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Generalized practical models of cylindrical plunge grinding processes
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International Journal of Machine Tools & Manufacture 48 (2008) 61–72 www.elsevier.com/locate/ijmactool
Generalized practical models of cylindrical plunge grinding processes T.J. Choi, N. Subrahmanya, H. Li, Y.C. Shin School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA Received 19 January 2007; received in revised form 16 July 2007; accepted 20 July 2007 Available online 3 August 2007
Abstract This paper presents generalized grinding process models developed for cylindrical grinding processes based on the systematic analysis and experiments. The generalized model forms are established to maintain the same model structures with a minimal number of parameters so that the model coefficients can be determined through a small number of experiments when applied to different grinding workpiece materials and wheels. The relationships for power, surface roughness, G-ratio and surface burning are established for various steel alloys and alumina grinding wheels. It is shown that the established models provide good predictive capabilities while maintaining simple structures. r 2007 Elsevier Ltd. All rights reserved. Keywords: Grinding; Process models; Cylindrical grinding; Grinding power; Surface roughness; Surface burning
1. Introduction Even though there have been a large amount of efforts to characterize and understand grinding processes, modeling of grinding processes still presents many difficulties because the shape of cutting elements cannot be well defined and cutting edges are irregularly shaped and distributed unlike in other manufacturing processes such as turning and milling processes. An early study in modeling of grinding processes was performed by Shaw and his colleagues [1,2]. They tried to model the cutting action by geometrically defining grits and grit spacing so that mean chip and chip width could be estimated and the wheel performance could be analyzed. Another noticeable work was presented by the CIRP grinding group [3] as a collaborated research by five European laboratories. The aim of their research was to evaluate the relationships among wheel wear, surface roughness, grinding forces, and grinding conditions by extensive grinding experiments. As a result of the cooperative research, a basic parameter called equivalent grinding thickness, heq, was defined, which was considered to be the thickness of the continuous material layer removed from the cutting area at a velocity equal to the Corresponding author.
E-mail address: [email protected] (Y.C. Shin). 0890-6955/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.07.010
wheel speed. Using this basic parameter, Snoeys et al. [4] formulated practical models for grinding force, surface roughness, grinding ratio, and stock removal. The major benefits of those models include: (i) process conditions can be directly related to the controllable operating variables such as infeed, wheel speed, work diameter, and table speed, and (ii) possible experimental efforts are reduced by the concise model forms. A different representation of chip thickness was given by Lindsay and Hahn [5] using grinding conditions and a surface finish model was proposed as a function of the chip thickness defined. The changes of cutting edges on the wheel surface due to wheel wear affect grinding performance. Therefore, modeling of wheel wear has been another focus of efforts. Davis [6] measured the development of grinding force and the progression of radial wheel wear with time and discussed the relationships with dressing conditions and various dressing techniques. The effects of dressing conditions on the changes in grinding forces were also discussed by Umino and Shinozaki [7]. Relationships between grinding ratio and grinding power or material removal rate were investigated by Lindsay [8] with three wheel grades for various grinding processes. Due to a better understanding on the physics of grinding processes, more comprehensive process models have been proposed, while the scope of modeling has been extended
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to various process conditions such as temperature [9], energy partition [10], and surface integrity [11]. More comprehensive grinding force and power models were proposed by Malkin [12] and Chen et al. [13] with the wheel wear effect considered. Recently, a more fundamental approach to grinding simulation was proposed by simulating the wheel topography formed by dressing [14], and there was an effort to simulate grinding processes by integrating various process models [15]. Despite this advance in the understanding and modeling of grinding processes, most models developed to date are seldom utilized in industrial or practical applications. The models are usually valid only for the specific set-ups or conditions under which they were developed. Besides, model forms are too complex and vary greatly from one to another. While complex models might provide insight into the underlying physics, the number of parameters and the effort to determine them usually increase with the complexity. In practice, simple and easy-to-handle models might often be preferable and sufficient as long as the models can predict the requisite output or process conditions with the accuracy commensurate with the application. To¨nshoff et al. [16] surveyed many existing process models for grinding processes and introduced the concept of ‘‘basic model’’ or ‘‘generalized model’’. Furthermore, Brinksmeier et al. [17] identified and summarized basic parameters that are fundamental to constructing process models for grinding processes. This means that a common functional form of various specific models can be captured and used to construct a generalized model. In other words, the principal relationships between input parameters and outputs commonly agree in various models and the influences of input parameters are described simply by the coefficients of the generalized models. The values of coefficients will vary with each application, usually depending on the workpiece material and the grinding wheel. Therefore, process modeling for a specific application can consist of two phases: model form development and determination of model coefficients as illustrated in Fig. 1. In the first phase, a model form is formulated through a combination of a systematic study, an extensive survey of existing specific models, experiments, and knowledge from experts. Once a suitable model form is developed, then in the second phase only model coefficients need to be determined for different specific applications with a relatively small amount of experimental effort. Using this concept, Shin et al. [18–21] proposed generalized grinding process models and used them for optimization of various grinding processes. The major advantage of this approach is that only a small number of experiments are needed to capture various process cause–effect relationships so that a systems-based approach can be used for optimization of various grinding processes. This paper will describe the development of generalized process models for cylindrical plunge grinding processes for practical utilization, and it will demonstrate the efficacy
Fig. 1. Process modeling procedure.
Table 1 Summary of grinding wheels and work material tested Grinding wheel
Work material
No. of data
32A-60-K-VBE
4140 alloy steel (Rc40)
94A-80-M-V18
4140 alloy steel (Rc55) 8620 alloy steel (Rc40) 4140 alloy steel (Rc55)
36 (18 for roughness measurement) 36 36 18
Total
126 (108 for roughness)
and effectiveness of this approach. This paper will present models for grinding power and surface roughness with time varying characteristics due to wheel wear, grinding ratio, and grinding burn. 2. Experimental set-up Cylindrical grinding experiments were conducted on an instrumented Supertec G20-50CNC grinding machine with the aim of developing process models for grinding power, surface roughness, grinding ratio, and burn threshold. The grinding wheels and work material used are summarized in Table 1. The wheels used have initial dimensions of 350 mm (14’’) in diameter, 25 mm in width and 125 mm bore. The workpieces were hardened with 75, 56 and 38 mm initial diameters. A single point diamond dresser was used with a water soluble coolant. The experimental design is divided into two parts. The first part, called the ‘‘Fresh Wheel Test’’, is used to build a model relating the process outputs to the operating conditions right after dressing. This is further subdivided into ‘‘Roughing’’ and ‘‘Finishing’’ conditions. As an example, Tables 2 and 3 show the operating ranges and the design of experiments (DOE) for the 4140 (HRC 40)
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Table 2 Range of operating parameters for grinding of 4140 alloy steel (HRC 40) with 32A-60-K-VBE wheel Parameter
Unit
Ds Dw vw vs a Deq ad sd heq vw/vs ts
Roughing conditions
mm mm m/min m/s mm mm mm mm/rev mm – s
Finishing conditions
Low (1)
Mid (2)
High (3)
Low (1)
Mid (2)
High (3)
355.6 38.1 20 28 0.012 34.42 0.025 0.1 0.143 0.0119 0
57.15 25 30.5 0.015 49.24 0.030 0.16 0.205 0.0137 0
76.2 30 33 0.018 62.75 0.036 0.22 0.273 0.0152 0
355.6 38.1 20 28 0.0013 34.42 0.015 0.04 0.0151 0.0119 1
57.15 25 30.5 0.0038 49.24 0.023 0.1 0.0520 0.0137 3
76.2 30 33 0.0064 62.75 0.030 0.16 0.0962 0.0152 5
Table 3 L18 design of experiments for 4140 alloy steel (HRC40) with 32A-60-KVBE wheel—seven factors at three levels (18 runs) Run
Deq
ad
sd
ts
vw
1/vs
a
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 2 1 1
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 2 1.25 2.33
1 2 3 1 2 3 2 3 1 3 1 2 2 3 1 3 1 2 2 2.75 2
1 2 3 2 3 1 1 2 3 3 1 2 3 1 2 2 3 1 2 2.5 1.75
1 2 3 2 3 1 3 1 2 2 3 1 1 2 3 3 1 2 2 1.5 2.5
1 2 3 3 1 2 2 3 1 2 3 1 3 1 2 1 2 3 2 2 2
1 2 3 3 1 2 3 1 2 1 2 3 2 3 1 2 3 1 2 2 2
Note: The test 19, 20, and 21 are verification tests.
workpiece with 32A60KVBE grinding wheel. Separation in the operating ranges for roughing and finishing is determined by the guideline from the machinability handbook and the constraints for grinding power and surface roughness. The first 18 tests are used to fit the model and the remaining three tests are for verification. Seven parameters have been considered in the L18 DOE, namely:
ad, dressing depth of cut , mm sd, dressing lead, mm/rev deq, equivalent diameter given as DsDw/(Ds+Dw), mm vw, surface speed of the workpiece, mm/s vs, surface speed of the grinding wheel, m/s a, depth of cut, mm/rev ts, spark-out time, s
where Ds and Dw are the wheel diameter and the work diameter, respectively. The second part, called the ‘‘Wheel Wear Test’’, is used to determine the rate of wheel wear and its effect on the process outputs. The range of parameters used in these experiments is given in Table 4 for 4140 steel (Rc40) with 32A60KVBE grinding wheel. In addition to these experiments, a set of experiments was also conducted to determine a suitable model for the burn threshold at the two hardness levels. It should be mentioned that based on some observations during the experiments the number of workpiece revolutions during spark-out was used as a control variable instead of the spark-out time to build the surface roughness model. A fast responsive power meter (Load Controls Incorporated) was connected to the wheel head spindle motor to measure the grinding power. The power and tangential force relationship was determined by measuring them directly at the same time with the preliminary calibration test. The setup fixtures for the calibration test were made to mount the compact spindle, a GII DTS1300 spindle, on a dynamometer as shown in Fig. 2. Two special workpieces of ANSI 4140 steel were designed, made and heat treated to hardness of HRC40 and HRC55, respectively. The initial diameters of the workpiece to be ground were around 20 mm. The same grinding wheel of 32A60KBV was used in the tests. Hence, the tangential force can be obtained accurately through the calibrated power measurement, which is much easier than direct force measurement in cylindrical grinding. The surface roughness was measured by a profilometer (Taylor Hobson, Surtronix 3 plus). Radial wear of grinding wheel was measured through the method depicted in Fig. 3. A 8.9 mm (0.35 in) grinding width was used for all the experiments. After removing a certain amount of work material, wheel wear measurements were made by grinding a cutter blade to get a negative impression of the wheel surface profile. For each experimental condition, the wheel profile was measured at a minimum of 6 (roughly) equally spaced intervals with respect to the volume of work material removed.
