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A New Model for Grinding Force Prediction and Analysis
ARTICLE IN PRESS International Journal of Machine Tools & Manufacture 50 (2010) 231–240
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A New Model for Grinding Force Prediction and Analysis U.S. Patnaik Durgumahanti n, Vijayender Singh, P. Venkateswara Rao Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India
a r t i c l e in fo
abstract
Article history: Received 4 July 2009 Received in revised form 4 December 2009 Accepted 4 December 2009 Available online 16 December 2009
In the present paper a new grinding force model is developed by incorporating the effects of variable coefficient of friction and ploughing force. This is based on the fact that chip formation during grinding consists of three stages: ploughing, cutting and rubbing. Equations for the total normal and tangential force components per unit width of the grinding, during these three stages, were established. These components were expressed in terms of the experimental coefficients and process parameters like wheel speed, table feed and depth of cut. All the coefficients were determined experimentally by performing grinding tests at specified conditions. The variation of the friction coefficient with process parameters such as wheel speed and work feed has been taken into consideration while calculating the frictional force components. The ploughing force components were modelled by performing single-grit tests. During these tests the grinding forces were measured by duplicating the cutting mechanism of grinding wheel using a dummy aluminium wheel and a diamond indenter. The predicted normal and tangential grinding forces were compared with those experimentally obtained and the results show reasonable agreement quantitatively. From the total force values the contributions of each component of force were obtained. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Grinding force Ploughing Variable coefficient of friction Single-grit test
1. Introduction Grinding is one of the main methods of precision manufacturing and the process quality depends to a large extent on the experience of the operator. The developments of wear resistant abrasives, powerful machinery and adequate machining technologies have led to a considerable increase in the efficiency of the grinding process. The economical advantages thus achieved consolidate and extend the position of grinding technology, the grinding processes being a quality defining finishing method. Higher productivity and higher quality require the optimum selection of process parameters in order to use the complete potential of this manufacturing process [1]. However, as grinding is a very complex process with a large number of characteristic quantities that influence each other, reproducibility is critical. In order to reduce the difficulty associated with experimental analysis, grinding process has been modelled theoretically. Models contribute significantly to the comprehension of the process itself and form the basis for the simulation of the grinding process [2]. They thus create a precondition for increased efficiency while ensuring high product quality at the same time. Every model represents a compromise between the obtained model quality on one hand, i.e. the accuracy of the model itself,
n
Corresponding author. Tel.: + 91 9720104770. E-mail address: [email protected] (U.S. Patnaik Durgumahanti).
0890-6955/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2009.12.004
and the efforts that were necessary to obtain the model on the other hand. Usually a technology study of the grinding processes results in the development of a model that is valid for a closely limited field, with given boundary conditions. Thus the model represents the kinematics of the grinding process and can be used to predict its working result. In the past considerable work has been done in order to develop a mathematical model for the grinding force, by considering the grinding forces as an interaction between the grinding wheel and the workpiece surfaces, rather than considering the shearing action as in the case of conventional cutting processes. But most of the recent work in grinding shows that the interaction is between the grit edges and the workpiece, which in turn develops the grinding forces [1,2]. Grinding forces are composed of cutting (or) chip formation force, frictional (or) rubbing force and the ploughing force. As the cutting edges of the abrasives come into contact with the workpiece, elastic deformation occurs and as they traverse further into the workpiece the deformation continues, steadily increasing the normal and tangential grinding forces. This phase of grinding is purely a frictional process, which in turn increases the temperature of the workpiece to a critical value, making the normal stress to exceed the yield stress of the material. The cutting edges then penetrate into the plastic matrix, displacing it sideways ahead of the grit, thus resulting in a surface up-heal, i.e. ploughing effect. If the deformed surface ahead of the grit comes into contact with the cutting edge profile then a transition from
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Nomenclature
Fenc
A A1
Fetr Fenr Fenp
A0 Ab a b C1 Cs d de ds e Fn0 Ft0 Fnp Ftp 0 Ftp 0 Fnp 0 Ftc 0 Fnc
Fnr Ftr 0 Ftr 0 Fnr
Fetc
fraction of wheel surface that has wear flats ratio of tangential ploughing force component to normal ploughing force component. area of contact, mm2 bearing area of scratch, mm2 depth of cut, mm width of the grinding, mm static cutting edge density, mm 3 grain distribution of the grinding wheel (number of grains per unit area), mm 2 diameter of the grit or grain, mm equivalent diameter, mm diameter of the grinding wheel, mm depth of penetration of the grit, mm normal grinding force per unit width of grinding, N/mm tangential grinding force per unit width of grinding, N/mm normal component of the ploughing force, N tangential component of the ploughing force, N tangential component of the ploughing force per unit width of grinding, N/mm normal component of the ploughing force per unit width of grinding, N/mm tangential component of chip formation force per unit width of grinding, N/mm normal component of the chip formation force per unit width of grinding, N/mm normal component of the frictional force, N tangential component of the frictional force, N tangential component of the frictional force per unit width of grinding, N/mm normal component of the frictional force per unit width of grinding, N/mm tangential chip formation force component of a single grain, N
ploughing to cutting will take place. The same mechanism may not occur for the dull grit; instead continuous rubbing of the wheel surface with the workpiece will take place due to increase in contact area of the grit, thereby increasing the grinding force components [1–3].
2. Literature review Grinding has been the object of technical research for some decades now. The necessity to exemplarily describe grinding process in models was recognized at an early stage so that there are numerous models today. There are various models based on different parameters, such as temperature model, force model, surface roughness model, specific energy model, etc. Grinding wheel wear, dynamic performance of the grinding equipment, geometric accuracy and surface quality of the workpiece are greatly influenced by the grinding forces and some considerable research developments in calculating the grinding force were made by various researchers. In early days no clear distinctions were made between the models describing the normal force and those of the tangential force, but rather differentiated by a multiplicative factor that differs considerably depending on the material of the workpiece.
