would be an increase in the various components of the grinding force which would result in an increase in shape errors. Other damaging effects on the course and the results of grinding are the prestressing of bearings which might be accompanied by their increased heating and therefore by a reduction in the permissible tool rotational speed. In order to obtain a high quality ISS it is necessary to choose values of spindle rotational speed, bearing rigidity and prestresses which are suitable for the required inner diameter. The course of an ISS grinding process with a relatively unfavourable rigidity in the wheelhead constructional elements is presented in Fig. 5. Phase I - the grinding wheel reaches the material (Fig. 5(a)): the grinding wheel, which rotates with rotational speed n, and moves with speed Ufr, must cover the distance Ed1so that its edge S can begin grinding the spherical socket of a workpiece turning with speed n,. Phase II - spherical surface grinding: removing the excess material of thickness q1 (Fig. 5(a)) leads to the attainment of the required diameter dkl for the spherical socket. If the grinding wheel were not being progressively worn away then its cutting edge S would move through the material for a distance Z,, . However, as the material is removed from the ground socket there is also wear on the edge of the grinding wheel reducing the height of the working ring by Ab, (Fig. 5(b)). The bigger the value of Ab, the greater are the grinding force components (F,, Ft and F,) and the lower is the supply of cutting fluid to the cutting zone. In consequence the intersection point of the X--X and y-y axes in the middle of the sphere Ok will change its position and move to the point Ok1 (Fig. 5(b)) which means that the ISS will assume the shape of an ellipsoid of revolution (shown in section as I-I in Fig. 5(b)). There comes a time when the working grinding wheel - workpiece interface can attain a value Ab, cat (catastrophic wear) such that as a consequence of the progressive restriction of the cutting fluid supply together with a simultaneous increase in the force components F, and Ft there will be a sudden indentation by the grinding wheel into the material which will leave a track in the form of a groove of depth g,. This phenomenon is connected with the self-dressing of the grinding wheel because a sudden increase in cutting resistance may result in a force sufficient to overcome the bonding power of the abrasive grains. The resulting violent loss of material from the grinding wheel causes it to lose contact with the workpiece. Phase III - repeated contact of the grinding wheel with a workpiece (Fig. 5(c)): to continue the process of grinding it is necessary to bring the grinding wheel wear from Abscat - by face-dressing - to a value Ab, < Ab Sperm< A& cat, where A&,,.,, is the maximum permissible size for the working surface of the grinding wheel, such that the grinding process is not dotted. The repeated contact of the grinding wheel with a socket entails covering the distance ld2, alternating with “grinding the air”. After grinding wheel-workpiece contact has been achieved the wheel goes into the material along the segment lq2 to grind out the groove which has been formed previously.
Traces of rapid grinding wheel cutting-in
into a workpiece in the form of groove
Fig. 5. ISS grinding with insufficient rigidity in the wheelhead elements: (a) grinding wheei before grinding (phase I); (b) phenomena that accompany griping-wheel spindle and mandrei bending (phase 11); (c) grinding wheel after self-dressing at a distance ictz from the ground spherical surface (phase III).
289
These phases of the course of the ISS grinding process illustrate the influence of unsuitable rigidity in the wheelhead elements both on the “outof-roundness” of the surface being formed and on its roughness, which resulted in loss of coherence in the cutting-track crossings.
5. The conditions for the investigation The investigation into the ISS grinding process was carried out during spherical socket machining on the face of a hydraulic pump shaft made of const~ction~ carburizing steel 20 H with a hardness of 57 - 62 HRC (Fig. 6). As well as the axial socket, on the circle of radius R there were 7 circumferential sockets equally spaced, the diameters of which were in the range dk = 26.5-o*1 - 2’7+‘*01mm. The spherical socket centres were on the shaft face.
Fig. 6. Hydraulic pump shaft with ground spherical sockets.
