Skip to main content

Advertisement

SpringerLink
Log in

Analysis and mathematical model of the circumferential accuracy of the groove cut on the surface of rotation

  • Review
  • Published:
Journal of the Brazilian Society of Mechanical Sciences and Engineering Aims and scope Submit manuscript

A Correction to this article was published on 11 October 2019

This article has been updated

Abstract

An analysis is made of the technology of vortex cutting method of groove for sealing rings provided on the lateral surfaces of rotation of a part of the sealing system and the dimensional chains thus formed, as well as accuracy issues by applying probability theories and dimensional chains. Features of the developed technology are presented; the factors influencing the accuracy of the circular diameter of the groove are analyzed, and mathematical dependencies are given for their determination; the mathematical model of the accuracy of the diameter of the groove is derived. The technological dimension chain, which is formed in the vortex cutting method of groove, is given its properties, and the new linear–circular dimensional chain, which is not found in the literature, and its mechanism of formation are presented. The features of the linear–circular dimensional chain and the management of the functional relation between the linear and circular feeds are used to determine the technological capabilities of the vortex method on the lateral surface of the groove. Also, families of possible single- and double-symmetrical forms of the grooves are identified. It is recommended to use the results obtained, when analyzing the technology of cutting of grooving and other similar surfaces with a vortex method, when solving accuracy problems, including the derivation of a mathematical model of accuracy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Change history

  • 11 October 2019

    The article Analysis and mathematical model of the circumferential accuracy of the groove cut on the surface of rotation written by Ugurlu M. Nadirov, Nariman M. Rasulov was originally published electronically on the publisher’s Internet portal (currently SpringerLink) on 30 August 2019 with open access.

Abbreviations

\(f_{t}\) :

Circular feed

\(f_{m}\) :

Linear feed

\(f{}_{r}\) :

Vertical feed

\(N_{s}\) :

Rotational motion

\(T_{m}\) :

Norms of the time

\(d_{\text{t}}\) :

Distance between the tool base surfaces of the tool head

\(b_{o}\) :

Distance between the base surface and the cutting edge of the tool

\(D\) :

Inner diameter of the groove

\(D_{0}\) :

Nominal diameter of the surface of rotation

\(a\) :

Growth of the groove diameter, in the cross section

\(D_{\text{t}}\) :

Size of the tool’s position at the tool head

\(L\) :

Size of linear displacement of the workpiece

\(B\) :

Length of the chord

\(\alpha\) :

Angle of rotation of the workpiece

\(D_{m}\) :

Outer diameter of the groove

\(b_{\text{t}}\) :

Width of the tool

\(\delta_{{\text{D}_{0} }}\) :

Initial error, resulting from the complex effect of tolerance of the diameter \(D_{0}\) and the error of the angle of turn of the workpiece

\(D_{\text{max} }\) :

Maximum diameter of the groove to be cut

\(D_{\text{min} }\) :

Minimum diameter of the groove to be cut

\(\delta_{a}\) :

Initial error resulting from the turn of the workpiece

\(\Delta \alpha\) :

Allowable angular error of turning the workpiece

\(\alpha_{\text{min} }\) :

Half of the minimum angle of the turn of the workpiece

\(T_{{\text{D}_{0} }}\) :

Tolerance of the diameter of the surface of rotation

\(\delta_{L}\) :

Initial error in the diameter \(D\) in the section with the angular position, expressed by the angle \(\beta\), associated with the error of only linear feed \(f_{m}\)

\(\delta_{{f_{t} }}\) :

Initial error in the diameter \(D\) in a section with an angular position, expressed by the angle \(\beta\), associated with the error of only circular feed \(f_{t}\)

\(\Delta L_{x}\) :

Permissible error of the drive of the linear feed

\(\Delta f_{t}\) :

Permissible error of the drive of the circular feed

\(\beta\) :

Angular position of the groove size

\(t\) :

Coefficient determining the percentage of risk of getting a reject when groove cutting