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Table 4 Design of experiments for wheel wear tests for cylindrical plunge grinding 4140 (HRC 40) with 32A-60-K-VBE wheel Test
Dressing feed
Dressing depth
Wheel speed
Work speed
Infeed
Heq
D_eq
Predicted Initial Power
G*
1/G*
unit High Heq Med Heq Low Heq Verif. 1 Verif. 2
mm/s 3.55 3.55 3.55 3.26 3.40
mm 0.0254 0.0254 0.0254 0.0254 0.0254
rpm 1638 1638 1638 1504 1571
rpm 108 108 108 127 114
mm/s 0.027 0.017 0.007 0.014 0.019
mm 0.208 0.124 0.052 0.106 0.149
mm 59.7 58.7 59.4 56.9 57.4
W 1820 1176 569 943 1315
68 180 401 237 125
0.0148 0.0056 0.0025 0.0042 0.0080
Dwell time ¼ 3 s, grinding width ¼ 8.9 mm.
Fig. 4. Workpiece after the burn experiments.
3. Model development 3.1. Fresh wheel tests Fig. 2. Setup of force calibration experiments.
Specific grinding power model: To find a generalized model form of grinding force, an extensive literature survey was performed as summarized in Table 5. Those models were developed and validated for various machines, grinding wheels, and work materials. Common variables among those models were identified as significant parameters affecting grinding force, which can be summarized as follows:
Fig. 3. Wear measurement of grinding wheel.
The volume of cutter blade material removed was extremely small and is not expected to substantially influence the wear of the wheel. The blade profile was measured using an optical microscope (Zeiss Axioskop). The burn conditions were labeled as ‘‘Burn’’ and ‘‘No Burn’’ based on visual inspection. Fig. 4 shows a picture of one of the workpieces, and it may be seen that visual inspection is sufficient to differentiate between ‘‘Burn’’ and ‘‘No Burn’’.
Single variables: ad, sd, vw, vs, deq, a. Product of variables: a deq, vw a. vw Ratio of variables: . vs vw a Mixed form: heq ¼ . vs
Among these parameters, some of them have been already identified as having physical meanings [17] as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffi a d eq geometrical contact length ðmmÞ 0 Q ¼ vw a material removal rate ðmm=sÞ vw a equivalent chip thickness ðmmÞ heq ¼ vs Ds Dw d eq ¼ equivalent wheel diameter ðmmÞ Ds þ Dw vw speed ratio q¼ vs
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Table 5 Summary of the existing grinding force model for the cylindrical plunge grinding Reference
Grinding wheel
Work material
Peters [22]
EKL7VX
100Cr6
Li [23]
Unknown
Bearing steel (GCr15)
Fielding [24]
Aluminium oxide (WA0LV) Aluminium oxide (BA46Q5V)
EN9 fully hand
Aluminium oxide (32A80L6)
AISI 52100
Younis [25]
Xiao [26]
R2
Grinding force or power model 0:56 vw a0:78 d 0:22 F 0n ¼ 10:34 eq vs 0:33 vw vw a0:665 aþc F 0n ¼ K vs vs F t ¼ 290:83v0:503 ðvw aÞ0:516 s0:244 s d
0.91 1.0 1.0
F 0t ¼ F 0tc þ F 0tr þ F 0tp vw F 0tc ¼ u a k1 ð1 La Þ vs vw F 0tp ¼ l c af ð1 La Þ vs 0 F tr ¼ l La K L 8 9 1 < = vw vs ða d eq Þ2 Aeff b P ¼ P1 vw a þ P2 vs þ P3 þ P4 : ; vw d eq
FN56b
0.89
0.93
Aeff ¼ 0:008A0 lnð1:4 104 mdÞ Werner [27]
EK80L7VX
1:75 d ¼ 1:1 1011 a0:75 d sd 2e1 vw ae d 1e F 0n ¼ K½C 1 g eq vs
Ck45N
0.99
Table 6 Correlation of the parameters with specific tangential force Correlation coeff.
ln(ad)
ln(sd)
ln(a)
ln(vw)
ln(vs)
ln(deq)
ln(vw.a)
ln(a.deq)
ln(vw/vs)
ln(heq)
ln(Ft0 )
0.11
0.44
0.73
0.42
0.37
0.32
0.85
0.72
0.57
0.94
Correlation coefficient is given as absolute value.
The number of parameters should be reduced to have a more simplified and condensed form, which could be done by inspecting the significance of parameters. A statistical analysis was carried out to determine the significance of parameters. The data were generated from the existing models listed in Table 5. The results of correlation analysis are given in Table 6. Among the parameters, the equivalent chip thickness heq can be chosen as the most significant parameter. Next, even though deq shows relatively low correlation and the combination a deq has higher correlation, the single variable deq is chosen to avoid the redundancy in the model parameters. For the dressing parameters, most of existing models have considered the distribution of the static cutting edges to represent the wheel topography, but did not directly explain the relationship with dressing parameters—ad and sd. However, direct consideration of the dressing parameters, ad and sd, will be reasonable for practical purpose rather than any theoretical parameter showing the wheel topography such as number of active cutting edges. Thus, the grinding power model is finally proposed as in Eq. (1). f
f
F 0t;0 ¼ F 0 ad1 sd2 d feq3 hfeq3 ; P00 ¼ F 0t;0 vs ;
(1)
where F0 t,0 and P0 0 are the specific tangential force and specific grinding power right after dressing, respectively. The data were generated from the existing models shown in Table 5 and then the proposed model in Eq. (1) was fitted to check its goodness of fitting. As the performance indicator, R-square values were calculated to determine the fitness of the model against the data generated by six existing models. The proposed model form has shown high R-square values as summarized in the last column of Table 5, thus proving its general applicability. After the model form is determined, the model coefficients for various grinding wheels and materials listed in Table 1 should be obtained. Using the linear least-squares method to fit the experimental data, the model coefficients are obtained and summarized in Table 7 with the goodness of fit. Surface roughness model: A same procedure has been applied to determine the model form for surface roughness. A literature survey was performed and selected representative models are summarized in Table 8. After performing a statistical analysis, the equivalent chip thickness is again adopted as a model parameter. Since the effect of the equivalent diameter Deq is found to be insignificant, it was removed from the model form. However, the work diameter is indirectly considered in the spark-out by
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replacing the spark-out time ts with the number of work revolutions during spark-out Ns. The model in Eq. (2) is proposed as a generalized surface roughness model for cylindrical plunge grinding:
Each test consisted of grinding multiple workpieces beginning with a newly dressed wheel under the specified operating condition. The test conditions for 4140 steel (Rc40) with 32A60KVBE grinding wheel are shown in
Table 4, in which the first three were used to fit the model coefficients and the last two were used to verify the model. For other work-wheel combinations, a similar procedure was repeated. The experimental measurement results and identified model forms of these conditions are summarized below. G-ratio model: A 8.9 mm grinding width was used for all the experiments. The grinding tests were conducted with certain intervals for measurements. The total wheel radial wear was over 20 mm for all the conditions and at least six measurements were made for each condition. At each measurement interval, a cutter blade was ground to get a negative impression of the wheel surface profile along the wheel width and the wheel wear is then measured by observing the ground portion of the blade. An example of the measurement is shown in Fig. 5 for the test condition with the highest equivalent chip thickness.
