JSSSJournal of Sensors and Sensor SystemsJSSSJ. Sens. Sens. Syst.2194-878XCopernicus PublicationsGöttingen, Germany10.5194/jsss-7-609-2018Precise linear measurements using a calibrated reference workpiece without
temperature measurementsPrecise linear measurements without control of the temperatureSuminDmytrod.sumin@tu-braunschweig.dehttps://orcid.org/0000-0001-8645-1774TutschRainerInstitute of Production Metrology, Technische Universität Braunschweig, Schleinitzstrasse 20, 38106 Braunschweig, GermanyDmytro Sumin (d.sumin@tu-braunschweig.de)23November20187260962029June201812October201816October2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://jsss.copernicus.org/articles/7/609/2018/jsss-7-609-2018.htmlThe full text article is available as a PDF file from https://jsss.copernicus.org/articles/7/609/2018/jsss-7-609-2018.pdf
We suggest a procedure for the correction of the errors caused by thermal
expansion of a workpiece and the scale of a linear measuring instrument (coordinate
measuring machines, length measuring machines, etc.) when linear measurements
are performed at nonstandard temperature. We use a calibrated reference
workpiece but do not require temperature measurements. An estimation of the
measurement uncertainty and application examples are given.
Introduction
Quality control loops in today's production in most cases rely on the
measurement of geometrical quantities. One problem that might arise is that
geometrical properties are specified at 20 ∘C, the so-called normal
temperature (ISO 1:2016-12, 2016). In practice, the temperature in a workshop
or a production line will not be adjusted to normal temperature but undergo
considerable fluctuations due to various factors which need to be studied
and taken into account (Baldo and Donatelli, 2012). Measurement values of
geometrical parameters taken at non-normal temperature will be erroneous due
to thermal expansion of the material (ISO/TR 16015:2003, 2003).
As a solution, production sites have to be equipped with measuring chambers
with specified and controlled equipment. Besides the high maintenance costs
of these measurement areas it takes time to bring the manufactured
workpieces (WPs) there and to get the required temperature balance between the
measuring instrument and the parts to be measured (ISO 15530-3:2011, 2011).
Control loops will thus become slow and changing influence factors might
therefore cause a certain number of workpieces to be produced out of tolerance
before the process can be stabilized again.
Nowadays, the temperature compensation problem is usually solved by a
detailed study of the metrological properties of the applied measuring
instrument in particular, the whole measuring process in general and hence by
a correction of errors due to these properties. This is performed either
indirectly – using an “ideal workpiece” that does not change its
geometrical properties due to temperature changes to make a comparison of
“what we expect” (nominal values) with “what we have” (measured values)
(Baldo and Donatelli, 2012; Ohnishi et al., 2010) or directly – when a
straight analysis of “what we have” is made (to discover possible factors
that might have affected the measured results and therefore to exclude or
minimize them) (Chenyang et al., 2011; Kruth et al., 2001). Despite these two
approaches being different according to the principles lying behind them, they
have something in common. They all require highly qualified personnel to
carry out all the tests and measuring temperature of all involved objects
(even in the case of using some ideal workpiece, the temperature of the
measuring instrument has to be determined).
Some precise complex measuring instruments like coordinate measuring machines
(CMMs) are equipped with thermal sensors for detecting temperature from both its
linear scales and the object to be measured. That will allow these measuring instruments to estimate the length of the
object reduced to a temperature of 20 ∘C. Manufacturers are expected to have enough funds for all required (the
described above) equipment to provide all necessary studies. However, in
spite of the potential availability of advanced technologies able to provide
automatic temperature compensation, they are not always affordable. This might
lead to a bigger uncertainty contribution during production processes and
post-production control. As an example, in Ukraine (where the author
Dmytro Sumin was born and raised) this is often the case when companies try to
provide some small production activity using obsolete (sometimes dozens of years
old) but cheap equipment.
In this research work a method is described which would allow companies like
these to have an affordable alternative to the expensive equipment, which at
the same time is comparably effective. The authors accept the challenge to
perform length measurements under non-normal temperature by using calibrated
reference workpieces (RWPs, therefore using the indirect approach) but without a
seemingly inevitable necessity to measure temperature, neither of the objects
to be measured nor of the measuring instrument. In Sect. 2 the procedure for
correction of thermal expansion of a workpiece without knowledge of the temperature
by referring to a calibrated reference workpiece is introduced. In Sect. 3 the uncertainty of the measurements is estimated and in
Sect. 4 application examples are given.
Correction of temperature effects during length measurements using a
reference workpiece
Thermal expansion is characterized with the coefficient of thermal expansion
(CTE) (ISO/TR 16015:2003, 2003):
α=1lst⋅ΔlΔt,
where Δl is the absolute length expansion of a body at the temperature
range Δt and lst is the length at standard
temperature.
