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Model of Inverse Envelope Cutter with Ring-Involute Tooth

  • Research Article - Mechanical Engineering
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Abstract

The design of a conical cutter is important in the manufacture of concave and convex gears. An inverse method is presented for determining the mathematical model of a rack cutter. The obtained rack cutter was used to generate concave and convex gears. The method proposed in this paper addressed design problems in the rack cutter that were used to generate a new type of concave and convex gears. Based on the two-parameter family of surfaces and direct envelope method, the conical cutter was used as the generating tool for the proposed gear type, and a mathematical model of gears with ring-involute teeth was developed according to gear theory. The contour of von-Mises stress distribution of the gear and the pinion of the proposed mathematical model is presented. Using CNC manufacturing technology, a gear with ring-involute teeth was manufactured by a conical cutter. Based on the inverse envelope concept, the mathematical model of the developed gear tooth was used to determine the geometrical and mathematical models of the rack cutter with ring-involute teeth. To illustrate the effectiveness of the method, a numerical example is presented to demonstrate the geometric model of a gear with a gear ratio of 3:2.

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Abbreviations

a c, a t, and b c :

Design parameters of a conical cutter

ℓ j :

Design parameters of the conical cutter where subscript j = c, d, h

m :

Module

r :

Radius of the root fillet surface of the teeth

r p1 :

Standard pitch radii of a convex gear

r p2 :

Standard pitch radii of a concave gear

M ij :

Co-ordinate transformation matrix from co-ordinate system S j to co-ordinate system S i

N 1 :

Number of teeth of convex gear

N 2 :

Number of teeth of concave gear

R gc :

Position vector where the upper sign indicates regions ab, bc, and cd of the conical cutter surface

R gi :

The family of the conical cutter surfaces, where the upper sign indicates regions ab, bc, and cd of the cutter surface. Subscript i is 1, and 2

S i (o i , x i , y i , z i ):

Co-ordinate systems where subscript i = c, 1, 2, r, n, c denotes conical cutter surface, 1 denotes convex gear, 2 denotes concave gear, f is rigidly connected to the frame of reference, r and n denote fixed reference coordinate systems

R ′i :

Position vectors where subscript i =  1, 2. 1 Denotes convex gear, 2 denotes concave gear

\({\phi _{\rm c}}\) :

Pressure angles

\({\phi _{1}}\) :

Rotate angle about x r -axis

\({\phi _{2}}\) :

Rotate angle about x n -axis

α :

A rotary angle and its region is 0 ~ 2π

Σ 1 :

Convex tooth of the gear

Σ 2 :

Concave tooth of the gear

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Authors and Affiliations

  1. Graduate Institute of Vehicle Engineering, National Changhua University of Education, Bao-Shan Campus, Number 2, Shi-Da Road, Changhua, 500, Taiwan, ROC

    Shyue-Cheng Yang

  2. Department of Industrial Education, National Taiwan Normal University, Taipei, Taiwan, ROC

    Chi-Chia Liu

  3. Department of Earth and Environment Sciences, Institute of Seismology, National Chung Cheng University, 168, University Road, Minhsiung, 62102, Chiayi County, Taiwan, ROC

    Tsun-Hui Huang

Authors
  1. Shyue-Cheng Yang
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  2. Chi-Chia Liu
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  3. Tsun-Hui Huang
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Corresponding author

Correspondence to Shyue-Cheng Yang.

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Yang, SC., Liu, CC. & Huang, TH. Model of Inverse Envelope Cutter with Ring-Involute Tooth. Arab J Sci Eng 37, 1119–1129 (2012). https://doi.org/10.1007/s13369-012-0232-3

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  • DOI: https://doi.org/10.1007/s13369-012-0232-3

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