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A Study of Grinding Force Mathematical Model
A Study of Grinding Force Mathematical Model Li Lichun, Fu Jizai. Hunan University/P. R. of China
- Submitted by J. Peklenik (1)
Surnmaq: 3rinaing forces are composed of chip formation forces and friction forces. Based upon this ccncept, we have developed a new grinding force model, which is composed of two terms, corresponding tc chip formation forces and friction forces respectively. The relationship between grinding forces and grinding kinematic process parameters with workpieces of different materials is determined experimentally. The experimental results are in close aqeement with the proposed urudel. .Ye tave also aqalysed the ratio between tangential fcrces and ncrmal forces with respect to both chir formation forces and friction forces. Tte ratio generally is within the range of 0.2-0.59. When workpieces cf different materials are grcund, the measured values of the ratio are within this range. Based upcn the grinding fcrce model we have developed, the problem of the significacce of equivalent thickness cf cut is discussed.
i.
The development of a qrindine force mathematical moiel
In recent years the study of grinding force mathematical models has becn gradually developiq. Through the study cf the distributiiLn of the active grains in the wheel periphery and kinematics in qrinding operations, 0 . lerner, 1. Konig and H. Cpitz have set up a grinding force c.odel / l / , / Z / . Their basic idea is based on two functionai descriptions. The first is the functional description of the number and distribbtion of the engaged cutting edges, giving the number of eri;aged cutting edges per unit of wheel surface N d y n (1) versus variable of contact length 1 as a function of the static edge density C , , the character of edge distribution described by the exponential coefficient a and p , the wheel speed VS , the work speed V. , the depth of cut a, and the equivalent wteei diamter 3 (Fig. 1):
where
-
An Proportionality factor I K contact length between wheel and workpiece The second is the functional descriptior.of magnitude and distribution of the individual chip cross sections in the contact zone of wheel and workpiece. The average chip cross section at the variable point 1 in the contact zone a(1) is
The exponential coefficients a a n d p refer to the edge distribution in the wheel periphery and can vary in the range of:
0.c
o(
J2- < B
<+
(1 3
Seing analogous to some formulas of turninf: force, the cutting force Pa of a sinsle edqe in grinding is defined as a porer function of the chip cross section
Q:
Fe -KQ"
o-=n
(3)
where K is a proportionality factor. They consider that the exponential coefficient n expresses the relaticnship between friction forces and deformation forces produced in the cutting process, The more n deviates from 1 towards 0, the greater is the friction part in the total grinding force; for a pure chip fcrmation process, n = 1, and for a pure friction process, n = C. The normal grinding force per unit cf contact area fcr a certain point 1 in the contact zone is expressed by (4)
By integration of this function with respect to the total contact length from 1 = 0 to 1 = l ~ : the normal grinding force per unit of grinding width is derived:
(5) Annals of the ClRP Vol. 29/1/1980
with
r-p
(I-n)
This model'has taken into consideration the stochastic peculiarities of the distribution of the cutting edges in the wheel periphery and the related theory of kinematics in grinding processes. However, it has its shortcomings. In spite of the fact that G. Werner et al. have recognized correctly that grinding forces come from two kinds of mechanisms chip formation and friction, they have not made a clear differentiation between them in the physical sense. The grinding force and turning force are similar in that their total 2utting force consists of some parts which essentially differ from one another. The formation mechanics and rules governing the various parts are different. In developing a cutting force model, one should deal with the different constituent parts in different way, so that the fundamental laws of cutting forces may be explored. ;mile they were researching turning forces, C. Hubenstein et al. tried to divide the cutting force into a chip formation f o r ce, a force component arising because of the finite radius of curvature of the cutting edge, and a friction force between the flank wear land and the /4/. Starting from such a point of view, .workpiece C. Hubenstein divided grinding forces into more detailed constituent parts in his researches on grinding mechanics. He took the following forces into consideration: the force component arising because of the finite radius of curvature of the cutting edge, the friction force between the wear flats of the active grains and the workpiece, the force for grains to cut the workpiece material, the force for grains to plough the workpiece material and the friction force between bond and workpiece material, etc /5/. However, such a division is too complicated to establish a practical formula for grinding forces. As compared with turning, the grinding process has its own characteristics. The tip area of grains sliding along the workpiece material is larger than the chip cross sections. Cf the total grinding force, the friction force between active grains and workpiece material accounts for a considerable part. With the wear of the wheel the rapid change in the grinding force is nainly $ue to the change of this friction force. The experimental results achieved by S. Malkin and h'. ii. Cook on the relation between the grinding force and the wear flat area of the wheel have proved that the following analysis is correct (Fig. 2) /6/.
/v,
Ft Fa
Fnc Ftc
+
Fns
Fts
(6)
(7)
'here Pn
- normal grinding force - tangential grinding force - normal grinding force due to chip formation F N - normal grinding force due to friction Ftc - tangential grinding force due to chip Ft
Plr
-
formation. Fts tangential grinding force due to friction. Through experiments they determined the mutual relationship between the grinding force and the wear flat area of the wheel under the condition that the kinematic process parameters are constant. The typical experimental curves are shown in Fig. 3. In the course of the normal grinding process, provided that burning does not oxur, the normal grinding force
245
increases linearly with the wear flat area of the wheel, indicating that the average contact pressure
3 between the wear flats and workpiece is constant
and proportional to the hardness of the 'workpiece material. Since the tangential force also increases linearly with wear flat area, +be ccefficientwof sliding friction between the wear flats and workpiece is also constant. The coefficient of sliding friction equals the ratio between the sioFe of the tangential force curve and that of the normal force curve in Fiq. 3. If C i R represents the real contact area between wheel and workpiece, tken
F;
- F&
+
F&
Each term in the above formlas is cciLposed of the corresponding forces acting on grains. From Eq. (12), Fen' = x,?, tte Chip fcrmation force of every grain is directly proportional to its chip cross sectior, 3, thus the chip formation part of the total normal ginding force per unit grinding .uidth 1s:
F& = Z K i Q i - K o E Q i For the cutting force imposedbyasingle qrain , similar result can be derived from the conclusions of S. Lalkin and N. H. Cook Fen-Few Fet 'Fetc.
