On the cylindricity measurement by the V-block method.
Stepien, K. ; Janecki, D. ; Adamczak, S. 等
1. Introduction
The knowledge of the measurement of cylindricity profiles is
relatively limited. The works available on the subject mainly concern
the application of radial methods. The radial methods can be divided
into two main groups: ones employing a rotary sensor, and ones employing
a rotary table (see Fig. 1).
[FIGURE 1 OMITTED]
The radial measurements of cylindricity profiles are of a high
metrological level. Characterised by high accuracy, they provide
complete information of the analysed surface (Zhao et al., 2006).
However, the radial measuring instruments are applied only if it is
possible to place the measured element on the measuring table.
In various branches of industry (for example, in paper, electric or
ship industry), large and heavy cylindrical elements play very important
role (Runje & Medic, 2011). Such parts should posses very good
quality of form and dimensions. Sometimes, they may be deformed under
certain operating conditions, e.g. due to a high temperature or great
external or internal loads. Consequently, wear and damage of the working
surfaces are observed. Therefore cylindricity deviations of such
cylinders should be measured during their manufacturing and work
(Beranek et al., 2011). However, some of the elements cannot be measured
accurately by existing measuring instruments because they are too large
or too heavy and therefore they cannot be placed on the measuring table
of the instrument. This is why the industries manufacturing or applying
cylinders expect that measurements of cylindricity profiles will be made
directly on the machine tool or in the work area (Gao & Kiyono,
1997; Muralikrishnan, 2005). Problem of in-situ roundness measurements
was successfully solved by applying so-called V-block method (Okuyama et
al., 2003). Therefore authors assumed that also problem of in-situ
cylindricity measurements can be solved this way. This is why the
concept of cylindricity measurements by the V-block method has been
developed.
In V-block methods a measuring signal is measured in relation to a
physical datum. The datum is constituted by contact points of the
V-blocks and the measuring sensor with the cylinder. Mutual location of
these points in relation to the assigned coordinate system is defined by
method parameters--angles [alpha] and [beta] (see Fig. 2). Traditional
V-block methods used for measurement of roundness deviation are not
accurate; they are suitable for rough evaluation only. However,
application of FFT algorithm can eliminate main shortcomings of methods
of this type.
In hitherto practice the V-block was used usually to measure
simplified cylindricity. The aim of the authors of this paper was
development of the V-block method that would allow evaluation of the
entire surface of the element (the number of the cross-sections would be
defined by the user).
2. Concept of the cylindricity measurement by the V-block method
The developed concept of cylindricity measurements by the V-block
method is shown in Fig. 2.
[FIGURE 2 OMITTED]
The proposed concept of cylindricity measurement by means of the
V-block method assumes that the measured cylinder is placed on a machine
tool (in a centering device). Two interconnected V-blocks adhere to its
surface. The connecting element of the V-block functions additionally as
a slideway for the measuring sensor. The object's angle of rotation
of the cylinder and the sensor's displacement are controlled by the
computer. The cylindricity measurement of an object implies appropriate
scanning of the cylinder's surface with a measuring sensor, along
the suitably designed trajectory, through appropriate steering of the
cylinder's angle of rotation and sensor's displacement. Values
a and p shown in Fig. 2 are the angular parameters of the V-block method
for cylindricity measurement. They are responsible for detecting
particular harmonic components of the measured cylindricity profile.
The developed concept requires a mathematical transformation of the
sensor readings.
2.1 Mathematical model of the method
The basis of the relationships presented in this section is the
mathematical model of transformation of sensor readings in roundness
measurements by the V-block method that was presented in (Adamczak et
al. 2010).
The equation of a profile deviation in a cross-section of the
cylinder can be best written into the polar co-ordinates system with a
centre on the cylinder axis. Let us assume that, if the axis of the
nominal cylinder coincides with the axis of the real cylinder, the
distance between any point of a profile and the surface of the nominal
cylinder is equal to R([phi], z), where [phi] and z are co-ordinates of
the profile point. If the deviation, R, is zero, then the sensor
readings do not depend on the co-ordinates [phi] and z. They are
proportional to the distance of a given point of a profile from the
surface of the nominal cylinder (Adamczak et al. 2011).
