Elsevier

Measurement

Volume 170, January 2021, 108692
Measurement

Measurement uncertainty of generalized stiffness of machine tools

https://doi.org/10.1016/j.measurement.2020.108692Get rights and content

Highlights

  • Generalized stiffness measurements of a machine tool using many displacement sensors.

  • Method errors of stiffness measurements in uncertainty analysis.

  • The measurement method of generalized stiffness based on the model of rigid body motion.

  • Reproducibility of mounting metrological equipment is the main source of uncertainty measuring machine tools stiffness.

Abstract

This paper deals with the formulation of a new method of measuring machine tool stiffness. The method enables measurements to be made in generalized coordinates, i.e. measurements of translational and torsional stiffness. It is based on geometrical transformations of a rigid body. It is universal, because it allows any positioning of displacement sensors on the surface of the tested object and to freely transform the excitation force. This allows testing the stiffness of various medium-sized machine tools - lathes, milling machines, etc. The validation of the measurement method (uncertainty budget) and validation of estimated measurement uncertainty are presented. The components of reproducibility and method-specific uncertainties have been shown to contribute significantly more to the budget than the components of metrological characteristics of the sensors used in the measurements. It was also shown which components of complex uncertainty can be omitted from the analysis without losing the accuracy of its estimation. The advantages of the proposed method have been demonstrated on the example of stiffness measurements on a milling machine.

Introduction

The stiffness of a machine tool reflects its resistance to deformation caused by cutting and gravity forces. It has a significant impact on the aggregated inaccuracy of the entire system. Even under stable machining conditions (i.e. without self-exciting vibrations), with finishing parameters for easily machined materials, geometric errors of workpieces will include, among others: SL elastic deformation effects and volumetric error effects. These effects are of the same order [1]. It is natural that the priority of errors caused by elastic deformations will increase as the cutting forces increase, i.e. when machining difficult-to-cut alloys and when maximizing productivity. This explains the increase (in recent years) in interest and the increase in the number of publications on the numerical compensation of geometric errors of workpieces caused by cutting forces [2], [3]. It is commonly claimed that this link is a tool holder with a tool. Slender tools are characterized by 15 times and short and thick ones by 5–7 times less stiffness than SL elements of the machine tool [4]. These proportions are significantly underestimated in relation to those mentioned in [5]. Quantitative data on this subject can be found in [6], [7]. It is also known that the increase in temperature caused by the spindle rotations has a slight effect on its stiffness [8]. An interesting fact is that the stiffness of 5 identical milling machines can vary by up to 40% [9], [10].

The stiffness distribution in the working space is modelled for a variable SL configuration of the machine tool. This phenomenon is called VolS [5]. Finite element method and the procedure to reduce the number of DoF were used by Gao et al. [11] to model VolS. Whereas in [5], [12] the variation method was used by Archenti to identify parameters of moving links. They were modeled with 3-DoF elastic links. The identification was carried out on the basis of stiffness measurement results using loaded double ball bar [13]. Loaded double ball bar stiffness measurements require the measuring system to be moved sequentially to different points in the machining space. In this way the possibility of indirect determination (modelling) of the VolS distribution of the milling machine was obtained. It would be appropriate today to speak of VolS as resistance to displacement of the functional point caused by the action of the equivalent of cutting force. In this sense, it is known that considering a force acting in one direction only (1-DoF) causes all six generalized movements (6-DoF) of the point under consideration to occur [11]. Therefore, in the following Sections QSSMT measurement method based on 6-DoF were analysed in detail. Special attention was paid to the issues of error identification, assumptions for the method and uncertainty analysis of QSSMT measurements.

QSSMT tests are generally conducted for two reasons. To compare the stiffness of different machine tools [1] and to obtain stiffness coefficients. Such coefficients, together with the knowledge of the value and direction of the cutting force, are used to predict geometric errors in workpieces. The former is of interest to machine tool manufacturers and the latter to users. The method of measuring the static stiffness proposed in ISO 230–1 [14] is, owing to its advantages, completely sufficient to carry out comparative tests of machines within one company. However, the prediction of geometric errors of machined workpieces is more interesting for practical reasons, because it allows to improve the aggregated accuracy of the machine tool. Today, there is no doubt that intelligent machine tools [15], [16], [17] (equipped with intelligent spindles and sensory systems) are able to monitor the value and direction of cutting load on-line. Knowing how to scale these forces for compensation corrections is an effective solution for improving machine tool accuracy. The scaling elements are the compliance coefficients of a machine tool (i.e. the inverse of stiffness coefficients).