Fetp Hs h K K0 K1 K2 K3 K4 K5 lc N p P0 Qi R R1 W Vc Vw
an a a0 b y
m d D
normal chip formation force component of a single grain, N tangential frictional force component of a single grain, N normal frictional force component of a single grain, N normal ploughing force component of a single grain or grit, N tangential ploughing force component of a single grain or grit, N scratch hardness, N/mm2 height of the pile up material, mm chip thickness coefficient, N/mm2 experimental coefficient, N/mm2 experimental coefficient, N/mm experimental coefficient, N/mm2 experimental coefficient, N/mm experimental coefficient experimental coefficient geometric contact length, mm active number of grains or grits average contact pressure between workpiece and the wheel, N/mm2 proportionality constant, N/mm chip cross-sectional area, mm2 radius of curvature of the cutting path, mm radial distance from the center of the grit, mm normal load on the workpiece, N wheel speed, mm/s workpiece feed or table speed, mm/s real contact area between the workpiece and the wheel, mm2 half apex angle of the grit, deg experimental coefficient, N/mm2 experimental coefficient half the tip angle of the grain, deg coefficient of friction tip area of the active grain, mm2 deviation of radius of curvature of cutting path and grinding wheel radius, mm 1
Salje [4] developed the initial grinding force model considering the shear strength as a specific parameter for the workpiece material and the model parameters are taken from a characteristic diagram that considers the influences of the respective combination of the material and the grinding wheel. Brach et al. [5] took the grinding wheel topography into consideration in twodimensional forms. Ono [6] considered the average grain distance and took the distance between the cutting edges into consideration for the modelling of the grinding forces. Lindsay [7] developed two grinding models for specific normal force, one for the materials that are easy to grind and the other for the materials that are difficult to grind. Werner [8] developed a grinding force model taking into account the combined effects of friction and chip formation. His model predicts the effect of workpiece properties on grinding. When the exponent n= 0, the phenomenon is purely frictional, while if it is 1 then the phenomenon is purely a chip formation force. Lichun and Jizai [9] developed a grinding force model from the Werner model that takes friction and chip formation into account; here they separated out the effects of frictional and chip formation forces. Younis et al. [10] extended Werner’s work taking the combined effect of ploughing, friction and chip formation forces for cylindrical grinding, where the loaded area is found out using a fiber optic system. Chang and Junz Wang [11] developed a
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stochastic grinding force model taking into account the random distribution of the grits in the grinding instead of assuming them to be uniform as done in the previous works. Tang et al. [12] developed a grinding force model for surface grinding considering friction and chip formation. Here the average contact pressure and the frictional coefficient are treated as variable parameters, unlike previous research, where these parameters were considered constant. Ghosh et al. [13] developed a mathematical model that predicts the specific energy consumed during HEDG of bearing steel by monolayer cBN wheel. The model successfully captures the mechanics of grinding under HEDG mode mainly consisting of chip formation, ploughing, primary and secondary rubbing phenomenon. Most of the proposed grinding force models neglected the effect of ploughing, considering it to be very low in comparison with the chip formation force. But if the model is to represent the actual grinding process the effect of ploughing has to be considered. The coefficient of friction in most of the models is taken as constant but in reality it varies with process parameters during the grinding process. On the basis of foundation of the achievements of these researchers, a new grinding force model was developed by incorporating the proposed improvements to the Werner model. The developed model considers the effect of the input process parameters and grain size on the ploughing force component and on the coefficient of friction
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grinding wheel [9]: Fetc ¼j Fenc
ð8Þ
where j = P/tan y Similar to the normal force component, the total tangential chip formation force component per unit width of the grinding can be determined as [9] 0 Ftc ¼
P
jFenc ¼ jK
P
Qi ¼ jK
Vw a Vc
Vw a Vc 0 K ¼ jK
0 ¼ K0 Ftc
ð9Þ
where K, K0 are the chip thickness coefficients, which are determined through experiments. 3.2. Frictional force components The rubbing phenomenon in grinding is because of the flat area of the grinding wheel, which is caused by the wear of the grains. Experiments prove that the normal force of each grain will vary directly with the wear area. [2,9] Hence the normal and tangential frictional force components of a single grain are
3. Grinding force model
Fenr ¼ dp
ð10Þ
Grinding forces can be separated into two parts, cutting deformation force and frictional force. The cutting deformation force is again sub-divided into ploughing force and cutting or chip formation force.
Fetr ¼ Fenr m ¼ mdp
ð11Þ
F ¼ Fplow þ Fchip þ Ffriction
ð1Þ
0 0 0 Fn0 ¼ Fnp þ Fnc þ Fnr
ð2Þ
Ft0
ð3Þ
where m is the coefficient of friction, d the tip area of the worn grain and p¯ the average contact pressure between the wheel and the workpiece. Hence the total normal and tangential frictional force components are Fnr ¼ an p ¼ ðN dÞp
ð12Þ
Ftr ¼ man p ¼ mðNdÞp
ð13Þ
3.1. Chip formation force components
an ¼ blc A
ð14Þ
In developing the chip formation force components the cutting action of a single grain of the grinding wheel is assumed to be similar to the action of a single point cutting tool in turning. Hence the normal component of chip formation force imposed by a single grain can be determined as a function of undeformed chip cross-sectional area, which can be written as [9,10]
lc ¼ ðde aÞ1=2
ð15Þ
0 0 0 ¼ Ftp þ Ftc þ Ftr
Fenc ¼ KQi
ð4Þ
The total normal chip formation force per unit width of the grinding is the total of the normal forces of all active grains within the contact area of wheel and workpiece [9,10]: X X 0 Fnc ¼ KQi ¼ K Qi ð5Þ P Qi is the total value of all simultaneous chip cross-sections per unit width of grinding and is given as [9,10] X
Qi ¼
0 Fnc ¼K
Vw a Vc
Vw a Vc
where an is the real contact area between the wheel and the workpiece and A the fraction of the wheel surface that has wear flats. N is the total number of active grains (i.e. grains contributing to rubbing action) and is given as [1,9] a1 Z lc An Vw ð1a1 Þ=2 Nydn ðlÞdl ¼ ðC1 Þb1 að1 þ a1 Þ=2 ds ð16Þ N¼ 1þ a Vc 1 0
ð6Þ
ð7Þ
In the case of pure chip formation, the ratio of tangential force component to normal force component of a single grain bears a ratio j that is dependent on the profile of the grains of the
Fig. 1. Schematic view of the scratch and pile up material in a single-grit test.
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where a1 and b1 are the exponential coefficients of edge distribution and Nydn is the number of dynamic grains (i.e. total grains participating in grinding).
Using the parabola function to approximate the cutting path, the deviation (D) between the grinding wheel radius (ds/2) and the radius of curvature of cutting path (R) [1] is
D¼ Table 1 Experimental set-up details of grinding force measurement. Grinder Grinding wheel Workpiece Dimensions Width of workpiece Dressing interval Dressing amount Charge amplifier Dynamometer Mechanical units/volt pC/N value
CHEVALIER CNC surface grinder Al2O3 wheel (A60 M6 VCNM) Mild steel (SAE 1018) 70 50 6 mm3 6 mm 1 mm 0.02 mm KISTLER 3-channel piezoelectric 5006 SN KISTLER piezoelectric 9257A 100 For X: 7.5 For Z: 3.5 National Instruments E-6210
Data acquisition card
D¼ 7
Shank with the diamond grit
Dummy Grinding Wheel
ð17Þ
4Vw Vc de
ð18Þ
The average contact pressure p¯ between the workpiece and the wear plane of the abrasive grains increases approximately linearly with the deviation (D) of the radius of curvature and this relationship is given by [1] p ¼ P0 D ¼
4P0 Vw de Vc
ð19Þ
The average contact pressure (p¯) varies with the processing parameters of grinding. Therefore most likely there exists an elastic contact, elasto-plastic contact or plastic contact. Hence the frictional coefficient m also varies with the average contact pressure [12]. According to the frictional binomial theorem, the variable frictional coefficient is given by the formula [12]
m¼ Dummy Aluminium Wheel
2 1 ds R
a0 A0 W
þ b ¼ a0 p þ b
ð20Þ
where a0 and b are the coefficients, which are dependent on the physical and mechanical properties of the contact interface. Substituting Eqs. (14), (15), (19) and (20) into Eqs. (12) and (13), the total tangential and normal frictional force components per unit width of grinding (F0 tr and F0 nr) are obtained as Ftr ¼ ða0 p þ bÞpbðde aÞ1=2 A
ð21Þ
Ftr ¼ ða0 þ bpÞbðde aÞ1=2 A
ð22Þ
Shank
4AP0 bVw 0 ðde aÞ1=2 ¼ Aa0 þ Ftr de Vc
ð23Þ
K3 Vw 0 ðde aÞ1=2 ¼ K2 þ Ftr de Vc
ð24Þ
Holding Screw Grit
K2 ¼ Aa0 K3 ¼ 4AP0 b
Workpiece Dynamometer
Fnr ¼
Fig. 2. Dummy grinding wheel with indenter and schematic view of experimental set-up for single-grit test.