Table 7 Summary of developed generalized grinding power model form and coefficients for cylindrical plunge grinding
Table 9 Summary of developed generalized surface roughness model form and coefficients for cylindrical plunge grinding
Combination
Model Form
Goodness of fit (R2)
Combination
Model form
Generic model form
P00 ¼ F 0 ad 1 sd 2 DFeq3 hFeq4 vs
N/A
Ra0 ¼ R0 ad1 sd2 hreq3 ð1 þ er4 N s Þ
N/A
4140 (HRC40) +32A-60K-VBE 4140 (HRC55) +32A-60K-VBE 8620 (HRC40) +32A-60K-VBE 4140 (HRC55) +94A-80M-V18
0:842 P00 ¼ 4:42a0:095 sd0:336 D0:163 eq heq vs d
0.98
0:2N s Ra0 ¼ 1:64ad0:021 s0:385 h0:284 Þ eq ð1 þ e d
0.93
0:701 P00 ¼ 6:91a0:053 s0:250 D0:097 vs eq heq d d
0.95
0:29N s Ra0 ¼ 1:30ad0:02 s0:205 h0:301 Þ eq ð1 þ e d
0.88
P00 ¼ 5:28a0:222 sd0:313 D0:031 h0:763 eq eq vs d
0.94
0:476N s Ra0 ¼ 2:28ad0:044 s0:250 h0:598 Þ eq ð1 þ e d
0.76
0:940 P00 ¼ 7:40a0:204 sd0:349 D0:028 eq heq vs d
0.95
Generic model form 4140 (HRC40) +32A-60K-VBE 4140 (HRC55) +32A-60K-VBE 8620 (HRC40) +32A-60K-VBE 4140 (HRC55) +94A-80M-V18
0392N s Ra0 ¼ 0:58ad0:243 s0:092 h0:497 Þ eq ð1 þ e d
0.90
r
r
Ra;0 ¼ R0 ad1 sd2 hreq3 ½1 þ expðr4 N s Þ,
(2)
where Ra,0 is the surface roughness right after dressing. The model coefficients for various grinding wheels and materials listed in Table 1 are obtained and summarized in Table 9 with the goodness of fit. 3.2. Wheel wear tests
F
F
r
Goodness of fit (R2) r
Table 8 Summary of the existing surface roughness model for the cylindrical plunge grinding Reference
Grinding wheel
Work material
Lindsay [5, 29]
2A80K4VFMB
AISI 52100
Grinding force or power model
Ra ¼
Snoeys [4]
Chiu [28]
EK60L7X EK60L7VX P60 6A54LSVA2 32A80M6V
4615 AISI 52100
Ko¨nig [30]
EK80JotKe
Riegger NSS 10710
Chen [31]
B56-3
100Cr6 X210CrW12
Si3N4
d 16=27 a19=27 g
ad 16=27 vw ð1 þ Þsd ð Þ16=27 8=27 sd vs Deq ( ) 0:4587T 0:30 for 0oT o0:254 ave ave for 0:254oT ave o2:54 0:7866T 0:72 ave
T ave ¼ 12:5 103
Ra ¼ R1 hreq
R2 0.98
1.0
Rda ¼ Rd1 hrd eq ðwith spakoutÞ 0:25 m Ra ¼ R1 s0:5 d ad heq 1 þ expð0:77tsp =tÞ ð1:02d g þ 0:76Þ 0:32 0 Q0 0:46 Rz ¼ 0:25s0:006 w ðV w þ 8:3Þ d
þ32:2s0:96 Q0 0:14 ðV 0w w d Ra ¼ g am
þ 16:4Þ
vw pffiffiffiffiffiffiffiffiffiffiffiffiffi am ¼ w a d eq vs 2
1.0 0.98
0:29
0.92
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As shown in Fig. 5, the worn wheel surface (left side) can be easily distinguished and measured using the unworn surface as reference (right side). The wheel radial wear values were obtained by averaging the measured radial wear along the wheel plunge width, and plotted as a function of work material removal (Fig. 6). In every case, there is initial transient rapid wear prior to a steady-state wear regime at a lower wear rate. Thus the steady-state G-ratio can be obtained by curve-fitting linear relationships in the volume–volume plane as shown in Fig. 6. The steady-state wear regime starting from the end of the transient after an accumulated material removal of workpiece Vw_st. The steady-state wheel wear can be characterized by a straight line with slope of 1/G*, which represent the reciprocal of the steady-state G-ratio, Gn ¼
DV w ; DV s
where V w 4V w_st .
(3)
The intercept wheel wear volume, Vs0, can be considered to represent that portion of the radial wear associated with the initial transient. The intercept, Vs0, is independent of grinding conditions, while the steady-state G-ratio is a strong function of equivalent chip thickness. Thus the constant Vs0 and 1/G*’s of the test conditions were obtained by linear least-squares fitting of the data in
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Fig. 7. The results for the intercept (equal to 36.0) and the steady-state G-ratios are plotted in Fig. 7 versus equivalent chip thickness, Heq, which can be represented by Gn ¼ 10:71h1:254 . eq
(4)
As shown in Fig. 7, the steady G-ratios can be well fitted using a log-linear model with Heq. The total steady-state wheel volume can be represented as Vs ¼
Vw Vw þ 36:0; n þ V s0 ¼ 1:254 G 10:71heq
where V w 4V w_st , (5)
where Vw_st is a small initial value during which the transient wheel wear occurs. Specific grinding power model: The grinding power measurements are plotted in Fig. 8 as a function of accumulated work material removal per unit plunge width (called the specific work removal from now on and has the unit of mm3/mm). The power data from all the tests shown in Fig. 8 exhibit the same trends with an initial transient region (generally decreasing), followed by a steady-state increase at a constant rate. To take into account the initial drop in power, a constant correction factor l is introduced as shown in Eq. (6).
Fig. 5. Measured wheel surfaces profiles along the wheel width (The legend shows the total work material removal in diameter).
Fig. 7. Steady-state G-ratio.
Fig. 6. Volumetric wheel wear vs. workpiece removal.
Fig. 8. Grinding power vs. specific work material removal.
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Let P0 0 represent the prediction of the power model just after dressing as specified in Eq. (1). Assume the power of the steady state has the following form: b
P0 ¼ lP00 ð1 þ aV 0 w Þ,
(6)
where a represents the power growth rate (and could be a function of the equivalent chip thickness), b denotes the non-linear growth rate and V0 w represents the specific work material removal. Using a linear least-square fitting method, the specific grinding power model coefficients are obtained. The growth rate a is plotted in Fig. 9 against equivalent chip thickness. It can be seen that a is approximately constant with an average value of 8 105. Therefore the coefficients are obtained as P0 ¼ 0:9P00 ð1 þ 8:0 105 V 0w Þ.
(7)
And the predictions using Eq. (7) are shown in Fig. 10 in comparison with corresponding experimental data. Although the coefficient a was found to be constant over the range of Heq considered in this set of experiments, in general, it may be a function of Heq. Hence, the generic model considers a as a function of the equivalent chip thickness. Surface roughness model: The surface roughness measurements are plotted in Fig. 11 as a function of specific work material removal. A constant spark-out time of 3 s
Fig. 11. Surface Roughness vs. specific work material removal.
was used for all the experiments. The roughness values for all the tests in Fig. 11 show similar trends with an initial rapid increase after which the growth rate is lower. Let Ra0 represent the prediction of the roughness model just after dressing as specified in Eq. (2). The variation of surface roughness with wheel wear is slightly more complex as compared to the power and hence the model form for surface roughness has to be flexible enough to explain these variations. Before stating the model form, some observations about the roughness variation are presented below. These observations are drawn from all the experiments from plunge grinding and cylindrical grinding. 1. For a given condition, surface roughness generally reaches a steady value as the specific work removal increases. 2. For the same value of the equivalent chip thickness, the roughness can show increasing or decreasing trends based on the initial dressing conditions. 3. The steady-state value seems to be independent of the dressing conditions. These observations suggest the following:
Fig. 9. Variation of a with equivalent chip thickness for specific grinding power (HRC 40).
Fig. 10. Specific power prediction for cylindrical plunge grinding (HRC 40).
1. The initial roughness is dependent on all the operating conditions as well as the dressing conditions. 2. The final roughness is dependent only on the operating conditions and does not depend on the dressing conditions. An example of the experiments conducted to validate the above assumptions is shown in Fig. 12. The experimental conditions are given in Table 10 and are considered as additional validation experiments. It may be seen that the only difference between the two experiments is the dressing condition. The aim of these tests was to observe both increasing and decreasing trends for the same equivalent chip thickness and at the same time verify if these two initial conditions resulted in the same steady-state roughness values. The observations confirm that both increasing and decreasing trends can be observed for the same
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equivalent chip thickness and that they both converge to the same steady-state values (around 0.55 mm in this case). Based on the above observations, the following generalized model is proposed: 0
Ra ¼ Ra0 þ ðRa1 Ra0 Þð1 eaV w Þ, Ra0 ¼ f 1 ðdressing conditions; operating conditionsÞ, Ra1 ¼ f 2 ðoperating conditionsÞ.
ð8Þ
The above model is general enough to hold for all grinding processes. The exact dressing parameters and operating parameters will vary from process to process. Given below is the model form for cylindrical plunge grinding with a single point dresser. R
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Fig. 13 shows a result of fitting a model of the form shown in (9) to the steady-state roughness values as measured after five spark-out revolutions. The model prediction is acceptable. It may also be noted that the model successfully predicts increasing and decreasing trends for the two additional verification conditions as shown in Fig. 12. The final model (with coefficients obtained using linear least squares) for the roughness model is given below. 0:2N s s0:385 h0:284 Þ; Ra0 ¼ 1:64a0:021 eq ð1 þ e d d 0:0624N s Ra1 ¼ 3:73h0:716 Þ; eq ð1 þ e;
Ra ¼ Ra0 þ ðRa1 Ra0 Þð1 e
0:00085V 0w
(10) Þ:
R
R4 N s 3 Ra0 ¼ R0 ad 1 sd 2 hR Þ; eq ð1 þ e
Ra1 ¼ r0 hreq1 ð1 þ er2 N s Þ;
(9)
And the predictions using Eq. (10) are shown in Fig. 14.
0
Ra ¼ Ra0 þ ðRa1 Ra0 Þð1 eaV w Þ: The advantage of using the above model form is that it has the same number of free parameters as a power model of the form used for predicting power, but it is more flexible as it separates the effect of the initial and final values and hence can incorporate observations like those shown in Fig. 12. Moreover, although the above model seems complicated, the curve-fitting process is decoupled and hence all the parameters may be obtained by repeated use of a simple linear least-squares fit. Since the model for Ra0 (the initial roughness) was already developed during the fresh wheel tests, only the steady-state roughness model and the growth rate parameter (alpha) have to be determined from the wear data.
