## 1 Introduction  
Burnishing is a method of smoothing and strengthening machine parts by surface plastic deformation. The method is implemented in modern machining centers, where a moving tool is pressed into the surface of a rotating workpiece. Traditionally, the indenter is made of diamond and has a hemispherical shape, but in recent years, other materials and forms of the indenter have been proposed, along with kinematic schemes for processing cylindrical and flat surfaces with the application of advanced methods of surface layer material nanocrystallization. In [1], a cylindrical polycrystalline diamond tool was used to burnish the LY12 aluminum alloy. The burnishing mechanism was investigated on the basis of the laws of contact mechanics. The simulation model brings into focus the interrelationship of the burnishing force, the tool geometry, the parameters of the technological process, and the mechanical properties of the material. [PERSON] et al. 's work [2] presents the burnishing of the shaft surface using a tool with a cylindrical indenter, the axis of which is perpendicular to the axis of the shaft being processed. The influence of burnishing parameters on the surface stereometric structure, surface layer strengthening, and residual stress distribution is investigated. It was found that the microhardness of the 42 CrMo4 alloy steel surface layer after burnishing increased by 29%, and the compression stresses in the surface layer increased up to 400 MPa. Studying the burnishing of 20 Cr4 steel by a spherical indenter with a radius of 2 mm, [PERSON] et al. [3] focused on the formation of a nanocrystalline structure in the burnished surface layers. In the previous works of [PERSON] et al. [4; 5], it has been shown that the formation of a nanocrystalline structure in structural steels is achieved by intense plastic shear deformation under hydrostatic compression with a pressure of more than 2 GPa and a value of cumulative shear strain of more than two. Similar data are reported in [3]. Based on the results of studies given in this paper, it is possible to calculate the wedge angle of the plastically displaced metal to the plane of the processing surface, depending on the pile-up height, the indenter insertion depth, and the contact zone length. With a burnishing force of 200 and 340 N, a feed of 0.01 and 0.04 mm/rev, and a friction coefficient of 0.13 and 0.34, this angle varied in the range from 1\({}^{\circ}\)6\({}^{\prime}\) to 2\({}^{\circ}\)36\({}^{\prime}\). The refinement of the structure of the quenched AISI 420 steel by nanostructuring burnishing with a spherical indenter made of polycrystalline synthetic diamond was studied in [6]. It is shown that at a burnishing force of 340 N, a feed of 0.04 mm/rev, and with a change in sliding speed from 6 up to 26 m/min, the plastic shear deformation ranges from 2.1 to 3.4. At the same time, a homogeneous nanocrystalline structure with grain sizes less than 100 nm is formed in the near-surface layer. As an interesting illustration, we can take a work in which an indenter made of cubic boron nitride has a cylindrical shape and is inclined at an angle to the processing surface [7]. This work describes experimental studies of the nanostructuring of the 1.3505 (AISI 52100) quenched steel surface layer by a sliding cylindrical indenter inclined to a flat surface being treated at an angle of 2 degrees. The studies revealed the influence of the burnishing force, applied to the tool and varying in the range from 100 to 340 N, on the change in the thickness and microhardness of the strengthened layer. It was established for the first time that with nanostructuring surface strengthening with a normal force of 250 N, the microhardness in the surface layer with a thickness of up to 20 \(\upmu\)m reaches more than 1000 HV\({}_{0.05}\). It is obvious that this method of burnishing with different inclination angles of the indenter needs further investigation with the application of various methods, including numerical analysis and experiment.