0 ¼ Fnr
4P0 Vw bðde aÞ1=2 A de Vc
ð25Þ
4AP0 Vw a 1=2 Vc de
ð26Þ
Table 2 Diameter of the grits used in the single-grit experimentation. Sl. number 1 2 3 4 5 6 7 8 9 10 11 12 13
l(mm)
b(mm)
h(mm)
Volume (mm3)
Grit radius (mm)
1.3 0.901 0.91 1.25 1.21 1.28 1.45 1.51 1.47 1.71 1.58 1.6 1.9
1.1 0.996 0.96 1.31 1.28 1.29 1.32 1.486 1.318 1.55 1.657 1.51 2.21
0.86 0.88 0.85 1.317 1.214 1.13 1.7 1.736 1.5123 2.19 2.112 2.101 2.52
0.410 0.263 0.247 0.718 0.626 0.622 1.084 1.298 0.976 1.934 1.843 1.692 3.527
0.460 0.397 0.389 0.555 0.531 0.529 0.637 0.677 0.615 0.773 0.760 0.739 0.944
Avg. grit diameter (mm)
0.83
1.07
1.29
1.52 1.90
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0 Fnr ¼
K1 Vw a 1=2 Vc de
ð27Þ
K1 ¼ 4AP0 where K1, K2, K3 are coefficients, which depend on the wheel and workpiece combination and are determined through experiments. 3.3. Ploughing force components Another mechanism associated with abrasive process is the ploughing. Ploughing energy is expended by deformation of workpiece material without removal. We normally associate ploughing with the side flow of material from the cutting path into ridges, but it can also include plastic deformation of the material passing under the cutting edge. Initially the grit makes elastic contact, which is assumed to make a negligible contribution to the total energy, followed by the plastic deformation (ploughing) of the workpiece. 3.3.1. Development of ploughing force model To determine the total normal and tangential ploughing force components per unit width of grinding, single-grit tests were performed using indenters of different sizes. Since the parameters on which the ploughing force is dependent are not known the basic ploughing force formulae developed by various researchers were analyzed and the parameters were found out. The ploughing effect in single-grit test is schematically shown in Fig. 1. De Vathaire et al. [14] proposed an upper bound model of ploughing by a pyramidal indenter and the experiments were Table 3 Levels of the parameters considered for single-grit experimentation.
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conducted on a low carbon steel workpiece. The resistance of the work material towards scratching by an indenter is defined as scratch hardness (Hs). It is the ratio of normal force of the grit (Fenp) during scratching to bearing area (Ab) of the scratch. [14] Fenp ¼ Hs Ab
ð28Þ
Ab ¼ eðe þ hÞtan2 a
ð29Þ
h Fenp ¼ Hs e2 tan2 a 1 þ ¼ Hs eðe þhÞtan2 a e
ð30Þ
Hs ¼
Fenp e2 tan2 að1þ HÞ
ð31Þ
e is the depth of penetration of grit, h the height of pile up material and H= (h/e) tana ¼
R1 ðe þ hÞ
ðe þ hÞ ¼
ð32Þ
R1 c0 d ¼ tana 2tana
ð33Þ
Table 5 Grain distribution of the grinding wheel (A60 M6 VCNM) for three different imprint areas. Sl. number 1 2 3
Area (mm2)
Number of grits
Grain distributionCs(mm 2)
25 49 100
65 150 320
2.60 3.06 3.20
Table 6 Levels of parameters considered for experimentation to determine the chip formation and frictional force model coefficients.
Parameters/levels
2
1
0
1
2
Wheel speed (m/min) Work feed (m/min) Depth of cut (mm) Grit diameter (mm)
600 3 0.01 0.83
840 6 0.015 1.07
1080 9 0.02 1.29
1320 12 0.025 1.52
1560 15 0.03 1.9
Table 4 Experimental constants of ploughing force component.
Parameters/levels
1
2
3
Wheel speed(m/min) Work feed (m/min) Depth of cut (mm)
708.6 3 0.01
1308.6 9 0.02
1908.6 15 0.03
Table 7 Experimental coefficients of chip formation and frictional force components.
K4
K5
a0
b0
c0
K
K’
K1
K2
K3
5.7
3.24
0.22
0.33
0.294
41,217
23,081
229,160
1.287
20,394
Fig. 3. Marked zones of the grinding wheel imprints for determining the grain distribution: (a) 25, (b) 49 and (c) 100 mm2.
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R1 = radial distance from the center of the grit, d= diameter of the 0 grit, a =half the apex angle of the grit and c is a multiplicative factor. Hence the normal ploughing force component of the single grit (Fenp) can be given as Fenp ¼ k4 tana Hs ed
ð34Þ
where k4 = c0 /2. The ratio of tangential ploughing force component (Fetp) to the normal ploughing force component (Fenp) of a single grit bears a ratio A1, which is dependent on the properties of the workpiece [14]: Fetp ¼ A1 Fenp
ð35Þ
Table 8 Design of experiments considered for validation of the grinding force model. Vc(m/min) 1200
600 900 1200 1500 1800 1200
Vw(m/min)
a(mm)
3 6 9 12 15 6
0.015
6
0.010 0.015 0.020 0.025 0.030
0.015
Fig. 4. Variation of grinding force per unit width with wheel speed: (a) normal grinding force and (b) tangential grinding force.
Fig. 5. Variation of grinding force per unit width with table feed: (a) normal grinding force and (b) tangential grinding force.
Fig. 6. Variation of grinding force per unit width with depth of cut: (a) normal grinding force and (b) tangential grinding force.
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Fetp ¼ A1 Fenp
ð36Þ
Fetp ¼ A1 k4 tana Hs ed
ð37Þ
of grits in the contact area: a0 Vw ðdÞb0 ðaÞc0 Cs ðblc Þ Fnp ¼ Fenp Cs ðblc Þ ¼ K4 Vc
Fetp ¼ k5 tana Hs ed
ð38Þ
Ftp ¼ Fetp Cs ðblc Þ ¼ K5
where k4, k5 are the constants and [14] Hs ed ¼ f ðVc ; Vw ; a; dÞ
ð39Þ
From this equation it is clear that the ploughing force is dependent on four parameters namely the wheel speed, table feed, depth of cut and the grit diameter. Since the relationship between these variables is not known the ploughing force components for a single grit are expressed as a0 Vw ðdÞb0 ðaÞc0 Fenp ¼ K4 Vc
Fetp ¼ K5
Vw Vc
a0
ðdÞb0 ðaÞc0
a0 Vw ðdÞb0 ðaÞc0 Cs ðblc Þ Vc
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ð42Þ
ð43Þ
Cs is the grain distribution of the grinding wheel and lc =(ade)1/2 is the geometric contact length. Hence the normal and tangential ploughing force components per unit width of grinding are a0 Vw 0 ¼ K4 ðdÞb0 ðaÞc0 Cs ðade Þ1=2 ð44Þ Fnp Vc 0 Ftp ¼ K5
a0 Vw ðdÞb0 ðaÞc0 Cs ðade Þ1=2 Vc
ð45Þ
ð40Þ 3.4. Final grinding force equations
ð41Þ
where K4 = k4(tan a1) and K5 = k5(tan a1). K4 and K5 are coefficients, which depend on the properties of the workpiece and on the orientation of the grits in the grinding wheel; a0, b0, c0 are constant exponents, which are determined from single-grit tests. The total normal and tangential ploughing force components were obtained by multiplying Eqs. (40) and (41) with the number
Substituting Eqs. (7), (9), (24), (27), (44) and (45) in Eqs. (2) and (3), the final equations for normal and tangential grinding force per unit width of grinding are a0 Vw K1 Vw a 1=2 Vw Fn0 ¼ K aþ þ K4 ðdÞb0 ðaÞc0 Cs ðade Þ1=2 ð46Þ Vc Vc Vc de Ft0 ¼ K 0
a0 Vw K3 Vw Vw ðde aÞ1=2 þ K5 a þ K2 þ ðdÞb0 ðaÞc0 Cs ðade Þ1=2 Vc de Vc Vc ð47Þ
Fig. 7. Contributions of each component of grinding force per unit width with wheel speed: (a) normal grinding force and (b) tangential grinding force.
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4. Experiments 4.1. Experimental set-up The experiments were conducted using CHEVALIER CNC surface grinder and the grinding wheel used for the experimentation was a medium grade alumina wheel with a grain size number of 60. The wheel has an outer diameter of 350 mm and width of 40 mm. The forces were measured using a 3-axis piezoelectric Kistler 9257A dynamometer, which is connected to Kistler 3channel piezoelectric 5006 SN charge amplifier. To record the data from the dynamometer, National Instruments Data Acquisition Card E-6210 was employed along with the Labview-7 software. An algorithm was developed for the recording and storage of the data and filtering of unwanted signals was performed using a low pass Butterworth filter by setting the filter frequency to 50 Hz. The workpiece used was mild steel (SAE 1018) with a rectangular cross-section and dimensions 70 50 6 mm3. The experimental set-up details are shown in Table 1.