Fig. 13. Variation of RaN (steady state roughness) with Heq.
Fig. 12. Variation of surface roughness with wheel wear for the same value of Heq with different dressing conditions.
Fig. 14. Surface roughness predictions for cylindrical plunge grinding.
Table 10 Experimental conditions for additional experiments conducted to validate roughness model form Test
Dressing feed
Dressing depth
Wheel speed
Work speed
Infeed
Dwell time
1 2
mm/rev 0.22 0.10
mm 0.0254 0.0125
rpm 1772 1772
rpm 114 114
mm/min 0.203 0.203
sec 3 3
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3.3. Burn tests For the burn threshold, the model proposed by Malkin et al. [12]. is adopted as given in Eq. (11). It was claimed that the combination of operating parameters shown in Eq. (11) clearly distinguishes between burn-free condition and burn condition in terms of specific grinding energy, u (J/mm3). Based on their observation, the intercept value u0 is obtained as 6.2 J/mm3 for alloy steels. u
n
0:75 0:5 ¼ u0 þ Bd 0:25 vw eq a n 0:25 0:5 ) P ¼ u0 vw a þ Bd 0:25 vw . eq a
determining the burn threshold using a linear support vector machine classifier [32], which is a supervised learning method used for regression and classification. The conditions were labeled as ‘‘Burn’’ and ‘‘No Burn’’ based on visual inspection. Based on the above plots, the following equation is recommended for the burn threshold: 0:75 0:5 un ¼ 6:2 þ 10:192d 0:25 vw eq a 0:5 ) Pn ¼ 6:2vw a þ 10:192d 0:25 eq vw .
ð12Þ
ð11Þ 4. Model summary for various wheels and work materials
The operating ranges for determining burn threshold are shown in Table 11. Although a large number of experiments were conducted in order to verify the model form at different diameters, it is not necessary to use the same number of experiments for every wheel/workpiece combination. During the experiments, the workspeed, infeed rate, dressing lead, dressing depth and the work diameter were varied to get different points as shown in Fig. 15. This data set was augmented with the data obtained during the ‘‘Fresh Wheel’’ finishing experiments which were all labeled as ‘‘No Burn’’ conditions. Fig. 15 shows the result of
Table 11 Range of operating parameters for burn test (4140 Rc40) with 32A-60-KVBE wheel Parameter
Operating range Low
Wheel diameter Work diameter Dressing feed Dressing depth Wheel speed Work speed Infeed Heq Grinding width
A similar set of experiments have been conducted to determine the model sets for the different grinding wheel and work material as summarized in Table 1. The model forms and the specific models with coefficient are summarized in Table 12–15 with the goodness of the fit. In the model forms like a[heq] denotes that a is a function of heq. The surface roughness model for the 4140 RC 40 workpiece model has been modified to use the newly proposed number of spark-out revolutions parameter, Ns.
Table 12 Summary of developed generalized wear model form of grinding power and coefficients for cylindrical plunge grinding
Unit Combination
Model form
Goodness of fit (R2)
Generic model form
P0 ¼ lP00 ð1 þ a½heq V 0 bw Þ
N/A
4140 (HRC40) +32A-60K-VBE 4140 (HRC55) +32A-60K-VBE 8620 (HRC40) +32A-60K-VBE
P0 ¼ 0:9P00 ð1 þ 8:0 105 V 0w Þ
0.86
P0 ¼ 0:95P00 ð1 þ 16:1 105 V 0w Þ
0.85
0 0 P0 ¼ 0:95h0:050 eq P0 ð1 þ aV w Þ
0.71
High 343
33 0.03 0.008
55 0.12 0.023 32
91 0.019 0.065
1167 0.051 0.231 6.35
mm mm mm/rev mm m/s mm/s mm/s mm mm
a¼
D2:418 0:51h0:470 eq eq
Table 13 Summary of developed generalized wear model form of surface roughness and coefficients for cylindrical plunge grinding
Fig. 15. Experimental data for burn and burn threshold determined using linear SVM.
Combination
Model form
Goodness of fit (R2)
Generic model form
Ra ¼ Ra0 þ ðRa1 Ra0 Þð1 eaV w Þ Ra1 ¼ r0 hreq1 ð1 þ er2 N s Þ
4140 (HRC40) +32A-60K-VBE
Ra ¼ Ra0 þ ðRa1 Ra0 Þð1 e0:00085V w Þ 0:0624N s Ra1 ¼ 3:73h0:716 Þ eq ð1 þ e
4140 (HRC55) +32A-60K-VBE
Ra ¼ Ra0 þ ðRa1 Ra0 Þð1 e0:00095V w Þ
8620 (HRC40) +32A-60K-VBE
Ra ¼ Ra0 þ ðRa1 Ra0 Þð1 e0:0018V w Þ
0
Ra1 ¼
3:47h0:689 eq ð1
þe
0:0596N s
N/A 0.87
0
0.92
Þ 0
0:075N s Ra1 ¼ 3:22h0:676 Þ eq ð1 þ e
0
0.72
ARTICLE IN PRESS T.J. Choi et al. / International Journal of Machine Tools & Manufacture 48 (2008) 61–72 Table 14 Summary of generalized grinding ratio model forms and coefficients Combination
Model form
Goodness of fit (R2)
Generic model form
1 G n ¼ G0 hG eq
N/A
4140 (HRC40) +32A-60K-VBE
G ¼
4140 (HRC55) +32A-60K-VBE
n
G ¼
8620 (HRC40) +32A-60K-VBE
n
n
G ¼
10:71h1:254 eq 9:48h1:051 eq 7:96h1:262 eq
0.96 0.97 0.96
Table 15 Summary of burn threshold model forms and coefficients Combination
Model form
Generic model form
0:25 0:5 vw P0 n ¼ B0 vw a þ B1 d 0:25 eq a
4140 (HRC40) +32A-60K-VBE
0:25 0:5 P0 n ¼ 6:2vw a þ 11:42d 0:25 vw eq a
4140 (HRC55) +32A-60K-VBE
0:25 0:5 vw P0 n ¼ 6:2vw a þ 10:19d 0:25 eq a
8620 (HRC40) +32A-60K-VBE
0:25 0:5 P0 n ¼ 6:2vw a þ 9:93d 0:25 vw eq a
5. Conclusions Simple and easy-to-use generalized grinding process models for cylindrical plunge grinding processes have been developed and presented in this paper. The major advantage of using these generalized models is that these models maintain the same functional forms regardless of different workpiece–wheel combinations used and hence can be readily applied to various grinding set-ups by determining their coefficients through a small number of designed experiments. The surface and power models were presented, including the terms to account for the effect of wheel wear. It has been shown that these models can predict process conditions over a wide range of grinding conditions. Furthermore, it has also been shown that steady-state surface roughness and steady-state G-ratio are primarily dependent only on the effective chip thickness. These models can be used for designing process operating conditions or process optimization without extensive experimental efforts.
Reference [1] E.R. Marshall, M.C. Shaw, Forces in dry surface grinding, Transactions of ASME 74/1 (1952) 51–58. [2] G.S. Reichenbach, J.E. Mayer, S. Kalpakcioglu, M.C. Shaw, Role of chip thickness in grinding, Transactions of ASME 78 (1956) 847–859. [3] R. Snoyes, A. Decneut, Review of results of the co-operative research program of the CIRP grinding group, Annals of the CIRP 19 (1971) 507–512. [4] R. Snoyes, J. Peters, A. Decneut, The significance of chip thickness in grinding, Annals of the CIRP 23/2 (1974) 227–237. [5] R.P. Lindsay, S. Hahn, On the surface finish-metal removal relationship in precision grinding, Annals of the CIRP 22/1 (1973) 105–106.
