EXPERIMENTAL TECHNIQUE TO MEASURE DEFLECTIONS AND
STIFFNESS IN SURFACE GRINDING
R. Bauer
Associate Professor
S. Lin
A. Warkentin
Graduate Student Associate Professor
Department of Mechanical Engineering, Dalhousie University, PO Box 1000, Halifax, Canada,
B3J 2X4
ABSTRACT
Deflections in surface grinding can lead to geometrical inaccuracies in the components being ground and can also
limit production rates in the grinding process. Such deflections depend on the stiffness of the grinding wheel,
workpiece, and machine itself due to the flexibility of the spindle and deformations of mechanical contacts inside
the machine. In this paper an experimental technique is developed and tested to indirectly measure the overall
system deflections in the surface grinding process by comparing the actual mass removed during a grinding pass
to the mass that would have been removed if there were no deflections. The relationships between the spindle
power, normal force and grinding deflections are then derived to provide estimates of individual grinding wheel,
workpiece and grinding machine deflections. Using this experimental technique, deflection and stiffness
estimates were obtained for a Blohm Planomat 408 CNC grinding machine in the grinding research laboratory at
Dalhousie University. The modulus of elasticity used in the grinding wheel stiffness calculations was determined
from natural frequency vibration measurements of the wheel. The results also showed that the relationship
between the spindle power and the deflections is nonlinear due to the presence of backlash in the spindle lead
screw.
Introduction:
Grinding is a material removal process in which the cutting medium consists of hard abrasive particles contained
in a bonded grinding wheel. Being a major manufacturing process, grinding accounts for about 20-25% of the
total expenditures on machining operations in industrialized countries [1]. Grinding is often used as a finishing
machining process which requires smooth surfaces and fine tolerances. Figure 1 shows a typical surface grinder
where the workpiece is mounted on a horizontal worktable that reciprocates past the grinding wheel. This
movement, called table speed, is usually very high in surface grinding. To achieve the cutting action, the grinding
wheel is fed down into the workpiece by an infeed motion. The typical infeed depth for surface grinding is less
than 13 µm. The transverse movement of the wheel is called crossfeed and it is usually ¼ of the width of the
wheel per pass [2].
In surface grinding, the actual depth of cut is usually less than the commanded depth of cut for a single pass.
This difference is caused by deflections in the workpiece-wheel-machine system due to the normal force
developed in the grinding process. Such deflections can be large in comparison to the accuracy required in the
grinding process [3]. Static deflections, therefore, can affect the geometrical accuracies of ground workpieces [4].
Static deflections can also influence the productivity of the grinding process because several spark-out passes
are often required to recover the deflections. A typical spark-out phase requires the grinding wheel to repeatedly
pass over the workpiece without any further infeed. Not only can this spark-out stage be very time consuming,
but the proper number of spark-out passes required to achieve the appropriate surface finish and geometric
accuracy for a given setup is often difficult to determine.
Figure 2 illustrates the static deflections of a surface grinding system. Such deflections are due to the following
sources: deflections in the grinding wheel εs, workpiece εw, and grinding machine itself εm due to the flexibility of
the spindle and deformations of mechanical contacts inside the machine [4]. Starting from the right-hand side of
Figure 2, the total static deflection ε can be written as:
ε = εm + εw + εs
(1)
Infeed
Table Speed
Grinding Wheel
Crossfeed
Workpiece
Worktable
Figure 1: Surface grinder with horizontal spindle and reciprocating table [5]
εw
Wheel
h
εm
Wheel
Workpiece
εs
a
Figure 2: Static deflections in surface grinding [4]
Various techniques have been used to measure static deflections in the grinding process. Hucker et al. [4] and
Brown et al. [6] used a profilometric trace taken perpendicular to the direction of grinding to determine the actual
depth of cut a of a grinding pass. Once the actual depth of cut is known, as shown in Figure 2, the total static
deflection ε can be calculated from the difference between the commanded depth of cut h and the actual depth of
cut a:
ε=h–a
(2)
Dhawan et al. [7] used dial gauges with a least count of 1µm suitably fixed on a grinding machine to measure the
grinding deflections in any given pass. Thomas et al. [3] and Trmal [8] used a diameter gauge to determine the
grinding deflections in the cylindrical grinding process. Trmal concluded that the power consumed by the
cylindrical grinding process can be used to accurately assess machine deflections more conveniently than
grinding force.
