Mathematical and Computer Simulation of Hex Head Screws for Implementation on a 3D Printer

APPLIED MATHEMATICS DOI: https://doi.org/10.15407/pmach2021.03.070 UDC 517.95+518.517 MATHEMATICAL AND COMPUTER SIMULATION OF HEX HEAD CREWS FOR IMPLEMENTATION ON A 3D PRINTER 1 Tetiana I. Sheiko, ORCID: 0000-0003-3295-5998 1, 2 Kyrylo V. Maksymenko-Sheiko, ORCID: 0000-0002-7064-2442 1 A. Pidhornyi Institute of Mechanical Engineering Problems of NASU, 2/10, Pozharskyi str., Kharkiv, 61046, Ukraine 2 V. N. Karazin Kharkiv National University, 4, Svobody sq., Kharkiv, 61022, Ukraine In this paper, based on the R-functions theory, methods have been developed and equations have been constructed for the 3D printing of hex-head screws with Bristol, Pentalobe, Polydrive and other types of screw slots. Such screws are used both in personal computers and other high-end equipment. The Bristol slot has four or six radial grooved beams. The advantage of the design of this slot is the correct perpendicular, rather than tangential, vector of force application when the slot is rotated by a tool, which minimizes the risk of stripping out the slot. For this reason, the Bristol slot is used in soft metal screws. Compared to the internal hex, the Bristol slot allows a noticeably higher torque, only slightly higher than that of the Torx slot. This type of slot is used in aviation, high-end telecommunications equipment, cameras, air brakes, agricultural equipment, astronomical equipment, and foreign military equipment. Variations with a pin in the center are found in game consoles to prevent the use of a flat-blade screwdriver as an improvised key. The Pentalobe slot is a five-point slot designed by Apple and used in its products to limit unauthorized disassembly. It was first used in mid 2009 to mount MacBook Pro batteries. Its miniature version was used in the iPhone 4 and later models, in the MacBook Air (available since late 2010 models), and the MacBook Pro with Retina screens. The Polydrive slot is a starlike slot with rounded star points, used in the automotive industry for applications requiring high tightening torque. The Torq-set slot is a cross slot for fasteners requiring high tightening torque. The grooves are slightly offset, not intersecting at one point. Fasteners with this type of slot are used in military aviation, for example, in E-3, P-3, F-16, Airbus, Embraer, and Bombardier Inc. The Phillips Screw Company owns the trademark and manufactures fasteners with this type of slot. The slot design standards are National Aerospace Standard NASM 3781 and NASM 4191 for the ribbed version. The resulting equations for the surfaces of screws were checked during the modeling of the screws before 3D printing. The 3D printing technology allows us to reduce the cost and labor intensity of manufacturing products, including complex slot screws. The analytical recording of designed objects makes it possible to use alphabetic geometric parameters, complex superposition of functions, which, in turn, allows us to quickly change their design elements. The positivity property of the constructed functions at the internal points of an object is very convenient for the implementation of 3D printing. Keywords: R-functions, mathematical model, screw, slot, 3D printing. Introduction One of the new technologies that has been gaining popularity in recent years is 3D printing. It allows us to create volumetric models of any objects using special equipment – a 3D printer. Advantages of using modern 3D printers are reducing the cost of manufacturing products and the timing of their emergence on the market, modeling objects of any shape and complexity, speed and high precision of manufacturing, the ability to use various materials. However, there is a problem with specifying the printing information. In the process of preparing products for 3D printing, it is necessary to create a 3D computer model of the desired object. In studies on the computer modeling of solid bodies, carried out in the works of A. G. Requicha [1–3], eight main presentation schemes have been identified: 1) engineering drawings; 2) wire frame representation; 3) representation with primitives; 4) voxel presentation; 5) discrete models; 6) Constructive Solid Geometry; 7) schematic sweep representation; 8) boundary representation (Brep). The analysis of the above classical representations showed that their practical application is limited, or requires significant efforts in constructing models of comThis work is licensed under a Creative Commons Attribution 4.0 International License.  Tetiana I. Sheiko, Kyrylo V. Maksymenko-Sheiko, 2021 70 ISSN 2709-2984. Journal of Mechanical Engineering – Problemy Mashynobuduvannia, 2021, vol. 24, no. 3 ПРИКЛАДНА МАТЕМАТИКА plex-shaped objects. From the point of view of universality, one of the most promising is the functional representation, which is based on the use of the language of implicit mathematical functions with the constructive capabilities of the R-functions theory developed by academician V. L. Rvachev [4]. The aim of this paper is to develop techniques, based on the R-functions theory, and construct equations for the surfaces of various types of slotted screws to be subsequently 3D printed. Fastener slots are recesses in the heads of threaded fasteners for transferring torque from the tool to the slots. The slots can be straight, cross, square, multi- square, inner hexagonal, five-pointed, star-shaped, combined, etc. In addition to the common, widely used types, there are less common ones used as anti-vandal or designed to prevent unauthorized access. Slots are anti-vandal if their task is both to complicate unauthorized access in public places and complicate the self-opening and repair of electronic devices. Anti-vandalism is often based on the fact that screwdrivers for the required slot are not available in standard tool sets, and it is rather difficult to find them on the market. In some cases, the manufacturer can replace a rare but standardized type of slot with its own, proprietary, patent-protected, which enables not only to make unauthorized untwisting as difficult as possible, but also prosecute any independent screwdriver manufacturer that does not have a manufacturing license from the patent holder. Examples of proprietary slots include T-Groove, Slot-Lok, Pentagon, Tork-Nut, T-Slope. Main Part When constructing mathematical models by the R-functions method, both the simplest R-operations  SR 2 2  x ∧ ρ y = x + y − x + y + 2 (SR + SR ) 8ρ x ∧ y ≡ x + y − x2 + y 2   0 SR  2 2 and R-operations  x ∨ ρ y = x + y + x 2 + y 2 + 2 (SR + SR ) will be used to x ∨0 y ≡ x + y + x + y 8ρ   x ≡ −x  x = −x SR = ρ2 − x 2 − y 2    smooth sharp edges and corners, where ρ is the radius of rounded corners. When constructing equations corresponding to geometric objects with a cyclic point symmetry, the results of the following theorem [5] will be used to reduce the number of R-operations. Theorem. Let the translation domain Σ 0 = [σ 0 (x, y ) ≥ 0] be symmetrica (...truncated)