+SP + Fets 'Fete + d P +
Fens -Fenc
where 6 is the tip area of an active gain, the real contact area between the active grain and the workpiece. The relationship between the chip formation force expressed in equations (10) and (11) and the chip cross section can be deduced from the results of experiments with turning force. Figure 4 shows the relationship between cutting force and undeformed chip thickness & determined by different authors /7/. Cwing to the squeezing of the round edge against the machined surface, the cutting force has a nonlinear relationship with the undeformed chip thickness when QC 0 , and has a linear relationship when Q is larger than the radius cf curvature of cutting edge, as shown'in Fig. 4(a). In Fig. 4(b), the cutting force always has a linear relationship with the undeformed chip thickness, but when U , = 0, a residual squeezing force exists. It follows that by subtracting the squeezing force of the round edge from the total cutting force, we obtain the ChiF formation force, which is proportional to the undeformed chip thickness, and consequently, to the undeformed chip cross-sectional area. The investigation by C. Rubenstein et al. also shows that the chip formation force as part of the turning force 1s proportional to the undefcrmed chip cross-sectional area /3/, /4/. As mentioned above, while developing the grinding force model, C. Nerner et al. hold that as far as the pure chip formation process is concerned, the following equation holds F. = KO. Using the results from experiments with turning forces, we obtain
-
Fenc
- KI Q
(12)
where K I is the specific chip formati0.i force of the. normal grinding force. As to the tangential force of chip formation, if only the normal force acting between the face of the grain and the chip is taken i - t o consideration, the friction force between grain and ?hip is neglected and it is supposed that grains are cone-shaped and the axes of the cones are arranged in the direction of the radius of the wheel, according to ttis geometric relationship, we then have /7/
where 8 Let then
- half of 9-2-
-
the tip angle of the grains.
4 tan@
Fetc
gK,Q
(14)
From Eqs. (lo), (111, (12) and (14)
can be derived by substitution. The,normal grinding force per unit of grinding width Fn and tangential grinding force per unit of grinfling width Fi , acting between a wheel and a workpiece, are equal to the total of the normal forces and the tangential forces of all active grains within the contact area between wheel and workpiece per unit of grinding width. Similarly, Fn and P; may be differentiated into,chip formation forces PAC, Pic and friction forces Fns , Fir Then
.
246
(191
&ere E Q i is the total value of all simultanecus chip cross secticns per m i t of grindinq width, kence
(1CI
(11)
(18)
~ n -;K I % . a
(20)
From Eq. (10)F w I s - ~ , ~the frictional part of t;e total grinding force per unit of grinding width = N x F , uhere N is Frrs is given by the formulas:'3 the number of all simult_anecuslyacting grains per unit of grinding width, 6 is the average tip area per active grain. Using Eq. (1) by C. 'Nerner et al. for engaged grains, we obtain
thereby
By adding together the chip formation part and the friction part of the normal grinding force per unit of grinding width, we obtain
Similarly, for tangential grinding force per unit of grinding width, we have
Thus, a grinding force model conposed of two terms is obtained. The first term corresponds to chip formation force and the second to friction force. Substituting o u r formla (12)Fen- K , Q + 6 P for G. 'Nerner's formula ( 3 ) Fe = KQ and using C. Nerner's calculating method, the normal grinding force per unit of contact area f o r a certain point 1 in the contact zoce ca? be expressed as:
Integrating this function with respect to the total contact length from 1 = C to 1 = l A :
we obtain the same result as formula (23). Comparing our grinding force formula (23) with G. Ne-qer's formula ( 5 ) we find that on the boundary values they are in agreement. F o r a pure chip formation process, the exponential coefficient n takes the valus of one in 5 . Nerner's formulas, thus E = 1 , I = 0, and therefore, formula ( 5 ) becomes Fk = K + Q and which is the same as the chip formation force part of our grinding force formula (23). For a pure friction process, the exponential coefficient n takes the value of 0, and formula ( 5 ) becomes
Comparing it to the friction part of our formula (23), find that they are the same except for *he different form of the proportionality factors. However, for a practical grindirg process in which both chip formation and friction-exist, the forms of the two formulas are different. a. Xerner et sl. take different values of the exponential coefficient n which is undefinable in a physical sense to express the relationship between the influence of chip formation and that of friction, however, we calculate the two force components separately and add them up. Each component has a more definite physical sense. WP
Furthermcre, in a grinding process, the grinding fcrce changes rapidly due to wheel wear. A grinding fcrce model should be able to reflect the change in the grinding force in a continual grinding process and shclild not be limited in a specific condition with a frestly dressed wheel. A n cur ZrindiK force model, the change of grinding fcrce due to wheel wear is mainly reflected by the change in tte average tip area of active grains. Based cn the grinding force model obtained in this mamer, it is pcssible t o ccnpletely discuss the ratio between tangential force and nornal force with respect to bcth chip formation force axxd friction force. From eqs. (231, (24) and (17), (1&) we get:
The ratio .between the chip formation component of tine tangential grinding force and that of the normal grinding force is decided by geometrical conditions, e.g. the tip angle etc., while the ratio between the frictional component of the tangential qrinding force and that cf the normal grinding force is decided by the coefficient of sliding friction between the tip fiats of active grains and the workpiece. The ratio between the tangential grinding force and the normal grinding force p is as follows:
p-s*++ Cnder the condition that the grinding force is a pure chip fonnaticn force, FA = P& , Pis= 0 and f =+. Cnder the condition that tte,grinding force is a pure friction force, FA = Fns , Fnc = 0 and P = r l ~ For a practical grindiq process, the ratio of the grinding force 'j is decided by 9 , Y and also by the relationship of the chip formation component and the friction component to the total normal grinding force. According $0 /g/, the tip angle of a grain is 28 = 104'- 108 .Taking the average value, we may have e = 53.. Thus,
.