4.2. Determination of experimental coefficients 4.2.1. Single-grit test The single-grit scratch tests were performed to develop the ploughing force model coefficients. In these tests the process of
grinding was duplicated with a dummy aluminium wheel and an indenter. The shanks with the diamond grit were held on to the wheel with the help of a holding screw and a spring. They are held in such a way that only the grit comes into contact with the workpiece. The wheel is rotated at the desired speed and the grit is allowed to make only a single scratch. The scratches were scanned by the probe of the Talysurf surface roughness measuring device to get the height of pile up material (h) and the depth of the scratch (e). The experimental set-up is shown in Fig. 2. Assuming the grits to be of pyramidal shape the diameter of the grits was found out by equating their volume to that of an equivalent sphere as shown in Table 2. The dimensions of the grits were obtained using a profile projector. Sets of 31 experiments were conducted to determine the coefficients of the ploughing force model. Each parameter, namely the wheel speed, table feed, depth of cut and grit diameter, was varied at five levels as shown in Table 3 and the designs of experiments were done using the CCD technique. Performing, regression analysis of the obtained force data, the coefficients were determined for mild steel (SAE 1018) workpiece and are shown in Table 4. The number of active grits or grains in the contact area is the product of the grit distribution of the grinding wheel (Cs) and the geometric contact length. The grain distribution of the grinding wheel was found out by taking the imprints of the grinding wheel surface and by marking areas of 25, 49 and 100 mm2 as shown in Fig. 3. The number of grits in the respective marked areas was first found out and from there the number of grits per unit
Fig. 8. Contributions of each component of grinding force per unit width with table feed: (a) normal grinding force and (b) tangential grinding force.
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area was calculated as shown in Table 5. The average grit or grain distribution obtained for the medium grade alumina wheel (A60 M6 VCNM) was 3 grits per unit area. 4.2.2. Chip formation force and frictional force coefficients The other five coefficients K, K0 , K1, K2 and K3 can be determined from five groups of experimental data by solving the linear equations. But, to make the model representative for all possible combinations of process parameters, twenty seven experiments were performed taking three levels of each parameter, namely the wheel speed, table feed and depth of cut, as shown in Table 6. The normal and tangential grinding forces were recorded with the help of the dynamometer and the ploughing force values per unit width of the grinding were subtracted from the obtained force values. The ploughing force components, were calculated from Eqs. (44) and (45) by substituting the constants obtained from the single-grit tests. The grain diameter (d) of the grinding wheel was found out from its mesh number. The grain diameter (d) and the mesh number (M) are related by d ðmmÞ ¼ ð15:2=MÞ
ð48Þ
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Hence the grain diameter for the alumina grinding wheel (A60 M6 VCNM) is approximately 0.250 mm. The experimental coefficients of chip formation and frictional force components for the alumina wheel (A60 M6 VCNM) and mild steel (SAE 1018) combination are shown in Table 7. These coefficients were determined through regression analysis of the resultant force data, which were obtained after subtracting the ploughing force components. Substituting the obtained coefficients in Eqs. (46) and (47), the final normal and tangential grinding force equations per unit width of grinding, obtained for the alumina wheel (A60 M6 VCNM) and mild steel workpiece (SAE 1018) combination were Vw 229; 160Vw a 1=2 aþ Fn0 ¼ 41; 217 Vc Vc de 0:22 Vw 0:33 0:294 þ5:7 ðdÞ ðaÞ Cs ðade Þ1=2 ð49Þ Vc Vw 203; 94Vw ðde aÞ1=2 a þ 1:287 þ Vc de Vc 0:22 Vw þ3:24 ðdÞ0:33 ðaÞ0:294 Cs ðade Þ1=2 Vc
Ft0 ¼ 23; 081
Fig. 9. Contributions of each component of grinding force per unit width with depth of cut: (a) normal grinding force and (b) tangential grinding force.
ð50Þ
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error in tangential force per unit width of grinding was 7.87%. The contribution of ploughing in the total grinding force is about 15–17% on average, while the contributions of chip formation and friction were 25–28% and 54–58%, respectively, for the mild steel (SAE 1018) and alumina wheel (A60 M6 VCNM) combination. The coefficient of friction varied from 0.27 to 0.86 depending on the process parameters considered. The ploughing force remained almost constant with changes in wheel speed and table feed but it showed an increase with increasing depth of cut.
6. Conclusion
Fig. 10. Variation of coefficient of friction with (a) wheel speed and (b) table feed.
5. Results and analysis 5.1. Validation of the model For the validation of model the basic values of the parameters considered were Vw = 6 m/min, Vc = 20 m/s and a= 0.015 mm. A total of 15 experiments were performed by varying each parameter for 5 levels keeping the other two parameters constant as shown in Table 8, and the corresponding experimental force values were recorded using a dynamometer. For the same values of the process parameters the theoretical normal and tangential grinding forces per unit width of grinding are calculated from the model equations (49) and (50). Both the experimental and theoretical grinding force values were compared in the plots of Fn0 Vc ; Ft0 Vc ; Fn0 Vw ; Ft0 Vw ; Fn0 a and Ft0 a as shown in Figs. 4–6. From the total tangential and normal grinding force values obtained from Eqs. (49) and (50), the contributions of ploughing, chip formation and friction were found out and are presented in the corresponding plots of Fn0 Vc ; Ft0 Vc ; Fn0 Vw ; Ft0 Vw ; Fn0 a and Ft0 aas shown in Figs. 7–9 5.2. Variable coefficient of friction Variation of the friction coefficient with process parameters like wheel speed and table feed for the alumina wheel (A60 M6 VCNM) and mild steel (SAE 1018) combination are shown in Fig. 10. The increase in the coefficient of friction with wheel speed was because of the increase in the wear flat area of the wheel with the wheel speed. On the other hand as the table feed increases less time was available for the grits in the contact area, to make contact with the workpiece, thereby decreasing the coefficient of friction. 5.3. Analysis The average percentage of error in the normal force per unit width of the grinding was 10.28%, while the average percentage of
The present research work models the total grinding force by incorporating the combined effects of variable coefficient of friction and the ploughing force. This model exhibits the importance of including the ploughing force, which becomes more predominant at very low depth of cuts. The coefficient of friction in the present model varies with process parameters like wheel speed and table feed, unlike the previous models, where it is a constant value throughout the process. This has considerably improved the model, making it more practical. A new method of developing the ploughing force model equations using the singlegrit test was adopted. Earlier schemes of execution of these tests were either by using an oscillating pendulum arrangement or by using an aluminium disc that was mounted on to a grinding machine. In both cases, the actual grinding process was not represented, as the scratches were made by moving the workpiece against a stationary grit held firmly onto the base of a dynamometer. It can be observed that the present scheme of executing the single-grit tests can take care of these limitations. Hence, the new grinding force model can be reliably used to predict the grinding forces and provide a certain theoretical basis for research on grinding force.
