Abstract
This paper studies the equations of rack cutters for generating involute gears with asymmetric teeth. The asymmetry means that different pressure angles are applied for the driving and coast sides. By applying the equations of the designed profile of the rack cutter, the principle of coordinate transformation, the theory of differential geometry, and the theory of gearing, the mathematical models of involute helical gear are given. Undercutting analysis is also investigated by considering the relative velocity and equation of meshing. Furthermore, a computer simulation program is developed to generate the tooth profile of asymmetric involute gears and to illustrate the effect of tool geometry on the generated surfaces. A CAD/FEM procedure for finite element modeling of asymmetric gear teeth is presented. This simulation should be helpful in the design and manufacture of asymmetric gears.
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Abbreviations
- a c :
-
Tool setting of the rack cutter
- a t :
-
Tool setting of the rack cutter
- b c :
-
Tool setting of the rack cutter
- e :
-
Amount of tool shifting (in mm)
- m n :
-
Normal module
- l j :
-
Curvilinear coordinates of rack cutter where subscript j = a, b, c, d, e, f
- [ M 1c ]:
-
Coordinate transformation matrix from coordinate system S c to S 1
- n c :
-
Unit normal vector of rack cutter surface
- \({{\bf R}_c^{i}}\) :
-
Position vector rack cutter surface for helical gear generation
- \({{\bf R}_{n}^{i}}\) :
-
Position vector of rack cutter normal section where superscript i represents regions \({\overline{ac}, \overline{bd}, \overline{ce}, \overline {df}, \overline{eg}}\) and \({\overline{fh}}\) of the rack cutter surface
- \({{\rm {\bf R}}_{1}^{i}}\) :
-
The family of the rack cutter surfaces
- r :
-
Left tip fillet radius of the cutter
- r 2 :
-
Right tip fillet radius of the cutter
- r p1 :
-
Radius of pitch circle of gear
- S c :
-
Rack cutter coordinate system
- S 1 :
-
Gear blank coordinate system
- \({V_{c}^{c1}}\) :
-
Relative velocity between the gear blank and the rack cutter represented in coordinate system S c
- β :
-
Helix angle
- \({\phi_{c1}}\) :
-
Normal pressure angle of the rack cutter (left side)
- \({\phi _{c2}}\) :
-
Normal pressure angle of the rack cutter (right side)
- \({\phi _{1}}\) :
-
Motion parameter
- ρ :
-
Surface parameter of rack cutter
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Fetvaci, C. Computer Simulation of Helical Gears Generated by Rack-Type Cutters. Arab J Sci Eng 36, 1321–1332 (2011). https://doi.org/10.1007/s13369-011-0116-y
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DOI: https://doi.org/10.1007/s13369-011-0116-y