As for the coefficient of slidine fricticn I ( , estimated according to the experimental curves of S. lalkin et al. about tangential and normal grinding forces versus wheel wear flat area, the values of different workpiece materials are different; the minimal value is that of high speed steel 118Cr4V, rccmin = 0.2. The values o f u with all other workpiece materials are less than 0.59, with the exception of grinding Titanium. Consequently it can be generally considered that the ratio between tangential grinding force and normal grindinq force should be within the range of 0.2-0.59. In other words, normal grinding force should be 1.7-5 times as large as tangential grinding force.
2.
Experiments and analysis of experimental results
Grinding experiments were performed to verify the grinding force model proposed with respect to the relationship between the kinematic process parameters and the grinding force .The experiments were conducted on an external cylindrical grinder using plunge grinding. Chanqing alternately each of the parameters such as peripheral work speed V, , peripheral wheel speed b and depth of cut a, while keeping the other parameters constant, we measured the normal and tangential grinding forces in each case. Ne then obtained the functional relationship between normal grinding force per unit of grinding width and the kinematic process parameters. 'Ne also obtained the ratio between tangential grinding force and normal grinding force. When V, , VS and a are constant, the basic set of parameters is VWO = 0.69 3f , VSO = 50 % and ao= O.CO2 mm. In order to keep the working Conditions of the wheel constant in all experiments, we used a GB 70 ZYI AP 4 0 0 x 40 X 127 grinding wheel with higher grade. For the purpose of keeping the working conditions of the dressing tool constant, two single diamond dressers were used for coarse and fine dressing respectively. l e used diamcnd for coarse dressing in order to maintain the cutting ability of the wheel and for fine dressing in order to keep the wheel surface conditions constant. The fine dressing parameters were
, dressing depth as follows: dressing lead 0.1 "%v of cut C.015, 0.015, 0.005 091 .iith two "spark-out" strokes. The experiments were performed under relatively small normal grinding forces per unit of grinding width so that a stationary wheel condition might be maintained, the values of which were less than 16 %. The coolant used in these experiments was of a soluble type. Three iifferefit sorkpiece materials were used in +.baeseexperiments. .They are listed in the following table together with their hardness at room temperature. Borkpiece Eaterial medium carbon steel 45(-0.45%C) bearing steel GCrl5 high speed steel iP18Cr4V
50
54
The grinding forces,were measured by a strain gage dynamometer with eight strain gages mounted on the tail centre, forming two bridges to measure the normal and tangential grinding force components respectively. The strain gage dyriamometer was connected to a light beam recorder. Disk-shaped specimens about 120 m in diameter were fixed one by one into a difinite place cn the axis. 3y calibration, the iynamometer centre gave C.25wc output for each N of normal o r tangential grinding force. The values of normal and tangential grinding force components measured by this dynamometer centre were undisturbed by one another. For each experiment, the wheel was dressed according to the dressing procedure mentioned above, and then the grinding process was started. m e n the grindine force attained its steady state, the grinding process was interrupted. Cnly the steady state values of the grinding force were taken into account. The results of experiments were interpreted in the following manner: Since we used the relationships between the grindiq force and grinding kinematic process parameters to check the grinding force model, Eq. (23) may be reduced to
i: bears a relation to the equivalent diame'.er of the wheel, the static edge density and the distribution of active grains, and is directly propcrtional to the average tip area of active grains and the average pressure on the contact area between the tip flats of activL grains and the workpiece fi. Since the accumulated metal removal after dressing-in each experiment was very small, the value of 6 may be considered 3s constant. Therefore, the value of C is a constant determined by the properties of the specimen materials (5). As the friction components of the grinding force are in direct proportion to C , C nay be called the specific friction force of the normal grinding force. There are three unknowns, R I , C and in formuia (29). By applying the method of least squares to the V.,, , Ph Vs three sets of experimental data, FA a, obtained for grinding bearing steel, we and F'n determined that the arameters are K I = 127000N/mm' , ci= 0.33, and C = 398 N / , n L 6 u In the grinding experiments with medium carbon steel and high speed steel, the same wheel and same dressing condition were maintained. Since c4 refers to the edge distribution in the wheel periphery from which the minding force formula is determined, dmay be considered constant for all specimen material. kiowever, the specific chip formation force and the specific friction force determined by the proporties of the specimen materials differ with different materials. Takingo!= 0.33, and applying the method of the le?.st squares to the various data sets for medium carbon steel and high speed steel, the values of parameters KI and C for grinding medium carbon steel and high speed steel were evaluated as follows:
-
-
$:$ctP; C (
-
.