The test stand was a grinding machine for spherical sockets of type SGK-1000 (Fig. 7) designed and manufactured at the Department of Machining and Machine Tools of the Technical University in Rzeszow by order of a hydraulic pump and engine manufacturer. Two kinds of grinding wheels were used in the process of grinding one made of Al:,Os with 220 grain size and ceramic bonding (89A 220 E7 AV2 T3 - made by Tyrolit manufacturers) and the other made of CBN (Elbor) with 150 grain size, medium hardness and ceramic bonding (made in the U.S.S.R.). Gl concentrate emulsion of 10% concentration was used as a cutting fluid with both cooling and washing effect on the cutting zone. 6. Results and discussion In the tests which were carried out particular attention was given to establishing the dependence of the grinding wheel axial wear rate A6, and the material removal rate (4 (linear) on the radial feed rate uf, and the grinding wheel peripheral speed u, as well as the effect on the quality of the ground surface of the spherical socket.
Fig. 7. General view of the grinding machine for spherical sockets of type SGK-1000: 1, bed; 2, workpiece headstock; 3, workpiece holder; 4, carriage; 5, angle plate; 6, crossslide; 7, wheelhead; 8, stepper motor; 9, harmonic drive.
“w
Fig. 8. The principle of the measurement of the material removal rate 4 and the grinding wheel axial wear rate Ad,: I,, length of tool displacement at its contact with the workpiece.
Figure 8 is a pictorial representation of the principles followed in measuring A& and 4 as functions of uIr and u, which could refer equally well to ESS grinding [43. The measurement of the tool (grinding wheel) displacement distance I, at its point of contact with the workpiece was made with a micrometer sensor; the depth q was measured directly in the socket with the help of a special gauge. During the tests the grinding wheel axial wear
291
Ab, was monitored and maintained in the range Ab, < 3 mm by dressing the grinding wheel from the surface. The value of the grinding wheel axial wear Ab, can be defined from the dependence Ab, = 1, - 1,
(13)
and after substitution of 1, from eqn. (5), the final form of eqn. (13) can be represented as Ab, = I, -
f [{(& - 2q)2 - d,2}l’* - (dk2 - d,2)“*]
04)
With the tool displacement distance Eskept constant, and using differwheel axial ent radial feed rates Ufr, the grinding time ts gave the ~ding wear rate Af;, and the material removal rate (4from the relations %
At;, = -
pm s-l
t,
(15)
Q
t=
pm s-l
ta
Curves showing the variation of Ab, and 4 as functions of ufr are plotted in Fig. 9. They show that the radial feed rate uti significantly influences both A6, and 4. Above ufI = 0.1 mm mm-i better results are gained when CBN grinding wheels are used to grind an ISS. The dii of A&, 4 = f(v,) is more complicated, and is presented in Fig. 10. Whexeas the influence of u, on A& and 4 is very slight and oscillates around a certain stable value when an Al,& grinding wheel is used, temporarily more advantageous working conditions are achieved if u, is increasing, e.g. at u, = 12 or 25 m s-l, when a CBN grinding wheel is used. This type of a)
b)
200 x10-2
0
50
loo
150
2w radial
feed
0 speed
vfr,
loo
150
i 100
)tm/mi”,
Fig. 9. Curves showing the grinding wheel axial wear rate Ab, and material removal rate 4 us. radial feed (in-feed) rate ufr for grinding wheels: (a) CBN; (b) AIzOJ; d, = 22 mm; us = 24.8 m s-l; n, = 40 rev min-‘.
292 a)
b) 60
x10-2
25
30
grinding
wheel
10
15
peripheral
speed
vs,
n&
Fig. 10. Curve showing the grinding wheel axial wear rate Ad, and material removal rate 4 us. grinding wheel peripheral speed us for grinding wheels: (a) CBN; (b) &OS; d, = 22 mm; Ufr = 34 pm s-l; n, = 40 rev min-‘.
