Features of Applying the Force Displacement Theory to the Contact Stiffness Calculation for the Spindle Angular Contact Ball Bearing Support

Abstract

Contact stiffness is an important parameter for describing the interaction of plenty precise engineering solutions, for example, for machine-tool manufacture, the most important element of the carrier system is the spindle assembly supports, largely determining the accuracy of machining parts on the machine. The article considers an example of a system approach to describing the variable stiffness of a high-precision spindle bearing, combining an analytical approach to calculating the variable quasi-static stiffness of an angular contact ball bearing and a methodological approach that takes into account the contact normal and tangential force displacements of the upper bearing ring while considering the friction force for tangential interactions. The proposed system approach to calculating the stiffness of a ball bearing support is based on fairly simple models, in particular, the apparatus proposed by Professor P.M. Chernyanskiy, describing the phenomena of changing the spindle supports stiffness. These changes were experimentally obtained and weren’t described in earlier works. It is shown that the experimental stiffness consists of elastic and contact components, depending on the friction forces, therefore, the concept of apparent stiffness is introduced for data that are not taken into account in the processes under study. The effect of the axial displacement of the spindle is explained on the basis of contact interactions in the support of the spindle unit. The resulting analytical system model was tested on well-known bearing supports and the obtained stiffness solutions were compared with validated experimental data from open sources

Please cite this article in English as:

Frolov A.V. Features of applying the force displacement theory to the contact stiffness calculation for the spindle angular contact ball bearing support. Herald of the Bauman Moscow State Technical University, Series Mechanical Engineering, 2022, no. 3 (142), pp. 100--128 (in Russ.). DOI: https://doi.org/10.18698/0236-3941-2022-3-100-128

References

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

[2] Chernyanskiy P.M. Osnovy proektirovaniya tochnykh stankov. Teoriya i raschet [Principles of precision machine tools design. Theory and calculation]. Moscow, Knorus Publ., 2020.

[3] Chernyanskiy P.M., Skhirtladze A.G. Proektirovanie i remont shpindel’nykh uzlov [Engineering and repair of spindle units]. Moscow, Infra-M Publ., 2014.

[4] Chernyanskiy P.M., ed. Proektirovanie avtomatizirovannykh stankov i kompleksov. T. 1 [Design of automated machine tools and complexes. Vol. 1]. Moscow, Bauman MSTU Publ., 2014.

[5] Volkova V.N., Denisov A.A. Osnovy teorii sistem i sistemnogo analiza [Basics of system theory and analysis]. St. Petersburg, SPbSTU Publ., 2001.

[6] Zverev I., Pyoun Y.S., Lee K.B., et al. An elastic deformation model of high speed spindles built into ball bearings. J. Mater. Process. Technol., 2005, vol. 170, no. 3, pp. 570--578. DOI: https://doi.org/10.1016/j.jmatprotec.2005.05.038

[7] Zverev I.A. Computational-experimental research of the stiffness of high-speed spindles. Vestnik mashinostroeniya, 2020, no. 9, pp. 33--39 (in Russ.). DOI: https://doi.org/10.36652/0042-4633-2020-9-33-39

[8] Jedrzejewski J., Kwasny W. Modelling of angular contact ball bearings and axial displacements for high-speed spindles. CIRP Annals, 2010, vol. 59, no. 1, pp. 377--382. DOI: https://doi.org/10.1016/j.cirp.2010.03.026

[9] Gorelik I.G. Razrabotka metodov rascheta i povysheniya kachestva vysokoskorostnykh shpindel’nykh uzlov. Dis. kand. tekh. nauk [Development of calculation method and raising efficiency of high speed spindle units. Cand. Sc. (Eng.). Diss.]. Moscow, ENIMS, 1987 (in Russ.).

[10] Frolov A.V., Smirnov S.V. Simulating variable quasistatic stiffness of machine tool spindle unit. Herald of the Bauman Moscow State Technical University, Series Mechanical Engineering, 2018, no. 6 (123), pp. 44--59 (in Russ.). DOI: http://dx.doi.org/10.18698/0236-3941-2018-6-44-59

[11] Chernyanskiy P.M. Residual effect of machine tool station mechanical system. Vestnik mashinostroeniya, 2013, no. 1, pp. 57--59 (in Russ.).

[12] Cao Y. Modeling of high-speed machine-tool spindle systems. University of British Columbia, 2006. DOI: http://dx.doi.org/10.14288/1.0080741

[13] Yang Z., Chen H., Yu T. Effects of rolling bearing configuration on stiffness of machine tool spindle. J. Mech. Eng. Sc., 2018, vol. 232, no. 5, pp. 775--785. DOI: https://doi.org/10.11770954406217693659

[14] Sokolovskiy A.P. Zhestkost’ v tekhnologii mashinostroeniya [Stiffness in mechanical engineering technology]. Moscow, MASHGIZ Publ., 1946.