Due to the fact that profile deviations are observed at the points
of contact of the cylinder's surface and the V-blocks, the real
cylinder axis, Z', will deviate slightly from the Z axis defined by
the nominal cylinder axis. We shall denote by [E.sub.x] ([phi], z) and
[E.sub.y] ([phi], z) the Cartesian co-ordinates of the intersection of
the axis of the cylinder turned through the angle [phi] with a plane
perpendicular to the Z axis with the coordinate z (see Fig. 3). Without
the cylinder axis evaluation of cylindricity would not be accurate (Lao
et al., 2003).
[FIGURE 3 OMITTED]
We can determine the values of
[E.sub.x0]([phi]) = [E.sub.x]([phi], 0), [E.sub.y0]([phi]) =
[E.sub.y] ([phi] ,0) (1)
and
[E.sub.xL]([phi]) = [E.sub.x] ([phi], L), [E.sub.yL]([phi]) =
[E.sub.y]([phi], L) (2)
Let us consider the cross-section of the cylinder defined by the
plane z = 0. The points of contact of the measured element and the
V-block coincide with the points of contact of the nominal cylinder. So,
in these points the distance of the profile from the nominal cylinder is
equal to zero. The cylinder rotated through the angle [phi] touches one
of the V-blocks at points of a profile with angular co-ordinates [alpha]
+ [phi] and [pi] - [alpha] + [phi]. Thus,
R([alpha] + [phi], 0) + [E.sub.x0] ([phi])cos[alpha] + [E.sub.y0]
([phi]) sin[alpha] = 0, (3)
R([pi] -[alpha] + [phi], 0) - [E.sub.x0] ([phi]) cos[alpha] +
[E.sub.y0] ([phi]) sin [alpha] = 0 (4)
Hence
[E.sub.xo] ([phi]) = - R([alpha] + [phi], 0)-R([pi] - [alpha] +
[phi], 0)/2cos[alpha] (5)
[E.sub.y0] ([phi]) = - R([alpha] + [phi], 0) + R([pi] - [alpha] +
[phi], 0)/2 sin [alpha] (6)
Similarly, analyzing the cylinder cross-section z = L, we obtain
[E.sub.xl]([phi]) = R([alpha] = [phi], L - R([pi] - [alpha] +
[phi], L)/2 cos[alpha] (7)
[E.sub.yL] ([phi]) = - R([alpha] + [phi], L) + R([pi] - [alpha] +
[phi], L)/2 sin[alpha] (8)
Now the values of the sensor readings F([phi], z) for given values
of the cylinder's rotation angle [phi] and the displacement of the
measuring sensor z can be determined. They are given by following
equation
F([phi], z) = R([phi] + [beta], z) + [E.sub.x] ([phi], z)cos[beta]
+ [E.sub.y] ([phi], z)sin [beta], (9)
where
[E.sub.x] ([phi], z) = [E.sub.x0] ([phi])(L - z) + [E.sub.xL]
([phi])z/L (10)
[E.sub.y] ([phi], z) = [E.sub.y0] ([phi])(L - z) +
[E.sub.yL]([phi])z/L (11)
From the obtained relationships it follows that the value of the
deviation can be determined on the basis of the sensor reading, if the
co-ordinates of the axis displacement, [E.sub.x0]([phi]),
[E.sub.y0]([phi]) and [E.sub.xL]([phi]), [E.sub.yL] ([phi]), are known.
This, accordingly, requires determining the profiles R(p,0) and R(p, L).
We can achieve this by two additional measurements of roundness profiles
in the cross-sections z = 0 and z = L. Then, from Eqs. (5), (6) and (9)
we have
F([phi], 0) = R([phi] + [beta],0) + [E.sub.x0]([phi]) cos[beta] +
[E.sub.y0]([phi]) sin[beta] =
= R([phi] + [beta], 0) - 1/2 R([phi] + [alpha],
0)[cos[beta]/cos[alpha] + sin[beta]/sin[alpha]]-1/2 R([phi] + [pi] -
[alpha], 0)[cos[beta]/cos[alpha] + sin[beta]/sin[alpha]] (12)
In the above equation, only the profile R([phi], 0) is unknown.