To simplify the measurement of the actual depth of cut a in the surface grinding process and, therefore, indirectly
measure the overall system stiffness and corresponding static deflections, this paper proposes the use of an
electronic balance. By measuring the mass of the workpiece before and after the grinding pass, the total volume
of material removed can be calculated. For a rectangular-shaped workpiece, it is then relatively straightforward to
calculate the actual depth of cut a by dividing the total volume removed by the length and width of cut. Equation
(2) can then be employed to calculate the total static deflection. This experimental technique is then used in this
paper to determine the relationships between the static deflections, spindle power and normal force for the Blohm
Planomat 408 grinding machine in the Grinding Research Lab at Dalhousie University.
Assessment of Static Deflections using Spindle Power and Normal Force
Deflections of the machine, grinding wheel and workpiece are directly caused by the grinding forces developed
during the grinding process. In surface grinding, the total force vector exerted by the workpiece against the wheel
can be separated into a tangential and normal component as shown in Figure 3.
vs
Spindle
Wheel
vw
Fn
Workpiece
Ft
ε
h
Worktable
Figure 3: Illustration of force components for surface grinding
For this grinding setup, the overall system stiffness ke can be modeled by
ke = Fn / ε
(3)
where Fn is the normal force and ε is the overall static deflection [1]. The spindle power P can be related to the
tangential cutting force as follows:
P = Ft (vs ± vw)
(4)
where P is the spindle power, Ft is the tangential force, vs is the wheel speed, and vw is the work speed. The plus
sign in Equation (4) is used for up-grinding where the wheel and workpiece velocities are in opposite directions at
the grinding zone, while the minus sign is for down grinding where both velocities are in the same direction.
Because vw is usually much smaller than vs, Equation (4) can often be simplified to:
P = Ft vs
(5)
Malkin [1] models a linear relationship between the normal force and the tangential force, which can be written as
Ft = µFn + F0
(6)
where µ is the grinding coefficient of friction or force-ratio, and F0 is a constant.
Combining Equations (3), (5) and (6) gives the following linear relationship between the spindle power and the
deflections in surface grinding:
(7)
P = µkevsε + P0
where P0 is a constant.
Research by Peters et al. [9] indicate that, for most wheel-material-coolant combinations, the force-ratio µ does
not change with the material removal rate, and the overall system stiffness ke is constant for a particular grinding
set-up. Given a known wheel speed vs, therefore, the measurement of the spindle power P could be used to
estimate the static deflections ε from Equation (7).
The following section describes the experimental setup and grinding parameters used to measure static
deflections, spindle power and grinding forces and determine if the above theoretical relationships can be applied
to the Blohm Planomat 408 surface grinding machine.
Experimental Setup
Grinding experiments were carried out on a Blohm Planomat 408 grinding machine. This machine is a Computer
Numerically Controlled (CNC) grinding machine and was set up to measure both spindle power (by a Load
Controls Inc. PH-3A power cell) and grinding forces (by a Kistler 9257B force transducer). The experimental
conditions are summarized in Table 1.
Table 1: Experimental parameters
Blohm Planomat 408
Infeed Control
CNC, up-grinding
Measurement Instruments
Power sensor and force transducer
Grinding Wheel
Aluminum Oxide WA46J Vitrified
Diameter × Thickness
335.4 mm × 25.4 mm
Surface Speed vs
30.5 m/s
Dressing Feed Rate vd
0.31 m/min
Dressing Infeed ad
20.32 µm
Dressing Lead sd
0.18 mm/rev
Coolant
Klenn-Kool 777 Synthetic Cutting & Grinding Fluid
Flow Rate
10.5 L/min
Brix#
2.4
Workpiece
1020 Low Carbon Steel density = 7841.8 kg/m3)
Length of Cut
166.6 mm
Width of Cut b
25.4 mm
Resolution
0.03 µm
Work speed
12.7 m/min
Commanded Depth of Cut
5 µm, 10 µm, 15µm, 20 µm, 25 µm, 30 µm
Up-grinding experiments were conducted with different commanded depths of cut while all other parameters,
including wheel speed, wheel dressing parameters, work speed and coolant, remained the same. There were six
commanded depths of cut as shown in Table 1. Because of the stochastic nature of the grinding process, ten
repeated experiments were carried out for each commanded depth of cut and the results were then averaged.
For each experiment, there was only one up-grinding pass of the wheel on the workpiece. For each depth of cut,
the grinding wheel was trued and dressed using a 1-Carat single-point diamond dressing tool to minimize the
effects of wheel wear on the experiments.