Using Eq. (1), Δl can be calculated as
Δl=lst⋅α⋅Δt.
Taking into account Eq. (2) the length lT (see Fig. 1) of a body at its
temperature can be estimated:
T=20∘C+ΔtlT=l20+Δl=l201+α⋅Δt,
where l20 is the length of the body at 20 ∘C.
Expansion of an object due to temperature change.
Equation (2) is a mathematical model of thermal expansion of an object. It is
a linear function. So the value of Δl is proportional to the active
parameter Δt.
The nominal length of the produced workpiece is defined at the standard
temperature 20 ∘C. If the measurement is executed at a different
temperature Δtwp, the workpiece has the actual length
lwpact that is related to the length value at
20∘C,l20wp, described by the following
equation:
l20wp=lwpact1+αwpΔtwp,
where αwp is the CTE of the workpiece and the value
1+αwpΔtwp-1 is the
characterization of the absolute expansion of the workpiece.
The main problem of this formula is that Δtwp and
lwpact must be known first. To estimate them a thermometer and a calliper, for
example, can be used. But the calliper is also a
material object, with a scale that has some CTE αsc and at
the temperature difference value Δtsc its length is
lscact. Also, equality of corresponding parameters
should not be expected, so it is assumed that Δtwp≠Δtsc and αwp≠αsc.
l20sc=lscact1+αscΔtsc,
where the value (1+αscΔtsc)-1
characterizes the absolute expansion of the scale.
Behavior of the object at different temperatures.
The actual value of the workpiece's length lwpact is
unknown and instead of it its measured (estimated) value lwp will
be operated with. Obviously, this value (as well as
lwpact) relates to the absolute expansion of the
workpiece. However, because it is measured (estimated) using the scale, it is
also related to the absolute expansion of the scale. For better understanding
of how this relation works and how it affects the results of measurements, two
special cases will be considered.
Changing temperature of the workpiece with constant temperature of the
scale:
In Fig. 2 it can be seen that the workpiece initially had the temperature
T1∘C, thus its actual length was
lwp1act, then its temperature became
T2∘C (and the length
lwp2act), whereas the scale had the same
temperature T∘C (the length
lscact=const) in both cases.
Estimated values lwp1 and lwp2 (the
readings from the scale) showed themselves to be directly proportional to the
parameters lwp1act and
lwp2act correspondingly: an increment of
lwpact leads to an increment of lwp:lwp1lwp1act=lwp2lwp2act.
Changing temperature of the scale with constant temperature of the
workpiece:
In Fig. 3 it is seen that the workpiece had the constant temperature T∘C (and the length
lwpact=const), whereas the scale initially had
the temperature T1∘C (the length
lsc1act), then its temperature became
T2∘C (the length lsc2act).
Estimated values lwp1 and lwp2 showed
themselves to be inversely proportional to the parameters
lsc1act and lsc2act
correspondingly: an increment of lscact leads to a decrement
of lwp:lwp1⋅lsc1act=lwp2⋅lsc2act.
In real life changing of both parameters lscact and
lwpact at the same time (see Fig. 4) is mostly the
case.
Behavior of the scale at different temperatures.
Behavior of both objects at different temperatures.
The workpiece had the temperature T1∘C (the length
lwp1act) and the scale had the temperature
T2∘C (the length lsc1act),
then they both changed their temperatures to the values
T3∘C (the length lwp2act)
and T4∘C (the length
lsc2act) correspondingly. The way that the
parameter lwp relates to the parameters
lwpact and lscact is already
known from Eqs. (6) and (7), so taking these into account, a new proportion
can be derived:
lwp1⋅lsc1actlwp1act=lwp2⋅lsc2actlwp2act.
If we assume all parameters with subscript “1” (the left side of Eq. 8) to
have been achieved at Δtwp=0K and Δtsc=0K, according to Eqs. (4) and (5), it can be
stated that
l20wp⋅l20scl20wp=lwp⋅lscactlwpact.
For further calculations it will be assumed that l20sc=l20wp (the scale is expected not to be deformed by temperature
effects at Δtsc=0K). After transformation of
Eq. (9), l20wp can be found as follows:
l20wp=lwp⋅lscactlwpact=lwp⋅1+αscΔtsc1+αwpΔtwp
or
l20wp=lwp⋅1+αscΔtscwp1+αwpΔtwp,
where Δtscwp is the temperature of the scale during
measurements of lwp.
