OBRABOTKAMETALLOV Vol. 27 No. 2 2025 technology ε 2 0 2 0 0 0 ; ( ) ( ) ( ), ( ) x x x d M dz EJ J M k z M z M z J z (3) where Ε is the coordinate of the tool axis bending, mm; z is the tool length coordinate, mm; M0 is the reduced bending moment, N·mm2; J0 x is the moment of inertia of the boring bar at the origin of the coordinate system, mm4; k(z) is the reduction factor; M(z) is the bending moment function; J x(z) is the function of inertia moment. The rigidity of this system changes according to the following relationship: π α 4 0 ( ) ( ) ( ) 2 tg , 64 x d z J z d z d z where d(z) is the diameter change function; d0 is the diameter of the boring bar at the origin of the coordinate system, mm; α is the angle of the conical surface of the boring bar, rad. The solution to the system of equations (3) consists of reducing the system with variable rigidity to a system with constant rigidity. To do this, we will compose a differential equation describing the function of change in the reduced bending moment: 0 dM = P k l z dz l ( )( ) , z where l is the boring tool length, mm. Taking into account that the reduction factor is found as the ratio of two moments of inertia in different sections, we obtain: α α 0 4 0 4 0 4 0 0 0 ( ) ; ( ) ( 2 tg ) ( ) . 2 tg x x J d k z J z d z d dM P l z dz l d z The solution of this differential equation by an analytical method will allow us to construct functions of reduced bending moments for each sample under study (Fig. 9). Fig. 9.Diagrams of reduced bending moments
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