References [1] S. Malkin, Theory and Application of Machining with Abrasives, Grinding Technology, Ellis Horwood Limited, 1989. [2] Chander Bhateja, Richard Lindsay, Techniques and Troubleshooting, Grinding: Theory, Society of Manufacturing Engineers, Deaborn, MI, 1992. [3] Mandeep Kumawat, 2000, Development of modified chip thickness model for estimation of surface roughness in grinding, M.Tech Thesis, Department of Mechanical Engineering, IIT Delhi. [4] E. Salje, Grundlagen des schleifvorganges, Werkstatt und Betrieb 86 (1953) 177–182. [5] K. Brach, D.M. Pai, E. Ratterson, M.C. Shaw, Grinding forces and energy, Transactions of ASME, Journal of Engineering for Industry 110 (1988) 25–31. [6] K. Ono, Analysis of the grinding force, Bulletin of the Japan Society of Grinding Engineers 1 (1961) 19–22. [7] Lindsay, R.P.: On the metal removal and wheel removal parameters—surface finish, geometry and thermal damage in precision grinding, Ph.D. Thesis, Worcester Polytechnic Institute, 1971. [8] G. Werner, Influence of work material on grinding forces, Annals of CIRP 27 (1978) 243–248. [9] Li Lichun, Fu Jizai, A study of grinding force mathematical model, Annals of CIRP 29 (1980) 245–249. [10] M. Younis, M.M. Sadek, T. El Wardani, A new approach to development of a grinding force model, Transactions of ASME 109 (1987) 306–313. [11] Huang-Cheng Chang, J.-J. Junz Wang, A stochastic grinding force model considering random grit distribution, International Journal of Machine Tools and Manufacture 48 (2008) 1336–1344. [12] Jinyuan Tang, Jin Du, Yongping Chen, Modelling and experimental study of grinding forces in surface grinding, Annals of CIRP (2008) 1–8. [13] S. Ghosh, A.B. Chattopadhyay, S. Paul, Modelling of specific energy requirement during high efficiency deep grinding, International Journal of Machine Tools and Manufacture 48 (11) (2008) 1242–1253. [14] M. De Vathaire, F. Delamare, E. Felder, An upper bound model of ploughing by a pyramidal indenter, Wear 66 (1981) 55–64.
Contents lists available at ScienceDirect
International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
A New Model for Grinding Force Prediction and Analysis U.S. Patnaik Durgumahanti n, Vijayender Singh, P. Venkateswara Rao Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India
a r t i c l e in fo
abstract
Article history: Received 4 July 2009 Received in revised form 4 December 2009 Accepted 4 December 2009 Available online 16 December 2009
In the present paper a new grinding force model is developed by incorporating the effects of variable coefficient of friction and ploughing force. This is based on the fact that chip formation during grinding consists of three stages: ploughing, cutting and rubbing. Equations for the total normal and tangential force components per unit width of the grinding, during these three stages, were established. These components were expressed in terms of the experimental coefficients and process parameters like wheel speed, table feed and depth of cut. All the coefficients were determined experimentally by performing grinding tests at specified conditions. The variation of the friction coefficient with process parameters such as wheel speed and work feed has been taken into consideration while calculating the frictional force components. The ploughing force components were modelled by performing single-grit tests. During these tests the grinding forces were measured by duplicating the cutting mechanism of grinding wheel using a dummy aluminium wheel and a diamond indenter. The predicted normal and tangential grinding forces were compared with those experimentally obtained and the results show reasonable agreement quantitatively. From the total force values the contributions of each component of force were obtained. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Grinding force Ploughing Variable coefficient of friction Single-grit test
1. Introduction Grinding is one of the main methods of precision manufacturing and the process quality depends to a large extent on the experience of the operator. The developments of wear resistant abrasives, powerful machinery and adequate machining technologies have led to a considerable increase in the efficiency of the grinding process. The economical advantages thus achieved consolidate and extend the position of grinding technology, the grinding processes being a quality defining finishing method. Higher productivity and higher quality require the optimum selection of process parameters in order to use the complete potential of this manufacturing process [1]. However, as grinding is a very complex process with a large number of characteristic quantities that influence each other, reproducibility is critical. In order to reduce the difficulty associated with experimental analysis, grinding process has been modelled theoretically. Models contribute significantly to the comprehension of the process itself and form the basis for the simulation of the grinding process [2]. They thus create a precondition for increased efficiency while ensuring high product quality at the same time. Every model represents a compromise between the obtained model quality on one hand, i.e. the accuracy of the model itself,
n
Corresponding author. Tel.: + 91 9720104770. E-mail address: [email protected] (U.S. Patnaik Durgumahanti).
0890-6955/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2009.12.004
and the efforts that were necessary to obtain the model on the other hand. Usually a technology study of the grinding processes results in the development of a model that is valid for a closely limited field, with given boundary conditions. Thus the model represents the kinematics of the grinding process and can be used to predict its working result. In the past considerable work has been done in order to develop a mathematical model for the grinding force, by considering the grinding forces as an interaction between the grinding wheel and the workpiece surfaces, rather than considering the shearing action as in the case of conventional cutting processes. But most of the recent work in grinding shows that the interaction is between the grit edges and the workpiece, which in turn develops the grinding forces [1,2]. Grinding forces are composed of cutting (or) chip formation force, frictional (or) rubbing force and the ploughing force. As the cutting edges of the abrasives come into contact with the workpiece, elastic deformation occurs and as they traverse further into the workpiece the deformation continues, steadily increasing the normal and tangential grinding forces. This phase of grinding is purely a frictional process, which in turn increases the temperature of the workpiece to a critical value, making the normal stress to exceed the yield stress of the material. The cutting edges then penetrate into the plastic matrix, displacing it sideways ahead of the grit, thus resulting in a surface up-heal, i.e. ploughing effect. If the deformed surface ahead of the grit comes into contact with the cutting edge profile then a transition from
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Nomenclature
Fenc
A A1
Fetr Fenr Fenp
A0 Ab a b C1 Cs d de ds e Fn0 Ft0 Fnp Ftp 0 Ftp 0 Fnp 0 Ftc 0 Fnc
Fnr Ftr 0 Ftr 0 Fnr
Fetc
fraction of wheel surface that has wear flats ratio of tangential ploughing force component to normal ploughing force component. area of contact, mm2 bearing area of scratch, mm2 depth of cut, mm width of the grinding, mm static cutting edge density, mm 3 grain distribution of the grinding wheel (number of grains per unit area), mm 2 diameter of the grit or grain, mm equivalent diameter, mm diameter of the grinding wheel, mm depth of penetration of the grit, mm normal grinding force per unit width of grinding, N/mm tangential grinding force per unit width of grinding, N/mm normal component of the ploughing force, N tangential component of the ploughing force, N tangential component of the ploughing force per unit width of grinding, N/mm normal component of the ploughing force per unit width of grinding, N/mm tangential component of chip formation force per unit width of grinding, N/mm normal component of the chip formation force per unit width of grinding, N/mm normal component of the frictional force, N tangential component of the frictional force, N tangential component of the frictional force per unit width of grinding, N/mm normal component of the frictional force per unit width of grinding, N/mm tangential chip formation force component of a single grain, N
ploughing to cutting will take place. The same mechanism may not occur for the dull grit; instead continuous rubbing of the wheel surface with the workpiece will take place due to increase in contact area of the grit, thereby increasing the grinding force components [1–3].
2. Literature review Grinding has been the object of technical research for some decades now. The necessity to exemplarily describe grinding process in models was recognized at an early stage so that there are numerous models today. There are various models based on different parameters, such as temperature model, force model, surface roughness model, specific energy model, etc. Grinding wheel wear, dynamic performance of the grinding equipment, geometric accuracy and surface quality of the workpiece are greatly influenced by the grinding forces and some considerable research developments in calculating the grinding force were made by various researchers. In early days no clear distinctions were made between the models describing the normal force and those of the tangential force, but rather differentiated by a multiplicative factor that differs considerably depending on the material of the workpiece.
