Equations of contact deformation for 2 spheres by indirect-fictitious boundary integral method (Modification of Hertz’s formula)

Abstract

Ball bearings are used for many mechanisms. For the precious linear motion mechanism, it is very important to know the ball bearing deformation because the driving load is supported by the bearing. Especially, for the focus control mechanism of the optical telescope, even a very small deformation is not negligible because a sub-micron precision is required for the driving stroke. The Hertz’s formula is very familiar to the relation between the load and the displacement in a small contact area. Hertz derived the formula by using a potential of the electricity distribution in the ellipsoid body. However, it is reported that an error becomes large when the contact area is large. It is a reason that the elliptic paraboloid is used for the ellipsoid body in the Hertzian contact theory. Some modified equations are reported by Nishihara et.al, Tatara and Villaggio. However, the modified equations by Nishihara et al. and Villaggio are a little different from the FEM result. Therefore, in this paper, a contact deformation for the 2 equivalent spheres is analyzed by the indirect-fictitious boundary integral method. For the relation between the load and the displacement, a new modified equation with the fractional expression is derived in this paper. And, for the relation between the load and the contact radius, it is confirmed that the Hertz’s formula is correct.

Keywords

• sphere
• contact
• deformation
• equation
• hertz’s formula
• indirect-fictitious boundary integral method
• papkovich-neuber
• fractional expression