MATHEMATICAL AND COMPUTER SIMULATION OF HEX HEAD CREWS FOR IMPLEMENTATION ON A 3D PRINTER

for the 3D printing of


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][2][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 com-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 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 be symmetrical about the abscissa axis, and let it be possible for the domain to be placed inside the sector n have been obtained as a result of rotating the domain around the origin by angles ).Then the equation of the boundary Ω ∂ of the domain , where ( ) .
For chamfering, we use the equation of a conical surface, for which the guide is described by the equation ( ) , and the vertex is at a point ) , , ( of the conical surface is obtained by changing the variables Let us build a mathematical model of a 6-32 (UTS) cross-slotted hex head screw used in personal computers ( ) ) ( ) ) Fig. 1 shows a hex washer head screw constructed by the formula ( ) ( ) . A rod with an external thread is attached to the screw head according to the formula ( ) ( ) (Fig. 2), the equation of which can be constructed according to the results of work [6].Substituting the formula for the cross slot (1), we finally get the screw model 6-32 (UTS) (Fig. 3).
we get a Torx hex head screw (Fig. 4).
Replacing (1) with we get a Pentalobe hex head screw (Fig. 5), which is used by Apple and Meizu in personal computers.
Replacing (1) with , we get a Polydrive hex head screw (Fig. 6).( ) ( ) Thus, having built a model of the screw base in the form of a hex washer head with external threaded rod, and then, by including in the formula of the main screw block only the logical formula of the required slot, we obtain the desired result.At the same time, construction and manufacturing costs are minimized.
Fig. 9 shows the results of the 3D printing of some of the constructed screw models.

Conclusions
Building a mathematical model is a central stage in studying or designing any system.All the subsequent analysis of the object depends on the quality of the model.The model must be sufficiently accurate, adequate, and easy to use.
Summing up, it should be said that in this work, for the first time, thanks to the R-functions theory, methods have been developed and equations have been constructed for various hex head slotted screws, which are used both in personal computers and in other high-class equipment, for the implementation on a 3D printer.The analytical recording of the designed objects makes it possible to use alphabetic geometric parameters, complex superposition of functions, which, in turn, allows us to quickly change their structural elements.
The proposed method for specifying the shape of products, using a limited number of parameters, can significantly reduce the complexity of work in cases where we need to view a large number of design options in search of an optimal solution.With parametric assignment, the change in the computational domains is carried out almost instantly.
With the help of R-functions, an algorithm for the step-by-step construction of the equations of screws has been developed and implemented, which allows us to check and make adjustments to the model at each stage of its construction.
The reliability of the results obtained, their adequacy to the designed objects is confirmed by visualization both in the operating conditions of the RFPreview program and by implementation on a 3D printer.

Література
to smooth sharp edges and corners, where ρ is the radius of rounded corners.