Correlation between shape errors of the tool and ceramic work-pieces are presented in the paper. A new tool, on which different shape errors of convexity or concavity can be set, was used during experiments. Results from flat grinding, such as the shape errors, are presented and analyzed. Computational calculations concerning the local shape errors of the tool and the technological effects such as surface roughness and waviness parameters as well as the workpiece plane-parallelism are also presented

M. Deja

English

Journal of Vibroengineering

  
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 2012. VOLUME 14, ISSUE 2. ISSN 1392-8716 
520 
776. Correlation between shape errors in flat grinding  
  
Mariusz Deja  
Department of Manufacturing Engineering and Automation, Mechanical Engineering Faculty  
Gdansk University of Technology, 11/12 Narutowicza Str., 80-233 Gdansk, Poland  
E-mail: mdeja@pg.gda.pl   
 (Received 8 September 2011; accepted 14 May 2012) 
 
Abstract. Correlation between shape errors of the tool and ceramic work-pieces are presented in 
the paper. A new tool, on which different shape errors of convexity or concavity can be set, was 
used during experiments. Results from flat grinding, such as the shape errors, are presented and 
analyzed. Computational calculations concerning the local shape errors of the tool and the 
technological effects such as surface roughness and waviness parameters as well as the work-
piece plane-parallelism are also presented.  
  
Keywords: flat grinding, shape errors, ceramics.  
  
Introduction  
  
Grinding of ceramics is widely used for reducing the workpiece roughness from preceding 
processes and can be carried out with different tool-workpiece configurations. Lapping is the 
basic flattening process which allows for achieving a high degree of flatness and parallelism of 
machined parts. It is still one of the most common processes to flatten e.g. the sliced silicon 
wafers [7]. Apart from lapping also grinding operations are carried out for parts flattening with 
two basic tool-workpiece configurations: single side grinding (SSG) and simultaneous double 
side grinding (SDSG) [13].  These processes have some advantages comparing with lapping e.g. 
lower cost and higher throughput although there is still a problem with effective waviness 
removal by SSG and with a dimple in the center of the wafer after SDSG [13]. Most suitable for 
processing the workpieces with coplanar functional areas are manufacturing technologies which 
allow for both areas to be processed at the same time. Grinding (flat honing, fine grinding) with 
the use of grinding wheels mounted on a double-disk lapping machine is widely used in practice 
[1, 5, 12]. Special machines for high  speed grinding are available now [2]. Single-disk machines 
can be also used for single side grinding [4] but its conventional configuration needs the 
modification - the leading ring rotational velocity should be higher and set independently unlike 
on conventional machines, where the conditioning (leading) rings rotate with workpieces 
without any additional drive. When grinding wheel is used on a lapping machine, dressing 
influences the grinding process significantly, depending on how it is done [11]. 
Grinding with lapping kinematics ensures a very good flatness due to the fact that the active 
surface of the tool is simultaneously the machining datum surface. In consequence of such 
configuration, there is a very close correlation between the tool and the machined part flatness 
errors. During the process both surfaces: active surface of the tool and the surface of the 
workpiece change the profile shape due to the abrasive wear. The active surface of the tool 
usually is not flat and can be either concave or convex. This exerts the main influence on the 
shape accuracy of the workpieces. Kinematical analysis of grinding marks was used in a 
mathematical model for the wafer shape prediction proposed by Wangping Sun et al. [13]. 
Influence of a double-side lapping kinematics on the face grinding was studied by Uhlmann and 
Ardelt [12]. The proposed and used in this paper model for the theoretical shape error estimation 
[7] is also based on the kinematical relations and additionally on local shape errors of the tool. 
This kind of relation must be considered when the tool surface is simultaneously the machining 
(fixturing) datum.  
Results of grinding of ceramics flat pieces when submitted to single-disc lapping kinematics 
are the subject of this paper. The grinding tools were electroplated with diamond grains D64 and 
D107. Different tool profiles were set for each tool to check the correlation between the shape 
  
776. CORRELATION BETWEEN SHAPE ERRORS IN FLAT GRINDING.   
M. DEJA 
  
  
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 2012. VOLUME 14, ISSUE 2. ISSN 1392-8716 
521 
 
errors. Results from the computational simulations concerning the local shape errors of the tool 
are presented to predict the correlation between shape errors in flat grinding.  Technological 
effects such as surface roughness and waviness parameters as well as the workpiece plane-
parallelism were also checked during experiments. 
 