GCrl5 127000
~
N/rnm'.MB
398
)
45 111600 250
11 8Cr4V
149000 2180
Substituting the values of KIand C for various specimen materials in formula (291, and takings= 0.33, the relationship of F'n- VW , F', VS and F'n a f o r various workpiece materials were obtained. Drawings are shown in Fig. 5-13. All the actual experimental points have been plotted. The deviations of the experimental data from the theoretical curves are small with the exception of the data set for high speed steel P h - a. This indicates that grinding formula (29) may well closely estimate the results of actual grinding experiments. In Fig. 5-13, the straight lines, F& K, a , have been drawn.
-
-
-
#
247
The portion below the straight line corresponds tc, chip formation force, and the portion above the straight line and under the curve corresponds to friction force. From the above results it may be seen that because of the "size effect", under the condition of light cutting, the specific chip formation force KIcf the normal grindkg force is very high. K, increase with the sequence: medium carbon steel 45, bearing steel and high speed steel. The specific friction force C of these three materials also increases with the same sequence. This is due to the increasing hardness of the workpiece materials and related values p under the conditions of high temperatures during the grinding process in the sequence mentioned above. The red hardness of high speed steel is especially high, and the value C is, tterefore, extremely qreat. The chip formation force plays a main part when easy-to-grind materials such as medium carbon steel o r bearing steel are ground. And for high speed steel which is difficult to grlnd, the frictional force is greater than the chip formation force in all our experimental conditions. Therefore the value of the total grinding force is extremely high. That is one of the reasons why high speed steel is difficult to grind. He have also experimentally determined the relationship between the ratio of tangential grinding force to normal force and the kinematic process parameters. The experimental results are shown in Fig.
14-16. For medium carbon steel 45 (-0.45$C), a material that is easy to grind, the valueofp is comparably great. The maximal value obtained experimentally is p,,= 0.49. This approximates the case when pure chip formation exists; for high speed steel, a material difficult to grind, the value of f is less, and the minimal value obtained ispnin= 0.25. This approximates the case when pure friction exists. In all cases, with the increase of the kinematic process parameters i.e. with the increase of the chip formation part of the grinding force, the ratio p of the tangential grinding force to the normal grinding force increases, as shown in Fig.14-16.
3. The significance of the equivalent thickness of cut heq W e n studying the relationship between input and output conditions of a complicated grinding process, people try their best to search for basic grinding parameters having influence on the grinding processes. In 1974 CIRP proposed to take the equivalent thickness of cut heq = h a as a basic grinding parameter VI
/8/. It has been several years since then, however,
this proposal has still not been widely adopted. To what degree is the equivalent thickness of out related to the physical nature of the grinding process? Taking the grinding force into consideration and according to our proposed grinding force model, we may transform formula (23) slightly into
FA -K,
heq + ~ [ c , ] ' @ . D ]
hagd
(30)
It is obvious that the component of the chip formation force as a part of the grinding force is proportional to the equivalent thickness of cut, while the frictional force component of the grinding force has only a slight relationship with it and is determined to a large degree by the condition of the grinding wheel. During the grinding process once the wheel is worn, the average tip area of active grains 6 varies, and the grinding force will change significantly. Cnly if an easy-to-grind material is ground, and the wheel remains sharp, with the result that the chip formation force plays a significant part in the grinding force, the equivalent chickness of cut together with the specific chip formation force approximetely determines the value of the grinding force. As for the other conditions, it seems that the equivalent thickness of cut is far from able to determine the value of the grinding force. S. Hahn et al. take the normal grinding force per unit of grinding width F h as a basic grinding parameter /lo/. This parameter further characterizes the essentials of the grinding process, however, for an ordinarJi constant infeed rate grinding process it is not covenient to use.
248
References 1. G. ilerner: "Influecce of York itaterial on Srinding Forces". Annals cf CIXP (1978) 2. H. Cpitz,l. Kcnig and G. 7erner: "Xinercatics and Mechanics in Grinding with Regard to the KaEhining Process", Proceedings of the International *rinding Conference, "Xew Development in :rindin#' (1972) 3. J. N. "Jeenhow and C. Rubenstein: "The Dependence of Cutting Force on Feed and Speed in Crthogonal cutting with llorn Tools" Int. u'. lach. Tool Des. Res. 9, 1 (1969) 4. R. Connolly and C. Hubenstein: "The Kechafiics of Continuous Chip Formation in Crthogcnal Cuttin I' Int. J. Ilach.Too1 Des. Res. 8, 159 (196f) 5. C. Subenstein: "The Eechanics of ?rir.ding" Int. J. Each. Tool Des. Res. 12, 127(1972) 6. S. Ealkin, N. H. Cook: "The Rear of Grinding #heels" Trans. AS123 "B" Vol. 93 N0.4 (1971) 7. Usuihideji "Technology of Cutting and Grinding". Japan (1971) R. Snocys: "The Significance of Chip Thickness in :rinding", Annals of CIRP (1974) Handbook 9. "The Part of bletal Cutting, Grinding" of Lechanical Engineering, Peop. Rep. of China 10 S . Hahn: "Principles of Grinding" Kachinery V o l . 77 (1971)
a.