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[6] C.E. Davis, The dependence of grinding wheel performance on dressing procedure, International Journal of Machine Tool Design Research 14 (1974) 33–52. [7] K. Umino, N. Shinozaki, One aspect of the variation of the grinding wheel surface based on the grinding force analysis studies on the wear and the redress life of grinding wheels (Part I), Bulletin of Japan Society of Precision Engineering 10/4 (1976) 139–144. [8] R.P. Lindsay, The effect of wheel wear rate on the grinding performance of three wheel grades, Annals of the CIRP 32/1 (1983) 247–249. [9] A.S. Lavine, S. Malkin, T. Jen, Thermal aspects of grinding with CBN, Annals of the CIRP 38/1 (1989) 557–560. [10] S. Kohli, C. Guo, S. Malkin, Energy partition to the workpiece for grinding with aluminium oxide and CBN abrasive wheels, Transactions of the ASME Journal of Engineering for Industry 117 (1995) 160–168. [11] B.W. Kruszyn´ski, R. Wo´jcik, Residual stress in grinding, Journal of Materials Processing Technology 109 (2001) 254–257. [12] S. Malkin, Grinding Technology: Theory and Applications of Machining with Abrasives, Ellis Horwood Ltd., Chichester, UK, 1989. [13] X. Chen, W.B. Rowe, D.R. Allanson, B. Mills, A grinding power model for selection of dressing and grinding conditions, Transactions of the ASME Journal of Manufacturing Science and Engineering 121 (1999) 632–637. [14] X. Chen, W.B. Rowe, Analysis and simulation of the grinding process. Part II: mechanics of grinding,, International Journal of Machine Tools and Manufacture 36/8 (1996) 883–896. [15] C. Guo, S. Malkin, Cylindrical grinding processes simulation, optimization, and control, SME fourth International Machining and Grinding Conference, 2001. [16] H.K. To¨nshoff, J. Peters, I. Inasaki, T. Paul, Modelling and simulation of grinding processes, Annals of the CIRP 41/2 (1992) 677–688. [17] E. Brinksmeier, H.K. To¨nshoff, I. Inasaki, J. Peddinghaus, Technical reports basic parameters in grinding report on a cooperative work in STC, Annals of the CIRP 42/2 (1993) 795–799. [18] P.S. Vishnupad, Y.C. Shin, Intelligent optimization of grinding processes using fuzzy logic, Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 212 (1998) 647–660. [19] C.W. Lee, Y.C. Shin, Evolutionary modeling and optimization of grinding processes, International Journal of Production Research 38 (12) (2000) 2787–2813. [20] C.W. Lee, T.J. Choi, Y.C. Shin, Intelligent model-based optimization of the surface grinding process for heat-treated 4140 steel alloys with aluminum oxide grinding wheels, Transactions ASME, Journal of Manufacturing Science and Engineering 125 (2003) 65–76. [21] T.J. Choi, Y.C. Shin, Generalized intelligent grinding advisory system, International Journal of Production Research 45/8 (2007) 1899–1932. [22] J. Perters, R. Snoeyes, A. Decneut, The proper selection of grinding conditions in cylindrical plunge grinding, Annals of the CIRP 26 (1) (1976) 387–394. [23] L. Li, J. Fu, A study of grinding force mathematical model, Annals of the CIRP 29 (1) (1980) 245–249. [24] E.R. Fielding, T.J. Vickerstaff, The prediction of grinding forces in cylindrical plunge grinding, Journal of Production Research 24 (1) (1986) 167–186. [25] M. Younis, M.M. Sadek, T. El-Wardani, A new approach to development of a grinding force model, ASME Journal of Engineering for Industry 109 (1987) 306–313. [26] G. Xiao, S. Malkin, K. Danai, Automated system for multi-stage cylindrical grinding, ASME Journal of Dynamic Systems, Measurement, and Control 115 (1993) 667–672. [27] G. Werner, Influence of work material on grinding forces, Annals of the CIRP 27 (1) (1978) 243–248. [28] N. Chiu, S. Malkin, Computer simulation for cylindrical plunge grinding, Annals of the CIRP 42 (1) (1993) 383–387.
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[29] X.M. Wen, A.A.O. Tay, A.Y.C. Nee, Micro-computer-based optimization of the surface grinding process, Journal of Material Processing Technology 29 (1992) 75–90. [30] W. Ko¨nig, K. Steffens, A numerical method to Describe the kinematics of grinding, Annals of the CIRP 31/1 (1982) 201–206.
[31] C. Chen, Y. Jung, I. Inasaki, Surface, cylindrical and internal grinding of advanced ceramics, ASME PED-V39 Grinding Fundamentals and Applications, 1989, pp. 201–211. [32] V. Vapnik, The Nature of Statistical Learning Theory, Springer, Berlin, 1999.
International Journal of Machine Tools & Manufacture 48 (2008) 61–72 www.elsevier.com/locate/ijmactool
Generalized practical models of cylindrical plunge grinding processes T.J. Choi, N. Subrahmanya, H. Li, Y.C. Shin School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA Received 19 January 2007; received in revised form 16 July 2007; accepted 20 July 2007 Available online 3 August 2007
Abstract This paper presents generalized grinding process models developed for cylindrical grinding processes based on the systematic analysis and experiments. The generalized model forms are established to maintain the same model structures with a minimal number of parameters so that the model coefficients can be determined through a small number of experiments when applied to different grinding workpiece materials and wheels. The relationships for power, surface roughness, G-ratio and surface burning are established for various steel alloys and alumina grinding wheels. It is shown that the established models provide good predictive capabilities while maintaining simple structures. r 2007 Elsevier Ltd. All rights reserved. Keywords: Grinding; Process models; Cylindrical grinding; Grinding power; Surface roughness; Surface burning
1. Introduction Even though there have been a large amount of efforts to characterize and understand grinding processes, modeling of grinding processes still presents many difficulties because the shape of cutting elements cannot be well defined and cutting edges are irregularly shaped and distributed unlike in other manufacturing processes such as turning and milling processes. An early study in modeling of grinding processes was performed by Shaw and his colleagues [1,2]. They tried to model the cutting action by geometrically defining grits and grit spacing so that mean chip and chip width could be estimated and the wheel performance could be analyzed. Another noticeable work was presented by the CIRP grinding group [3] as a collaborated research by five European laboratories. The aim of their research was to evaluate the relationships among wheel wear, surface roughness, grinding forces, and grinding conditions by extensive grinding experiments. As a result of the cooperative research, a basic parameter called equivalent grinding thickness, heq, was defined, which was considered to be the thickness of the continuous material layer removed from the cutting area at a velocity equal to the Corresponding author.
E-mail address: [email protected] (Y.C. Shin). 0890-6955/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.07.010
wheel speed. Using this basic parameter, Snoeys et al. [4] formulated practical models for grinding force, surface roughness, grinding ratio, and stock removal. The major benefits of those models include: (i) process conditions can be directly related to the controllable operating variables such as infeed, wheel speed, work diameter, and table speed, and (ii) possible experimental efforts are reduced by the concise model forms. A different representation of chip thickness was given by Lindsay and Hahn [5] using grinding conditions and a surface finish model was proposed as a function of the chip thickness defined. The changes of cutting edges on the wheel surface due to wheel wear affect grinding performance. Therefore, modeling of wheel wear has been another focus of efforts. Davis [6] measured the development of grinding force and the progression of radial wheel wear with time and discussed the relationships with dressing conditions and various dressing techniques. The effects of dressing conditions on the changes in grinding forces were also discussed by Umino and Shinozaki [7]. Relationships between grinding ratio and grinding power or material removal rate were investigated by Lindsay [8] with three wheel grades for various grinding processes. Due to a better understanding on the physics of grinding processes, more comprehensive process models have been proposed, while the scope of modeling has been extended
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to various process conditions such as temperature [9], energy partition [10], and surface integrity [11]. More comprehensive grinding force and power models were proposed by Malkin [12] and Chen et al. [13] with the wheel wear effect considered. Recently, a more fundamental approach to grinding simulation was proposed by simulating the wheel topography formed by dressing [14], and there was an effort to simulate grinding processes by integrating various process models [15]. Despite this advance in the understanding and modeling of grinding processes, most models developed to date are seldom utilized in industrial or practical applications. The models are usually valid only for the specific set-ups or conditions under which they were developed. Besides, model forms are too complex and vary greatly from one to another. While complex models might provide insight into the underlying physics, the number of parameters and the effort to determine them usually increase with the complexity. In practice, simple and easy-to-handle models might often be preferable and sufficient as long as the models can predict the requisite output or process conditions with the accuracy commensurate with the application. To¨nshoff et al. [16] surveyed many existing process models for grinding processes and introduced the concept of ‘‘basic model’’ or ‘‘generalized model’’. Furthermore, Brinksmeier et al. [17] identified and summarized basic parameters that are fundamental to constructing process models for grinding processes. This means that a common functional form of various specific models can be captured and used to construct a generalized model. In other words, the principal relationships between input parameters and outputs commonly agree in various models and the influences of input parameters are described simply by the coefficients of the generalized models. The values of coefficients will vary with each application, usually depending on the workpiece material and the grinding wheel. Therefore, process modeling for a specific application can consist of two phases: model form development and determination of model coefficients as illustrated in Fig. 1. In the first phase, a model form is formulated through a combination of a systematic study, an extensive survey of existing specific models, experiments, and knowledge from experts. Once a suitable model form is developed, then in the second phase only model coefficients need to be determined for different specific applications with a relatively small amount of experimental effort. Using this concept, Shin et al. [18–21] proposed generalized grinding process models and used them for optimization of various grinding processes. The major advantage of this approach is that only a small number of experiments are needed to capture various process cause–effect relationships so that a systems-based approach can be used for optimization of various grinding processes. This paper will describe the development of generalized process models for cylindrical plunge grinding processes for practical utilization, and it will demonstrate the efficacy
Fig. 1. Process modeling procedure.
