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Research Papers: Applications

Complete Analytical Expression of the Stiffness Matrix of Angular Contact Ball Bearings

[+] Author and Article Information
Benoit Furet

Institut de Recherche en Communications
et en Cybernetique de Nantes (IRCCyN),
UMR CNRS 6597,
1 rue de la Noe,
44321 Nantes, France;
LUNAM Universite,
Universite de Nantes,
1 quai de Tourville,
44035 Nantes, France

Sebastien Le Loch

LUNAM Universite,
Universite de Nantes,
1 quai de Tourville,
44035 Nantes, France

Contributed by the Tribology Division of ASME for publication in the Journal of Tribology. Manuscript received March 6, 2012; final manuscript received October 5, 2012; published online June 4, 2013. Assoc. Editor: Xiaolan Ai.

J. Tribol 135(4), 041101 (Jun 04, 2013) (8 pages) Paper No: TRIB-12-1032; doi: 10.1115/1.4024109 History: Received March 06, 2012; Revised October 05, 2012

Angular contact ball bearings are predominantly used for guiding high speed rotors such as machining spindles. For an accurate modeling, dynamic effects have to be considered, most notably in the bearings model. The paper is based on a dynamic model of angular contact ball bearings. Different kinematic hypotheses are discussed. A new method is proposed for the computation of the stiffness matrix: a complete analytical expression including dynamic effects is presented in order to ensure accuracy at high shaft speed. It is demonstrated that the new method leads to the exact solution, contrary to the previous ones. Besides, the computational cost is similar. The new method is then used to investigate the consequence of the kinematic hypotheses on bearing stiffness values. Last, the relevance of this work is illustrated through the computation of the dynamic behavior of a high speed milling spindle. The impact of this new computation method on the accuracy of a finite element spindle model is quantified.

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Figures

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Fig. 1

Principle of the bearing model

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Fig. 2

Dynamic effects on the balls

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Fig. 3

Position of the ball center and raceway groove curvature centers at angular position ψ with and without applied load

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Fig. 7

Global displacement determination

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Fig. 6

Example of the inner and outer-race control regions for the uni-axial case, according to Jones's theory

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Fig. 5

Spinning and rolling motion of the ball

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Fig. 8

Relative error on the axial stiffness ΔK11 considering the uni-axial case under different loads Fx and speed ω

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Fig. 11

Gyroscopic moment Mg versus shaft speed for different kinematic hypotheses

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Fig. 12

CAD model of the spindle rotor

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Fig. 9

Axial stiffness versus shaft speed in relation to the kinematic hypotheses

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Fig. 10

Pitch angle β versus shaft speed in relation to the kinematic hypotheses

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