It is clearly seen that in the case of equality of absolute expansion characterization values of the workpiece and the scale (1+αscΔtscwp=1+αwpΔtwp) even if Δtwp≠Δtscwp and αwp≠αsc, the temperature effect is
cancelled and the desired value l20wp
therefore can be achieved without any additional action. However, such an
instance is a rare case. To use Eq. (10) we should figure out each of these
four unknown parameters: αwp, αsc, Δtwp and Δtscwp. Even if a thermometer was
used so that Δtwp and Δtscwp
could be found, the CTEs nevertheless may not be calculated directly, but
only achieved after a series of measurements (Amatuni, 1972).
Analyzing Eq. (10), it can be discovered that lwp and l20wp are related to each other through a term which contains all
four unknown parameters. This means that if a checked workpiece were used as the
reference standard (ISO/TR 16015:2003, 2003; ISO 15530-3:2011, 2011)
(a so-called reference workpiece), Eq. (10) would be as follows:
l20rwp=lrwp⋅1+αscΔtscrwp1+αrwpΔtrwp,
where lrwp and l20rwp are the length of the
reference workpiece at Δtrwp≠0K and
Δtrwp=0K correspondingly, Δtscrwp is the temperature difference at the scale
during measurements of lrwp, αrwp is the CTE of
the reference workpiece and αsc remains unchanged as we are
using the same measuring instrument.
Now it is necessary to check these workpieces using the same measuring
instrument as a comparator (ISO/TR 16015:2003, 2003). If the reference
workpiece's CTE and its temperature were αrwp=αwp and Δtrwp=Δtwp
correspondingly and the temperature of the scale during the reference
workpiece measurements were Δtscrwp=Δtscwp (assuming that all objects' temperature distribution is uniform), it
would be able to decrease the number of four unknown parameters to only one,
combining Eqs. (10) and (11) to a proportion
l20wplwp=l20rwplrwp.
Only l20rwp is now needed. As long as the value l20rwp is measured at 20 ∘C in a laboratory and
provided to us, Eq. (12) will not have any unknown parameters (out of the four
stated above) anymore.
The desired value l20wp in this case could be found as
l20wp=l20rwp⋅lwplrwp.
Using the reference workpiece according to Eq. (13), we can find out l20wp for any other workpiece of the same type (with the same
desired geometrical parameter) without knowing αwp,
αsc, twp and tscwp. The
components lwp and lrwp in Eq. (13) are estimated
(not actual!) values achieved using the scale of a measuring instrument.
Because usually the scale's CTE has a nonzero value and expands too, the
differences lrwp-l20rwp and
lwp-l20wp show the so-called relative
expansion of the workpiece compared to the scale which will be smaller than
the expected absolute expansion of the workpiece (e.g., lwpact-l20wp in Eq. 4).
Estimation of uncertainty
It is assumed that three of the four corresponding parameters are equal: αrwp=αwp, Δtrwp=Δtwp and Δtscrwp=Δtscwp (αsc=const). But, it is clear,
that in real life they never are; even CTEs of two different objects made
from one material may be slightly different. Also, despite the fact that it is required
that the reference workpiece and the workpiece to be measured are similar
(within certain tolerances) (ISO 15530-3:2011, 2011), there might be some
difference between geometrical and physical properties of the workpiece and
the reference workpiece.
Using Eqs. (10), (11) and (13) without making simplifying assumptions, it can
be stated that
l20wp=l20rwplwp1+αscΔtscwp1+αrwpΔtrwplrwp1+αwpΔtwp1+αscΔtscrwp,
where Δtscwp and Δtscrwp are the
temperature differences at the scale during measurements of lwp
and lrwp.
Uncertainty of the measurements can be estimated using Eq. (14) as a
mathematical model of measurements (JCGM 100:2008, 2008). After a series of
transformations it will look like
l20wp=l20rwplwp1+αrwpΔtrwp+αscΔtscwp+αrwpαscΔtrwpΔtscwplrwp1+αwpΔtwp+αscΔtscrwp+αwpαscΔtwpΔtscrwp.
In Eq. (15) αrwpαscΔtrwpΔtscwp→0 and
αwpαscΔtwpΔtscrwp→0 at any natural conditions, so the
formula can be shown as
l20wp=l20rwplwplrwp⋅1+αrwpΔtrwp+αscΔtscwp1+αwpΔtwp+αscΔtscrwp+….
Defining Δtscwp=Δtrwp+δtscwp, Δtscrwp=Δtrwp+δtscrwp, Δtwp=Δtrwp+δtwp and
αwp=αrwp+δαwp, Eq. (16) can be shown as
l20wp=fl20rwp,lwp,lrwp,Δtrwp,αrwp,αsc,δtscwp,δtscrwp,δtwp,δαwp=l20rwplwplrwp⋅1+αrwpΔtrwp+αscΔtrwp+δtscwp1+αrwp+δαwpΔtrwp+δtwp+αscΔtrwp+δtscrwp.