\(\lambda_{1} ,\,\lambda_{2} ,\,\,\lambda_{3} ,\,\,\lambda_{4}\) and \(\lambda_{5}\) :

Coefficients that take into account the distribution forms of the corresponding primary errors

\(\delta_{{\text{w}}}\) :

Error related to the accuracy of manufacture and tool wears

\(\delta_{{\text{D}_{\text{t}} }}\) :

Error associated with the position of the tool in the head

\(T_{\text{w}}\) :

Tolerances for tool manufacture and wear

\(\Delta f_{t}\) :

Formed the errors of the device parameters

\(T_{\text{d}}\) :

Tolerance of the diameter \(D\)

\(f_{m\text{max} }\), \(f_{m\text{min} }\) :

Largest and smallest values of the linear feed, respectively, provided by the drive of the machine when it installed

\(T_{{\text{D}_{\text{t}} }}\) :

Tolerance of the diameter \(D_{\text{t}}\)

References

  1. Calleja A, Fernández A, Rodriguez A, de Lacalle LNL, Lamikiz A (2013) A new approach for the production of blades by hybrid processes. In: Davim JP (ed) Nontraditional machining processes research advances. Springer, London, pp 205–230

    Chapter  Google Scholar 

  2. Kief HB, Roschiwal HA (2009) CNC-Handbuch 2009/2010. Carl Hanser, Munchen

    Google Scholar 

  3. Dalsky AM, Kosilova AG, Meshcheryakova RK et al (eds) (2003) Handbook of the technologist of the machine builder. Mechanical Engineering, vol 2, pp 944

  4. DeGarmo EP, Black JT, Kohser RA (2003) Materials and processes in manufacturing, 9th edn. Wiley, pp 1168

  5. de Oliveira AL, Donatelli GD (2017) Historical measurement data reuse and similarity analysis for dimensional production tolerancing of injected plastic parts. J Braz Soc Mech Sci Eng 39(10):4161–4175

    Article  Google Scholar 

  6. Aydin M, Ucar M, Cengiz A et al (2015) Identification of static surface form errors from cutting force distribution in flat-end milling processes. J Braz Soc Mech Sci Eng 37(3):1001–1013

    Article  Google Scholar 

  7. Yanyukina MV, Bolotov MA, Ruzanov NV (2018) Interrelated dimensional chains in predicting accuracy of turbine wheel assembly parameters. In: 11th International conference on mechanical engineering, automation and control systems (MEACS). IOP conference series: materials science and engineering, vol 327, Russia

  8. Zhao C, Cheung CF (2018) Theoretical and experimental investigation of the effect of the machining process chain on surface generation in ultra-precision fly cutting. Int J Adv Manuf Technol 99(9–12):2819–2831

    Article  Google Scholar 

  9. Singh PK, Jain SC, Jain PK (2005) Advanced optimal tolerance design of mechanical assemblies with interrelated dimension chains and process precision limits. Comput Ind 56(2):179–194

    Article  Google Scholar 

  10. Jawale HP, Jaiswal A (2018) Investigation of mechanical error in four-bar mechanism under the effects of link tolerance. J Braz Soc Mech Sci Eng 40(8):UNSP 383

    Article  Google Scholar 

  11. Hu M, Ma J, Zhao W (2016) Variable dimensional chain of stroke-related mechanical assemblies. Proc Inst Mech Eng Part B J Eng Manuf 230(5):909–922

    Article  Google Scholar 

  12. Rasulov NM, Nadirov UM (2015) Accuracy research of the diametrical sizes forming at gear shaping by stepped cutter. Sci Tech J Inf Technol Mech Opt 15(5):893–899

    Google Scholar 

  13. Shrivastava Y, Singh B (2018) Estimation of stable cutting zone in turning based on empirical mode decomposition and statistical approach. J Braz Soc Mech Sci Eng 40(2):UNSP 77

    Article  Google Scholar 

  14. Patil RA, Gombi SL (2018) Experimental study of cutting force on a cutting tool during machining using inverse problem analysis. J Braz Soc Mech Sci Eng 40(10):UNSP 494

