Using an Imaginary Planar Rack Cutter to Create a Spherical Gear Pair with Continue Involute Teeth

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Abstract

This paper presents an imaginary planar rack cutter having two parameters of motion to create a spherical gear pair with two degrees of freedom. The working region of the imaginary planar rack cutter is that of symmetrical straight lines. The relationship between the coordinate system of the imaginary planar rack cutter and that of the spherical gear pair having two parameters of motion is illustrated. The two-parameter family of imaginary planar rack cutter surfaces is obtained by using the homogeneous coordinate transformation matrix to transfer the coordinate system of the planar rack cutter to that of the spherical pinion or the spherical gear. Based on the two-parameter enveloping theory, the mathematical models of the spherical gear pair with two degrees of freedom were proposed. The developed mathematical model of the gear pair with two degrees of freedom can be used to analyze undercutting analysis of the imaginary planar rack cutter. Based on the undercutting equation, a minimum number of teeth of the spherical pinion are determined. A line of singular points on the spherical pinion’s teeth are displayed. Results show that a minimum number of teeth of the spherical pinion can be determined in order to avoid undercutting.

Original languageEnglish
Pages (from-to)4725-4735
Number of pages11
JournalArabian Journal for Science and Engineering
Volume42
Issue number11
DOIs
Publication statusPublished - 2017 Nov 1

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cutters
racks
teeth
degrees of freedom
mathematical models
coordinate transformations

All Science Journal Classification (ASJC) codes

  • General

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title = "Using an Imaginary Planar Rack Cutter to Create a Spherical Gear Pair with Continue Involute Teeth",
abstract = "This paper presents an imaginary planar rack cutter having two parameters of motion to create a spherical gear pair with two degrees of freedom. The working region of the imaginary planar rack cutter is that of symmetrical straight lines. The relationship between the coordinate system of the imaginary planar rack cutter and that of the spherical gear pair having two parameters of motion is illustrated. The two-parameter family of imaginary planar rack cutter surfaces is obtained by using the homogeneous coordinate transformation matrix to transfer the coordinate system of the planar rack cutter to that of the spherical pinion or the spherical gear. Based on the two-parameter enveloping theory, the mathematical models of the spherical gear pair with two degrees of freedom were proposed. The developed mathematical model of the gear pair with two degrees of freedom can be used to analyze undercutting analysis of the imaginary planar rack cutter. Based on the undercutting equation, a minimum number of teeth of the spherical pinion are determined. A line of singular points on the spherical pinion’s teeth are displayed. Results show that a minimum number of teeth of the spherical pinion can be determined in order to avoid undercutting.",
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N2 - This paper presents an imaginary planar rack cutter having two parameters of motion to create a spherical gear pair with two degrees of freedom. The working region of the imaginary planar rack cutter is that of symmetrical straight lines. The relationship between the coordinate system of the imaginary planar rack cutter and that of the spherical gear pair having two parameters of motion is illustrated. The two-parameter family of imaginary planar rack cutter surfaces is obtained by using the homogeneous coordinate transformation matrix to transfer the coordinate system of the planar rack cutter to that of the spherical pinion or the spherical gear. Based on the two-parameter enveloping theory, the mathematical models of the spherical gear pair with two degrees of freedom were proposed. The developed mathematical model of the gear pair with two degrees of freedom can be used to analyze undercutting analysis of the imaginary planar rack cutter. Based on the undercutting equation, a minimum number of teeth of the spherical pinion are determined. A line of singular points on the spherical pinion’s teeth are displayed. Results show that a minimum number of teeth of the spherical pinion can be determined in order to avoid undercutting.

AB - This paper presents an imaginary planar rack cutter having two parameters of motion to create a spherical gear pair with two degrees of freedom. The working region of the imaginary planar rack cutter is that of symmetrical straight lines. The relationship between the coordinate system of the imaginary planar rack cutter and that of the spherical gear pair having two parameters of motion is illustrated. The two-parameter family of imaginary planar rack cutter surfaces is obtained by using the homogeneous coordinate transformation matrix to transfer the coordinate system of the planar rack cutter to that of the spherical pinion or the spherical gear. Based on the two-parameter enveloping theory, the mathematical models of the spherical gear pair with two degrees of freedom were proposed. The developed mathematical model of the gear pair with two degrees of freedom can be used to analyze undercutting analysis of the imaginary planar rack cutter. Based on the undercutting equation, a minimum number of teeth of the spherical pinion are determined. A line of singular points on the spherical pinion’s teeth are displayed. Results show that a minimum number of teeth of the spherical pinion can be determined in order to avoid undercutting.

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