Simulating Variable Quasistatic Stiffness of Machine Tool Spindle Unit

Precision of a machine tool directly depends on the rotation precision of the spindle unit, which, in turn, is determined by the stiffness of the bearings that support it. The bearing is a complex mechanical device, the stiffness of which is not constant either radially or axially and depends on multiple factors, such as purely geometrical considerations, operating conditions, initial mechanical and thermal loads. Variable stiffness computation involves developing an adequate model that accounts for existing loads and conditions. The paper presents a method for computing variable stiffness of angular contact ball bearings, since they are the most widely used type in spindle units. The variable stiffness model takes into account the distribution of external loads between bearing balls via a function describing the variation of the contact angle between balls and bearing races. It ensures adequate accuracy for the majority of spindle unit stiffness problems. We validated our model by comparing our simulation results to experimental data and the results derived from models published previously. This variable bearing stiffness model is used for fast computations in both research and development. It was tested by computing displacements of a spindle nose in an experimental spindle unit

References

[1] Abele E., Altintas Y., Brecher C. Machine tool spindle units. CIRP Annals, 2010, vol. 59, no. 2, pp. 781–802. DOI: 10.1016/j.cirp.2010.05.002

[2] Holkup T., Cao H., Kolar P., Altintas Y., Zeleny J. Thermo-mechanical model of spindles. CIRP Annals, 2010, vol. 59, no. 1, pp. 365–368. DOI: 10.1016/j.cirp.2010.03.021

[3] Marsh E.R. Precision spindle metrology. DEStech Publications, 2010. 165 p.

[4] Gorelik I.G. Razrabotka metodov rascheta i povyshenie kachestva vysokoskorostnykh shpindelnykh uzlov. Diss. kand. tekh. nauk [Development of calculation method and raising efficiency of high speed spindle units. Сand. tech. sc. diss.]. Moscow, ENIMS Publ., 1987. 252 p.

[5] Chernyanskiy P.M., Skhirtladze A.G. Proektirovanie i remont shpindelnykh uzlov [Engineering and repair of spindle units]. Moscow, INFRA-M Publ., 2014. 272 p.

[6] Jones A.B. Analysis of stress and deflections. New departure engineering data. Vol. 1. General Motors, 1946. Pp. 12–24.

[7] De Mul J., Vree J., Maas D. Equilibrium and associated load distribution in ball and roller bearings loaded in five degrees of freedom while neglecting friction. Part 1: General theory and application to ball bearings. Journal of Tribology, 1989, vol. 111, no. 1, pp. 142–148. DOI: 10.1115/1.3261864

[8] Hert H. On the contact of elastic solids. J. Reine Angew. Math., 1881, vol. 92, pp. 156–171.

[9] Zverev I.A. Mnogokriterialnoe proektirovanie shpindelnykh uzlov na oporakh kacheniya. Diss. dok. tekh. nauk [Multicriteria engineering of spindle units on rolling-contact bearing. Doc. tech. sc. diss.]. Moscow, STANKIN Publ., 1997. 227 p.

[10] Timoshenko G. Theory of elasticity. McGraw-Hill, 1970. 608 p.

[11] Cao Y. Modeling of high-speed machine-tool spindle system. Ph.D. thesis. The University of British Columbia, 2006. 140 p.

[12] Adams G.G., Nosonovsky M. Contact modelling—forces. Tribol. Int. 2000, vol. 33, no. 5-6, pp. 431–442. DOI: 10.1016/S0301-679X(00)00063-3

[13] Hernot X., Sarto M., Guillot J. Calculation of the stiffness matrix of angular contact ball bearings by using the analytical approach. ASME J. Mech. Des., 2000, vol. 122, no. 1, pp. 83–80. DOI: 10.1115/1.533548

[14] Kurvinen E., Sopanen J., Mikkola A. Ball bearing model performance on various sized rotors with and without centrifugal and gyroscopic forces. Mech. Mach. Theory, 2015, vol. 90, pp. 240–260. DOI: 10.1016/j.mechmachtheory.2015.03.017

[15] Holkup T., Holy S. Complex modelling of spindle rolling bearings. Journal of Machine Engineering, 2006, vol. 6, no. 3, pp. 48–61.