Table 1 Summary of grinding wheels and work material tested Grinding wheel
Work material
No. of data
32A-60-K-VBE
4140 alloy steel (Rc40)
94A-80-M-V18
4140 alloy steel (Rc55) 8620 alloy steel (Rc40) 4140 alloy steel (Rc55)
36 (18 for roughness measurement) 36 36 18
Total
126 (108 for roughness)
and effectiveness of this approach. This paper will present models for grinding power and surface roughness with time varying characteristics due to wheel wear, grinding ratio, and grinding burn. 2. Experimental set-up Cylindrical grinding experiments were conducted on an instrumented Supertec G20-50CNC grinding machine with the aim of developing process models for grinding power, surface roughness, grinding ratio, and burn threshold. The grinding wheels and work material used are summarized in Table 1. The wheels used have initial dimensions of 350 mm (14’’) in diameter, 25 mm in width and 125 mm bore. The workpieces were hardened with 75, 56 and 38 mm initial diameters. A single point diamond dresser was used with a water soluble coolant. The experimental design is divided into two parts. The first part, called the ‘‘Fresh Wheel Test’’, is used to build a model relating the process outputs to the operating conditions right after dressing. This is further subdivided into ‘‘Roughing’’ and ‘‘Finishing’’ conditions. As an example, Tables 2 and 3 show the operating ranges and the design of experiments (DOE) for the 4140 (HRC 40)
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Table 2 Range of operating parameters for grinding of 4140 alloy steel (HRC 40) with 32A-60-K-VBE wheel Parameter
Unit
Ds Dw vw vs a Deq ad sd heq vw/vs ts
Roughing conditions
mm mm m/min m/s mm mm mm mm/rev mm – s
Finishing conditions
Low (1)
Mid (2)
High (3)
Low (1)
Mid (2)
High (3)
355.6 38.1 20 28 0.012 34.42 0.025 0.1 0.143 0.0119 0
57.15 25 30.5 0.015 49.24 0.030 0.16 0.205 0.0137 0
76.2 30 33 0.018 62.75 0.036 0.22 0.273 0.0152 0
355.6 38.1 20 28 0.0013 34.42 0.015 0.04 0.0151 0.0119 1
57.15 25 30.5 0.0038 49.24 0.023 0.1 0.0520 0.0137 3
76.2 30 33 0.0064 62.75 0.030 0.16 0.0962 0.0152 5
Table 3 L18 design of experiments for 4140 alloy steel (HRC40) with 32A-60-KVBE wheel—seven factors at three levels (18 runs) Run
Deq
ad
sd
ts
vw
1/vs
a
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 2 1 1
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 2 1.25 2.33
1 2 3 1 2 3 2 3 1 3 1 2 2 3 1 3 1 2 2 2.75 2
1 2 3 2 3 1 1 2 3 3 1 2 3 1 2 2 3 1 2 2.5 1.75
1 2 3 2 3 1 3 1 2 2 3 1 1 2 3 3 1 2 2 1.5 2.5
1 2 3 3 1 2 2 3 1 2 3 1 3 1 2 1 2 3 2 2 2
1 2 3 3 1 2 3 1 2 1 2 3 2 3 1 2 3 1 2 2 2
Note: The test 19, 20, and 21 are verification tests.
workpiece with 32A60KVBE grinding wheel. Separation in the operating ranges for roughing and finishing is determined by the guideline from the machinability handbook and the constraints for grinding power and surface roughness. The first 18 tests are used to fit the model and the remaining three tests are for verification. Seven parameters have been considered in the L18 DOE, namely:
ad, dressing depth of cut , mm sd, dressing lead, mm/rev deq, equivalent diameter given as DsDw/(Ds+Dw), mm vw, surface speed of the workpiece, mm/s vs, surface speed of the grinding wheel, m/s a, depth of cut, mm/rev ts, spark-out time, s
where Ds and Dw are the wheel diameter and the work diameter, respectively. The second part, called the ‘‘Wheel Wear Test’’, is used to determine the rate of wheel wear and its effect on the process outputs. The range of parameters used in these experiments is given in Table 4 for 4140 steel (Rc40) with 32A60KVBE grinding wheel. In addition to these experiments, a set of experiments was also conducted to determine a suitable model for the burn threshold at the two hardness levels. It should be mentioned that based on some observations during the experiments the number of workpiece revolutions during spark-out was used as a control variable instead of the spark-out time to build the surface roughness model. A fast responsive power meter (Load Controls Incorporated) was connected to the wheel head spindle motor to measure the grinding power. The power and tangential force relationship was determined by measuring them directly at the same time with the preliminary calibration test. The setup fixtures for the calibration test were made to mount the compact spindle, a GII DTS1300 spindle, on a dynamometer as shown in Fig. 2. Two special workpieces of ANSI 4140 steel were designed, made and heat treated to hardness of HRC40 and HRC55, respectively. The initial diameters of the workpiece to be ground were around 20 mm. The same grinding wheel of 32A60KBV was used in the tests. Hence, the tangential force can be obtained accurately through the calibrated power measurement, which is much easier than direct force measurement in cylindrical grinding. The surface roughness was measured by a profilometer (Taylor Hobson, Surtronix 3 plus). Radial wear of grinding wheel was measured through the method depicted in Fig. 3. A 8.9 mm (0.35 in) grinding width was used for all the experiments. After removing a certain amount of work material, wheel wear measurements were made by grinding a cutter blade to get a negative impression of the wheel surface profile. For each experimental condition, the wheel profile was measured at a minimum of 6 (roughly) equally spaced intervals with respect to the volume of work material removed.
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Table 4 Design of experiments for wheel wear tests for cylindrical plunge grinding 4140 (HRC 40) with 32A-60-K-VBE wheel Test
Dressing feed
Dressing depth
Wheel speed
Work speed
Infeed
Heq
D_eq
Predicted Initial Power
G*
1/G*
unit High Heq Med Heq Low Heq Verif. 1 Verif. 2
mm/s 3.55 3.55 3.55 3.26 3.40
mm 0.0254 0.0254 0.0254 0.0254 0.0254
rpm 1638 1638 1638 1504 1571
rpm 108 108 108 127 114
mm/s 0.027 0.017 0.007 0.014 0.019
mm 0.208 0.124 0.052 0.106 0.149
mm 59.7 58.7 59.4 56.9 57.4
W 1820 1176 569 943 1315
68 180 401 237 125
0.0148 0.0056 0.0025 0.0042 0.0080
Dwell time ¼ 3 s, grinding width ¼ 8.9 mm.
Fig. 4. Workpiece after the burn experiments.
3. Model development 3.1. Fresh wheel tests Fig. 2. Setup of force calibration experiments.
Specific grinding power model: To find a generalized model form of grinding force, an extensive literature survey was performed as summarized in Table 5. Those models were developed and validated for various machines, grinding wheels, and work materials. Common variables among those models were identified as significant parameters affecting grinding force, which can be summarized as follows:
Fig. 3. Wear measurement of grinding wheel.
The volume of cutter blade material removed was extremely small and is not expected to substantially influence the wear of the wheel. The blade profile was measured using an optical microscope (Zeiss Axioskop). The burn conditions were labeled as ‘‘Burn’’ and ‘‘No Burn’’ based on visual inspection. Fig. 4 shows a picture of one of the workpieces, and it may be seen that visual inspection is sufficient to differentiate between ‘‘Burn’’ and ‘‘No Burn’’.
Single variables: ad, sd, vw, vs, deq, a. Product of variables: a deq, vw a. vw Ratio of variables: . vs vw a Mixed form: heq ¼ . vs
Among these parameters, some of them have been already identified as having physical meanings [17] as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffi a d eq geometrical contact length ðmmÞ 0 Q ¼ vw a material removal rate ðmm=sÞ vw a equivalent chip thickness ðmmÞ heq ¼ vs Ds Dw d eq ¼ equivalent wheel diameter ðmmÞ Ds þ Dw vw speed ratio q¼ vs
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Table 5 Summary of the existing grinding force model for the cylindrical plunge grinding Reference
Grinding wheel
Work material
Peters [22]
EKL7VX
100Cr6
Li [23]
Unknown
Bearing steel (GCr15)
Fielding [24]
Aluminium oxide (WA0LV) Aluminium oxide (BA46Q5V)
EN9 fully hand
Aluminium oxide (32A80L6)
AISI 52100
Younis [25]
Xiao [26]
R2
Grinding force or power model 0:56 vw a0:78 d 0:22 F 0n ¼ 10:34 eq vs 0:33 vw vw a0:665 aþc F 0n ¼ K vs vs F t ¼ 290:83v0:503 ðvw aÞ0:516 s0:244 s d
0.91 1.0 1.0
F 0t ¼ F 0tc þ F 0tr þ F 0tp vw F 0tc ¼ u a k1 ð1 La Þ vs vw F 0tp ¼ l c af ð1 La Þ vs 0 F tr ¼ l La K L 8 9 1 < = vw vs ða d eq Þ2 Aeff b P ¼ P1 vw a þ P2 vs þ P3 þ P4 : ; vw d eq
FN56b
0.89
0.93
Aeff ¼ 0:008A0 lnð1:4 104 mdÞ Werner [27]
EK80L7VX
1:75 d ¼ 1:1 1011 a0:75 d sd 2e1 vw ae d 1e F 0n ¼ K½C 1 g eq vs
Ck45N
0.99
Table 6 Correlation of the parameters with specific tangential force Correlation coeff.
ln(ad)
ln(sd)
ln(a)
ln(vw)
ln(vs)
ln(deq)
ln(vw.a)
ln(a.deq)
ln(vw/vs)
ln(heq)
ln(Ft0 )
0.11
0.44
0.73
0.42
0.37
0.32
0.85
0.72
0.57
0.94
Correlation coefficient is given as absolute value.