Initially, all six sides of the rectangular-shaped 1020 low carbon steel workpiece were ground flat, making it
straightforward to compute the workpiece volume from length, width and height measurements. The workpiece
was then weighed and its density was calculated. Before every experiment, the workpiece was prepared first by
sparking out the surface to ensure that there was no residual material remaining on the ground surface of the
workpiece from previous experiments. The width of the spark-out surface was also larger than the width of the
wheel to ensure that only the bottom surface of the wheel was involved in the grinding action.
A Sartorius Master LP1200S electronic precision balance was used to measure the mass of the material removed
in the experiments. The mass capacity of the electronic balance is 1200 g, and the readability is 0.001 g. The
workpiece was chosen to be 166.6 mm to match the length of the force transducer on the Blohm Planomat 408
grinding machine The width of cut in these experiments was 25.4 mm; therefore, the measurement resolution of
the actual depth of cut using the electronic balance was 0.03 µm. Sequentially, the static deflection was obtained
by subtracting the actual depth of cut measured by the electronic balance from the commanded depth of cut set
on the machines.
The following section summarizes the results of these grinding experiments.
Experimental Results and Discussion
Figure 4 plots the actual depth of cut as a function of the commanded depth of cut. The solid lines have been
fitted to the data using linear regression and the R-squared values shown in the figures indicate that the linear
regression closely follows the data. The dotted lines represent the boundaries within which 95% of the data lie.
Actual Depth of Cut ( µm )
30.0
Actual Depth of Cut
25.0
Average Actual Depth
of Cut
20.0
Linear (Actual Depth
of Cut)
15.0
slope=0.98
10.0
2
R = 0.983
5.0
0.0
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
Commanded Depth of Cut (µm)
Figure 4: Actual depth of cut vs. commanded depth of cut
vs = 30.5 m/s, ds = 335.4 mm, vw = 12.7 m/min, b = 25.4 mm,
WA46J Vitrified Aluminum Oxide wheel, AISI 1020 steel workpiece
Malkin [1] suggests a linear relationship between the actual depth of cut and the commanded depth of cut and
such a linear relationship has been observed in the data shown in Figure 4. Malkin proposes that the normal
force Fn is proportional to the material removal rate Qw, as follows:
Fn = F0 Qw
where Qw = vwab, and b is the grinding width.
(8)
For a given work speed vw, and grinding width b, the normal force is proportional to the actual depth of cut a:
F n = kc a
where kc is the cutting stiffness given by:
(9)
kc =F0 bvw
(10)
Combining Equations (2), (3) and (9) gives:
a=
1
h
kc
1+
ke
(11)
Using the slopes of the curve fits in Figures (3) and (4), the ratio kc / ke was calculated to be 0.02. This result
suggests that the overall system stiffness ke is much larger compared with the cutting stiffness kc on the Blohm
Planomat.
Figures 5 shows the resulting static deflections plotted as a function of the commanded depth of cut. Note that the
static deflections do not change significantly with the commanded depth of cut and are almost a constant 5 µm.
This result suggests that the Blohm Planomat grinding machine is very rigid as there is virtually no change in the
static deflections as the commanded depth of cut increases.
Figures 6 shows the relationship between the spindle power and the static deflections while Figure 7 plots the
relationship between normal force and static deflections. Evidently the the static deflections do not change
significantly with spindle power or normal force. Furthermore the fitted line in Figure 7 does not pass through the
origin in as predicted by Equation (3). A likely reason for these results is that backlash exists in the Blohm Planomat
408 grinding machine. Figure 8 shows the typical mechanism for backlash on a grinding machine. The infeed is
achieved by a rotating lead screw. The nut for the lead screw is usually attached to a spindle bearing, where the
grinding wheel spindle is mounted. All lead screw assemblies have some backlash at assembly because of the
tolerance required between the screw and the nut.
20.0
Static Deflections (µm).
Static Deflections
Average Static Deflections
2
R = 0.023
15.0
10.0
5.0
0.0
0.0
5.0
10.0
15.0
20.0
25.0
30.0
Commanded Depth of Cut (µm)
Figure 5 Static deflections vs. commanded depth of cut
vs = 30.5 m/s, ds = 335.4 mm, vw = 12.7 m/min, b = 25.4 mm,
WA46J Vitrified Aluminum Oxide Wheel, AISI 1020 steel workpiece
35.0
7.0
Spindle Power
6.0
Average Spindle Power
5.0
4.0
2
R = 0.008
3.0
2.0
1.0
0.0
0.0
2.0
4.0
6.0
8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0
Static Deflections (µm )
Figure 6: Spindle power vs. static deflections
vs = 30.5 m/s, ds = 335.4 mm, vw = 12.7 m/min, b = 25.4 mm,
WA46J Vitrified Aluminum Oxide Wheel, AISI 1020 steel workpiece
400.0
Normal Force
350.0
Average Normal Force
300.0
Normal Force (N)
Spindle Power (KW) .