Thus, the equation should be solved in relation to R([phi], 0). The
easiest way is to do this in the domain of the coefficients is by
expansion of the profile into an exponential Fourier series. Let
[[??].sub.n0] and [[??].sub.n0] be the n--th components of the expansion
of the profiles F ([phi], 0) and R([phi], 0) into an exponential Fourier
series n = -[infinity], ..., -1,0,1, ..., [infinity] , i.e.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
Then, we get from (12)
[[??].sub.n0] = [[??].sub.n0]/[K.sub.n] (14)
where [[??].sub.n] is the so-called coefficient of detectability
defined by
[[??].sub.n] = [e.sup.in[beta]] - 1/2[e.sup.in[alpha]]
[cos[beta]/cos[alpha] + sin[beta]/sin[alpha]] - 1/2 [(- 1).sup.n]
[e.sup.-in[alpha]] [cos[beta]/cos[alpha] + sin[beta]/sin[alpha]]. (15)
Similarly, for the profile z = L, we get
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
Finally, rewriting Eq. (9) in the Fourier coefficients domain, we
get the relationship between Fourier expansion coefficients of the
sensor readings [[??].sub.n](z) and the real signal [[??].sub.n](z)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
where
[[??].sub.n] = 1/2[e.sup.in[alpha]] [cos[beta]/cos[alpha] +
sin[beta]/sin[alpha]] + 1/2 [(- 1).sup.n] [e.sup.- in[alpha]]
[cos[beta]/cos[alpha] + sin[beta]/sin[alpha]].
Equations (14)-(18) allow precise calculation of values of Fourier
coefficients of the real signal [[??].sub.n] (z).)(18)
Values of real signal R([phi], z) can be obtained by inverse
Fourier transform of coefficients [[??].sub.n] (z). It can be easily
performed by means of IFFT (Inverse Fast Fourier Transform) algorithm.
3. Simulations
On the grounds of the mathematical model of V-block cylindricity
measurements, presented in previous section, a set of the computer
procedures using the MATLAB package was developed. The developed
procedures help prepare qualitative and quantitative evaluation of the
cylindricity profiles obtained by V block methods. In order to verify
the correctness of the procedures, computer simulations were conducted.
Computer simulations of the V-block cylindricity measurements were
conducted in the following stages:
-- the generation of an exemplary cylindricity profile R([phi], z),
-- the calculation of the measured profile F ([phi], z) for given
method parameters [alpha], [beta] and L,
-- the transformation of profile F ([phi], z) into the
reconstructed profile [R.sub.p]([phi], z),
-- the determination of the mean cylinder axis for profiles
R([phi], z) and [R.sub.p]([phi], z),
-- the comparison of the deviations of profiles R([phi], z) and
[R.sub.p] ([phi], z) from the mean axis.
Profile R([phi], z) corresponds to the real profile of the
workpiece, F ([phi], z) is the profile obtained directly from the
measuring sensor by a V-block method, and [R.sub.p] ([phi], z) is the
profile calculated from profile F ([phi], z) through its mathematical
transformation.
3.1 Determining the real cylindricity profile
The real cylindricity profile R([phi], z), being the basis for the
calculation of profiles F([phi], z) and [R.sub.p]([phi], z), was
designed in order to include all the cylindricity components, described
by the standard ISO 12180, i.e. the diameter change in the subsequent
cross-sections, the straightness deviation of the cylinder axis and the
roundness deviation in individual cross-sections of the cylinder. The
profile is described by the relationship
R([phi], z) = [R.sub.0] + 0.001 x [R.sub.0] x (sin([pi]z/L) + 10z/L
sin([pi]z/L)sin[phi] + cos([pi]z/L)cos[phi] + sin2[phi] + 5cos3[phi] +
2sin6[phi] + cos10[phi] + cos11[phi])(19) (19)
For the purposes of the simulation, the following values of the
parameters were assumed:
-- the V-block angle [alpha] = 60[degrees],
-- the angle between the X axis and the sensor axis [beta] =
90[degrees],
-- the distance between the prisms L = 200 mm,
-- the nominal cylinder radius [R.sub.0] = 20 mm,
-- the number of measured cross-sections (including the
cross-section of the prisms) K = 11,
-- the number of sampled points per one cross-section N = 1024,
-- sampling angle [phi] = 2 x [pi] x n/N, where n = 0,1,2,...N.