The number of parameters should be reduced to have a more simplified and condensed form, which could be done by inspecting the significance of parameters. A statistical analysis was carried out to determine the significance of parameters. The data were generated from the existing models listed in Table 5. The results of correlation analysis are given in Table 6. Among the parameters, the equivalent chip thickness heq can be chosen as the most significant parameter. Next, even though deq shows relatively low correlation and the combination a deq has higher correlation, the single variable deq is chosen to avoid the redundancy in the model parameters. For the dressing parameters, most of existing models have considered the distribution of the static cutting edges to represent the wheel topography, but did not directly explain the relationship with dressing parameters—ad and sd. However, direct consideration of the dressing parameters, ad and sd, will be reasonable for practical purpose rather than any theoretical parameter showing the wheel topography such as number of active cutting edges. Thus, the grinding power model is finally proposed as in Eq. (1). f
f
F 0t;0 ¼ F 0 ad1 sd2 d feq3 hfeq3 ; P00 ¼ F 0t;0 vs ;
(1)
where F0 t,0 and P0 0 are the specific tangential force and specific grinding power right after dressing, respectively. The data were generated from the existing models shown in Table 5 and then the proposed model in Eq. (1) was fitted to check its goodness of fitting. As the performance indicator, R-square values were calculated to determine the fitness of the model against the data generated by six existing models. The proposed model form has shown high R-square values as summarized in the last column of Table 5, thus proving its general applicability. After the model form is determined, the model coefficients for various grinding wheels and materials listed in Table 1 should be obtained. Using the linear least-squares method to fit the experimental data, the model coefficients are obtained and summarized in Table 7 with the goodness of fit. Surface roughness model: A same procedure has been applied to determine the model form for surface roughness. A literature survey was performed and selected representative models are summarized in Table 8. After performing a statistical analysis, the equivalent chip thickness is again adopted as a model parameter. Since the effect of the equivalent diameter Deq is found to be insignificant, it was removed from the model form. However, the work diameter is indirectly considered in the spark-out by
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66
replacing the spark-out time ts with the number of work revolutions during spark-out Ns. The model in Eq. (2) is proposed as a generalized surface roughness model for cylindrical plunge grinding:
Each test consisted of grinding multiple workpieces beginning with a newly dressed wheel under the specified operating condition. The test conditions for 4140 steel (Rc40) with 32A60KVBE grinding wheel are shown in
Table 4, in which the first three were used to fit the model coefficients and the last two were used to verify the model. For other work-wheel combinations, a similar procedure was repeated. The experimental measurement results and identified model forms of these conditions are summarized below. G-ratio model: A 8.9 mm grinding width was used for all the experiments. The grinding tests were conducted with certain intervals for measurements. The total wheel radial wear was over 20 mm for all the conditions and at least six measurements were made for each condition. At each measurement interval, a cutter blade was ground to get a negative impression of the wheel surface profile along the wheel width and the wheel wear is then measured by observing the ground portion of the blade. An example of the measurement is shown in Fig. 5 for the test condition with the highest equivalent chip thickness.
Table 7 Summary of developed generalized grinding power model form and coefficients for cylindrical plunge grinding
Table 9 Summary of developed generalized surface roughness model form and coefficients for cylindrical plunge grinding
Combination
Model Form
Goodness of fit (R2)
Combination
Model form
Generic model form
P00 ¼ F 0 ad 1 sd 2 DFeq3 hFeq4 vs
N/A
Ra0 ¼ R0 ad1 sd2 hreq3 ð1 þ er4 N s Þ
N/A
4140 (HRC40) +32A-60K-VBE 4140 (HRC55) +32A-60K-VBE 8620 (HRC40) +32A-60K-VBE 4140 (HRC55) +94A-80M-V18
0:842 P00 ¼ 4:42a0:095 sd0:336 D0:163 eq heq vs d
0.98
0:2N s Ra0 ¼ 1:64ad0:021 s0:385 h0:284 Þ eq ð1 þ e d
0.93
0:701 P00 ¼ 6:91a0:053 s0:250 D0:097 vs eq heq d d
0.95
0:29N s Ra0 ¼ 1:30ad0:02 s0:205 h0:301 Þ eq ð1 þ e d
0.88
P00 ¼ 5:28a0:222 sd0:313 D0:031 h0:763 eq eq vs d
0.94
0:476N s Ra0 ¼ 2:28ad0:044 s0:250 h0:598 Þ eq ð1 þ e d
0.76
0:940 P00 ¼ 7:40a0:204 sd0:349 D0:028 eq heq vs d
0.95
Generic model form 4140 (HRC40) +32A-60K-VBE 4140 (HRC55) +32A-60K-VBE 8620 (HRC40) +32A-60K-VBE 4140 (HRC55) +94A-80M-V18
0392N s Ra0 ¼ 0:58ad0:243 s0:092 h0:497 Þ eq ð1 þ e d
0.90
r
r
Ra;0 ¼ R0 ad1 sd2 hreq3 ½1 þ expðr4 N s Þ,
(2)
where Ra,0 is the surface roughness right after dressing. The model coefficients for various grinding wheels and materials listed in Table 1 are obtained and summarized in Table 9 with the goodness of fit. 3.2. Wheel wear tests
F
F
r
Goodness of fit (R2) r
Table 8 Summary of the existing surface roughness model for the cylindrical plunge grinding Reference
Grinding wheel
Work material
Lindsay [5, 29]
2A80K4VFMB
AISI 52100
Grinding force or power model
Ra ¼
Snoeys [4]
Chiu [28]
EK60L7X EK60L7VX P60 6A54LSVA2 32A80M6V
4615 AISI 52100
Ko¨nig [30]
EK80JotKe
Riegger NSS 10710
Chen [31]
B56-3
100Cr6 X210CrW12
Si3N4
d 16=27 a19=27 g
ad 16=27 vw ð1 þ Þsd ð Þ16=27 8=27 sd vs Deq ( ) 0:4587T 0:30 for 0oT o0:254 ave ave for 0:254oT ave o2:54 0:7866T 0:72 ave
T ave ¼ 12:5 103
Ra ¼ R1 hreq
R2 0.98
1.0
Rda ¼ Rd1 hrd eq ðwith spakoutÞ 0:25 m Ra ¼ R1 s0:5 d ad heq 1 þ expð0:77tsp =tÞ ð1:02d g þ 0:76Þ 0:32 0 Q0 0:46 Rz ¼ 0:25s0:006 w ðV w þ 8:3Þ d
þ32:2s0:96 Q0 0:14 ðV 0w w d Ra ¼ g am
þ 16:4Þ
vw pffiffiffiffiffiffiffiffiffiffiffiffiffi am ¼ w a d eq vs 2
1.0 0.98
0:29
0.92
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As shown in Fig. 5, the worn wheel surface (left side) can be easily distinguished and measured using the unworn surface as reference (right side). The wheel radial wear values were obtained by averaging the measured radial wear along the wheel plunge width, and plotted as a function of work material removal (Fig. 6). In every case, there is initial transient rapid wear prior to a steady-state wear regime at a lower wear rate. Thus the steady-state G-ratio can be obtained by curve-fitting linear relationships in the volume–volume plane as shown in Fig. 6. The steady-state wear regime starting from the end of the transient after an accumulated material removal of workpiece Vw_st. The steady-state wheel wear can be characterized by a straight line with slope of 1/G*, which represent the reciprocal of the steady-state G-ratio, Gn ¼
DV w ; DV s
where V w 4V w_st .
(3)
The intercept wheel wear volume, Vs0, can be considered to represent that portion of the radial wear associated with the initial transient. The intercept, Vs0, is independent of grinding conditions, while the steady-state G-ratio is a strong function of equivalent chip thickness. Thus the constant Vs0 and 1/G*’s of the test conditions were obtained by linear least-squares fitting of the data in
67
Fig. 7. The results for the intercept (equal to 36.0) and the steady-state G-ratios are plotted in Fig. 7 versus equivalent chip thickness, Heq, which can be represented by Gn ¼ 10:71h1:254 . eq
(4)
As shown in Fig. 7, the steady G-ratios can be well fitted using a log-linear model with Heq. The total steady-state wheel volume can be represented as Vs ¼
Vw Vw þ 36:0; n þ V s0 ¼ 1:254 G 10:71heq
where V w 4V w_st , (5)
where Vw_st is a small initial value during which the transient wheel wear occurs. Specific grinding power model: The grinding power measurements are plotted in Fig. 8 as a function of accumulated work material removal per unit plunge width (called the specific work removal from now on and has the unit of mm3/mm). The power data from all the tests shown in Fig. 8 exhibit the same trends with an initial transient region (generally decreasing), followed by a steady-state increase at a constant rate. To take into account the initial drop in power, a constant correction factor l is introduced as shown in Eq. (6).
Fig. 5. Measured wheel surfaces profiles along the wheel width (The legend shows the total work material removal in diameter).
Fig. 7. Steady-state G-ratio.
Fig. 6. Volumetric wheel wear vs. workpiece removal.
Fig. 8. Grinding power vs. specific work material removal.
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Let P0 0 represent the prediction of the power model just after dressing as specified in Eq. (1). Assume the power of the steady state has the following form: b
P0 ¼ lP00 ð1 þ aV 0 w Þ,
(6)
where a represents the power growth rate (and could be a function of the equivalent chip thickness), b denotes the non-linear growth rate and V0 w represents the specific work material removal. Using a linear least-square fitting method, the specific grinding power model coefficients are obtained. The growth rate a is plotted in Fig. 9 against equivalent chip thickness. It can be seen that a is approximately constant with an average value of 8 105. Therefore the coefficients are obtained as P0 ¼ 0:9P00 ð1 þ 8:0 105 V 0w Þ.
(7)
And the predictions using Eq. (7) are shown in Fig. 10 in comparison with corresponding experimental data. Although the coefficient a was found to be constant over the range of Heq considered in this set of experiments, in general, it may be a function of Heq. Hence, the generic model considers a as a function of the equivalent chip thickness. Surface roughness model: The surface roughness measurements are plotted in Fig. 11 as a function of specific work material removal. A constant spark-out time of 3 s
Fig. 11. Surface Roughness vs. specific work material removal.
was used for all the experiments. The roughness values for all the tests in Fig. 11 show similar trends with an initial rapid increase after which the growth rate is lower. Let Ra0 represent the prediction of the roughness model just after dressing as specified in Eq. (2). The variation of surface roughness with wheel wear is slightly more complex as compared to the power and hence the model form for surface roughness has to be flexible enough to explain these variations. Before stating the model form, some observations about the roughness variation are presented below. These observations are drawn from all the experiments from plunge grinding and cylindrical grinding. 1. For a given condition, surface roughness generally reaches a steady value as the specific work removal increases. 2. For the same value of the equivalent chip thickness, the roughness can show increasing or decreasing trends based on the initial dressing conditions. 3. The steady-state value seems to be independent of the dressing conditions. These observations suggest the following:
Fig. 9. Variation of a with equivalent chip thickness for specific grinding power (HRC 40).