FIG. 7
FIG. 6 1
a = 0.002 mm
FIG. 2 FIG. 9
FIG. I 0
STEEL Wl8t.4 V V p 0 . 6 9 9$ = 0.002 nm
4
0
O.MI
FIG. I I
0.W2
FIG 13
0&13a PW
mn
i 0
o
0.4
0.6~aeaslC%M
FIG 15
0
Odol
Mo2
QWanrn
FIG 16
FIG y4
249
- Submitted by J. Peklenik (1)
Surnmaq: 3rinaing forces are composed of chip formation forces and friction forces. Based upon this ccncept, we have developed a new grinding force model, which is composed of two terms, corresponding tc chip formation forces and friction forces respectively. The relationship between grinding forces and grinding kinematic process parameters with workpieces of different materials is determined experimentally. The experimental results are in close aqeement with the proposed urudel. .Ye tave also aqalysed the ratio between tangential fcrces and ncrmal forces with respect to both chir formation forces and friction forces. Tte ratio generally is within the range of 0.2-0.59. When workpieces cf different materials are grcund, the measured values of the ratio are within this range. Based upcn the grinding fcrce model we have developed, the problem of the significacce of equivalent thickness cf cut is discussed.
i.
The development of a qrindine force mathematical moiel
In recent years the study of grinding force mathematical models has becn gradually developiq. Through the study cf the distributiiLn of the active grains in the wheel periphery and kinematics in qrinding operations, 0 . lerner, 1. Konig and H. Cpitz have set up a grinding force c.odel / l / , / Z / . Their basic idea is based on two functionai descriptions. The first is the functional description of the number and distribbtion of the engaged cutting edges, giving the number of eri;aged cutting edges per unit of wheel surface N d y n (1) versus variable of contact length 1 as a function of the static edge density C , , the character of edge distribution described by the exponential coefficient a and p , the wheel speed VS , the work speed V. , the depth of cut a, and the equivalent wteei diamter 3 (Fig. 1):
where
-
An Proportionality factor I K contact length between wheel and workpiece The second is the functional descriptior.of magnitude and distribution of the individual chip cross sections in the contact zone of wheel and workpiece. The average chip cross section at the variable point 1 in the contact zone a(1) is
The exponential coefficients a a n d p refer to the edge distribution in the wheel periphery and can vary in the range of:
0.c
o(
J2- < B
<+
(1 3
Seing analogous to some formulas of turninf: force, the cutting force Pa of a sinsle edqe in grinding is defined as a porer function of the chip cross section
Q:
Fe -KQ"
o-=n
(3)
where K is a proportionality factor. They consider that the exponential coefficient n expresses the relaticnship between friction forces and deformation forces produced in the cutting process, The more n deviates from 1 towards 0, the greater is the friction part in the total grinding force; for a pure chip fcrmation process, n = 1, and for a pure friction process, n = C. The normal grinding force per unit cf contact area fcr a certain point 1 in the contact zone is expressed by (4)
By integration of this function with respect to the total contact length from 1 = 0 to 1 = l ~ : the normal grinding force per unit of grinding width is derived:
(5) Annals of the ClRP Vol. 29/1/1980
with
r-p
(I-n)
This model'has taken into consideration the stochastic peculiarities of the distribution of the cutting edges in the wheel periphery and the related theory of kinematics in grinding processes. However, it has its shortcomings. In spite of the fact that G. Werner et al. have recognized correctly that grinding forces come from two kinds of mechanisms chip formation and friction, they have not made a clear differentiation between them in the physical sense. The grinding force and turning force are similar in that their total 2utting force consists of some parts which essentially differ from one another. The formation mechanics and rules governing the various parts are different. In developing a cutting force model, one should deal with the different constituent parts in different way, so that the fundamental laws of cutting forces may be explored. ;mile they were researching turning forces, C. Hubenstein et al. tried to divide the cutting force into a chip formation f o r ce, a force component arising because of the finite radius of curvature of the cutting edge, and a friction force between the flank wear land and the /4/. Starting from such a point of view, .workpiece C. Hubenstein divided grinding forces into more detailed constituent parts in his researches on grinding mechanics. He took the following forces into consideration: the force component arising because of the finite radius of curvature of the cutting edge, the friction force between the wear flats of the active grains and the workpiece, the force for grains to cut the workpiece material, the force for grains to plough the workpiece material and the friction force between bond and workpiece material, etc /5/. However, such a division is too complicated to establish a practical formula for grinding forces. As compared with turning, the grinding process has its own characteristics. The tip area of grains sliding along the workpiece material is larger than the chip cross sections. Cf the total grinding force, the friction force between active grains and workpiece material accounts for a considerable part. With the wear of the wheel the rapid change in the grinding force is nainly $ue to the change of this friction force. The experimental results achieved by S. Malkin and h'. ii. Cook on the relation between the grinding force and the wear flat area of the wheel have proved that the following analysis is correct (Fig. 2) /6/.
/v,
Ft Fa
Fnc Ftc
+
Fns
Fts
(6)
(7)
'here Pn
- normal grinding force - tangential grinding force - normal grinding force due to chip formation F N - normal grinding force due to friction Ftc - tangential grinding force due to chip Ft
Plr
-
formation. Fts tangential grinding force due to friction. Through experiments they determined the mutual relationship between the grinding force and the wear flat area of the wheel under the condition that the kinematic process parameters are constant. The typical experimental curves are shown in Fig. 3. In the course of the normal grinding process, provided that burning does not oxur, the normal grinding force
245
increases linearly with the wear flat area of the wheel, indicating that the average contact pressure
3 between the wear flats and workpiece is constant
and proportional to the hardness of the 'workpiece material. Since the tangential force also increases linearly with wear flat area, +be ccefficientwof sliding friction between the wear flats and workpiece is also constant. The coefficient of sliding friction equals the ratio between the sioFe of the tangential force curve and that of the normal force curve in Fiq. 3. If C i R represents the real contact area between wheel and workpiece, tken
F;
- F&
+
F&
Each term in the above formlas is cciLposed of the corresponding forces acting on grains. From Eq. (12), Fen' = x,?, tte Chip fcrmation force of every grain is directly proportional to its chip cross sectior, 3, thus the chip formation part of the total normal ginding force per unit grinding .uidth 1s:
F& = Z K i Q i - K o E Q i For the cutting force imposedbyasingle qrain , similar result can be derived from the conclusions of S. Lalkin and N. H. Cook Fen-Few Fet 'Fetc.