The differences δtscwp, δtscrwp, δtwp and δαwp (but not their uncertainties) are estimated to be zero.
The values l20rwp, lwp, lrwp,
Δtrwp, αrwp and αsc are
assumed to be uncorrelated. Taking this into account, the combined standard
uncertainty can be expressed as
uc2l20wp=c2l20rwpu2l20rwp+c2lwpu2lwp+c2lrwpu2lrwp+c2Δtrwpu2Δtrwp+c2αrwpu2αrwp+c2αscu2αsc+c2δtscwpu2δtscwp+c2δtscrwpu2δtscrwp+c2δtwpu2δtwp+c2δαwpu2δαwp,
where the sensitivity factors cxi=∂fx1,x2,…,xn∂(xi),i=1,2,…,n are
cl20rwp=lwp1+αrwpΔtrwp+αscΔtrwp+δtscwplrwp1+αrwp+δαwpΔtrwp+δtwp+αscΔtrwp+δtscrwp,
cδtwp=-l20rwplwp1+αrwpΔtrwp+αscΔtrwp+δtscwpαrwp+δαwplrwp1+αrwp+δαwpΔtrwp+δtwp+αscΔtrwp+δtscrwp2
and
cδαwp=-l20rwplwp1+αrwpΔtrwp+αscΔtrwp+δtscwpΔtrwp+δtwplrwp1+αrwp+δαwpΔtrwp+δtwp+αscΔtrwp+δtscrwp2.
Uncertainty of the calibration of the reference workpiece ul20rwp should be taken from the calibration certificate,
which states the value and uncertainty of l20rwp for the
reference workpiece.
The uncertainties of the measured lengths ulwp,
ulrwp and temperature difference uΔtrwp could be calculated as a standard uncertainty of type
A and/or type B after the values lwp, lrwp and
Δtrwp were measured by corresponding measurement
instruments.
Uncertainty of the reference workpiece's thermal expansion coefficient
u(αrwp) should be taken from the calibration certificate,
which states the value and uncertainty of αrwp for the
reference workpiece. The uncertainty for the values l20rwp
and αrwp (as well as the values themselves) is provided in
the same certificate (a CTE investigation should be specifically carried out
for the given object prior to the method application).
Regardless of what the value of cαsc in Eq. (24)
is, it was decided to use the same measuring instrument for defining lengths
of the reference workpiece and the workpiece, so αsc itself
can not (or, at least, should not) affect values lrwp and
lwp, and so it will be assumed that cαsc=0 (it is estimated that the CTE of the scale αsc
remains the same during our measurement session). Thus, the standard
uncertainty uδαsc (whatever its value
is) can be neglected.
The uncertainties of the differences of temperature uδtscwp, uδtscrwp, u(δtwp) and the difference of CTEs u(δαwp) should be considered specifically for the circumstances according to which the values δtscwp, δtscrwp, δtwp and δαwp were assumed.
Expanded uncertainty Up with a level of confidence p and coverage
factor kp can be calculated using Eq. (18) as follows:
Up=kp⋅ucl20wp.
The final estimation l20wpact of the value of
l20wp therefore can be expressed as
l20wp=l20wp±Up.
The uncertainty of measuring humidity and atmospheric pressure was
neglected, as α=f(t) (Amatuni, 1972).
All given formulae are well known and used here as an adaptation for a specific task solution.
Practical applicationExample 1
There are two gauge blocks
calibrated beforehand which will be used as the WP to be
measured using a CMM and the RWP (shown in Fig. 5). The
task is to determine the length of a WP (as if it were unknown) at the normal
temperature l20wp, evaluate the uncertainty of
measurements and compare calculated results with actual ones.
The workpiece (upper) and the reference workpiece (lower) on
the CMM's table in example 1.
The RWP's properties are known; its length at normal temperature and the CTE
are
l20rwp=125.0000 mm ±0.0002 mm and
αrwp=10.52×10-6K-1±1×10-6K-1, respectively.
Measurements which have been carried out assuming that αrwp=αwp give the following values of length of RWP and WP at a
current temperature lrwp=125.0043 mm and lwp=125.0048 mm.
Using Eq. (13) the desired value can be calculated:
l20wp=125.0000mm⋅125.0048mm125.0043mm=125.0005mm.
In this way, the first step of the task is done – l20wp is
known. Now, all possible uncertainty factors must be considered.
Evaluation of the uncertainty will be done in three steps:
calculation of sensitivity factors cxi,
consideration of standard uncertainty values uxi,
calculation of combined standard uncertainty ucfx and expanded uncertainty Up.