dependence probably arises from a difference in self-excited vibration and its influence on the course of the grinding process. In the range of ufr and u, illustrated in Figs. 9 and 10 the most advantageous results in respect of the working capacity were achieved when CBN grinding wheels with u, = 25 m 5-l and ufr > 0.15 mm min-’ were used for ISS grinding. The spherical socket surface quality can basically be defined in terms of two parameters: roughness and “out-of-roundness”. Both parameters are affected by the correctness or otherwise of the choice of grinding kinematics as well as by wheelhead rigidity, the specific character of the grinding wheel and cutting fluid, and the continuous control of tool rotation and workpiece axes so that they intersect in the middle of the sphere at Ok. When an ISS is ground with Al,03 grinding wheels its surface roughness is generally lower than that achieved when CBN grinding wheels are used. The influence of the grinding wheel speed u, on the value of the parameter R, (Fig. 11) shows that the best results for an A1203 grinding wheel occur when u, = 18ms-1,andforaCBNgrindingwheelwhenu,~225ms-1. From R, = f(tsp) in the diagram presented in Fig. 11(b), for both grinding wheels, it is possible to see that a value for the time up to sparkingbb
al
SO X1O-2
Xl0‘(:
s40
5
40
2
i*
30
30
B p H -
10
:: 3
$
20
& p
10
t: 0
10
15
20
2s grinding
30 sptui
35 V$’
m/s
B
0
0
‘3
6
9 time
I2
of spark-out
t
sp’
p
Fig. 11. Curve for ground surface roughness R, vs. : (a) grinding wheel peripheral speed us; (b) time of spark-out tsp.
293
Fig. 12. Micromorphology (a) CBN; (b) Al203.
of a spherical surface after being ground with grinding wheels:
out t,, = 6 s is the most advantageous from the point of view of surface roughness. The smaller R, parameter values achieved during ISS grinding with an AIIOj grinding wheel might seem at odds with the ~~o~opic~y undulating finish of that surface. This ~croscopi~~y more ~d~at~g fmisb on a spherical socket made by an A1203 grinding wheel (Fig. 12) can be beneficial from the tribological point of view and thus significantly extend the life of a spherical joint. The “out-of-roundness” of the spherical surfaces did not exceed 3 pm in any cross-section plane in any of the tests.
7. Conclusions On the basis of the conside~~ons presented here on ISS grinding and of the invest~atio~ which were carried out, it is possible to formulate the following conclusions. (1) The quality of a ground ISS obtained with the usuaI grinding kinematics “along the chord” depends mainly on the grinding wheel characteristics, wheelhead rigidity, the intersection of the tool and workpiece axes of rotation in the middle of the sphere at Ok (particularly in the final phase of grinding), the properties of the cutting fluid and the size of the gap W. (2) The rigidity of the wheelhead constructional elements should be chosen in view of the tool (grinding wheel) rotational speed, but choosing an optimum value from the point of view of acquired macro- and micromorphology of the ground ISS (with regard to the conditions mentioned in the preceding conclusion) should also assure improved eff~~ven~ of the machining process.
294
(3) In defining the total deflection Y of the grinding wheel it is possible Ymb which appears as an effect. of the force component F’,, because its value is very small compared with the sum of the remaining terms in eqn. (8 ) . (4) The material removal rate 4 is greatly influenced by the radial feed rate uEr; however, above the value uy, = 0.1 mm mm-’ more favourable results are obtained when CBN grinding wheels are used for ISS grinding. (5) With the grinding wheels examined, one essential fur a relatively favourable surface roughness was a time period to sparking-out. Qp of about 6 s. (6) With the same grinding con~tio~ greater roughness but less microscopic undulation in the surface was obtained with a CBN grinding wheel, compared with that produced by an Al,& grinding wheel. This results chiefly from the differences in the rigidity and thermal cu~~ucti~ty of the grinding wheels. A better surface state, from the tribological point of view, is acquired when an A&Q, grinding wheel is used for ISS grinding. to disregard the deflection
References I K. Qczo~ and T. Dzioch, Grinding of spherical surfaces, Mechanik, 52 (11) (1979) 591 - f&E (in Polish). 2 A. Bats&, W. Brzozowski, T. Dzioch aad J. Nowak, A method of g~ndi~~ of spherical socket surfaces in mult~*piston pump shafts, Potish Patmt 139803, 1988. 3 E. %ljb, H. Garnung and H.-G. v. Mackensen, Beitrag zur ~ormfa~le~in~mie~ng beii Innen~ud~hleifen, TZ Prakt. Me~ai~~ear~.~ 74 (4) (1980) 13 - 15. 4 K. OczoS and T. Dzioch, Phenomena in the spherical surface grinding of prostheses, lVear, $6 (1) (1984) 45 ” 59.