Fetp Hs h K K0 K1 K2 K3 K4 K5 lc N p P0 Qi R R1 W Vc Vw
an a a0 b y
m d D
normal chip formation force component of a single grain, N tangential frictional force component of a single grain, N normal frictional force component of a single grain, N normal ploughing force component of a single grain or grit, N tangential ploughing force component of a single grain or grit, N scratch hardness, N/mm2 height of the pile up material, mm chip thickness coefficient, N/mm2 experimental coefficient, N/mm2 experimental coefficient, N/mm experimental coefficient, N/mm2 experimental coefficient, N/mm experimental coefficient experimental coefficient geometric contact length, mm active number of grains or grits average contact pressure between workpiece and the wheel, N/mm2 proportionality constant, N/mm chip cross-sectional area, mm2 radius of curvature of the cutting path, mm radial distance from the center of the grit, mm normal load on the workpiece, N wheel speed, mm/s workpiece feed or table speed, mm/s real contact area between the workpiece and the wheel, mm2 half apex angle of the grit, deg experimental coefficient, N/mm2 experimental coefficient half the tip angle of the grain, deg coefficient of friction tip area of the active grain, mm2 deviation of radius of curvature of cutting path and grinding wheel radius, mm 1
Salje [4] developed the initial grinding force model considering the shear strength as a specific parameter for the workpiece material and the model parameters are taken from a characteristic diagram that considers the influences of the respective combination of the material and the grinding wheel. Brach et al. [5] took the grinding wheel topography into consideration in twodimensional forms. Ono [6] considered the average grain distance and took the distance between the cutting edges into consideration for the modelling of the grinding forces. Lindsay [7] developed two grinding models for specific normal force, one for the materials that are easy to grind and the other for the materials that are difficult to grind. Werner [8] developed a grinding force model taking into account the combined effects of friction and chip formation. His model predicts the effect of workpiece properties on grinding. When the exponent n= 0, the phenomenon is purely frictional, while if it is 1 then the phenomenon is purely a chip formation force. Lichun and Jizai [9] developed a grinding force model from the Werner model that takes friction and chip formation into account; here they separated out the effects of frictional and chip formation forces. Younis et al. [10] extended Werner’s work taking the combined effect of ploughing, friction and chip formation forces for cylindrical grinding, where the loaded area is found out using a fiber optic system. Chang and Junz Wang [11] developed a
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stochastic grinding force model taking into account the random distribution of the grits in the grinding instead of assuming them to be uniform as done in the previous works. Tang et al. [12] developed a grinding force model for surface grinding considering friction and chip formation. Here the average contact pressure and the frictional coefficient are treated as variable parameters, unlike previous research, where these parameters were considered constant. Ghosh et al. [13] developed a mathematical model that predicts the specific energy consumed during HEDG of bearing steel by monolayer cBN wheel. The model successfully captures the mechanics of grinding under HEDG mode mainly consisting of chip formation, ploughing, primary and secondary rubbing phenomenon. Most of the proposed grinding force models neglected the effect of ploughing, considering it to be very low in comparison with the chip formation force. But if the model is to represent the actual grinding process the effect of ploughing has to be considered. The coefficient of friction in most of the models is taken as constant but in reality it varies with process parameters during the grinding process. On the basis of foundation of the achievements of these researchers, a new grinding force model was developed by incorporating the proposed improvements to the Werner model. The developed model considers the effect of the input process parameters and grain size on the ploughing force component and on the coefficient of friction
233
grinding wheel [9]: Fetc ¼j Fenc
ð8Þ
where j = P/tan y Similar to the normal force component, the total tangential chip formation force component per unit width of the grinding can be determined as [9] 0 Ftc ¼
P
jFenc ¼ jK
P
Qi ¼ jK
Vw a Vc
Vw a Vc 0 K ¼ jK
0 ¼ K0 Ftc
ð9Þ
where K, K0 are the chip thickness coefficients, which are determined through experiments. 3.2. Frictional force components The rubbing phenomenon in grinding is because of the flat area of the grinding wheel, which is caused by the wear of the grains. Experiments prove that the normal force of each grain will vary directly with the wear area. [2,9] Hence the normal and tangential frictional force components of a single grain are
3. Grinding force model
Fenr ¼ dp
ð10Þ
Grinding forces can be separated into two parts, cutting deformation force and frictional force. The cutting deformation force is again sub-divided into ploughing force and cutting or chip formation force.
Fetr ¼ Fenr m ¼ mdp
ð11Þ
F ¼ Fplow þ Fchip þ Ffriction
ð1Þ
0 0 0 Fn0 ¼ Fnp þ Fnc þ Fnr
ð2Þ
Ft0
ð3Þ
where m is the coefficient of friction, d the tip area of the worn grain and p¯ the average contact pressure between the wheel and the workpiece. Hence the total normal and tangential frictional force components are Fnr ¼ an p ¼ ðN dÞp
ð12Þ
Ftr ¼ man p ¼ mðNdÞp
ð13Þ
3.1. Chip formation force components
an ¼ blc A
ð14Þ
In developing the chip formation force components the cutting action of a single grain of the grinding wheel is assumed to be similar to the action of a single point cutting tool in turning. Hence the normal component of chip formation force imposed by a single grain can be determined as a function of undeformed chip cross-sectional area, which can be written as [9,10]
lc ¼ ðde aÞ1=2
ð15Þ
0 0 0 ¼ Ftp þ Ftc þ Ftr
Fenc ¼ KQi
ð4Þ
The total normal chip formation force per unit width of the grinding is the total of the normal forces of all active grains within the contact area of wheel and workpiece [9,10]: X X 0 Fnc ¼ KQi ¼ K Qi ð5Þ P Qi is the total value of all simultaneous chip cross-sections per unit width of grinding and is given as [9,10] X
Qi ¼
0 Fnc ¼K
Vw a Vc
Vw a Vc
where an is the real contact area between the wheel and the workpiece and A the fraction of the wheel surface that has wear flats. N is the total number of active grains (i.e. grains contributing to rubbing action) and is given as [1,9] a1 Z lc An Vw ð1a1 Þ=2 Nydn ðlÞdl ¼ ðC1 Þb1 að1 þ a1 Þ=2 ds ð16Þ N¼ 1þ a Vc 1 0
ð6Þ
ð7Þ
In the case of pure chip formation, the ratio of tangential force component to normal force component of a single grain bears a ratio j that is dependent on the profile of the grains of the
Fig. 1. Schematic view of the scratch and pile up material in a single-grit test.
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where a1 and b1 are the exponential coefficients of edge distribution and Nydn is the number of dynamic grains (i.e. total grains participating in grinding).
Using the parabola function to approximate the cutting path, the deviation (D) between the grinding wheel radius (ds/2) and the radius of curvature of cutting path (R) [1] is
D¼ Table 1 Experimental set-up details of grinding force measurement. Grinder Grinding wheel Workpiece Dimensions Width of workpiece Dressing interval Dressing amount Charge amplifier Dynamometer Mechanical units/volt pC/N value
CHEVALIER CNC surface grinder Al2O3 wheel (A60 M6 VCNM) Mild steel (SAE 1018) 70 50 6 mm3 6 mm 1 mm 0.02 mm KISTLER 3-channel piezoelectric 5006 SN KISTLER piezoelectric 9257A 100 For X: 7.5 For Z: 3.5 National Instruments E-6210
Data acquisition card
D¼ 7
Shank with the diamond grit
Dummy Grinding Wheel
ð17Þ
4Vw Vc de
ð18Þ
The average contact pressure p¯ between the workpiece and the wear plane of the abrasive grains increases approximately linearly with the deviation (D) of the radius of curvature and this relationship is given by [1] p ¼ P0 D ¼
4P0 Vw de Vc
ð19Þ
The average contact pressure (p¯) varies with the processing parameters of grinding. Therefore most likely there exists an elastic contact, elasto-plastic contact or plastic contact. Hence the frictional coefficient m also varies with the average contact pressure [12]. According to the frictional binomial theorem, the variable frictional coefficient is given by the formula [12]
m¼ Dummy Aluminium Wheel
2 1 ds R
a0 A0 W
þ b ¼ a0 p þ b
ð20Þ
where a0 and b are the coefficients, which are dependent on the physical and mechanical properties of the contact interface. Substituting Eqs. (14), (15), (19) and (20) into Eqs. (12) and (13), the total tangential and normal frictional force components per unit width of grinding (F0 tr and F0 nr) are obtained as Ftr ¼ ða0 p þ bÞpbðde aÞ1=2 A
ð21Þ
Ftr ¼ ða0 þ bpÞbðde aÞ1=2 A
ð22Þ
Shank
4AP0 bVw 0 ðde aÞ1=2 ¼ Aa0 þ Ftr de Vc
ð23Þ
K3 Vw 0 ðde aÞ1=2 ¼ K2 þ Ftr de Vc
ð24Þ
Holding Screw Grit
K2 ¼ Aa0 K3 ¼ 4AP0 b
Workpiece Dynamometer
Fnr ¼
Fig. 2. Dummy grinding wheel with indenter and schematic view of experimental set-up for single-grit test.
