Smoothness assurance of the restorable surface of the cones of drill bits by cubic splines

Abstract

The article covers the issues of the application of the spline functions in engineering practice, in particular, in the design of cone drill bits. The polynomial splines, which most commonly used in computational methods are also described. It is also expedient to indicate three important advantages of the spline function for describing the design of cone drill bits. Obtaining a unified mathematical equation for the technical surface of the cones of drill bits is a routine work. The construction of a cubic spline causes difficulties due to the determination of the parameters of the spline, which requires the solution of a system of linear algebraic equations. At the same time, the constructed splines provide second-order smoothness in any direction of the surface line. Further splines of the fifth degree which is reduced to solving a complex system of equations have been developed.