Wear, 133 (1989) 281 - 294
INTERNAL SPHERICAL SURFACE GRADING KAZIMIERZ
E. OCZOS and TADEUSZ S. DZIOCH
Mechanical Faculty, Department of Machining and Machine Tools, Technical University, P.O. Box 85, Rzeszitw 35-959 (Poland) (Received August 8,1988;
revised February 9,1989;
accepted March 23,198Q)
Summary Roth internal spherical surface grinding and external spherical surface grinding require the application of specific kinematics of the grinding process. In this paper a study is made of the dimensional interrelations between the ground surface diameter and the crossing angle of the tool and workpiece axes as well as the grinding wheel dimensions, A definition is also proposed for the effect on the grinding wheel axial wear of the size of the slot between the grinding wheel and the ground socket. The effect is shown of the rigidity of the wheelhead on the quality of the ground spherical surface. In test results presented here particular attention has been given to defining the dependence of the grinding wheel axial wear rate and the material removal rate (linear) on the radial feed rate and the grinding wheel peripheral speed as well as the effect on the quality of the spherical socket ground surface.
1. Introduction Constructional elements, with both external spherical surfaces (ESSs) and internal spherical surfaces (ISSs) are becoming more and more widely used in various fields of technology. Spherical surfaces mated as kinematic pairs ensure - besides a maximum number of degrees of freedom - great durability and reliability in such motor couplings. These features are dependent not only on the material properties of the mating elements but also on the workmanship of their spherical surface macro- and micromorphology. The quality of the latter is mainly limited by the conditions during machining. Different methods of machining [l] are applied depending on the spherical surface diameter, the properties of the material being ground and the required accuracy and quality of production. The higher the required quality and usability and the lower the degree of mach~abili~ inherent in the material, the more complex is the process of machining a spherical surface. The final shape and usability properties of a spherical surface are produced primarily by abrasive machining in which grinding predominates 0043-1648/$9/$3.50
@ Elsevier Sequoia/Printed in The Netherlands
282
and particularly generation grinding (“along the chord”) with the grinding wheel face. This paper presents the problems connected with ISS shaping conditions as well as some experimental results from the grinding process, giving particular attention to the problem of grinding wheel wear, and using spherical socket production for hydraulic pump shafts as an illustration. 2. Internal spherical surface (ISS) shaping Let an arbitrary point G (see Fig. 1) on the grinding wheel cutting edge S turn around the y-y axis; then on contact with the wurkpieee it will describe a circular arc with the diameter of the grinding wheel d,, By coordinating the rotational speeds of the grinding wheel n, and the workpiece n, one obtains a circular annulus P of diameter &. The width of this armulus depends on the grinding wheel diameter d, but its position is dependent on the angle (x. The nature of ISS shaping kinematics is illustrated in Fig. 1. Since the position of the sphere centre Ok is invariant it is possible to shape ISSs of different diameters &, changing only the distance E and the diameter d,, as expressed by the dependence cd, = (d,2 + 412)“2
(11
An incrementation in t of Al produces an increment in dk of Adk. A necessary condition for proper ISS shaping is the intersection of the X-X and y-y axes at the point Ok as well as that the direction of the radial
Fig. 1, Kinematic-geometrical
dependence in internal sphericat surface shaping.
283
feed f, (in-feed) should be parallel to the plane intersecting the x-x and y-y axes. Failure to meet this condition results in “out-of-roundness” in the plane intersecting the x-x axis. 3. Dimensional interrelations between the ground surface diameter dk, the angle (Yand the grinding wheel dimensions ISS grinding according to the kinematics followed here - “along the chord” [l, 21 imposes geometrical dependence between the spherical (socket) surface diameter dk, the grinding wheel diameter d, and its width b, at the specified angle (Y. The chosen variants in ISS shaping with a space at the bottom of the spherical socket are presented in Fig. 2. (Fig. 2(a)) is theoretically the smallest grinding The diameter d,,i, wheel diameter which does not ensure the presence of a gap W for conveying the cutting fluid to the contact zone and carrying it away, together with the cutting products, from the contact zone of a tool and a workpiece. The presence, and any change in the size of the gap W will require adjustment of the width b, of the grinding wheel, the mandrel diameter d, and the angle CY. Because of its function the presence of W is of great importance; therefore it is essential to treat it as one of the output quantities determined by the choice of the grinding wheel. Having assumed a value for the size of the gap W it is possible to set the grinding wheel diameter d, in the range d,, & d, < ds2
(2)
To establish the necessary grinding wheel dimensions together with those of the holder used during specialized machine tool designing it it possible to assume a nominal grinding wheel setting as in Fig. 3.