Fig. 10. Specific power prediction for cylindrical plunge grinding (HRC 40).
1. The initial roughness is dependent on all the operating conditions as well as the dressing conditions. 2. The final roughness is dependent only on the operating conditions and does not depend on the dressing conditions. An example of the experiments conducted to validate the above assumptions is shown in Fig. 12. The experimental conditions are given in Table 10 and are considered as additional validation experiments. It may be seen that the only difference between the two experiments is the dressing condition. The aim of these tests was to observe both increasing and decreasing trends for the same equivalent chip thickness and at the same time verify if these two initial conditions resulted in the same steady-state roughness values. The observations confirm that both increasing and decreasing trends can be observed for the same
ARTICLE IN PRESS T.J. Choi et al. / International Journal of Machine Tools & Manufacture 48 (2008) 61–72
equivalent chip thickness and that they both converge to the same steady-state values (around 0.55 mm in this case). Based on the above observations, the following generalized model is proposed: 0
Ra ¼ Ra0 þ ðRa1 Ra0 Þð1 eaV w Þ, Ra0 ¼ f 1 ðdressing conditions; operating conditionsÞ, Ra1 ¼ f 2 ðoperating conditionsÞ.
ð8Þ
The above model is general enough to hold for all grinding processes. The exact dressing parameters and operating parameters will vary from process to process. Given below is the model form for cylindrical plunge grinding with a single point dresser. R
69
Fig. 13 shows a result of fitting a model of the form shown in (9) to the steady-state roughness values as measured after five spark-out revolutions. The model prediction is acceptable. It may also be noted that the model successfully predicts increasing and decreasing trends for the two additional verification conditions as shown in Fig. 12. The final model (with coefficients obtained using linear least squares) for the roughness model is given below. 0:2N s s0:385 h0:284 Þ; Ra0 ¼ 1:64a0:021 eq ð1 þ e d d 0:0624N s Ra1 ¼ 3:73h0:716 Þ; eq ð1 þ e;
Ra ¼ Ra0 þ ðRa1 Ra0 Þð1 e
0:00085V 0w
(10) Þ:
R
R4 N s 3 Ra0 ¼ R0 ad 1 sd 2 hR Þ; eq ð1 þ e
Ra1 ¼ r0 hreq1 ð1 þ er2 N s Þ;
(9)
And the predictions using Eq. (10) are shown in Fig. 14.
0
Ra ¼ Ra0 þ ðRa1 Ra0 Þð1 eaV w Þ: The advantage of using the above model form is that it has the same number of free parameters as a power model of the form used for predicting power, but it is more flexible as it separates the effect of the initial and final values and hence can incorporate observations like those shown in Fig. 12. Moreover, although the above model seems complicated, the curve-fitting process is decoupled and hence all the parameters may be obtained by repeated use of a simple linear least-squares fit. Since the model for Ra0 (the initial roughness) was already developed during the fresh wheel tests, only the steady-state roughness model and the growth rate parameter (alpha) have to be determined from the wear data.
Fig. 13. Variation of RaN (steady state roughness) with Heq.
Fig. 12. Variation of surface roughness with wheel wear for the same value of Heq with different dressing conditions.
Fig. 14. Surface roughness predictions for cylindrical plunge grinding.
Table 10 Experimental conditions for additional experiments conducted to validate roughness model form Test
Dressing feed
Dressing depth
Wheel speed
Work speed
Infeed
Dwell time
1 2
mm/rev 0.22 0.10
mm 0.0254 0.0125
rpm 1772 1772
rpm 114 114
mm/min 0.203 0.203
sec 3 3
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3.3. Burn tests For the burn threshold, the model proposed by Malkin et al. [12]. is adopted as given in Eq. (11). It was claimed that the combination of operating parameters shown in Eq. (11) clearly distinguishes between burn-free condition and burn condition in terms of specific grinding energy, u (J/mm3). Based on their observation, the intercept value u0 is obtained as 6.2 J/mm3 for alloy steels. u
n
0:75 0:5 ¼ u0 þ Bd 0:25 vw eq a n 0:25 0:5 ) P ¼ u0 vw a þ Bd 0:25 vw . eq a
determining the burn threshold using a linear support vector machine classifier [32], which is a supervised learning method used for regression and classification. The conditions were labeled as ‘‘Burn’’ and ‘‘No Burn’’ based on visual inspection. Based on the above plots, the following equation is recommended for the burn threshold: 0:75 0:5 un ¼ 6:2 þ 10:192d 0:25 vw eq a 0:5 ) Pn ¼ 6:2vw a þ 10:192d 0:25 eq vw .
ð12Þ
ð11Þ 4. Model summary for various wheels and work materials
The operating ranges for determining burn threshold are shown in Table 11. Although a large number of experiments were conducted in order to verify the model form at different diameters, it is not necessary to use the same number of experiments for every wheel/workpiece combination. During the experiments, the workspeed, infeed rate, dressing lead, dressing depth and the work diameter were varied to get different points as shown in Fig. 15. This data set was augmented with the data obtained during the ‘‘Fresh Wheel’’ finishing experiments which were all labeled as ‘‘No Burn’’ conditions. Fig. 15 shows the result of
Table 11 Range of operating parameters for burn test (4140 Rc40) with 32A-60-KVBE wheel Parameter
Operating range Low
Wheel diameter Work diameter Dressing feed Dressing depth Wheel speed Work speed Infeed Heq Grinding width
A similar set of experiments have been conducted to determine the model sets for the different grinding wheel and work material as summarized in Table 1. The model forms and the specific models with coefficient are summarized in Table 12–15 with the goodness of the fit. In the model forms like a[heq] denotes that a is a function of heq. The surface roughness model for the 4140 RC 40 workpiece model has been modified to use the newly proposed number of spark-out revolutions parameter, Ns.
Table 12 Summary of developed generalized wear model form of grinding power and coefficients for cylindrical plunge grinding
Unit Combination
Model form
Goodness of fit (R2)
Generic model form
P0 ¼ lP00 ð1 þ a½heq V 0 bw Þ
N/A
4140 (HRC40) +32A-60K-VBE 4140 (HRC55) +32A-60K-VBE 8620 (HRC40) +32A-60K-VBE
P0 ¼ 0:9P00 ð1 þ 8:0 105 V 0w Þ
0.86
P0 ¼ 0:95P00 ð1 þ 16:1 105 V 0w Þ
0.85
0 0 P0 ¼ 0:95h0:050 eq P0 ð1 þ aV w Þ
0.71
High 343
33 0.03 0.008
55 0.12 0.023 32
91 0.019 0.065
1167 0.051 0.231 6.35
mm mm mm/rev mm m/s mm/s mm/s mm mm
a¼
D2:418 0:51h0:470 eq eq
Table 13 Summary of developed generalized wear model form of surface roughness and coefficients for cylindrical plunge grinding
Fig. 15. Experimental data for burn and burn threshold determined using linear SVM.
Combination
Model form
Goodness of fit (R2)
Generic model form
Ra ¼ Ra0 þ ðRa1 Ra0 Þð1 eaV w Þ Ra1 ¼ r0 hreq1 ð1 þ er2 N s Þ
4140 (HRC40) +32A-60K-VBE
Ra ¼ Ra0 þ ðRa1 Ra0 Þð1 e0:00085V w Þ 0:0624N s Ra1 ¼ 3:73h0:716 Þ eq ð1 þ e
4140 (HRC55) +32A-60K-VBE
Ra ¼ Ra0 þ ðRa1 Ra0 Þð1 e0:00095V w Þ
8620 (HRC40) +32A-60K-VBE
Ra ¼ Ra0 þ ðRa1 Ra0 Þð1 e0:0018V w Þ
0
Ra1 ¼
3:47h0:689 eq ð1
þe
0:0596N s
N/A 0.87
0
0.92
Þ 0
0:075N s Ra1 ¼ 3:22h0:676 Þ eq ð1 þ e
0
0.72
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Model form
Goodness of fit (R2)
Generic model form
1 G n ¼ G0 hG eq
N/A
4140 (HRC40) +32A-60K-VBE
G ¼
4140 (HRC55) +32A-60K-VBE
n
G ¼
8620 (HRC40) +32A-60K-VBE
n
n
G ¼
10:71h1:254 eq 9:48h1:051 eq 7:96h1:262 eq
0.96 0.97 0.96
Table 15 Summary of burn threshold model forms and coefficients Combination
Model form
Generic model form
0:25 0:5 vw P0 n ¼ B0 vw a þ B1 d 0:25 eq a
4140 (HRC40) +32A-60K-VBE
0:25 0:5 P0 n ¼ 6:2vw a þ 11:42d 0:25 vw eq a
4140 (HRC55) +32A-60K-VBE
0:25 0:5 vw P0 n ¼ 6:2vw a þ 10:19d 0:25 eq a
8620 (HRC40) +32A-60K-VBE
0:25 0:5 P0 n ¼ 6:2vw a þ 9:93d 0:25 vw eq a
5. Conclusions Simple and easy-to-use generalized grinding process models for cylindrical plunge grinding processes have been developed and presented in this paper. The major advantage of using these generalized models is that these models maintain the same functional forms regardless of different workpiece–wheel combinations used and hence can be readily applied to various grinding set-ups by determining their coefficients through a small number of designed experiments. The surface and power models were presented, including the terms to account for the effect of wheel wear. It has been shown that these models can predict process conditions over a wide range of grinding conditions. Furthermore, it has also been shown that steady-state surface roughness and steady-state G-ratio are primarily dependent only on the effective chip thickness. These models can be used for designing process operating conditions or process optimization without extensive experimental efforts.
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