0 ¼ Fnr
4P0 Vw bðde aÞ1=2 A de Vc
ð25Þ
4AP0 Vw a 1=2 Vc de
ð26Þ
Table 2 Diameter of the grits used in the single-grit experimentation. Sl. number 1 2 3 4 5 6 7 8 9 10 11 12 13
l(mm)
b(mm)
h(mm)
Volume (mm3)
Grit radius (mm)
1.3 0.901 0.91 1.25 1.21 1.28 1.45 1.51 1.47 1.71 1.58 1.6 1.9
1.1 0.996 0.96 1.31 1.28 1.29 1.32 1.486 1.318 1.55 1.657 1.51 2.21
0.86 0.88 0.85 1.317 1.214 1.13 1.7 1.736 1.5123 2.19 2.112 2.101 2.52
0.410 0.263 0.247 0.718 0.626 0.622 1.084 1.298 0.976 1.934 1.843 1.692 3.527
0.460 0.397 0.389 0.555 0.531 0.529 0.637 0.677 0.615 0.773 0.760 0.739 0.944
Avg. grit diameter (mm)
0.83
1.07
1.29
1.52 1.90
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0 Fnr ¼
K1 Vw a 1=2 Vc de
ð27Þ
K1 ¼ 4AP0 where K1, K2, K3 are coefficients, which depend on the wheel and workpiece combination and are determined through experiments. 3.3. Ploughing force components Another mechanism associated with abrasive process is the ploughing. Ploughing energy is expended by deformation of workpiece material without removal. We normally associate ploughing with the side flow of material from the cutting path into ridges, but it can also include plastic deformation of the material passing under the cutting edge. Initially the grit makes elastic contact, which is assumed to make a negligible contribution to the total energy, followed by the plastic deformation (ploughing) of the workpiece. 3.3.1. Development of ploughing force model To determine the total normal and tangential ploughing force components per unit width of grinding, single-grit tests were performed using indenters of different sizes. Since the parameters on which the ploughing force is dependent are not known the basic ploughing force formulae developed by various researchers were analyzed and the parameters were found out. The ploughing effect in single-grit test is schematically shown in Fig. 1. De Vathaire et al. [14] proposed an upper bound model of ploughing by a pyramidal indenter and the experiments were Table 3 Levels of the parameters considered for single-grit experimentation.
235
conducted on a low carbon steel workpiece. The resistance of the work material towards scratching by an indenter is defined as scratch hardness (Hs). It is the ratio of normal force of the grit (Fenp) during scratching to bearing area (Ab) of the scratch. [14] Fenp ¼ Hs Ab
ð28Þ
Ab ¼ eðe þ hÞtan2 a
ð29Þ
h Fenp ¼ Hs e2 tan2 a 1 þ ¼ Hs eðe þhÞtan2 a e
ð30Þ
Hs ¼
Fenp e2 tan2 að1þ HÞ
ð31Þ
e is the depth of penetration of grit, h the height of pile up material and H= (h/e) tana ¼
R1 ðe þ hÞ
ðe þ hÞ ¼
ð32Þ
R1 c0 d ¼ tana 2tana
ð33Þ
Table 5 Grain distribution of the grinding wheel (A60 M6 VCNM) for three different imprint areas. Sl. number 1 2 3
Area (mm2)
Number of grits
Grain distributionCs(mm 2)
25 49 100
65 150 320
2.60 3.06 3.20
Table 6 Levels of parameters considered for experimentation to determine the chip formation and frictional force model coefficients.
Parameters/levels
2
1
0
1
2
Wheel speed (m/min) Work feed (m/min) Depth of cut (mm) Grit diameter (mm)
600 3 0.01 0.83
840 6 0.015 1.07
1080 9 0.02 1.29
1320 12 0.025 1.52
1560 15 0.03 1.9
Table 4 Experimental constants of ploughing force component.
Parameters/levels
1
2
3
Wheel speed(m/min) Work feed (m/min) Depth of cut (mm)
708.6 3 0.01
1308.6 9 0.02
1908.6 15 0.03
Table 7 Experimental coefficients of chip formation and frictional force components.
K4
K5
a0
b0
c0
K
K’
K1
K2
K3
5.7
3.24
0.22
0.33
0.294
41,217
23,081
229,160
1.287
20,394
Fig. 3. Marked zones of the grinding wheel imprints for determining the grain distribution: (a) 25, (b) 49 and (c) 100 mm2.
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R1 = radial distance from the center of the grit, d= diameter of the 0 grit, a =half the apex angle of the grit and c is a multiplicative factor. Hence the normal ploughing force component of the single grit (Fenp) can be given as Fenp ¼ k4 tana Hs ed
ð34Þ
where k4 = c0 /2. The ratio of tangential ploughing force component (Fetp) to the normal ploughing force component (Fenp) of a single grit bears a ratio A1, which is dependent on the properties of the workpiece [14]: Fetp ¼ A1 Fenp
ð35Þ
Table 8 Design of experiments considered for validation of the grinding force model. Vc(m/min) 1200
600 900 1200 1500 1800 1200
Vw(m/min)
a(mm)
3 6 9 12 15 6
0.015
6
0.010 0.015 0.020 0.025 0.030
0.015
Fig. 4. Variation of grinding force per unit width with wheel speed: (a) normal grinding force and (b) tangential grinding force.
Fig. 5. Variation of grinding force per unit width with table feed: (a) normal grinding force and (b) tangential grinding force.
Fig. 6. Variation of grinding force per unit width with depth of cut: (a) normal grinding force and (b) tangential grinding force.
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Fetp ¼ A1 Fenp
ð36Þ
Fetp ¼ A1 k4 tana Hs ed
ð37Þ
of grits in the contact area: a0 Vw ðdÞb0 ðaÞc0 Cs ðblc Þ Fnp ¼ Fenp Cs ðblc Þ ¼ K4 Vc
Fetp ¼ k5 tana Hs ed
ð38Þ
Ftp ¼ Fetp Cs ðblc Þ ¼ K5
where k4, k5 are the constants and [14] Hs ed ¼ f ðVc ; Vw ; a; dÞ
ð39Þ
From this equation it is clear that the ploughing force is dependent on four parameters namely the wheel speed, table feed, depth of cut and the grit diameter. Since the relationship between these variables is not known the ploughing force components for a single grit are expressed as a0 Vw ðdÞb0 ðaÞc0 Fenp ¼ K4 Vc
Fetp ¼ K5
Vw Vc
a0
ðdÞb0 ðaÞc0
a0 Vw ðdÞb0 ðaÞc0 Cs ðblc Þ Vc
237
ð42Þ
ð43Þ
Cs is the grain distribution of the grinding wheel and lc =(ade)1/2 is the geometric contact length. Hence the normal and tangential ploughing force components per unit width of grinding are a0 Vw 0 ¼ K4 ðdÞb0 ðaÞc0 Cs ðade Þ1=2 ð44Þ Fnp Vc 0 Ftp ¼ K5
a0 Vw ðdÞb0 ðaÞc0 Cs ðade Þ1=2 Vc
ð45Þ
ð40Þ 3.4. Final grinding force equations
ð41Þ
where K4 = k4(tan a1) and K5 = k5(tan a1). K4 and K5 are coefficients, which depend on the properties of the workpiece and on the orientation of the grits in the grinding wheel; a0, b0, c0 are constant exponents, which are determined from single-grit tests. The total normal and tangential ploughing force components were obtained by multiplying Eqs. (40) and (41) with the number
Substituting Eqs. (7), (9), (24), (27), (44) and (45) in Eqs. (2) and (3), the final equations for normal and tangential grinding force per unit width of grinding are a0 Vw K1 Vw a 1=2 Vw Fn0 ¼ K aþ þ K4 ðdÞb0 ðaÞc0 Cs ðade Þ1=2 ð46Þ Vc Vc Vc de Ft0 ¼ K 0
a0 Vw K3 Vw Vw ðde aÞ1=2 þ K5 a þ K2 þ ðdÞb0 ðaÞc0 Cs ðade Þ1=2 Vc de Vc Vc ð47Þ
Fig. 7. Contributions of each component of grinding force per unit width with wheel speed: (a) normal grinding force and (b) tangential grinding force.