Fig. 2. Geometrical interdependence between a grinding wheel and a ground workpiece: (a) minimum grinding wheel diameter dsdn with possible absence of gap W; (b) maximum mandrel diameter d,, - the influence of angle a! on d, and W values.
284
Fig. 3. Geometrical interpretation of the choice of grinding wheel dimensions us. dk, grinding-off allowance q and W: (a) initial size of gap We; (b) final size of gap Wk; Ic, length of grinding wheel travel before making contact; Zr, length of radial displacement of the grinding wheel; I,, length of grinding wheel displacement inside the allowance; I,, assumed distance of rear grinding wheel face from the ground surface.
Assuming a spherical surface diameter dk, grinding-off allowance value IJand initial gap size Ws, it is possible to define the following quantities: (a) grinding wheel diameter d,
dk2
ds= (dk2 +(dk -
(3)
2&)2}1’2
(b) the angle of inclination (x of the tool axis to the socket axis d 0. = arcsin 2
(4)
dk
(c) the height allowance along the generating grinding wheel axis I, 1, = $ [(dk’ - d,2)“2
-
{(dk-
2q)’
- d,2)l”]
(5)
(d) maximum grinding wheel with b, max(Fig. 3(a)) b Smax =dkCOSQ!-21q--ld-lt
(6)
the effect of the value of Wk on the grinding wheel axial wear Ab, (Fig. 3(b)) (e)
Ab, = It is essential to assume a permissible grinding wheel axial wear Ab, such that Wk > Wkperm. The amount of this wear significantly influences the macro- and micromorphology of the ISS being ground. An increase in the value of Ab, causes an increase in the contact area between a grinding wheel and a workpiece which hinders the access of the cutting fluid to the working zone. An excessive increase in Ab,, depending on the characteristics
285
of the grinding wheel, may result in burning of the ISS or crushing of the grinding wheel and may even cause it to break in its holder.
4. The influence of the rigidity of the wheelhead on the quality of the ground spherical surface ISS grinding is also constrained by machine rigidity. As with any internal grinding the rigidity of the various constructional elements of a machine tool on the side of the wheelhead si~ific~tly influences the quality of the shaped spherical surface. In~prop~a~ly chosen rigidity values in those elements may result in (i) an increase in “out-of-roundness”; (ii) the grinding wheel rapidly cutting into the workpiece material; (iii) violent self-dressing of the grinding wheel; (iv) loss of coherence in cutting-track crossings; (v) excessive grinding wheel wear. A tabulation of constructional element deflections of a wheelhead (Fig. 4) enables the total value of the grinding wheel deflection to be defined from the dependence Y=Yb+Ys+Ym-Ymb
(8)
If it is assumed that the spindle between the bearings in section B and the mandrel with the grinding wheel in section A (Fig. 4(a)) have stable and relatively large moments of inertia around their common axis of rotation, then it is possible to express the particular deflections (flexibility elements) in eqn. (8) by the following [ 31:
(11) YC
Ymb = cos(F.JEmIm)1’2a- yc
(12)
where F,, is the normal component of the grinding force, F, is the axial component of the grinding force, a is the length of reach of the grinding wheel mandrel, b is the distance between the spindle bearings, kA (kB) is the rigidity of the bearing A (B), Es (E,) is the modulus of elasticity of the spindel (mandrel) material, Is (I,) is the moment of inertia of the spindle (mandrel) and yc is the eccentricity of the force F,.
286
a)
YC
b
Fig. 4. Deflections of wheelhead constructional elements at the point of application of the force: (a) load model; (b) deflection yb caused by flexibility of the bearings; (c)deflection ys caused by spindle bending; (d) deflection ym caused by grinding-wheel mandrel bending; (e) deflection yu,r, caused by grinding-wheel mandrel buckling; (f) total deflection y.