Experimental procedure  
  
Experiments were carried out on a modified single-disk lapping machine with the 
independent drive  of a leading ring realised with the use of a stepper motor [3]. Placed in a 
separator ceramic workpieces were loaded against a rotating tool and rotated together with a 
leading ring. The rotational velocity of the tool was set to nt = 63 rpm and the rotational velocity 
of a ring was set to n2 = 120 rpm in the opposite direction of the tool rotation. Cutting fluid 
(based on kerosene and machine oil) was introduced drop by drop into the contact region 
between the workpieces and the diamond electroplated tool. To check the correlation between 
the tool and the workpiece shape errors, tool radial profiles were measured along eight paths on 
the CNC measuring machine as well as with the use of a digital indicator - Fig. 1 [6]. The design 
of the tool allows for setting required profiles and is a subject for the polish patent law. After the 
measurements of two tools, the average profiles were calculated (Fig. 2) and used for numerical 
analysis. The tool #1 was covered with diamond grains D107, and the tool #2 with grains D64.    
    
 
Fig. 1. Measurement of the tool shape 
  
Computational results 
 
Numerical simulations are based on the developed model for the shape error estimation 
during machining with flat lapping kinematics [7, 8, 10].  During the simulations, the theoretical 
distance  between the workpiece and the tool is calculated [9]. The analyzed tool shapes, set and 
measured before experiments (Fig. 2) and for which the calculations were carried out are 
presented in Fig. 3. The workpiece area is divided into elementary areas and the orientation of an 
adjacent plane containing the workpiece is found. This orientation depends not only on the shape 
error of the lap but also on the shape and size of the workpiece [7]. During the computational 
calculations the active tool surface is limited to the area containing the analytical points for each 
discrete location of the workpiece. The movement of the circular workpiece is analyzed and 
some its discrete locations are shown in Fig. 3. 
  
776. CORRELATION BETWEEN SHAPE ERRORS IN FLAT GRINDING.   
M. DEJA 
  
  
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 2012. VOLUME 14, ISSUE 2. ISSN 1392-8716 
522 
 
0
0,2
0,4
0,6
0,8
1
1,2
0 20 40 60 80 100 120 140 160 180 200
tool radius [mm]
z 
[m
m
]
Tool #1
Tool #2
 
Fig. 2. Measured and analyzed tool profiles  
 
 
Fig. 3. Workpiece orientation on the tool surface: 
a) tool #1 - one discrete location, b) tool #2 - three workpiece discrete locations 
 
The average distance (during the cycle time Tz) between the elementary work-piece area A1i 
and the tool surface is the main variable in a computer model developed for the estimation of the 
work-piece shape [7]. Distributions of the distance d2 for a circular work-piece (φ34) and for 
analyzed tool shapes (Fig. 3a, b) are shown in Fig. 4. The average distribution along the work-
piece radius is presented in Fig. 4c, d. The results indicates that the work-piece shape errors 
should be higher when the tool #2 (Fig. 2 and Fig. 3b) is applied.   
Grinding results 
 
Ceramic work-pieces were pre-grounded before experiments with the initial average surface 
roughness parameters: Ra = 1,96 µm, Rz = 22,7 µm, measured within the central area of the 
radius R = 6 mm, with the initial and final biggest shape errors. The average initial plane-
parallelism was Tp = 0.062 µm. Surface roughness and waviness parameters were measured with 
the profilometers Hommel Tester T500 and ME10 on each sample after each test along three 
paths. Additionally these parameters were measured and the work-piece surface was inspected 
with the FRT non-contact measuring instrument (from Fries Research & Technology Company, 
Germany) - Fig. 5. Plane-parallelism was checked with a micrometer of sensitivity 1 µm (from 
Mitutoyo Company, Japan, Kawasaki). The height of the samples ∆hw reduced by grinding was 
measured individually after each test.  
The first three tests were carried out on ceramics work-pieces (Al2O3) with the use of a 
tool #1 with diamond grains D107 and with the radial profile presented in Fig. 2 and in Fig. 3a. 
The required velocity of the tool (nt = 62 rpm) and of the work-piece holder (n2 = 120 rpm) as 
  
776. CORRELATION BETWEEN SHAPE ERRORS IN FLAT GRINDING.   
M. DEJA 
  
  
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 2012. VOLUME 14, ISSUE 2. ISSN 1392-8716 
523 
 
well as the load on the contact were set (p = 2.7 kPa), and tests (T1 ÷ T3) were run for total 
machining time of 3 minutes for each test. Every three minutes, five work-pieces were removed 
from the separator and, after cleaning, the plane-parallelism, surface roughness and waviness 
parameters were measured. Results show the reduction in the shape errors from the initial state 
(Fig. 6, Fig. 7) to the final one (Fig. 7 and Fig. 8) for each work-piece. 
 