+SP + Fets 'Fete + d P +
Fens -Fenc
where 6 is the tip area of an active gain, the real contact area between the active grain and the workpiece. The relationship between the chip formation force expressed in equations (10) and (11) and the chip cross section can be deduced from the results of experiments with turning force. Figure 4 shows the relationship between cutting force and undeformed chip thickness & determined by different authors /7/. Cwing to the squeezing of the round edge against the machined surface, the cutting force has a nonlinear relationship with the undeformed chip thickness when QC 0 , and has a linear relationship when Q is larger than the radius cf curvature of cutting edge, as shown'in Fig. 4(a). In Fig. 4(b), the cutting force always has a linear relationship with the undeformed chip thickness, but when U , = 0, a residual squeezing force exists. It follows that by subtracting the squeezing force of the round edge from the total cutting force, we obtain the ChiF formation force, which is proportional to the undeformed chip thickness, and consequently, to the undeformed chip cross-sectional area. The investigation by C. Rubenstein et al. also shows that the chip formation force as part of the turning force 1s proportional to the undefcrmed chip cross-sectional area /3/, /4/. As mentioned above, while developing the grinding force model, C. Nerner et al. hold that as far as the pure chip formation process is concerned, the following equation holds F. = KO. Using the results from experiments with turning forces, we obtain
-
Fenc
- KI Q
(12)
where K I is the specific chip formati0.i force of the. normal grinding force. As to the tangential force of chip formation, if only the normal force acting between the face of the grain and the chip is taken i - t o consideration, the friction force between grain and ?hip is neglected and it is supposed that grains are cone-shaped and the axes of the cones are arranged in the direction of the radius of the wheel, according to ttis geometric relationship, we then have /7/
where 8 Let then
- half of 9-2-
-
the tip angle of the grains.
4 tan@
Fetc
gK,Q
(14)
From Eqs. (lo), (111, (12) and (14)
can be derived by substitution. The,normal grinding force per unit of grinding width Fn and tangential grinding force per unit of grinfling width Fi , acting between a wheel and a workpiece, are equal to the total of the normal forces and the tangential forces of all active grains within the contact area between wheel and workpiece per unit of grinding width. Similarly, Fn and P; may be differentiated into,chip formation forces PAC, Pic and friction forces Fns , Fir Then
.
246
(191
&ere E Q i is the total value of all simultanecus chip cross secticns per m i t of grindinq width, kence
(1CI
(11)
(18)
~ n -;K I % . a
(20)
From Eq. (10)F w I s - ~ , ~the frictional part of t;e total grinding force per unit of grinding width = N x F , uhere N is Frrs is given by the formulas:'3 the number of all simult_anecuslyacting grains per unit of grinding width, 6 is the average tip area per active grain. Using Eq. (1) by C. 'Nerner et al. for engaged grains, we obtain
thereby
By adding together the chip formation part and the friction part of the normal grinding force per unit of grinding width, we obtain
Similarly, for tangential grinding force per unit of grinding width, we have
Thus, a grinding force model conposed of two terms is obtained. The first term corresponds to chip formation force and the second to friction force. Substituting o u r formla (12)Fen- K , Q + 6 P for G. 'Nerner's formula ( 3 ) Fe = KQ and using C. Nerner's calculating method, the normal grinding force per unit of contact area f o r a certain point 1 in the contact zoce ca? be expressed as:
Integrating this function with respect to the total contact length from 1 = C to 1 = l A :
we obtain the same result as formula (23). Comparing our grinding force formula (23) with G. Ne-qer's formula ( 5 ) we find that on the boundary values they are in agreement. F o r a pure chip formation process, the exponential coefficient n takes the valus of one in 5 . Nerner's formulas, thus E = 1 , I = 0, and therefore, formula ( 5 ) becomes Fk = K + Q and which is the same as the chip formation force part of our grinding force formula (23). For a pure friction process, the exponential coefficient n takes the value of 0, and formula ( 5 ) becomes
Comparing it to the friction part of our formula (23), find that they are the same except for *he different form of the proportionality factors. However, for a practical grindirg process in which both chip formation and friction-exist, the forms of the two formulas are different. a. Xerner et sl. take different values of the exponential coefficient n which is undefinable in a physical sense to express the relationship between the influence of chip formation and that of friction, however, we calculate the two force components separately and add them up. Each component has a more definite physical sense. WP
Furthermcre, in a grinding process, the grinding fcrce changes rapidly due to wheel wear. A grinding fcrce model should be able to reflect the change in the grinding force in a continual grinding process and shclild not be limited in a specific condition with a frestly dressed wheel. A n cur ZrindiK force model, the change of grinding fcrce due to wheel wear is mainly reflected by the change in tte average tip area of active grains. Based cn the grinding force model obtained in this mamer, it is pcssible t o ccnpletely discuss the ratio between tangential force and nornal force with respect to bcth chip formation force axxd friction force. From eqs. (231, (24) and (17), (1&) we get:
The ratio .between the chip formation component of tine tangential grinding force and that of the normal grinding force is decided by geometrical conditions, e.g. the tip angle etc., while the ratio between the frictional component of the tangential qrinding force and that cf the normal grinding force is decided by the coefficient of sliding friction between the tip fiats of active grains and the workpiece. The ratio between the tangential grinding force and the normal grinding force p is as follows:
p-s*++ Cnder the condition that the grinding force is a pure chip fonnaticn force, FA = P& , Pis= 0 and f =+. Cnder the condition that tte,grinding force is a pure friction force, FA = Fns , Fnc = 0 and P = r l ~ For a practical grindiq process, the ratio of the grinding force 'j is decided by 9 , Y and also by the relationship of the chip formation component and the friction component to the total normal grinding force. According $0 /g/, the tip angle of a grain is 28 = 104'- 108 .Taking the average value, we may have e = 53.. Thus,
.