Before starting this procedure, we will summarize all known arguments and assume all
unknown arguments of the function
l20wp=fl20rwp,lwp,lrwp,Δtrwp,αrwp,αsc,δtscwp,δtscrwp,δtwp,δαwp,
where
l20wp=125.0005 mm,
l20rwp=125.0000 mm,
lwp=125.0048 mm,
lrwp=125.0043 mm,
Δtrwp=15 K,
αrwp=10.52×10-6K-1,
αsc=8×10-6K-1δtscwp=δtscrwp=δtwp=0K and
δαwp=0×10-6K-1.
If the parameter αsc is unknown it can be estimated as any
reasonable value; in this very instance it is provided though.
The calibration certificate gives the expanded uncertainty of the l20rwpU95=0.0002 mm with
coverage factor k=2. The standard uncertainty is then
ul20rwp=U95k=0.0001mm=0.1µm.
The value lwp is achieved as a mean value of five measurements. The
standard uncertainty of the measured value can be calculated as a type A
standard uncertainty:
ulwp‾=∑inlwpi-lwp2n-1=0.9µm.
According to the calibration certificate of the measuring instrument, the
expanded uncertainty of measurements with it is given as a distribution range
±1.5+L/500µm, where L (in mm) is the nominal size of the object to be
measured, and because the actual value lwpact can be
anywhere within this range at equal probability, a rectangular law of a
random value distribution is to be considered. The standard uncertainty due
to a random error of the measuring instrument (mi) is then
ulwpmi=1.5+125500µm3=1.0µm.
The standard uncertainty for the lwp will be
ulwp=u2lwp‾+u2lwpmi=0.9µm2+1.0µm2=1.4µm.
The standard uncertainty for the value lrwp can be calculated
similarly as for the lwp:
ulrwp‾=∑inlrwpi-lrwp2n-1=0.6µm,ulrwpmi=1.5+125500µm3=1.0µm,ulrwp=u2lrwp‾+u2lrwpmi=0.6µm2+1.0µm2=1.2µm.
The reported environmental temperature was ∼35∘C.
The temperature at the time of measurements was not recorded. By this, it is
assumed that Δtrwp=15 K (trwp=35∘C), and because the RWP, WP and the measuring instrument were in the
same chamber, all temperature differences δtscwp,
δtscrwp and δtwp are
estimated to be zero. The measurements took around 20 min during which
the environmental temperature might have changed by ±2 K. So,
taking into account the conditions described above and the expanded
uncertainty of the thermometer, which was used for environmental temperature
measurements U95=1.0 K (k=2), it is possible to
estimate the standard uncertainty as follows:
uΔtrwp‾=2K3=1.2K,uΔtrwpmi=U95k=0.5K,uΔtrwp=u2Δtrwp+u2Δtrwpmi=1.2K2+0.5K2=1.3K.
The calibration certificate gives the value of the αrwp with
the distribution range ±1×10-6K-1, and
because actual value αrwpact can be anywhere
within this range at equal probability, a rectangular law of a random value
distribution is to be considered. The standard uncertainty is then
uαrwp=1×10-6K-13=0.6×10-6K-1.
As was said before, the temperatures of the RWP, WP and the measuring
instrument are expected to be equal, but the differences δtscwp, δtscrwp and δtwp should be within the estimated range ±1.0K.
The standard uncertainty is
uδtscwp=uδtscrwp=uδtwp=1.0K3=0.6K.
Similarly to the speculations regarding αrwp, the
difference δαwp is estimated to be within the range
±1×10-6K-1. The standard uncertainty is
then
uδαwp=1×10-6K-13=0.6×10-6K-1.
Now it is possible to express the combined standard uncertainty
uc2l20wp according to Eq. (18):
uc2l20wp=1.02⋅0.1µm2+1.02⋅1.4µm2+-1.02⋅1.2µm2+0mmK-12⋅1.3K2+0mmK2⋅0.6×10-6K-12+1.0µmK-12⋅0.6K2+-1.0µmK-12⋅0.6K2+-1.3mmK-12⋅0.6K2+-1874416.2µmK2⋅0.6×10-6K-12=0.1µm2+1.4µm2+-1.2µm2+0µm2+0µm2+0.6µm2+-0.6µm2+-0.8µm2+-1.1µm2=6.0µm2.
The crucial contributors here are the values of lwp,
lrwp and δαwp – each of them contributed
≥1µm. These major contributors in the example are the
sources of the uncertainty that come from the measuring instrument and
the difference between CTEs of the objects to be measured. So, more accurate and
precise results can be obtained using a measuring instrument with a lower
uncertainty of measurements and workpieces made of the same material.