The total deflection (flexibility) y (Fig. 4(f)) is dependent not only on the kind of material used for the mandrel and the spindle of the grinding wheel, on their method of connection and on the variable parameters kA, k,, I,, Inn,a and b, but also, to a small degree, on the angle cy. These considerations might suggest that a direct increase in the rigidity of the wheelhead could be acquired by envy the spindle and bearing diameters. However, the greater the diameter of the bearings the lower is the
287
limiting rotational speed and, indirectly, the permissible grinding speed. In consequence, for any particular constant radial feed rate ufr there would be an increase in the various components of the grinding force which would result in an increase in shape errors. Other damaging effects on the course and the results of grinding are the prestressing of bearings which might be accompanied by their increased heating and therefore by a reduction in the permissible tool rotational speed. In order to obtain a high quality ISS it is necessary to choose values of spindle rotational speed, bearing rigidity and prestresses which are suitable for the required inner diameter. The course of an ISS grinding process with a relatively unfavourable rigidity in the wheelhead constructional elements is presented in Fig. 5. Phase I - the grinding wheel reaches the material (Fig. 5(a)): the grinding wheel, which rotates with rotational speed n, and moves with speed Ufr, must cover the distance Ed1so that its edge S can begin grinding the spherical socket of a workpiece turning with speed n,. Phase II - spherical surface grinding: removing the excess material of thickness q1 (Fig. 5(a)) leads to the attainment of the required diameter dkl for the spherical socket. If the grinding wheel were not being progressively worn away then its cutting edge S would move through the material for a distance Z,, . However, as the material is removed from the ground socket there is also wear on the edge of the grinding wheel reducing the height of the working ring by Ab, (Fig. 5(b)). The bigger the value of Ab, the greater are the grinding force components (F,, Ft and F,) and the lower is the supply of cutting fluid to the cutting zone. In consequence the intersection point of the X--X and y-y axes in the middle of the sphere Ok will change its position and move to the point Ok1 (Fig. 5(b)) which means that the ISS will assume the shape of an ellipsoid of revolution (shown in section as I-I in Fig. 5(b)). There comes a time when the working grinding wheel - workpiece interface can attain a value Ab, cat (catastrophic wear) such that as a consequence of the progressive restriction of the cutting fluid supply together with a simultaneous increase in the force components F, and Ft there will be a sudden indentation by the grinding wheel into the material which will leave a track in the form of a groove of depth g,. This phenomenon is connected with the self-dressing of the grinding wheel because a sudden increase in cutting resistance may result in a force sufficient to overcome the bonding power of the abrasive grains. The resulting violent loss of material from the grinding wheel causes it to lose contact with the workpiece. Phase III - repeated contact of the grinding wheel with a workpiece (Fig. 5(c)): to continue the process of grinding it is necessary to bring the grinding wheel wear from Abscat - by face-dressing - to a value Ab, < Ab Sperm< A& cat, where A&,,.,, is the maximum permissible size for the working surface of the grinding wheel, such that the grinding process is not dotted. The repeated contact of the grinding wheel with a socket entails covering the distance ld2, alternating with “grinding the air”. After grinding wheel-workpiece contact has been achieved the wheel goes into the material along the segment lq2 to grind out the groove which has been formed previously.
Traces of rapid grinding wheel cutting-in
into a workpiece in the form of groove
Fig. 5. ISS grinding with insufficient rigidity in the wheelhead elements: (a) grinding wheei before grinding (phase I); (b) phenomena that accompany griping-wheel spindle and mandrei bending (phase 11); (c) grinding wheel after self-dressing at a distance ictz from the ground spherical surface (phase III).
289
These phases of the course of the ISS grinding process illustrate the influence of unsuitable rigidity in the wheelhead elements both on the “outof-roundness” of the surface being formed and on its roughness, which resulted in loss of coherence in the cutting-track crossings.
5. The conditions for the investigation The investigation into the ISS grinding process was carried out during spherical socket machining on the face of a hydraulic pump shaft made of const~ction~ carburizing steel 20 H with a hardness of 57 - 62 HRC (Fig. 6). As well as the axial socket, on the circle of radius R there were 7 circumferential sockets equally spaced, the diameters of which were in the range dk = 26.5-o*1 - 2’7+‘*01mm. The spherical socket centres were on the shaft face.
Fig. 6. Hydraulic pump shaft with ground spherical sockets.