To the best of authors' knowledge, there is a research gap in the literature on estimating uncertainty of QSSMT measurements. This gap concerns in particular the reproducibility studies and the complete omission in the uncertainty analysis of the budgeting issues of the components brought by the measurement method. The main purpose of writing this paper is to fill this gap. The confirmation of the correctness of the above statement is the fact that in the studies quoted here we do not find budgets or even numerical values for the machine tool stiffness coefficients measurement uncertainty ranges [18]. However, repeatability ranges for stiffness coefficients can be found e.g. in [5], [12]. In one of the recent papers [19], an uncertainty budget for estimating the torsional stiffness coefficients of an industrial robot was presented. It takes into account the components introduced by the metrological equipment and the reproducibility of the displacement measurement due to the change of the measurement operator. However, the components introduced by the measurement method are completely omitted. The estimation of uncertainty in relation to the manufacturing tolerances of the dimensions is considered in [20]. It can be emphasized that this work is currently one of the most advanced in the field of uncertainty analysis of machine tool stiffness measurements. Whereas, it can be approached critically to the reliability of obtaining information about the stiffness of machine tools by Pimenov et al. in [21], [22]. There are published research results devoted to the prediction of machining errors with the use of models based on the knowledge of stiffness coefficient values. It is clearly visible that the level of conviction of the authors of these studies about the effectiveness of predictions of machining errors is disproportionately higher than the level of justification for obtaining the values of machine stiffness coefficients (sometimes using very primitive measurement methods). It is significant that stiffness coefficients are treated as determined values and not as random values as it should be. With regard to measuring and testing the stiffness of machine tools, this is a dogma, not a scientific consideration. For this reason, it is not possible to carry out experimental falsification of models for prediction of machining errors. Therefore, this article is the first time that the results of stiffness measurements are presented together with the expanded uncertainty. Uncertainty estimation comprehensively takes into account errors of environmental effects, operator's errors, errors of metrological equipment and in particular method errors. In this respect, the method (presented in Section 2) and budgeting for uncertainty (in Section 5) is an original contribution to the development of QSSMT measurement methods. The results obtained can be the basis for validation and comparison with other measurement methods. Unfortunately, at this stage, it is not possible to make such a comparison due to the fact that the estimation procedures and expanded uncertainty of QSSMT measurements are not widely published.

Section snippets

Measurement model of generalized stiffness and estimation of its uncertainty

As mentioned earlier, one of the problems considered in this work is the reliability of stiffness measurements in the context of compensating for deformations of SL components of machine tools during the machining process. This directly results in boundary conditions for the QSSMT measurement method. Namely, if deviations from deflections are compensated for, then under finishing conditions it is to improve the geometric accuracy of the final product. For machining other than finishing, this

Error sources identification in QSSMT measurement

This Section presents the identification of error sources for QSSMT measurement. This stage is essential for obtaining data for uncertainty analysis. The QSSMT measurement is performed under the conditions of quasi-static load, e.g. a low-frequency periodical waveform. A low frequency is one where the effect of inertia forces on displacement measurements can be neglected. This means the frequency is much lower than when rocking vibrations occur. According to [33] for typical machine tools the

Validation of estimation of measurement uncertainty

Due to the accuracy of uncertainty calculation the increase in number n causes an intensive increase in the demand for the number of repetitions of the MCM iteration loop [27]. To establish this accuracy, it is recommended to validate the methods [27]. This comes down to establishing a minimum number of MCM iterations for the acceptable accuracy threshold for uncertainty calculations. Fig. 8 presents a convergence graph of the relative error of uncertainty of measurement of K, B, L and τ.

After

Uncertainty budget for generalized stiffness measurement

The uncertainty budget is a summary statement of the considered standard uncertainties and/or their effect on the value of the complex uncertainty of measurement uc. The value of the share of the considered standard uncertainty U.Bud.(xi) in the combined standard uncertainty was calculated from the Eq. (30).U.Bud.xi=ucx1,...,xn2-uc*x1,...,xn-12ucx1,...,xn2·100%

where: uc – combined standard measurement uncertainty, uc* – combined uncertainty calculated without the considered source of error xi, n

Measuring the generalized stiffness of milling machines – Practical considerations

The greatest advantage of the method of generalized stiffness measurement analysed in this paper is that it allows to determine individual effects of translational and rotational displacements. Stiffness measurements by other methods usually include in the displacement measurement signal the equivalent of translational and rotational displacements. This constitutes a disadvantage of such methods. The method presented here is free of this disadvantage. Therefore, for demonstrative as well as

Discussion

This work shows that the uncertainties related to the reproducibility of the assembly of metrological equipment components on the machine tool and the uncertainties of the characteristic methods of measurement are a major contributor to the uncertainty budget of the quasi-static stiffness measurement methods considered here. This work has shown that the scientific studies indicate only the uncertainty components for the sensors used in the measurements are insufficient to reliably assess the

Conclusions

In accordance with international standards, the uncertainty of measurement must be stated for all measurements. including the results of machine tool tests. Detailed procedures for estimating the uncertainty of measurement when testing the geometric accuracy of machine tools can be found in the ISO standards. For other types of tests, i.e. the measurement of stiffness or thermal drift, such guidelines have not been formulated yet. Therefore, this obligation rests with the person performing the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was carried out on the research apparatus purchased under the project No. RPZP.01.03.00-32-0004/17. The project is co-financed by the European Union from the European Regional Development Fund within the framework of the Regional Operational Programme of the Zachodniopomorskie Voivodeship for the years 2014-2020. The project is co-financed by the Polish Ministry of Science and Higher Education.

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