When we have generated profile R([phi], z) for given method
parameters, i.e. angles [alpha], [beta] and the distance between the
prisms L, we can calculate the profile F([phi], z), measured by the
sensor, from the relationship (9). As it was mentioned in the previous
sections, in order to obtain correct cylindricity parameters by a
V-block method, it is necessary to transform the measured profile F
([phi], z) into the reconstructed profile [R.sub.p]([phi], z) using
relationships (14)-(18).
In order to compare inspected profiles correctly, the orientation
of the mean cylinders for the profiles R([phi], z) and [R.sub.p]([phi],
z) has to be calculated (which is tantamount to the calculation of
co-ordinates [E.sub.x0], [E.sub.xL], [E.sub.y0] [E.sub.yL] of the
orientation of the mean cylinder axis in the initial and final
cross-section of the inspected cylinder), and then both profiles values
in relation to the calculated axes have to be determined. In order to
determine the orientation of the mean cylinders axes for profiles
R([phi], z) and [R.sub.p]([phi], z), the values of the first harmonic
component in all the cross-sections of the cylinders were calculated.
Then, co-ordinates [E.sub.x0], [E.sub.xL], [E.sub.y0], [E.sub.yL] were
calculated using relationships (5)-(8). The next step was the
calculation of the axis orientation in each cross-section from
relationships (10) and (11). The last step was the removing of run-out
components from the inspected profile, which is the result of the axis
shift. After conducting these operations for both profiles, they can be
compared.
In order to investigate the conformity of both profiles accurately,
the comparison of their subsequent cross-sections was carried out.
Figure 4 represents superimposed roundness profiles in the first
cross-section of inspected cylinders in Cartesian coordinates. The
diagram shown in Fig. 4 confirms that there is a difference between the
inspected profiles after considering the orientation of the associated
cylinders axes. In order to represent this difference better, the
diagram in Fig. 5 represents the bar chart of the amplitudes of
subsequent harmonic components (for the range 1 4 / 15) of the profiles
in the first cross-section of the inspected profiles.
As we can see in Fig. 6, the eleventh harmonic component occurs in
the generated profile R([phi], z),whereas it does not occur in the
processed profile [R.sub.p]([phi], z). It is caused by the fact that the
selected values of angular parameters [alpha] = 60[degrees] and [beta]
90[degrees] do not allow detection the eleventh harmonic component in
the generated profile.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
The results of additional computer simulations showed that
difference between the values of profiles R([phi], z) and
[R.sub.p]([phi], z) does not change on the entire surface of the
inspected cylinder.
Therefore, we can assume that after considering the orientation of
the associated cylinder axes, the only significant difference between
the profiles [R.sub.p] ([phi], z) and R([phi], z) is the occurrence of
the harmonic components which cannot be detected by selected values of
the method parameters [alpha] and [beta]. The results of additional
computer simulations, show that if generated profile R([phi], z)
contains only harmonic components detectable by given angular method
parameters [alpha] and [beta], then the difference between inspected
profiles is negligible.
4. Experiment
4.1 A measuring instrument for V-block cylindricity measurements
After successful finish of the theoretical research work efforts
were taken aiming at practical verification of the developed concept.
Therefore it was necessary to design and construct the measuring system
allowing cylindricity measurements by the V-block method. Such system
has been finally designed and constructed at the Rolling Bearings
Factory "Krasnik" (Poland) in the framework of the research
project sponsored by Polish Ministry of Science. The system is shown in
Fig. 6.
[FIGURE 6 OMITTED]
The testpiece (5) is fixed in the tailstock and in the spindle. The
rotation of the workpiece is controlled by the computer. The system of
two connected V-blocks (4) lies on the surface of the workpiece. The
V-blocks are fixed to the beam (2). The beam (2) is suspended on the
system of springs (1). The measuring sensor (6) is fixed to the support
(3) that can move along the beam (2). The system was adjusted to the
strategy of measurements of subsequent cross-sections. So, roundness
profiles in predefined cross-sections of the workpiece are measured.
After the measurement the system automatically joins them into
cylindrical data.