The test stand was a grinding machine for spherical sockets of type SGK-1000 (Fig. 7) designed and manufactured at the Department of Machining and Machine Tools of the Technical University in Rzeszow by order of a hydraulic pump and engine manufacturer. Two kinds of grinding wheels were used in the process of grinding one made of Al:,Os with 220 grain size and ceramic bonding (89A 220 E7 AV2 T3 - made by Tyrolit manufacturers) and the other made of CBN (Elbor) with 150 grain size, medium hardness and ceramic bonding (made in the U.S.S.R.). Gl concentrate emulsion of 10% concentration was used as a cutting fluid with both cooling and washing effect on the cutting zone. 6. Results and discussion In the tests which were carried out particular attention was given to establishing the dependence of the grinding wheel axial wear rate A6, and the material removal rate (4 (linear) on the radial feed rate uf, and the grinding wheel peripheral speed u, as well as the effect on the quality of the ground surface of the spherical socket.
Fig. 7. General view of the grinding machine for spherical sockets of type SGK-1000: 1, bed; 2, workpiece headstock; 3, workpiece holder; 4, carriage; 5, angle plate; 6, crossslide; 7, wheelhead; 8, stepper motor; 9, harmonic drive.
“w
Fig. 8. The principle of the measurement of the material removal rate 4 and the grinding wheel axial wear rate Ad,: I,, length of tool displacement at its contact with the workpiece.
Figure 8 is a pictorial representation of the principles followed in measuring A& and 4 as functions of uIr and u, which could refer equally well to ESS grinding [43. The measurement of the tool (grinding wheel) displacement distance I, at its point of contact with the workpiece was made with a micrometer sensor; the depth q was measured directly in the socket with the help of a special gauge. During the tests the grinding wheel axial wear
291
Ab, was monitored and maintained in the range Ab, < 3 mm by dressing the grinding wheel from the surface. The value of the grinding wheel axial wear Ab, can be defined from the dependence Ab, = 1, - 1,
(13)
and after substitution of 1, from eqn. (5), the final form of eqn. (13) can be represented as Ab, = I, -
f [{(& - 2q)2 - d,2}l’* - (dk2 - d,2)“*]
04)
With the tool displacement distance Eskept constant, and using differwheel axial ent radial feed rates Ufr, the grinding time ts gave the ~ding wear rate Af;, and the material removal rate (4from the relations %
At;, = -
pm s-l
t,
(15)
Q
t=
pm s-l
ta
Curves showing the variation of Ab, and 4 as functions of ufr are plotted in Fig. 9. They show that the radial feed rate uti significantly influences both A6, and 4. Above ufI = 0.1 mm mm-i better results are gained when CBN grinding wheels are used to grind an ISS. The dii of A&, 4 = f(v,) is more complicated, and is presented in Fig. 10. Whexeas the influence of u, on A& and 4 is very slight and oscillates around a certain stable value when an Al,& grinding wheel is used, temporarily more advantageous working conditions are achieved if u, is increasing, e.g. at u, = 12 or 25 m s-l, when a CBN grinding wheel is used. This type of a)
b)
200 x10-2
0
50
loo
150
2w radial
feed
0 speed
vfr,
loo
150
i 100
)tm/mi”,
Fig. 9. Curves showing the grinding wheel axial wear rate Ab, and material removal rate 4 us. radial feed (in-feed) rate ufr for grinding wheels: (a) CBN; (b) AIzOJ; d, = 22 mm; us = 24.8 m s-l; n, = 40 rev min-‘.
292 a)
b) 60
x10-2
25
30
grinding
wheel
10
15
peripheral
speed
vs,
n&
Fig. 10. Curve showing the grinding wheel axial wear rate Ad, and material removal rate 4 us. grinding wheel peripheral speed us for grinding wheels: (a) CBN; (b) &OS; d, = 22 mm; Ufr = 34 pm s-l; n, = 40 rev min-‘.
