Kinematic performance of a parabolic gear tooth with two parabolic coefficients

Hsueh Cheng Yang, P. Frank Pai

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

This paper presents design imaginary skew rack cutters having teeth of two parabolic coefficients to generate spur and helical gears. First, a mathematical model of the imaginary rack cutter having teeth of two parabolic coefficients is derived using geometric relations and coordinate transformation. After the relationship between the coordinate system of the imaginary rack cutter and that of the helical gear pair is set up, a family of imaginary rack cutter surfaces is obtained using the homogeneous coordinate transformation matrix to transfer the coordinate system of the rack cutters to that of the helical gear. Based on the gear theory, the equations of meshing between the rack cutter and the helical gear are determined. Substituting the equations of meshing into the family of imaginary rack cutter surfaces, a pinion and a gear are generated and their geometries are plotted using an in-house software package. Based on the assembly errors, tooth contact analysis is performed to determine the influences of kinematic assembly errors and the parabolic coefficients. Results show that maximum parabolic coefficient value of the imaginary skew rack cutter exists in order to avoid undercutting and it can be determined.

Original languageEnglish
Pages (from-to)4431-4440
Number of pages10
JournalProceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science
Volume231
Issue number23
DOIs
Publication statusPublished - 2017 Dec 1

Fingerprint

Helical gears
Gear teeth
Kinematics
Gears
Spur gears
Software packages
Mathematical models
Geometry

All Science Journal Classification (ASJC) codes

  • Mechanical Engineering

Cite this

@article{4cb2b72eb3614dde946fd554b12c9165,
title = "Kinematic performance of a parabolic gear tooth with two parabolic coefficients",
abstract = "This paper presents design imaginary skew rack cutters having teeth of two parabolic coefficients to generate spur and helical gears. First, a mathematical model of the imaginary rack cutter having teeth of two parabolic coefficients is derived using geometric relations and coordinate transformation. After the relationship between the coordinate system of the imaginary rack cutter and that of the helical gear pair is set up, a family of imaginary rack cutter surfaces is obtained using the homogeneous coordinate transformation matrix to transfer the coordinate system of the rack cutters to that of the helical gear. Based on the gear theory, the equations of meshing between the rack cutter and the helical gear are determined. Substituting the equations of meshing into the family of imaginary rack cutter surfaces, a pinion and a gear are generated and their geometries are plotted using an in-house software package. Based on the assembly errors, tooth contact analysis is performed to determine the influences of kinematic assembly errors and the parabolic coefficients. Results show that maximum parabolic coefficient value of the imaginary skew rack cutter exists in order to avoid undercutting and it can be determined.",
author = "Yang, {Hsueh Cheng} and Pai, {P. Frank}",
year = "2017",
month = "12",
day = "1",
doi = "10.1177/0954406216668205",
language = "English",
volume = "231",
pages = "4431--4440",
journal = "Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science",
issn = "0954-4062",
publisher = "SAGE Publications Ltd",
number = "23",

}

TY - JOUR

T1 - Kinematic performance of a parabolic gear tooth with two parabolic coefficients

AU - Yang, Hsueh Cheng

AU - Pai, P. Frank

PY - 2017/12/1

Y1 - 2017/12/1

N2 - This paper presents design imaginary skew rack cutters having teeth of two parabolic coefficients to generate spur and helical gears. First, a mathematical model of the imaginary rack cutter having teeth of two parabolic coefficients is derived using geometric relations and coordinate transformation. After the relationship between the coordinate system of the imaginary rack cutter and that of the helical gear pair is set up, a family of imaginary rack cutter surfaces is obtained using the homogeneous coordinate transformation matrix to transfer the coordinate system of the rack cutters to that of the helical gear. Based on the gear theory, the equations of meshing between the rack cutter and the helical gear are determined. Substituting the equations of meshing into the family of imaginary rack cutter surfaces, a pinion and a gear are generated and their geometries are plotted using an in-house software package. Based on the assembly errors, tooth contact analysis is performed to determine the influences of kinematic assembly errors and the parabolic coefficients. Results show that maximum parabolic coefficient value of the imaginary skew rack cutter exists in order to avoid undercutting and it can be determined.

AB - This paper presents design imaginary skew rack cutters having teeth of two parabolic coefficients to generate spur and helical gears. First, a mathematical model of the imaginary rack cutter having teeth of two parabolic coefficients is derived using geometric relations and coordinate transformation. After the relationship between the coordinate system of the imaginary rack cutter and that of the helical gear pair is set up, a family of imaginary rack cutter surfaces is obtained using the homogeneous coordinate transformation matrix to transfer the coordinate system of the rack cutters to that of the helical gear. Based on the gear theory, the equations of meshing between the rack cutter and the helical gear are determined. Substituting the equations of meshing into the family of imaginary rack cutter surfaces, a pinion and a gear are generated and their geometries are plotted using an in-house software package. Based on the assembly errors, tooth contact analysis is performed to determine the influences of kinematic assembly errors and the parabolic coefficients. Results show that maximum parabolic coefficient value of the imaginary skew rack cutter exists in order to avoid undercutting and it can be determined.

UR - http://www.scopus.com/inward/record.url?scp=85034612468&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85034612468&partnerID=8YFLogxK

U2 - 10.1177/0954406216668205

DO - 10.1177/0954406216668205

M3 - Article

AN - SCOPUS:85034612468

VL - 231

SP - 4431

EP - 4440

JO - Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science

JF - Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science

SN - 0954-4062

IS - 23

ER -