The device shown in Fig. 6 was used in the experiments aiming at
evaluation of the accuracy of the cylindricity measurement by the
V-block method.
The experiments involved the comparison of measurement results of
the group of cylindrical elements. The results were obtained by the
proposed V-block method, using the stand PSA6 and by the radial method,
using computer-aided instrument for straightness, roundness and
cylindricity measurements Talycenta by Taylor Hobson.
4.2 Experimental verification of the concept
Experimental verification of V-block cylindricity measurements
involved the comparison of measurement results of the set of eight
cylindrical elements, obtained by the V-block method, using the stand
PSA6 and by the radial method, using Talycenta. The obtained measurement
results were compared using the software, described in the previous
subsection.
When comparing the profiles, the authors focused on observing the
occurrence of systematic errors of the V-block method. The comparison
was made qualitatively (visually by superimposition compared profiles)
and quantitatively--by the calculation of the coincidence coefficient
between the profiles and by the comparison of selected cylindricity
parameters. In the experiment the following parameters were analyzed:
-- a cylindricity deviation [CYL.sub.t],
-- a maximum peak over the associated cylinder [CYL.sub.p],
-- a minimum valley below the associated cylinder [CYL.sub.v],
-- a conicity angle [CYL.sub.at] and
-- a straightness deviation of the cylinder mean line [STR.sub.t]).
Table 1 gives the above mentioned parameters calculated for the
measured profiles of the set of seven cylindrical elements.
The analysis of the results obtained by the reference and radial
methods shows that they are similar. In most cases the difference
between values of cylindricity deviation is equal to about a few
micrometers. Taking to account that the V-block method is investigated
as the method that could be applied under industrial condition, such
difference is sufficient.
In Fig. 7 diagrams useful for qualitative comparison of the
profiles are shown. The diagrams were generated using the software based
on the procedures developed fort he purpose of computer simulations.
Each compared pair of profiles was represented in a 3D diagram at first.
If more accurate analysis was required, the diagrams for individual
cross-sections were plotted, e.g. polar and Cartesian diagrams of
roundness profiles.
[FIGURE 7 OMITTED]
4.3 Statistical research
In order to establish the accuracy of the V-block method of
cylindricity measurement quantitatively a statistical research was
carried out. The statistical verification of the proposed method under
industrial conditions was carried out at "Krasnik" Rolling
Bearings Factory. The aim of the test was to determine the method error
of V-block measurement of cylindricity profiles for selected sample of
elements by means of statistical parameters (taking into account the
values of the cylindricity deviation CYLt), and to estimate the
coefficient of coincidence of the compared profiles defined on the basis
of the standardized function of cross correlation.
The relative error of the measurement of the cylindricity deviation
w CYLt was calculated for each element using this relationship:
W[CYL.sub.t][absolute value of [[DELTA].sub.o]] -
[DELTA][C.sub.p]/[DELTA] [C.sub.p]] x 100% (20)
where:
[DELTA][C.sub.o] - cylindricity deviation calculated from the
profile obtained with the V-block method,
[DELTA][C.sub.p] - cylindricity deviation calculated from the
profile obtained with the accurate radial method.
On the basis of relative error the method accuracy (MA) can be
calculated from the formula:
MA = [[bar.w].sub.CYLt] [+ or -] u x s (21)
where
MA--method accuracy,
[[bar.w].sub.CYLt]--mean value of the relative error [w.sub.CYLt],
u--standardized variable (dependent on reliability level),
s--standard deviation of the relative error [w.sub.CYLt].
The number of samples was 25. Thus, twenty five specially machined
cylinders and twenty five randomly selected cylinders were used for
the laboratory tests and factory verification, respectively. In the
calculations reliability level P = 0,95 was assumed. For this value, u =
1,96. The results of the calculations are provided in the Table 2.
Results of measurements, given in Table 1 show that the
cylindricity profiles obtained with the V-block and accurate radial
method are characterized by high correlation. This testifies to their
good coincidence, illustrated also in the diagrams showing two compared
profiles (see Fig. 7).
Statistical investigation of the accuracy of the method, whose
results are given in Table 2, show that the result of the cylindricity
deviation measurement taken by means of the proposed method lies within
the range 19% in relation to result obtained by the highly accurate
radial method (for assumed probability level P=0,95).
