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

Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher A. Pidhornyi Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN  2709-2984 (Print), 2709-2992 (Online)
Issue Vol. 24, no. 3, 2021 (September)
Pages 70-75
Cited by J. of Mech. Eng., 2021, vol. 24, no. 3, pp. 70-75



Tetiana I. Sheiko, A. Pidhornyi Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi St., Kharkiv, 61046, Ukraine), e-mail:, ORCID: 0000-0003-3295-5998

Kyrylo V. Maksymenko-Sheiko, A. Pidhornyi Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi St., Kharkiv, 61046, Ukraine), V. N. Karazin Kharkiv National University (4, Svobody sq., Kharkiv, 61022, Ukraine), e-mail:, ORCID: 0000-0002-7064-2442



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.


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Received 26 July 2021

Published 30 September 2021