 
Fig. 4. Distribution of the distance d2 between the work-piece and the active surface of the: a) tool #1,  
b) tool #2, c) average radial distribution for the tool #1, d) average radial distribution for the tool #2 
 
   
a)                                                                                      b) 
Fig. 5. Machined ceramic workpiece: a) analyzed total surface b) filtered image with porosity 
  
Next tests (T4 ÷ T6) were carried out also on ceramics work-pieces (Al2O3) with the use of 
the electroplated diamond tool #2 with grains D64 and with the radial profile presented in Fig. 2 
and in Fig. 3b. The required velocity of the tool (nt = 62 rpm) and of the work-piece holder      
  
776. CORRELATION BETWEEN SHAPE ERRORS IN FLAT GRINDING.   
M. DEJA 
  
  
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 2012. VOLUME 14, ISSUE 2. ISSN 1392-8716 
524 
 
(n2 = 120 rpm) as well as the load on the contact (p = 2.7 kPa) were set. The first test (T4) was 
run for total machining time of 3 minutes and next tests T5 and T6 for 2 minutes. The machining 
time was reduced in order to minimise the wear of a single-layer electroplated diamond tool 
which was higher comparing to the previous set of tests, because of smaller grains - D64. 
Results show the increase in the shape errors from the initial state (after the test T3) to the final 
one (Fig. 9 and Fig. 10) for each work-pieces. That was expected after the numerical 
calculations (Fig. 4), which indicated the higher work-piece shape errors for the tool #2 applied. 
   
 
Fig. 6. Exemplary results from the measurement of the initial state of a ceramic sample 
 
0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
0,16
W1 W2 W3 W4 W5
Workpiece #
[m
m
] Before
After
 
Fig. 7. Plane-parallelism errors recorded after grinding tests T1 ÷ T3 
  
776. CORRELATION BETWEEN SHAPE ERRORS IN FLAT GRINDING.   
M. DEJA 
  
  
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 2012. VOLUME 14, ISSUE 2. ISSN 1392-8716 
525 
 
 
               a)           b) 
Fig. 8. Exemplary results from the roughness and waviness measurements of the ceramic samples after 
grinding tests T3: a) workpiece #4, b) workpiece #5 
 
0
0,02
0,04
0,06
0,08
0,1
0,12
W1' W2' W3' W4' W5'
Workpiece #
[m
m
] Before
After
 
Fig. 9. Plane-parallelism errors recorded after grinding tests T4 ÷ T6 
 
Surface roughness and waviness parameters achieved after grinding tests and measured 
within the central area of the radius R = 6 mm, with the initial and final biggest shape errors, are 
presented in Fig. 11. All parameters are below the initial values but the change in the grain size 
from D107 to D64 does not influence considerably the surface finish - Ra, Rz, Rt. The increase 
in the waviness parameters (Wt, Wa), which are also below the initial values as it is seen from 
Fig. 11, is caused by the bigger tool shape error set for the tool #2. This shape results in higher 
values of the distance between the work-piece elementary areas and the active tool surface what 
is seen in Fig. 4.   
 
Conclusions 
 
The results of experiments enable formulation of the following conclusions: 
(i) There is a close relationship between shape errors of the tool and the work-pieces in flat 
grinding with lapping kinematics. As expected from computational calculations, higher 
workpiece shape errors were induced by grinding with the tool of bigger shape error.   
  
776. CORRELATION BETWEEN SHAPE ERRORS IN FLAT GRINDING.   
M. DEJA 
  
  
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 2012. VOLUME 14, ISSUE 2. ISSN 1392-8716 
526 
 
(ii) Surface finish was improved by flat grinding with the roughness and waviness parameters 
lower than the initial values, although the grain size from D64 to D107 does not influence 
considerably the surface finish - Ra, Rz, Rt. The increase in the waviness parameters (Wt, Wa), 
which are also below the initial values, is caused by the bigger tool shape error set for the tool 
#2. This shape results in higher values of the distance between the work-piece elementary areas 
and the active tool surface. 
(iii) Plane-parallelism was also improved by single-side flat grinding with the correlation to the 
tool shape error.  
(iv) Presented in the computer simulations assumptions could be included in a more 
sophisticated model to predict the work-piece shape machined with grinding or lapping 
configurations. 
 
 
a)                                                                        b) 
Fig. 10. Exemplary results from the roughness and waviness measurements of the ceramic samples after 
grinding tests T6: a) workpiece #1', b) workpiece #2' 
 
 
Fig. 11. Average surface roughness and waviness parameters achieved after grinding experiments 
  
776. CORRELATION BETWEEN SHAPE ERRORS IN FLAT GRINDING.   
M. DEJA 
  
  
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 2012. VOLUME 14, ISSUE 2. ISSN 1392-8716 
527 
 
 
Acknowledgments 
 
Scientific work financed from Polish Budget Funds for Science for the years 2010-2011 as a 
Research Project N N503 157638.  
The author would like to thank Dr. Jens König, Tom Hoghé and Patrick John from IWF, 
Technische Universität Berlin for the possibility to use the FRT non-contact measuring 
instrument. Computations were carried out with the use of software and computers from 
Academic Computer Centre in Gdansk - TASK (http://www.task.gda.pl). 
 
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Correlation between shape errors in flat grinding

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Correlation between shape errors in flat grinding

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