5. Specific problems about the V-block method of cylindricity
measurements
5.1 Problems relating to the measurement concept
The first problem relating to the measurement concept is that some
harmonic components cannot be detected by the measuring system. This
shortcoming can be compensated by such selection of the method
parameters (angles [alpha] and [beta] in Fig.2) that would permit
detection these harmonic components that are most important for the
user. Guidelines for selection desired method parameters can be found,
for example, in (Sonozaki & Fujiwara, 1989). Another solution of
this problem is such design of the measuring system that allows
adjustment of the angles [alpha] and [beta]. Unfortunately, the system
shown in Fig. 2 does not allow adjustment of the method parameters. The
second problem concerning the practical application of the method is the
fact that the mathematical transformation of the measured signal into a
processed one is necessary to obtain reliable measurement results. As
far as the authors know there is not any commonly available commercial
software that for this purpose. Therefore authors developed their own
software for such transformation. The software was based on the
procedures written in Matlab environment that had been developed on the
fundamental of equations given in the section 2. Thus, the measuring
system shown in Fig. 6 was used only for data acquisition. Then the
measurement data were saved as a text file and transferred to the
software operating in the Matlab environment. The software allows
measurement data processing, an evaluation of the profile by values of
cylindricity deviations and plotting different types of diagrams.
5.2 Problems relating to the design of the measurement system
Another practical difficulties that occurred were problems relating
to the design of the measuring system. The first one concerned the shape
of the contact surfaces of the V-blocks. At the beginning of experiments
this shape was sharp in order to assure area of contact of the V-blocks
and measured workpiece to be as small as possible. However, it turned
out that such design of V-blocks results in damaging of surface of
workpieces made of soft steels. Therefore, it was necessary to modify
the shape of the contact surfaces of the V-blocks. In final experiments
their shape was spherical. The second factor that had negative influence
on measurement results was too stiff suspension of the system:
V-blocks--the beam (the guideway)--the sensor. At the beginning of the
experiment the beam was suspended by the set of two flat springs.
However, it turned out that such suspension is too stiff and was the
reason of incorrect sensor readings. Therefore, the design of the system
has been modified and stiff flat springs have been replaced by softer
coil springs (see elements no. 1 in Fig. 6).
5.3 Errors of measuring system components
Important problems from the point of view of practical application
of V-block methods are errors of components of measuring system. Such
errors sometimes influence very significantly on measurement results.
For example, during experiments it turned out that profiles measured by
the V-block method are characterized by conical error. However,
measurements of these profiles by the standard, highly accurate radial
method showed that this conicity error is only virtual
-- the real workpiece surface is not conical. Therefore research on
the source of this conicity error was done. The result of the research
was that virtual conicity of measured profile may appear due to:
-- unequal angles of V-blocks,
-- the slope of the guideway in relation to the surface of measured
workpiece (for example, caused by unequal fixing the V-blocks to the
guideway).
In order to investigate the first issue, a computer simulation was
made. It showed that even small difference of V-blocks angles influences
very significantly on measurement result (Janecki & Stepieh, 2008).
For example, difference of the V-blocks angles equal to 0,02[degrees]
causes appearance of virtual conicity deviation equal to 11,7 [micro]m
(simulation was made for the nominal cylinder defined by following
values: the radius: 20 mm and the height 100 mm).
Taking to account results of the research the angles of the
V-blocks were measured by coordinate measuring machine. It was
established that:
-- the angle of the left V-block [[alpha].sub.0] =
59[degrees]51'27",
-- the angle of the right V--block.: [[alpha].sub.L] =
59[degrees]51'58".
Measured values of the V-blocks angles were used to calculate
appropriate correction coefficients that were put into the software.
Besides the difference of the V-blocks angles and the guideway
slope, influence of other factors on measurement results was also
investigated. For example, following problems were investigated:
straightness deviation of the guideway, off nominal orientation of the
measuring sensor and calibration error of the sensor. It turned out that
in the case of the measuring system shown in Fig. 8 these factors hardly
influence measurements results.