Uncertainty budget for example 1.
ErrorTypeStandardSensitivityUncertaintysourceuncertaintycoefficientcontributionxiu(xi)c(xi)u(xi)⋅c(xi), µmCalibrated length of the RWP at 20 ∘C(l20rwp=125.0000 mm)B0.1 µm10.1Measured length of the WP(lwp=125.0048 mm)B1.4 µm11.4– random effects during measurementsA0.9 µm– measuring instrumentB1.0 µmMeasured length of the RWP(lrwp=125.0043 mm)B1.2 µm-1-1.2– random effects during measurementsA0.6 µm– measuring instrumentB1.0 µmMeasured temperature of the RWP(trwp=Δtrwp+20∘C = 35 ∘C)B1.3 K0 µm K-10.0– random effects during measurementsA1.2 K– measuring instrumentB0.5 KCalibrated CTE of the RWP(αrwp=10.52×10-6 K-1)B0.6×10-6 K-10 µm K0.0Known/assumed CTE of the scale(αrwp=8×10-6 K-1)–Can be neglected0 µm K-10.0Possible temperature difference between the RWP and the scale during measurements of length of the WP(δtscwp=0 K)B0.6 K1.0 µm K-10.6Possible temperature difference between RWP and the scale temperature during measurements of length of the RWP(δtscrwp=0 K)B0.6 K-1.0 µm K-1-0.6Possible temperature difference between the RWP and the WP(δtwp=0 K)B0.6 K-1.3 µm K-1-0.8Possible CTE difference between RWP and WP(δαwp=0×10-6 K-1)B0.6×10-6 K-1-1 874 416.2 µm K-1.1Calculated length of the WP at 20 ∘Cl20wp=125.0005 mmCombined standard uncertaintyuc(l20wp)=2.5µmCoverage factor k95=2 expanded uncertaintyU95=k95⋅uc(l20wp)=2⋅2.5µm = 5.0 µm(confidence level p=95 %)
Therefore,
ucl20wp=2.5µm.
Using Eqs. (29) and (30), we can express the final value l20wpact
at the level of confidence p=95%:
Up=2⋅2.5µm=5.0µm,l20wp=(125.0005±0.0050)mm.
All calculated results according to Eqs. (18)–(28) are shown in Table 1.
The properties of the WP according to the calibration certificate are
l20wp=124.9968±0.0002mm
and αwp=11.60×10-6±1×10-6K-1 (the gauge blocks are from different
manufacturers). Now, the En criterion (Wöger, 1999) can be applied
to check if the calculated value is compatible with the value given in the
certificate (JCGM 200:2012, 2012):
En=ldet-lrefUdet2+Uref2,
where ldet is the measured result and lref is the reference value
that compatibility is to be checked with. The values Udet and Uref
are their expanded uncertainties correspondingly.
Compatibility is considered to be confirmed if En≤1.
In this example ldet and Udet are the values that were determined
during the measurements and uncertainty evaluation. The values lref and
Uref are given in the certificate. Therefore
ldet=125.0005mm,
Udet=0.0050mm,
lref=124.9968mm and
Uref=0.0002mm.
Applying Eq. (31), the criterion is
En=125.0005mm-124.9968mm0.0050mm2+0.0002mm2=0.0037mm0.0050mm≈0.8≤1.
Technically, the compatibility is confirmed (En≤1); however, the achieved criterion is very close to its boundary
value (at which the compatibility is disproved). Is there any way to make the
measured result ldet more reliable? As was stated above, in this
example the main uncertainty contributors are lwp,
lrwp and δαwp. The first two are due to
the measuring instrument and cannot be dramatically affected (unless the
measuring instrument is changed for a better one), whereas the last is up to
the material of the workpieces to be measured so that it can be greatly reduced.
Example 2
For this example, two actual workpieces are used. They are made of the
same material, so now it is known that their CTEs should be completely the
same (αrwp=αwp). However the workpieces have
different shape and length (along the measurement axis). Despite the fact that the
workpieces had not been officially calibrated, they were both measured using
the CMM at 20 ∘C multiple times with the uncertainty estimation
beforehand so that their geometrical parameters were known and could be checked.
The measurements will be made using a length gauge (see Fig. 6).
The reference workpiece (left) and the workpiece (right) and the length gauge in example 2.
The RWP has the following properties:
l20rwp=160.0013mm±0.0020mm and αrwp=23.6×10-6K-1.
Measurements that have been carried out give the following values of length
of RWP and WP at a current temperature:
lrwp=160.0418mm and lwp=150.0015mm.
Using Eq. (13), the desired value can be calculated
l20wp=160.0013mm⋅150.0015mm160.0418mm=149.9635mm.