[16] Li X., Lu Yi., Yan Ke, Liu J., Hong J. Study on the influence of thermal characteristics of rolling bearings and spindle resulted in condition of improper assembly. Applied Thermal Engineering, 2017, vol. 114, pp. 221–233. DOI: 10.1016/j.applthermaleng.2016.11.194

[17] Holkup T., Kolar P. Influence of thermo-mechanical conditions of rolling bearings on the dynamics of machine tool spindles. MATAR PRAHA 2008. Part 1: Drives & Control, Design, Models & Simulation. 2008, pp. 141–146.

[18] Harris T.A. Rolling bearing analysis. CRC Press, 2006. 760 p.

[19] Zverev I.A., Averyanova I.O. Complex mathematical model of highspeed spindle unit on rolling-contact bearing. STIN, 1995, no. 1, pp. 7–9 (in Russ.).

[20] Segida A.P. Raschet i issledovanie temperaturnykh poley i temperaturnykh deformatsiy metallorezhushchikh stankov. Diss. kand. tekh. nauk [Calculation and research on temperature fields and temperature deformations of metal-cutting machines. Cand. tech. sc. diss.]. Moscow, ENIMS, 1984. 196 p.

[21] Dmitriev B.M. Issledovanie prichin izmeneniya tochnosti stanka i razrabotka metoda stabilizatsii tochnosti na printsipakh samoregulirovaniya. Diss. dok. tekh. nauk [Study on cause of change in machine accuracy and development of stabilization method based on self-regulation principles. Cand. tech. sci. diss.]. Moscow, Bauman MSTU Publ., 2007. 218 p.

[22] Kovalev M.P., Narodetskiy M.Z. Raschet vysokotochnykh sharikopodshipnikov [Calculation of high-precision ball bearings ]. Moscow, Mashinostroenie Publ., 1975. 280 p.

[23] Sheng X., Li B., Wu Zh., Li H. Calculation of ball bearing speed-varying stiffness. Mech. Mach. Theory, 2014, vol. 81, pp. 166–180. DOI: 10.1016/j.mechmachtheory.2014.07.003

[24] Gargiulo Jr. A simple way to estimate bearing stiffness. Machine Design, 1980, vol. 52, no. 17, pp. 107–110.

[25] Guo Y., Parker R.G. Stiffness matrix calculation of rolling element bearings using a finite element/contact mechanics model. Mech. Mach. Theory, 2012, vol. 51, pp. 32–45. DOI: 10.1016/j.mechmachtheory.2011.12.006

[26] Zhang J., Fang B., Zhu Yo., Hong J. A comparative study and stiffness analysis of angular contact ball bearings under different preload mechanisms. Mech. Mach. Theory, 2017, vol. 115, pp. 1–17. DOI: 10.1016/j.mechmachtheory.2017.03.012

[27] Kraus J., Blech J., Braun S. In situ determination of rolling bearing stiffness and damping by modal analysis. J. Vib. Acoust. Stress. Reliab. Des., 1987, vol. 109, no. 3, pp. 235–240. DOI: 10.1115/1.3269426

[28] Zahedi A., Movahhedy M.R. Thermo-mechanical modeling of high speed spindles. Scientia Iranica, 2012, vol. 19, no. 2, pp. 282–293. DOI: 10.1016/j.scient.2012.01.004

[29] Chi-Wei Lin, Yang-Kuei Lin, Chih-Hsing Chu. Dynamic models and design of spindle-bearing systems of machine tools: a review. International Journal of Precision Engineering and Manufacturing, 2013, vol. 14, no. 3, pp. 513–521. DOI: 10.1007/s12541-013-0070-6

[30] Perel L.Ya., Filatov A.A. Podshipniki kacheniya: raschet, proektirovanie i obsluzhivanie opor [Ball bearings: calculation, engineering and supports maintainance]. Moscow, Mashinostroenie Publ., 1992. 608 p.

[31] Frolov A.V. Povyshenie tochnosti shpindelnykh uzlov pretsizionnykh stankov metodami termouprugogo modelirovaniya pri zadannoy ikh teploustoychivosti. Diss. kand. tekhn. nauk [Raising precision of spindle units of precision machines at given thermal stability]. Moscow, Bauman MSTU Publ., 2007. 284 p.