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4. Experiments 4.1. Experimental set-up The experiments were conducted using CHEVALIER CNC surface grinder and the grinding wheel used for the experimentation was a medium grade alumina wheel with a grain size number of 60. The wheel has an outer diameter of 350 mm and width of 40 mm. The forces were measured using a 3-axis piezoelectric Kistler 9257A dynamometer, which is connected to Kistler 3channel piezoelectric 5006 SN charge amplifier. To record the data from the dynamometer, National Instruments Data Acquisition Card E-6210 was employed along with the Labview-7 software. An algorithm was developed for the recording and storage of the data and filtering of unwanted signals was performed using a low pass Butterworth filter by setting the filter frequency to 50 Hz. The workpiece used was mild steel (SAE 1018) with a rectangular cross-section and dimensions 70 50 6 mm3. The experimental set-up details are shown in Table 1.
4.2. Determination of experimental coefficients 4.2.1. Single-grit test The single-grit scratch tests were performed to develop the ploughing force model coefficients. In these tests the process of
grinding was duplicated with a dummy aluminium wheel and an indenter. The shanks with the diamond grit were held on to the wheel with the help of a holding screw and a spring. They are held in such a way that only the grit comes into contact with the workpiece. The wheel is rotated at the desired speed and the grit is allowed to make only a single scratch. The scratches were scanned by the probe of the Talysurf surface roughness measuring device to get the height of pile up material (h) and the depth of the scratch (e). The experimental set-up is shown in Fig. 2. Assuming the grits to be of pyramidal shape the diameter of the grits was found out by equating their volume to that of an equivalent sphere as shown in Table 2. The dimensions of the grits were obtained using a profile projector. Sets of 31 experiments were conducted to determine the coefficients of the ploughing force model. Each parameter, namely the wheel speed, table feed, depth of cut and grit diameter, was varied at five levels as shown in Table 3 and the designs of experiments were done using the CCD technique. Performing, regression analysis of the obtained force data, the coefficients were determined for mild steel (SAE 1018) workpiece and are shown in Table 4. The number of active grits or grains in the contact area is the product of the grit distribution of the grinding wheel (Cs) and the geometric contact length. The grain distribution of the grinding wheel was found out by taking the imprints of the grinding wheel surface and by marking areas of 25, 49 and 100 mm2 as shown in Fig. 3. The number of grits in the respective marked areas was first found out and from there the number of grits per unit
Fig. 8. Contributions of each component of grinding force per unit width with table feed: (a) normal grinding force and (b) tangential grinding force.
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area was calculated as shown in Table 5. The average grit or grain distribution obtained for the medium grade alumina wheel (A60 M6 VCNM) was 3 grits per unit area. 4.2.2. Chip formation force and frictional force coefficients The other five coefficients K, K0 , K1, K2 and K3 can be determined from five groups of experimental data by solving the linear equations. But, to make the model representative for all possible combinations of process parameters, twenty seven experiments were performed taking three levels of each parameter, namely the wheel speed, table feed and depth of cut, as shown in Table 6. The normal and tangential grinding forces were recorded with the help of the dynamometer and the ploughing force values per unit width of the grinding were subtracted from the obtained force values. The ploughing force components, were calculated from Eqs. (44) and (45) by substituting the constants obtained from the single-grit tests. The grain diameter (d) of the grinding wheel was found out from its mesh number. The grain diameter (d) and the mesh number (M) are related by d ðmmÞ ¼ ð15:2=MÞ
ð48Þ
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Hence the grain diameter for the alumina grinding wheel (A60 M6 VCNM) is approximately 0.250 mm. The experimental coefficients of chip formation and frictional force components for the alumina wheel (A60 M6 VCNM) and mild steel (SAE 1018) combination are shown in Table 7. These coefficients were determined through regression analysis of the resultant force data, which were obtained after subtracting the ploughing force components. Substituting the obtained coefficients in Eqs. (46) and (47), the final normal and tangential grinding force equations per unit width of grinding, obtained for the alumina wheel (A60 M6 VCNM) and mild steel workpiece (SAE 1018) combination were Vw 229; 160Vw a 1=2 aþ Fn0 ¼ 41; 217 Vc Vc de 0:22 Vw 0:33 0:294 þ5:7 ðdÞ ðaÞ Cs ðade Þ1=2 ð49Þ Vc Vw 203; 94Vw ðde aÞ1=2 a þ 1:287 þ Vc de Vc 0:22 Vw þ3:24 ðdÞ0:33 ðaÞ0:294 Cs ðade Þ1=2 Vc
Ft0 ¼ 23; 081
Fig. 9. Contributions of each component of grinding force per unit width with depth of cut: (a) normal grinding force and (b) tangential grinding force.
ð50Þ
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error in tangential force per unit width of grinding was 7.87%. The contribution of ploughing in the total grinding force is about 15–17% on average, while the contributions of chip formation and friction were 25–28% and 54–58%, respectively, for the mild steel (SAE 1018) and alumina wheel (A60 M6 VCNM) combination. The coefficient of friction varied from 0.27 to 0.86 depending on the process parameters considered. The ploughing force remained almost constant with changes in wheel speed and table feed but it showed an increase with increasing depth of cut.
6. Conclusion
Fig. 10. Variation of coefficient of friction with (a) wheel speed and (b) table feed.
5. Results and analysis 5.1. Validation of the model For the validation of model the basic values of the parameters considered were Vw = 6 m/min, Vc = 20 m/s and a= 0.015 mm. A total of 15 experiments were performed by varying each parameter for 5 levels keeping the other two parameters constant as shown in Table 8, and the corresponding experimental force values were recorded using a dynamometer. For the same values of the process parameters the theoretical normal and tangential grinding forces per unit width of grinding are calculated from the model equations (49) and (50). Both the experimental and theoretical grinding force values were compared in the plots of Fn0 Vc ; Ft0 Vc ; Fn0 Vw ; Ft0 Vw ; Fn0 a and Ft0 a as shown in Figs. 4–6. From the total tangential and normal grinding force values obtained from Eqs. (49) and (50), the contributions of ploughing, chip formation and friction were found out and are presented in the corresponding plots of Fn0 Vc ; Ft0 Vc ; Fn0 Vw ; Ft0 Vw ; Fn0 a and Ft0 aas shown in Figs. 7–9 5.2. Variable coefficient of friction Variation of the friction coefficient with process parameters like wheel speed and table feed for the alumina wheel (A60 M6 VCNM) and mild steel (SAE 1018) combination are shown in Fig. 10. The increase in the coefficient of friction with wheel speed was because of the increase in the wear flat area of the wheel with the wheel speed. On the other hand as the table feed increases less time was available for the grits in the contact area, to make contact with the workpiece, thereby decreasing the coefficient of friction. 5.3. Analysis The average percentage of error in the normal force per unit width of the grinding was 10.28%, while the average percentage of
The present research work models the total grinding force by incorporating the combined effects of variable coefficient of friction and the ploughing force. This model exhibits the importance of including the ploughing force, which becomes more predominant at very low depth of cuts. The coefficient of friction in the present model varies with process parameters like wheel speed and table feed, unlike the previous models, where it is a constant value throughout the process. This has considerably improved the model, making it more practical. A new method of developing the ploughing force model equations using the singlegrit test was adopted. Earlier schemes of execution of these tests were either by using an oscillating pendulum arrangement or by using an aluminium disc that was mounted on to a grinding machine. In both cases, the actual grinding process was not represented, as the scratches were made by moving the workpiece against a stationary grit held firmly onto the base of a dynamometer. It can be observed that the present scheme of executing the single-grit tests can take care of these limitations. Hence, the new grinding force model can be reliably used to predict the grinding forces and provide a certain theoretical basis for research on grinding force.
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