Similar to the first example, we will summarize all known arguments and assume
all unknown arguments:
l20wp=149.9635mm,
l20rwp=160.0013mm,
lwp=150.0015mm,
lrwp=160.0418mm,
Δtrwp=23.5K,
αrwp=23.6×10-6K-1,
αsc=12.6×10-6K-1,
δtscwp=δtscrwp=δtwp=0K and
δαwp=0×10-6K-1.
In this example the parameter αsc according to the measuring
instrument's manual should be close to zero (made of Zerodur). However,
the instrument is mounted on a steel stand which has some nonzero CTE. So
the stand's CTE (a table value for steel) was taken as the scale's CTE.
The uncertainty of the l20rwpU95=0.0020mm with coverage factor k=2. The standard uncertainty is then
ul20rwp=U95k=0.0010mm=1.0µm.
The value lwp is achieved as a mean value of six measurements. The
standard uncertainty of the measured value can be calculated as a type A
standard uncertainty:
ulwp‾=∑inlwpi-lwp2n-1=0.2µm.
The calibration certificate for the measuring instrument is not available;
the manual states the accuracy of the device to be ±0.2µm, which we
will use as the expanded uncertainty of measurements, and because the actual
value lwpact can be anywhere within this range at equal
probability, a rectangular law of a random value distribution is to be
considered. The standard uncertainty due to a random error of the measuring
instrument is then
ulwpmi=0.2µm3=0.1µm.
The standard uncertainty for the lwp will be
ulwp=u2lwp‾+u2lwpmi=0.2µm2+0.1µm2=0.2µm.
The standard uncertainty for the value lrwp can be calculated
similarly as for the lwp:
ulrwp‾=∑inlrwpi-lrwp2n-1=0.2µm,ulrwpmi=0.2µm3=0.1µm.ulrwp=u2lrwp‾+u2lrwpmi=0.1µm2+0.2µm2=0.2µm.
The measured environmental temperature was 43–44 ∘C. The temperature at the time of measurements was not recorded. In this way, it is assumed that Δtrwp=23.5K
(trwp=43.5∘C), and because the RWP, WP and the
measuring instrument were in the same chamber, all temperature differences,
δtscwp, δtscrwp and
δtwp, are estimated to be zero. The measurements took around
10 min during which the environmental temperature might have changed
by ±1K. So, taking into account the conditions described
above and the expanded uncertainty of the thermometer, which was used for the
environmental temperature measurements U95=0.1K (k=2),
it is possible to estimate the standard uncertainty as following
uΔtrwp‾=1K3=0.6K,uΔtrwpmi=U95k=0.1K,uΔtrwp=u2Δtrwp+u2Δtrwpmi=0.6K2+0.1K2=0.6K.
The manufacturer of the aluminium billet (the one that the workpieces are
made of) only states the value of the CTE itself without any distribution
range; that is why a typical range of ±1×10-6K-1 is assumed. Because the actual value
αrwpact can be anywhere within this range at equal
probability, a rectangular law of a random value distribution is to be
considered. The standard uncertainty is then
uαrwp=1×10-6K-13=0.6×10-6K-1.
The temperatures of the RWP, WP and the measuring instrument are expected to
be equal, but the differences δtscwp, δtscrwp and δtwp should be within
the estimated range ±0.5K. The standard uncertainty is
uδtscwp=uδtscrwp=uδtwp=0.5K3=0.3K.
The difference δαwp is estimated to be 0×10-6K-1. The standard uncertainty is then
uδαwp=0×10-6K-13=0×10-6K-1.
Now it is possible to express the combined standard uncertainty
uc2l20wp according to Eq. (18):
uc2l20wp=0.92⋅1.0µm2+1.02⋅0.2µm2+-0.92⋅0.2µm2+0mmK-12⋅0.6K2+0mmK2⋅0.6×10-6K-12+1.9µmK-12⋅0.3K2+-1.9µmK-12⋅0.3K2+-3.5µmK-12⋅0.3K2+-3520237.2µmK2⋅0×10-6K-12=0.9µm2+0.2µm2+-0.2µm2+0µm2+0µm2+0.5µm2+-0.5µm2+-1.0µm2+0µm2=2.4µm2.
In this example crucial contributors are only the values of l20rwp and δtwp – each of them contributed
∼1µm. This means that using a measuring instrument
that is even more precise is not profitable. Technically speaking, better
accuracy in this case is hardly possible; one of the contributors depends on
the accuracy of the RWP's calibration (which is already expected to be
higher), and the other on the way that the environmental temperature is distributed.
Example 2 shows that the equality of the workpiece's CTEs is much more
important than the equality of the geometrical properties.