    Article  Google Scholar 

  15. Heling B, Aschenbrenner A, Walter MSJ et al (2016) On connected tolerances in statistical tolerance-cost-optimization of assemblies with interrelated dimension chains. In: 14th CIRP conference on computer aided tolerancing (CAT). Procedia CIRP, Wingquist Lab, Gothenburg, Sweden, vol 43, pp 262–267

  16. Li KL, Wang ZY, Li JF et al (2008) The parallel-chain in the machining-process-dimension-chain and application. In: 6th International conference on e-engineering and digital enterprise. Applied mechanics and materials, vol 10–12, pp 615–620

  17. Rasulov NM (2001) Technological dimensional communications when the thread-rolling. Machine builder, No 8, pp 12–16, ISSN 0025-4568

  18. Nadirov UM (2017) Testing and implementing the process of groove cutting on the part «insert» using vortex method. In: Proceedings of higher educational institutions. Machine building, vol. 1, pp 79–85

  19. Rasulov NM, Nadirov UM (2015) The specifics of vortex machining of ring profiles on lateral surfaces of rotation. In: Proceedings of higher educational institutions. Machine building, vol. 12, pp 45–51

  20. Nadirov UM, Rasulov NM (2016) Fundamentals of quality assurance of the grooves on side rotational surfaces machined by vortex method. In: Proceedings of higher educational institutions. Machine building, vol. 3, pp 65–73

  21. Lauro CH, Brandão LC, Ribeiro Filho SM, Davim JP (2015) Quality in the machining: characteristics and techniques to obtain good results. In: Davim JP (ed) Manufacturing engineering: new research, chapter 5. Nova Publishers, New York, pp 51–75. ISBN 978-1-63463-378-9

    Google Scholar 

  22. Liu H, Xue X, Tan G (2010) Backlash error measurement and compensation on the vertical machining center. Engineering 2(6):403–407

    Article  Google Scholar 

  23. Majda P (2012) Relation between kinematic straightness errors and angular errors of machine tool. Adv Manuf Sci Technol 36(1):47–53

    Google Scholar 

  24. Okafor AC, Ertekin YM (2000) Derivation of machine tool error models and error compensation procedure for three axes vertical machining center using rigid body kinematics. Int J Mach Tools Manuf 40:1120–1221

    Article  Google Scholar 

  25. Schwenke H, Knapp W, Haitjema H, Weckenmann A, Schmitte R, Delbressine F (2008) Geometric error measurement and compensation of machines: an update. CIRP Ann Manuf Technol 57(2):660–675

    Article  Google Scholar 

  26. Korn GA, Korn TM (2000) Mathematical handbook for scientists and engineers: definitions, theorems, and formulas for reference and review (Dover civil and mechanical engineering), 2 revised edn. Dover Publications, New York

    MATH  Google Scholar 

  27. McQuarrie DA (2003) Mathematical methods for scientists and engineers. University Science Books, Sausalito

    MATH  Google Scholar 

  28. Venttsel ES, Ovcharov LA (2000) Theory of probability and its engineering applications. Vysshaya shkola, Moscow

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Engineering Graphics, Azerbaijan Technical University, Baku, Azerbaijan

    Ugurlu M. Nadirov

  2. Mechanical Engineering Department, Azerbaijan Technical University, Baku, Azerbaijan

    Nariman M. Rasulov

Authors
  1. Ugurlu M. Nadirov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nariman M. Rasulov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ugurlu M. Nadirov.

Additional information

Technical Editor: Lincoln Cardoso Brandao.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The original version of this article was revised due to a retrospective Open Access cancellation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nadirov, U.M., Rasulov, N.M. Analysis and mathematical model of the circumferential accuracy of the groove cut on the surface of rotation. J Braz. Soc. Mech. Sci. Eng. 41, 384 (2019). https://doi.org/10.1007/s40430-019-1883-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40430-019-1883-8

Keywords

Search

Navigation