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.

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

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

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.

Thermal expansion is characterized with the coefficient of thermal expansion
(CTE) (ISO/TR 16015:2003, 2003):

where

Using Eq. (1),

Taking into account Eq. (2) the length

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

The nominal length of the produced workpiece is defined at the standard
temperature 20

The main problem of this formula is that

Behavior of the object at different temperatures.

The actual value of the workpiece's length

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

Estimated values

Changing temperature of the scale with constant temperature of the workpiece:

In Fig. 3 it is seen that the workpiece had the constant temperature

Estimated values

In real life changing of both parameters

Behavior of the scale at different temperatures.

Behavior of both objects at different temperatures.

The workpiece had the temperature

If we assume all parameters with subscript “1” (the left side of Eq. 8) to
have been achieved at

It is clearly seen that in the case of equality of absolute expansion characterization values of the workpiece and the scale (

Analyzing Eq. (10), it can be discovered that

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

The desired value

Using the reference workpiece according to Eq. (13), we can find out

It is assumed that three of the four corresponding parameters are equal:

Using Eqs. (10), (11) and (13) without making simplifying assumptions, it can
be stated that

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

In Eq. (15)

Defining

The differences

Uncertainty of the calibration of the reference workpiece

The uncertainties of the measured lengths

Uncertainty of the reference workpiece's thermal expansion coefficient

Regardless of what the value of

The uncertainties of the differences of temperature

Expanded uncertainty

The final estimation

The uncertainty of measuring humidity and atmospheric pressure was
neglected, as

All given formulae are well known and used here as an adaptation for a specific task solution.

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

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

Measurements which have been carried out assuming that

Using Eq. (13) the desired value can be calculated:

Evaluation of the uncertainty will be done in three steps:

calculation of sensitivity factors

consideration of standard uncertainty values

calculation of combined standard uncertainty

If the parameter

The calibration certificate gives the expanded uncertainty of the

Uncertainty budget for example 1.

Therefore,

The properties of the WP according to the calibration certificate are

Compatibility is considered to be confirmed if

In this example

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 (

The reference workpiece (left) and the workpiece (right) and the length gauge in example 2.

The RWP has the following properties:

Measurements that have been carried out give the following values of length
of RWP and WP at a current temperature:

Using Eq. (13), the desired value can be calculated

In this example the parameter

The uncertainty of the

Uncertainty budget for example 2.

Therefore,

The established length of the WP using precise measurements is

The

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

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.

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