dependence probably arises from a difference in self-excited vibration and its influence on the course of the grinding process. In the range of ufr and u, illustrated in Figs. 9 and 10 the most advantageous results in respect of the working capacity were achieved when CBN grinding wheels with u, = 25 m 5-l and ufr > 0.15 mm min-’ were used for ISS grinding. The spherical socket surface quality can basically be defined in terms of two parameters: roughness and “out-of-roundness”. Both parameters are affected by the correctness or otherwise of the choice of grinding kinematics as well as by wheelhead rigidity, the specific character of the grinding wheel and cutting fluid, and the continuous control of tool rotation and workpiece axes so that they intersect in the middle of the sphere at Ok. When an ISS is ground with Al,03 grinding wheels its surface roughness is generally lower than that achieved when CBN grinding wheels are used. The influence of the grinding wheel speed u, on the value of the parameter R, (Fig. 11) shows that the best results for an A1203 grinding wheel occur when u, = 18ms-1,andforaCBNgrindingwheelwhenu,~225ms-1. From R, = f(tsp) in the diagram presented in Fig. 11(b), for both grinding wheels, it is possible to see that a value for the time up to sparkingbb
al
SO X1O-2
Xl0‘(:
s40
5
40
2
i*
30
30
B p H -
10
:: 3
$
20
& p
10
t: 0
10
15
20
2s grinding
30 sptui
35 V$’
m/s
B
0
0
‘3
6
9 time
I2
of spark-out
t
sp’
p
Fig. 11. Curve for ground surface roughness R, vs. : (a) grinding wheel peripheral speed us; (b) time of spark-out tsp.
293
Fig. 12. Micromorphology (a) CBN; (b) Al203.
of a spherical surface after being ground with grinding wheels:
out t,, = 6 s is the most advantageous from the point of view of surface roughness. The smaller R, parameter values achieved during ISS grinding with an AIIOj grinding wheel might seem at odds with the ~~o~opic~y undulating finish of that surface. This ~croscopi~~y more ~d~at~g fmisb on a spherical socket made by an A1203 grinding wheel (Fig. 12) can be beneficial from the tribological point of view and thus significantly extend the life of a spherical joint. The “out-of-roundness” of the spherical surfaces did not exceed 3 pm in any cross-section plane in any of the tests.
7. Conclusions On the basis of the conside~~ons presented here on ISS grinding and of the invest~atio~ which were carried out, it is possible to formulate the following conclusions. (1) The quality of a ground ISS obtained with the usuaI grinding kinematics “along the chord” depends mainly on the grinding wheel characteristics, wheelhead rigidity, the intersection of the tool and workpiece axes of rotation in the middle of the sphere at Ok (particularly in the final phase of grinding), the properties of the cutting fluid and the size of the gap W. (2) The rigidity of the wheelhead constructional elements should be chosen in view of the tool (grinding wheel) rotational speed, but choosing an optimum value from the point of view of acquired macro- and micromorphology of the ground ISS (with regard to the conditions mentioned in the preceding conclusion) should also assure improved eff~~ven~ of the machining process.
294
(3) In defining the total deflection Y of the grinding wheel it is possible Ymb which appears as an effect. of the force component F’,, because its value is very small compared with the sum of the remaining terms in eqn. (8 ) . (4) The material removal rate 4 is greatly influenced by the radial feed rate uEr; however, above the value uy, = 0.1 mm mm-’ more favourable results are obtained when CBN grinding wheels are used for ISS grinding. (5) With the grinding wheels examined, one essential fur a relatively favourable surface roughness was a time period to sparking-out. Qp of about 6 s. (6) With the same grinding con~tio~ greater roughness but less microscopic undulation in the surface was obtained with a CBN grinding wheel, compared with that produced by an Al,& grinding wheel. This results chiefly from the differences in the rigidity and thermal cu~~ucti~ty of the grinding wheels. A better surface state, from the tribological point of view, is acquired when an A&Q, grinding wheel is used for ISS grinding. to disregard the deflection
References I K. Qczo~ and T. Dzioch, Grinding of spherical surfaces, Mechanik, 52 (11) (1979) 591 - f&E (in Polish). 2 A. Bats&, W. Brzozowski, T. Dzioch aad J. Nowak, A method of g~ndi~~ of spherical socket surfaces in mult~*piston pump shafts, Potish Patmt 139803, 1988. 3 E. %ljb, H. Garnung and H.-G. v. Mackensen, Beitrag zur ~ormfa~le~in~mie~ng beii Innen~ud~hleifen, TZ Prakt. Me~ai~~ear~.~ 74 (4) (1980) 13 - 15. 4 K. OczoS and T. Dzioch, Phenomena in the spherical surface grinding of prostheses, lVear, $6 (1) (1984) 45 ” 59.