As for the coefficient of slidine fricticn I ( , estimated according to the experimental curves of S. lalkin et al. about tangential and normal grinding forces versus wheel wear flat area, the values of different workpiece materials are different; the minimal value is that of high speed steel 118Cr4V, rccmin = 0.2. The values o f u with all other workpiece materials are less than 0.59, with the exception of grinding Titanium. Consequently it can be generally considered that the ratio between tangential grinding force and normal grindinq force should be within the range of 0.2-0.59. In other words, normal grinding force should be 1.7-5 times as large as tangential grinding force.
2.
Experiments and analysis of experimental results
Grinding experiments were performed to verify the grinding force model proposed with respect to the relationship between the kinematic process parameters and the grinding force .The experiments were conducted on an external cylindrical grinder using plunge grinding. Chanqing alternately each of the parameters such as peripheral work speed V, , peripheral wheel speed b and depth of cut a, while keeping the other parameters constant, we measured the normal and tangential grinding forces in each case. Ne then obtained the functional relationship between normal grinding force per unit of grinding width and the kinematic process parameters. 'Ne also obtained the ratio between tangential grinding force and normal grinding force. When V, , VS and a are constant, the basic set of parameters is VWO = 0.69 3f , VSO = 50 % and ao= O.CO2 mm. In order to keep the working Conditions of the wheel constant in all experiments, we used a GB 70 ZYI AP 4 0 0 x 40 X 127 grinding wheel with higher grade. For the purpose of keeping the working conditions of the dressing tool constant, two single diamond dressers were used for coarse and fine dressing respectively. l e used diamcnd for coarse dressing in order to maintain the cutting ability of the wheel and for fine dressing in order to keep the wheel surface conditions constant. The fine dressing parameters were
, dressing depth as follows: dressing lead 0.1 "%v of cut C.015, 0.015, 0.005 091 .iith two "spark-out" strokes. The experiments were performed under relatively small normal grinding forces per unit of grinding width so that a stationary wheel condition might be maintained, the values of which were less than 16 %. The coolant used in these experiments was of a soluble type. Three iifferefit sorkpiece materials were used in +.baeseexperiments. .They are listed in the following table together with their hardness at room temperature. Borkpiece Eaterial medium carbon steel 45(-0.45%C) bearing steel GCrl5 high speed steel iP18Cr4V
50
54
The grinding forces,were measured by a strain gage dynamometer with eight strain gages mounted on the tail centre, forming two bridges to measure the normal and tangential grinding force components respectively. The strain gage dyriamometer was connected to a light beam recorder. Disk-shaped specimens about 120 m in diameter were fixed one by one into a difinite place cn the axis. 3y calibration, the iynamometer centre gave C.25wc output for each N of normal o r tangential grinding force. The values of normal and tangential grinding force components measured by this dynamometer centre were undisturbed by one another. For each experiment, the wheel was dressed according to the dressing procedure mentioned above, and then the grinding process was started. m e n the grindine force attained its steady state, the grinding process was interrupted. Cnly the steady state values of the grinding force were taken into account. The results of experiments were interpreted in the following manner: Since we used the relationships between the grindiq force and grinding kinematic process parameters to check the grinding force model, Eq. (23) may be reduced to
i: bears a relation to the equivalent diame'.er of the wheel, the static edge density and the distribution of active grains, and is directly propcrtional to the average tip area of active grains and the average pressure on the contact area between the tip flats of activL grains and the workpiece fi. Since the accumulated metal removal after dressing-in each experiment was very small, the value of 6 may be considered 3s constant. Therefore, the value of C is a constant determined by the properties of the specimen materials (5). As the friction components of the grinding force are in direct proportion to C , C nay be called the specific friction force of the normal grinding force. There are three unknowns, R I , C and in formuia (29). By applying the method of least squares to the V.,, , Ph Vs three sets of experimental data, FA a, obtained for grinding bearing steel, we and F'n determined that the arameters are K I = 127000N/mm' , ci= 0.33, and C = 398 N / , n L 6 u In the grinding experiments with medium carbon steel and high speed steel, the same wheel and same dressing condition were maintained. Since c4 refers to the edge distribution in the wheel periphery from which the minding force formula is determined, dmay be considered constant for all specimen material. kiowever, the specific chip formation force and the specific friction force determined by the proporties of the specimen materials differ with different materials. Takingo!= 0.33, and applying the method of the le?.st squares to the various data sets for medium carbon steel and high speed steel, the values of parameters KI and C for grinding medium carbon steel and high speed steel were evaluated as follows:
-
-
$:$ctP; C (
-
.
