Within the fast advancement of the aviation defense  attempt and deep-sea operation gear, the transmission framework with ball screw is influenced by  the deep-sea tall weight and the affect working conditions, such as the docking of shuttle, the  opening and closing of the carry cabin entryway, the spreading and withdrawing of air ship landing  equip, and the swing of the deep-sea mechanical arm. In this way, the necessities are displayed  on the bearing capacity, affect resistance, exactness sustainability, and execution reliability of ball  screw within the particular transmission framework. 

The equivalent elastic Hertz stresses are a common approach for the static load rating  determination of ball bearings in all bearing industries. This theory can be used to determine the  limit equivalent static stress for elastic contact stress between two curved bodies. The Hertzian  contact mechanism gives an accurate non-linear model for the elastic contact problem of two  different bodies, which is the basis for analyzing the elastic–perfectly plastic contact problem. The  elastic approach (deformation) of ball screw in the contact points can be obtained where r is the sum of principal curvature; ma is the coefficient of semi-major axes of contact ellipse; d is the contact deformation between the ball and the raceway; k(e) is the eccentricity of ellipse; P is the normal contact force; E1 and E2 are effective Young’s moduli of two  different contact surfaces, respectively; and m1 and m2 are effective Poisson’s ratio of two different contact sur- faces, respectively.

The contact between two general contact bodies with four different curvature radii has to be considered. The contact stress distribution of the full contact surface can be obtained as followswhere a and b are the semi-major axis and the semi-minor axis on the contact ellipse,  respectively. The maximum stress can be expressed as The elastic deformation loading formula can be obtained by equation where A is the coefficient of deformation, E* is the equivalent Young’s modulus, R* is the reciprocal value of the sum of the principal curvature, and ma is the coefficient of semi-major  axes of contact ellipse. The contact elliptical area can be obtaine.

When the contact area of ball screw subjected to the normal load is to yield, the semimajor axis and the semi-minor axis of the contact ellipse are a = as and b = bs, respectively. And, smax = ss/(1.73 * kst) is obtained based on the von Mises yield criterion, where kst is  the assurance coefficient dependent on material properties.

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