Important part of the experimental verification of the concept of
was visual comparison of profiles obtained by the V-block method and by
the accurate radial method regarded as a reference. The visual
comparison showed that there were quite large fluctuations of the
profiles obtained by the V-block method, which was not observed in the
profiles obtained by the radial method (see Fig. 7b)). The source of
these fluctuations is not clear. In authors' opinion they may be
the result of vibrations of the screw thread mechanism of the measuring
sensor drive. However, this problem should be investigated in the
nearest future.
6. Conclusion
Taking into consideration the requirements of modern technological
processes, presented method of the V-block cylindricity measurements
seems to be very interesting. Practical advantages of the computer-aided
V-block method were shown in the case of roundness measurements.
Research work on the application of the V-block method for roundness
measurements has resulted in developing of measurement systems allowing
in-situ roundness measurements. Such systems work in industrial plants
and universities, such as, for example, Alstom (England) and ABB Brno
(Czech Republic). In authors' opinion it is possible to design and
construct similar system that could be used also for cylindricity
measurements.
In order to make it possible to apply the V-block method to
measurements of cylindricity it was necessary to develop a mathematical
model of such measurement. Very important part of the model are
equations allowing transformation of the measured profile into the
reconstructed one. Computer simulations showed that developed model is
correct, however, some harmonic components of the profile cannot be
detected by the measuring system. Except for this shortcoming, the model
is not the source of any other systematic errors.
On the fundamentals of the developed concept a special-purpose
measuring system was designed and constructed. The system was used in
the experimental verification of the method. Results of measurements,
given in Table 1 show that the cylindricity profiles obtained with the
V-block and accurate radial method are characterized by high
correlation. This testifies to their good coincidence, illustrated also
in the diagrams showing two compared profiles (see Fig. 7).Statistical
investigation of the accuracy of the method, whose results are given in
Table 2, show that the result of the cylindricity deviation measurement
taken by means of the proposed method lies within the range 19% in
relation to result obtained by the highly accurate radial method (for
assumed probability level P = 0,95). Assuming that the calculated
experimental method error is a sufficient measure of accuracy, and that
the accuracy of an instrument to be used for measurement of the
geometrical surface structure during the product quality control ranges
10%-25%, it is clear that the proposed concept of reference measurement
of cylindricity profiles is characterized by sufficient accuracy. The
test results indicate that the V-block method is applicable to
cylindricity measurement, and that the specially constructed measuring
system can be used under industrial conditions. Note, that workpieces,
whose cylindricity deviation was equal to about 30 / 50 um were
measured. In authors' opinion, taking to account the current
state-of the-art on the V-block cylindricity measurement, this method
should not be applied in measurements of workpieces, whose deviation is
smaller. There were a lot of practical problems that occured in the
experimental part of the research work.
The first one was necessity to develop suitable software allowing
transformation of the sensor readings into the reconstructed profile.
Without such transformation measurement results would be completely
wrong.
The second problem was that some harmonic components could not be
detected by the measuring system. This shortcoming, characteristic
feature of all V-block methods, can be compensated, for example, by
adjustment of the angular method parameters (angles [alpha] and [beta]
from Fig. 2). Unfortunately, the measuring system used in the
experiments does not allow adjustment of the method parameters.
Another practical difficulties concerned the mass and stiffness of
the measuring system. It turned out in the case of measurement of
workpieces made of soft steels, contact surfaces of the V-blocks can
damage the surface of the workpiece. It may be caused, for example, by
too large mass of the V-blocks system, sharp shape of the contact
surfaces of the V-blocks and too stiff suspension of the measuring
system.
Therefore, engineers, who would design V-block systems in future,
should take into consideration following questions:
-- shape of measuring surfaces of V-blocks should not be sharp,
-- the mass of the V-blocks system should be rather small,
-- the suspension of the system should not be too stiff.
Other factors that influenced measurement results were errors of
the components of the measuring system. For, example, when analyzing
measurement results it turned out that profiles measured by the V-block
method were characterized by a conicity deviation. However, the same
profiles measured by the radial method used as a reference, were not
conical. Therefore, authors assumed that the deviation of conicity is
only virtual and it is caused by geometrical deviations of the
components of the V-block measuring system. Investigation on this
problem showed that virtual conicity deviation can be caused , for
example, by the difference of the V-blocks angles. Therefore, very
important issue is measurement of real values of the V-blocks and
putting proper correction coefficients into the software.