GCrl5 127000
~
N/rnm'.MB
398
)
45 111600 250
11 8Cr4V
149000 2180
Substituting the values of KIand C for various specimen materials in formula (291, and takings= 0.33, the relationship of F'n- VW , F', VS and F'n a f o r various workpiece materials were obtained. Drawings are shown in Fig. 5-13. All the actual experimental points have been plotted. The deviations of the experimental data from the theoretical curves are small with the exception of the data set for high speed steel P h - a. This indicates that grinding formula (29) may well closely estimate the results of actual grinding experiments. In Fig. 5-13, the straight lines, F& K, a , have been drawn.
-
-
-
#
247
The portion below the straight line corresponds tc, chip formation force, and the portion above the straight line and under the curve corresponds to friction force. From the above results it may be seen that because of the "size effect", under the condition of light cutting, the specific chip formation force KIcf the normal grindkg force is very high. K, increase with the sequence: medium carbon steel 45, bearing steel and high speed steel. The specific friction force C of these three materials also increases with the same sequence. This is due to the increasing hardness of the workpiece materials and related values p under the conditions of high temperatures during the grinding process in the sequence mentioned above. The red hardness of high speed steel is especially high, and the value C is, tterefore, extremely qreat. The chip formation force plays a main part when easy-to-grind materials such as medium carbon steel o r bearing steel are ground. And for high speed steel which is difficult to grlnd, the frictional force is greater than the chip formation force in all our experimental conditions. Therefore the value of the total grinding force is extremely high. That is one of the reasons why high speed steel is difficult to grind. He have also experimentally determined the relationship between the ratio of tangential grinding force to normal force and the kinematic process parameters. The experimental results are shown in Fig.
14-16. For medium carbon steel 45 (-0.45$C), a material that is easy to grind, the valueofp is comparably great. The maximal value obtained experimentally is p,,= 0.49. This approximates the case when pure chip formation exists; for high speed steel, a material difficult to grind, the value of f is less, and the minimal value obtained ispnin= 0.25. This approximates the case when pure friction exists. In all cases, with the increase of the kinematic process parameters i.e. with the increase of the chip formation part of the grinding force, the ratio p of the tangential grinding force to the normal grinding force increases, as shown in Fig.14-16.
3. The significance of the equivalent thickness of cut heq W e n studying the relationship between input and output conditions of a complicated grinding process, people try their best to search for basic grinding parameters having influence on the grinding processes. In 1974 CIRP proposed to take the equivalent thickness of cut heq = h a as a basic grinding parameter VI
/8/. It has been several years since then, however,
this proposal has still not been widely adopted. To what degree is the equivalent thickness of out related to the physical nature of the grinding process? Taking the grinding force into consideration and according to our proposed grinding force model, we may transform formula (23) slightly into
FA -K,
heq + ~ [ c , ] ' @ . D ]
hagd
(30)
It is obvious that the component of the chip formation force as a part of the grinding force is proportional to the equivalent thickness of cut, while the frictional force component of the grinding force has only a slight relationship with it and is determined to a large degree by the condition of the grinding wheel. During the grinding process once the wheel is worn, the average tip area of active grains 6 varies, and the grinding force will change significantly. Cnly if an easy-to-grind material is ground, and the wheel remains sharp, with the result that the chip formation force plays a significant part in the grinding force, the equivalent chickness of cut together with the specific chip formation force approximetely determines the value of the grinding force. As for the other conditions, it seems that the equivalent thickness of cut is far from able to determine the value of the grinding force. S. Hahn et al. take the normal grinding force per unit of grinding width F h as a basic grinding parameter /lo/. This parameter further characterizes the essentials of the grinding process, however, for an ordinarJi constant infeed rate grinding process it is not covenient to use.
248
References 1. G. ilerner: "Influecce of York itaterial on Srinding Forces". Annals cf CIXP (1978) 2. H. Cpitz,l. Kcnig and G. 7erner: "Xinercatics and Mechanics in Grinding with Regard to the KaEhining Process", Proceedings of the International *rinding Conference, "Xew Development in :rindin#' (1972) 3. J. N. "Jeenhow and C. Rubenstein: "The Dependence of Cutting Force on Feed and Speed in Crthogonal cutting with llorn Tools" Int. u'. lach. Tool Des. Res. 9, 1 (1969) 4. R. Connolly and C. Hubenstein: "The Kechafiics of Continuous Chip Formation in Crthogcnal Cuttin I' Int. J. Ilach.Too1 Des. Res. 8, 159 (196f) 5. C. Subenstein: "The Eechanics of ?rir.ding" Int. J. Each. Tool Des. Res. 12, 127(1972) 6. S. Ealkin, N. H. Cook: "The Rear of Grinding #heels" Trans. AS123 "B" Vol. 93 N0.4 (1971) 7. Usuihideji "Technology of Cutting and Grinding". Japan (1971) R. Snocys: "The Significance of Chip Thickness in :rinding", Annals of CIRP (1974) Handbook 9. "The Part of bletal Cutting, Grinding" of Lechanical Engineering, Peop. Rep. of China 10 S . Hahn: "Principles of Grinding" Kachinery V o l . 77 (1971)
a.
FIG. 7
FIG. 6 1
a = 0.002 mm
FIG. 2 FIG. 9
FIG. I 0
STEEL Wl8t.4 V V p 0 . 6 9 9$ = 0.002 nm
4
0
O.MI
FIG. I I
0.W2
FIG 13
0&13a PW
mn
i 0
o
0.4
0.6~aeaslC%M
FIG 15
0
Odol
Mo2
QWanrn
FIG 16
FIG y4
249