[15] Demkin N.B., Ryzhov E.V. Kachestvo poverkhnosti i kontakt detaley mashin [Surface quality and contact of machine parts]. Moscow, Mashinostroenie Publ., 1981.

[16] Levina Z.M., Reshetov D.N. Kontaktnaya zhestkost’ mashin [Contact stiffness of the machines]. Moscow, Mashinostroenie Publ., 1971.

[17] Timoshenko S.P., Goodier J.N. Theory of elasticity. New York, McGrawHill, 1951.

[18] Greenwood J.A., Williamson J.B.P. Contact of nominally flat surfaces. Proc. R. Soc. Lond., 1966, vol. 295, no. 1442, pp. 300--311. DOI: https://doi.org/10.1098/rspa.1966.0242

[19] Pronikov A.S., ed. Proektirovanie metallorezhushchikh stankov i stanochnykh sistem. T. 1. Proektirovanie stankov [Design of metal-cutting machines and machine tools systems. Vol. 1. Machine tool design]. Moscow, Bauman MSTU Publ., 1994.

[20] Mindlin R.D. Compliance of elastic bodies in contact. J. Appl. Mech., 1949, vol. 16, no. 3, pp. 259--268. DOI: https://doi.org/10.1115/1.4009973

[21] Sherif H.A., Kossa S.S. Relationship between normal and tangential contact stiffness of nominally flat surfaces. Wear, 1991, vol. 151, no. 1, pp. 49--62. DOI: https://doi.org/10.1016/0043-1648(91)90345-U

[22] Krolikowski J., Szczepek J. Assessment of tangential and normal stiffness of contact between rough surfaces using ultrasonic method. Wear, 1993, vol. 160, no. 2, pp. 253--258. DOI: https://doi.org/10.1016/0043-1648(93)90428-O

[23] Baltazar A., Rokhlin S.I., Pecorari C. On the relationship between ultrasonic and micromechanical properties of contacting rough surfaces. J. Mech. Phys. Solids, 2002, vol. 50, no. 7, pp. 1397--1416. DOI: https://doi.org/10.1016/S0022-5096(01)00119-3

[24] Raffa M.L., Lebon F., Vairo G. Normal and tangential stiffnesses of rough surfaces in contact via an imperfect interface model. Int. J. Solids Struct., 2016, vol. 87, pp. 245--253. DOI: https://doi.org/10.1016/j.ijsolstr.2016.01.025

[25] Goodman L.E., Brown C.B. Energy dissipation in contact friction: constant normal and cyclic tangential loading. J. Appl. Mech., 1962, vol. 29, no. 1, pp. 17--22. DOI: https://doi.org/10.1115/1.3636453

[26] Ruderman M., Bertram T. Modified Maxwell-slip model of presliding friction. IFAC Proceedings Volumes, 2011, vol. 44, no. 1, pp. 10764--10769. DOI: https://doi.org/10.3182/20110828-6-IT-1002.00309

[27] Verkhovskiy A.V. The phenomenon of preliminary displacements when starting non-greased surfaces from the spot. Zhurn. prikl. fiziki, 1926, vol. 13, no. 3-4, pp. 311--315 (in Russ.).

[28] Sevostianov I., Kachanov M. Normal and tangential compliances of interface of rough surfaces with contacts of elliptic shape. Int. J. Solids Struct., 2008, vol. 45, no. 9, pp. 2723--2736. DOI: https://doi.org/10.1016/j.ijsolstr.2007.12.024

[29] Johnson K.L. Contact mechanics. Cambridge, University Press, 1987.

[30] Shtaerman I.Ya. Hertz theory of local deformations under compression of elastic bodies. Dokl. AN SSSR, 1939, vol. 25, no. 5, pp. 360--362 (in Russ.).

[31] Perel’ L.Ya., Filatov A.A. Podshipniki kacheniya. Raschet, proektirovanie i obsluzhivanie opor [Ball bearings. Calculation, engineering and supports maintenance]. Moscow, Mashinostroenie Publ., 1992.

[32] Zhang J., Fang B., Hong J., et al. A general model for preload calculation and stiffness analysis for combined angular contact ball bearings. J. Sound Vib., 2017, vol. 411, pp. 435--449. DOI: https://doi.org/10.1016/j.jsv.2017.09.019