An investigation on cage instability based on dynamic model considering guiding surface three-dimensional contact

Abstract

The cage instability significantly affects the bearing dynamic behavior and weakens the rotating system performance. To predict this instability, however, a bearing dynamic model with detailed consideration of the interface and velocity between the cage and the other components is still lacking. To this end, this paper proposes a new bearing dynamic model that places all moving components in the inertial coordinate frame. The geometrical interactions and velocities between the cage and other components are considered in detail. The slicing method is employed to derive the interaction between the deflected cage and the guiding ring. The influence mechanism of two typical operating conditions on cage dynamic behavior is investigated from the perspective of ball/pocket collisions. On this basis, new indictors are defined to study the thresholds for operating condition leading to cage instability. Attempts are further made to improve cage stability by optimizing bearing contact angle. The results show that heavy radial loads significantly enhance the cage instability and the shock exerted on the cage. Interestingly, the smaller the contact angle, the more pronounced the improvement in the cage dynamic performance.

Appendix: Hertzian point-contact theory

The Hertzian deflection coefficient for point contact can be defined as:

$$ K = \pi \kappa E^{\prime } \left( {\frac{{{\text{E}}\Sigma \rho }}{{4.5{\text{F}}^{3} }}} \right)^{0.5} $$

(64)

The contact half-width of the semi-major axis is given as follows:

$$ a = \left( {\frac{{3\kappa^{2} {\text{E}}Q}}{{\pi E^{\prime } \Sigma \rho }}} \right)^{\frac{1}{3}} $$

(65)

The contact half-width of the semi-minor axis is given as follows:

$$ b = \left( {\frac{{3{\text{E}}Q}}{{\pi \kappa E^{\prime } \Sigma \rho }}} \right)^{\frac{1}{3}} $$

(66)

The maximum contact stress in the contact ellipse can be expressed as:

$$ p_{0} = \frac{2Q}{{\pi ab}} $$

(67)

The contact stress at any point in the contact ellipse is represented as:

$$ p{}_{{\left( {x,y} \right)}} = p_{0} \sqrt {1 - \left( \frac{x}{a} \right)^{2} - \left( \frac{y}{b} \right)^{2} } $$

(68)

where

$$ \Sigma \rho = \rho_{{{\text{I1}}}} + \rho_{{{\text{I2}}}} + \rho_{{{\text{II1}}}} + \rho_{{{\text{II}}2}} $$

(69)

$$ E^{\prime} = \left( {\frac{{1 - \nu_{1}^{2} }}{{E_{1} }} + \frac{{1 - \nu_{2}^{2} }}{{E_{2} }}} \right) $$

(70)

$$ {\text{F = }}\int_{0}^{{\frac{\pi }{2}}} {\left[ {1 - \left( {1 - \frac{1}{{\kappa^{2} }}} \right)\sin^{2} \phi } \right]}^{{ - \frac{1}{2}}} d\phi $$

(71)

$$ {\text{E = }}\int_{0}^{{\frac{\pi }{2}}} {\left[ {1 - \left( {1 - \frac{1}{{\kappa^{2} }}} \right)\sin^{2} \phi } \right]}^{\frac{1}{2}} d\phi $$

(72)

and κ denotes the ratio of semi-major axis to semi-minor axis, which can be obtained by solving following formula.

$$ \frac{{\left( {\kappa^{2} + 1} \right){\text{E}} - 2{\text{F}}}}{{\left( {\kappa^{2} - 1} \right){\text{E}}}} = \frac{{\left| {\rho_{{{\text{I1}}}} - \rho_{{{\text{II2}}}} } \right| + \left| {\rho_{{{\text{II1}}}} - \rho_{{{\text{II}}2}} } \right|}}{\Sigma \rho } $$

(73)

ρ is the curvature of the contact body surface. v and E represent the material-related coefficients.