Abstract
This research attempts to develop spindle deflection error models for high-speed machining systems. A model for determining total spindle deflection at the tool-end is presented. The model incorporates spindle bearing characteristics, shifts in ball contact angles, and centrifugal force and gyroscopic moment effects at high speeds. It uses the transfer matrix method to determine the total deflections at the tool-end based upon the point contact deformations at the individual balls of an angular contact ball-bearing assembly. A simulator is also developed for simulating spindle end deflections for various spindle rotational speeds. The results of the simulation show contact angle variations and peak deflections at particular spindle rotational speeds. Important research issues are also presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- AF :
-
final position, inner raceway groove centre
- RF :
-
initial position, inner raceway groove centre
- W :
-
final position of ball centre
- V :
-
initial position of ball centre
- D :
-
ball diameter, mm
- r o :
-
inner raceway groove radius, mm
- r i :
-
inner raceway groove radius, mm
- M :
-
gyroscopic moment, N-mm
- FO :
-
r o/D
- FI :
-
r i/D
- P :
-
bearing pitch diameter, mm
- K o :
-
outer race load-deflection constant, N/mm1.5
- K i :
-
inner race load-deflection constant, N/mm1.5
- CF :
-
centrifugal force, N
- J :
-
mass moment of inertia, N.mm2
- l :
-
length of spindle, mm
- E :
-
modulus of elasticity, N/mm2
- I :
-
moment of inertia of spindle, mm4
- Y :
-
deflection of spindle alongy-direction, mm
- z :
-
deflection of spindle alongz-direction, mm
- M :
-
moment at spindle end, N.mm
- V :
-
shear force at spindle end, N
- m :
-
spindle mass, kg
- ρ:
-
material density
- βo :
-
outer race contact angle
- βi :
-
inner race contact angle
- β:
-
nominal contact angle
- δi :
-
inner race deformation
- δo :
-
outer race deformation
- Γ:
-
angle between ball centre of rotation and the horizontal
- ξ:
-
mis-alignment (in degrees) of shaft assembly measured in a plane perpendicular to shaft axis (x-direction)
- W1:
-
ball and raceway angular raceway velocity ratio for outer raceway control
- W2:
-
ball orbital and angular raceway velocity ratio for rotating inner raceway and outer raceway control
- ϕ:
-
circumferential ball position
- λ:
-
raceway control parameter
References
R. I. King (ed.),Handbook of High-Speed Machining Technology Chapman and Hall, New York, pp. 1–26, 1985.
Y. C. Shin, K. W. Wang and C. H. Chen, “Dynamic analysis of a high speed spindle system”,Transactions of the NAMRI/SME, pp. 298–304, 1990.
Y. C. Shin, H. Chin and M. J. Brink, “Characterization of CNC machining centers”,Journal of Manufacturing Systems 10(5), pp. 407–421, 1991.
R. J. Hocken,Technology of Machine Tools, vol. 5, Lawrence Livermore Laboratory CA, UCRL-52960-5, 1980.
E. Rivin, “Trends in tooling for CNC machine tools: machine system stiffness”,Manufacturing Review 4(4), pp. 257–263, 1991.
E. Rivin, “Trends in tooling for CNC machine tools: tool—spindle interfaces”,Manufacturing Review 4(4), pp. 264–274, 1991.
G. Chryssolouris, “Effects of machine-tool—workpiece stiffness on the wear behavior of superhard cutting materials”,Annals of the CIRP 31(1), pp. 65–69, 1982.
R. R. Donaldson, “Error budgets”,Technology of Machine Tools vol. 5, Lawrence Livermore Laboratory CA, UCRL-52960-5, pp. 9.14-1–9.14-14, 1980.
G. Zhang, R. Ouyang, B. Lu, R. Hocken, R. Veale and A. Donmez, “A displacement method for machine geometry calibration”,Annals of the CIRP 37(1), pp. 515–518, 1988.
R. Hocken, J. A. Simpson, B. Borchardt, L. Lazar, C. Reeve and P. Stein, “Three dimensional metrology”,Annals of the CIRP 26(2), pp. 403–408, 1977.
P. M. Ferreira and C. R. Liu, “A method for estimating and compensating quasistatic errors of machine tools”,Journal of Engineering for Industry 115 pp. 149–159, 1993.
R. Schultschik, “The components of the volumetric accuracy”,Annals of the CIRP 25(1), pp. 223–228, 1977.
J. Mou and C. R. Liu, “A method for enhancing the accuracy of CNC machine tools for on-line inspection”,Journal of Manufacturing Systems 11(4), pp. 229–237, 1992.
J. Bryan, “International status of thermal research”,Annals of the CIRP 39(2), pp. 645–656, 1990.
J. Jedrzejewski, J. Kaczmarek and Z. Kowal, “Numerical optimization of thermal behavior of machine tools”,Annals of the CIRP 39(1), pp. 379–382, 1990.
R. Venugopal and M. Barash, “Thermal effects on the accuracy of numerically controlled machine tools”,Annals of the CIRP 35(1), pp. 255–258, 1986.
M. Rahman, M. A. Mansur and K. H. Chua, “Evaluation of advanced cementitious composities for machine-tool structures”,Annals of the CIRP 37(1), pp. 373–376, 1988.
D. G. Lee, H.-C. Sin and N. P. Suh, “Manufacturing of a graphite epoxy composite spindle for a machine tool”,Annals of the CIRP 34(1), pp. 365–369, 1985.
A. B. Jones, “A general theory for elastically constrained ball and radial roller bearings under arbitrary load and speed conditions”,Journal of Basic Engineering, pp. 309–320, June 1960.
T. A. Harris,Rolling Bearing Analysis, 3rd edn, John Wiley and Sons, 1991.
A. Bhattacharya,Principles of Machine Tools 2nd edn, New Central Book Agency, Calcutta, India, pp. 605–609, 1975.
B. S. Gottfried and J. Weismann,Introduction to Optimization Theory Prentice-Hall, Englewoods Cliffs, NJ, pp. 113–130, 1973.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chatterjee, S. Spindle deflections in high-speed machine tools — Modelling and simulation. Int J Adv Manuf Technol 11, 232–239 (1996). https://doi.org/10.1007/BF01351280
Issue Date:
DOI: https://doi.org/10.1007/BF01351280