8.0
250.0
200.0
150.0
Fn = ke ε
100.0
2
R = 0.008
50.0
0.0
0.0
2.0
4.0
6.0
8.0
10.0 12.0
14.0
16.0 18.0
Static Deflections (µm )
Figure 7: Normal force vs. static deflections
vs = 30.5 m/s, ds = 335.4 mm, vw = 12.7 m/min, b = 25.4 mm,
WA46J Vitrified Aluminum Oxide wheel, AISI 1020 steel workpiece
20.0
Lead screw
Nut attached to spindle bearing
Infeed
Backlash
Figure 8: Backlash
With the measured force data from the Blohm Planomat machine, it is possible to estimate the individual static
deflections of the grinding wheel, workpiece and machine system to determine the amount of the backlash in this
grinding machine.
Individual Deflections of Grinding Wheel, Workpiece and Machine
Peters et al. [10] developed a method of determining the elastic modulus based on measuring the natural
frequency of a grinding wheel excited by a force impact. Once the elastic modulus of a grinding wheel is
obtained, the deflection and corresponding stiffness of the grinding wheel can be estimated by measuring the
normal force applied. The relationship between the elastic modulus E and the natural frequency ωn for the two
nodal diameter vibration modes is given by the following approximate relationship:
E=
P1ρd s4ω n2
(12)
4 × 1012 b 2
where ρ is the density of wheel, ds is the outer diameter of wheel, and b is the width of wheel. The values of P1
depend on the ratio of the inner hole diameter d0 to the outer diameter of wheel ds. Such values of P1 have been
measured for d0/ds ratios ranging from 0 to 0.7 [10].
A spectrum analyzer (HP 3582A) and accelerometer (Kistler 8776A50) were used to measure the natural
frequency of the wheel as shown in Figure 9.
The grinding wheel to be tested was placed on a cone to produce the pinned boundary condition required to apply
Equation (12), with the accelerometer attached to the periphery surface of the wheel. When an impulsive force on
the wheel was applied, the accelerometer detected the vibrations and sent the vibration signal to the spectrum
analyzer. The corresponding experimental results are summarized in Table 2.
ρ (g/cm3)
1.948
ds (mm)
330.2
Table 2: Parameters in Equation (12)
d0(mm)
b(mm)
P1
127.0
25.4
4.85
ωn (Hz)
880
E (MPa)
39.37×103
Spectrum Analyzer
(HP 3582A)
Wooden Cone
Accelerometer
(Kistler 8776A50)
Grinding Wheel
Figure 9: Grinding wheel natural frequency measurement
Among the sixty sets of data measured, the largest static deflection of the wheel occurred when the largest
contact force of 360 N was experienced. For this worst case, the deflection of the grinding wheel was calculated
to be 1.16 µm using the modulus of elasticity E in Table 2 and finite element analysis, yielding an effective wheel
stiffness of 0.31 KN/µm. The deflection of the workpiece was calculated to be 0.18 µm using the modulus of
elasticity of 1020 low carbon steel. The corresponding deflection of the dynamometer was calculated to be
0.27µm using the 2kN/µm product specification for its stiffness. Given that the total static deflection was
approximately 5µm (Figure 6), the machine deflection, which is dominated by backlash, is approximately 3.4µm.
Conclusions
This paper characterized the static deflections in the Blohm Planomat 408 grinding machines in the Grinding
Research Lab at Dalhousie University. Using the difference between the mass of the unground and ground
workpieces was found to be a straightforward method of indirectly determining the total static deflections of the
grinding machines. The static deflections for the Blohm Planomat machine were approximately a constant of 5µm
because of the high rigidity of the machine and a constant offset was observed in the spindle power, normal force
and static deflection relationships. This offset is likely due, in part, to backlash in the spindle lead screw. For the
worst case among all the experiments, the deflections of workpiece and force dynamometer were estimated to be
0.45 µm, the wheel deflection was estimated to be 1.16 µm, and the backlash in the Blohm Planomat machine
was estimated to be 3.4 µm.
Acknowledgements
The authors would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) and
the Canadian Foundation for Innovation (CFI) who provided financial support for this work.
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