Uncertainty budget for example 2.
ErrorTypeStandardSensitivityUncertaintysourceuncertaintycoefficientcontributionxiu(xi)c(xi)u(xi)⋅c(xi), µmCalibrated length of the RWP at 20 ∘C(l20rwp=160.0013 mm)B1.0 µm0.90.9Measured length of the WP(lwp=150.0015 mm)B0.2 µm1.00.2– random effects during measurementsA0.2 µm– measuring instrumentB0.1 µmMeasured length of the RWP(lrwp=160.0418 mm)B0.2 µm-0.9-0.2– random effects during measurementsA0.2 µm– measuring instrumentB0.1 µmMeasured temperature of the RWP(trwp=Δtrwp+20∘C = 43.5 ∘C)B0.6 K0 µm K-10.0– random effects during measurementsA0.1 K– measuring instrumentB0.6 KCalibrated CTE of the RWP(αrwp=23.6×10-6 K-1)B0.6×10-6 K-10 µm K0.0Known/assumed CTE of the scale(αrwp=12.6×10-6 K-1)–Can be neglected0 µm K-10.0Possible temperature difference between the RWP and the scale during measurements of length of the WP(δtscwp=0 K)B0.3 K1.9 µm K-10.5Possible temperature difference between RWP and the scale temperature during measurements of length of the RWP(δtscrwp=0 K)B0.3 K-1.9µm K-1-0.5Possible temperature difference between the RWP and the WP(δtwp=0 K)B0.3 K-3.5µm K-1-1.0Possible CTE difference between RWP and WP(δαwp=0×10-6 K-1)B0×10-6 K-1-3520237.2µm K0.0Calculated length of the WP at 20 ∘Cl20wp=149.9635 mmCombined standard uncertaintyuc(l20wp)=1.6µmCoverage factor k95=2 expanded uncertaintyU95=k95⋅uc(l20wp)=2⋅1.6µm = 3.2 µm(confidence level p=95 %)
Therefore,
ucl20wp=1.6µm.
Using Eqs. (29) and (30), we can express the final value l20wpact at the level of confidence p=95%:
Up=2⋅1.6µm=3.2µm,l20wp=(149.9635±0.0032)mm.
All calculated results are shown in Table 2.
The established length of the WP using precise measurements is l20wp=149.9617mm±0.0020mm. So, for this example,
according to Eq. (31):
ldet=149.9635mmUdet=0.0032mmlref=149.9617mm,Uref=0.0020mm,En=149.9635mm-149.9617mm0.0032mm2+0.0020mm2=0.0018mm0.0038mm≈0.5≤1.
Here, compatibility is also confirmed. But, in this example, due to a lesser
value of En, reliability of the determined value ldet
(the probability that ldet and lref are really compatible) is
higher.
The En criterion is unfortunately inapplicable for the real
measurements as the value lref is usually not known. However, the
uncertainty contributors' analysis (which for both examples was performed
after calculation of the combined standard uncertainty according to
Eq. 18) can give enough information about the compatibility of the determined
value with some unknown reference value.
Note that all calculations for examples 1 and 2 were performed
using MS Excel. Due to a higher accuracy, some of the results might be
negligibly different than if they had been obtained using a conventional
calculator. The final errors were rounded up (e.g., 2.401 to 2.5 µm).
Conclusions
The suggested method does not provide a level of accuracy reachable with
methods which require temperature measurements; however, it is universal
(can be applied to any linear measuring instrument without any hardware
modifications), it does not slow down the measuring process for temperature
stabilization (does not create a so-called bottleneck at the production
conveyor) and it can be used by a measuring instrument's operator without
additional qualification training. The method can be applicable in production areas where no submicron accuracy is required.
The best results are achievable with shortening of the measurements' duration
(so the temperature does not change significantly) and the RWP should at least be
made of a similar material to the WP. During the measurements,
draughts and proximity to warming sources should be avoided (to prevent
inequality in the temperature distribution for the RWP and the WP under test).
No data sets were used in this article.
DS has developed the correction algorithm and derived the
uncertainty estimation as a part of his thesis work. RT has
initiated the project of correction of thermal expansion by reference to
calibrated workpieces and is supervising the thesis work.
Author Rainer Tutsch is a member of the editorial board
of the journal. Role of the funding source: the authors declare that they
do not have any personal or other forms of material interest and no subsequent
potential conflicts existing in the presented research work.
Acknowledgements
The author Dmytro Sumin gratefully acknowledges the funding of his scholarship by the Federal Ministry for Economic Cooperation and Development of Germany and
support of the Braunschweig International Graduate School of Metrology
B-IGSM. Edited by: Rosario Morello
Reviewed by: two anonymous referees
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