Another good solution allowing calculation of the correction
coefficients could be measurement of the element, whose profile is known
(for example, if it was measured earlier by the measurement instrument
of high accuracy). Difference between the measured and the reference
profile would allow identification of such errors as slope and
straightness deviation of the guide-way or calibration error of the
measuring sensor and other ones. Of course, such error as the
straightness deviation of the guideway could be also identified by
direct measurement with an interferometer or other measuring instrument
for straightness measurement.
Visual comparison of profiles measured by the V-block and the
radial method showed that the profile obtained by the V-block is
characterized by larger fluctuations that the one measured by the radial
method. The reason of this phenomenon is not quite clear. In
authors' opinion it is probably caused by the vibrations of the
screw thread drive of the sensor. However, this problem should be
investigated in near future.
Despite the problems with the implementation, the V-block method of
cylindricity measurements is very promising. In authors' opinion
most of the problems can be solved, which would make it possible to
design and construct the V-block system for cylindricity measurement.
Such system would allow accurate in-situ measurement of form deviations
of large cylinders that are used, for example, in paper or electric
power industry. In authors' opinion design, building up and testing
of such system should be the next stage of the research work on V-block
cylindricity measurements.
DOI: 10.2507/daaam.scibook.2012.03
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Authors' data: Dipl.-Ing. Dr. techn. Stepien, K[KrzysztofJ*;
Univ. Prof. Dipl.-Ing. Dr. techn. Janecki, D[ariusz]*; Univ.Prof.
Dipl.-Ing. Dr.h.c.mult. Dr.techn. Adamczak, S[tanislawJ*, *Kielce
University of Technology, Al. 1000-lecia P. P. 7, 25-314 Kielce, Poland,
kstepien@tu.kielce.pl, djanecki@tu.kielce.pl, adamczak@tu.kielce.pl
Tab. 1. The results of comparative cylindricity measurements
conducted by the V- block and the radial method
No. Parameters
Measuring
device [CYL.sub.t] [CYL.sub.p] [CYL.sub.v]
[micro]m [micro]m [micro]m
1 PSA 6 32,7 13,9 18,8
Talycenta 29,4 10,5 18,9
2 PSA 6 50,6 17,2 33,4
Talycenta 44,6 20 24,6
3 PSA 6 41,3 20 21,3
Talycenta 37,6 14,9 22,7
4 PSA 6 38,7 17 21,7
Talycenta 34,5 14,6 19,9
5 PSA 6 42,3 16,7 25,6
Talycenta 37,5 12,5 -25
6 PSA 6 34,4 10,3 24
Talycenta 32,8 10,8 22
7 PSA 6 31,4 8,2 23,2
Talycenta 31,4 9,1 22,3
Parameters
No. Measuring
device [CYL.sub.at] [STR.subt]
(deg) [micro]m
1 PSA 6 0,29 [approximately equal to] 0
Talycenta -0,07 [approximately equal to] 0
2 PSA 6 -0,13 [approximately equal to] 0
Talycenta -0,25 [approximately equal to] 0
3 PSA 6 0,29 [approximately equal to] 0
Talycenta 0,02 [approximately equal to] 0
4 PSA 6 0,6 [approximately equal to] 0
Talycenta -0,04 [approximately equal to] 0
5 PSA 6 0,32 [approximately equal to] 0
Talycenta -0,13 [approximately equal to] 0
6 PSA 6 0,17 [approximately equal to] 0
Talycenta -0,12 [approximately equal to] 0
7 PSA 6 0,1 [approximately equal to] 0
Talycenta -0,35 [approximately equal to] 0
Tab. 2. Results of statistical testing of the experimental
method error for V-block cylindricity measurement under
industrial conditions and the values of the correlation
coefficients
Number of samples [n.sub.s] 25
[sup.w][CYL.sub.t] min -0.142
Obseived value of the relative eiror [sup.w][CYL.sub.t] max 0.124
Mean value of relative error ir [sup.w][CYL.sub.t] 0,020
Standard deviation s of the relative error 0.084
Method accuracy. MA [%] 19.4
Correlation coefficient r[tau]
Mean value 0.954
Minimum value 0.793
Maximum value 0.994
