Abstract
A simple and robust method to simulate spiral bevel gears generating and meshing processes is proposed. In a first part, a mathematical model of universal hypoid tooth surfaces generator is formulated. It is based on Fong’s approach. The model takes into account all the kinematic motions of common CNC machine tools dedicated to hypoid gears machining. It is general enough to enable the simulation of various hypoid gears cutting methods such as face-hobbing, face-milling, plunge cutting and bevel-worm-shaped-hobbing processes. In this paper, only developments related to face-milled spiral bevel gear generation are presented. We show that the results obtained are in good agreement with those of certified software. In a second part, a simple and numerically stable algorithm is proposed for unloaded tooth contact analysis. The simulation method is based on a discretization of one of the two tooth flank surfaces in contact and a specific projection of the points on the opposite flank. It gives a good approximation of the contact pattern location. The accuracy of the contact point locations and computing time is directly dependent on the mesh density. However, this approach enables obtaining in a very short time sufficiently accurate results to meet the needs of designers, particularly in the preliminary stages of design. The relative displacements of the gears can be taken into consideration. The robustness of the proposed computing process and the adjustable accuracy of the results are the two main advantages of the presented approaches.
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Abbreviations
- A :
-
Machine center to cross point
- A m , A s :
-
Axial displacement of the master part of the slave part
- B :
-
Sliding base
- E :
-
Work offset
- G w :
-
Outer cone distance of the tooth
- i :
-
Tilt angle
- j :
-
Swivel angle
- L pt :
-
3 × 3 upper-left submatrix of M wt system
- \({\vec {n}_t }\) :
-
Unit normal vector to tool surface
- \({\vec {n}_p }\) :
-
Unit normal vector to workpiece surface
- O :
-
Offset displacement
- p m , p s :
-
Angular pitch of the master part, of the Slave part
- R f :
-
Filet radius of the tool
- R p :
-
Profile radius of the tool
- R t :
-
Mean radius of the tool
- R w , R s :
-
Sphere radius of the workpiece
- \({\vec {r}_t }\) :
-
Position vector in tool system
- \({\vec {r}_p, \vec {r}_m, \vec {r}_s }\) :
-
Position vector in workpiece system
- \({\vec {r}_{\theta s}}\) :
-
Master point position vector in slave part system
- S :
-
Radial distance
- s t :
-
Curvilinear abscissa along the tool section
- XW f :
-
Point radius of the tool
- W f :
-
Point width of the tool
- W w :
-
Face width of the tooth
- α p :
-
Profile angle of the tool
- γ :
-
Shaft angle between the master and the Slave part
- γ f :
-
Root cone angle of the tooth profile
- γ h :
-
Head cone angle of the tooth profile
- γ w , γ s :
-
Cone angle of the workpiece
- Γ :
-
Machine root angle
- δ c :
-
Distance between master part point and Slave part flank
- θ m :
-
Rotation angle of the master part in the Meshing process
- \({\phi _c }\) :
-
Rotation angle of the cradle in the generation process
- \({\phi _t }\) :
-
Rotation angle of the tool in the generation process
- \({\phi _p, \phi _s }\) :
-
Rotation angle of the workpiece in the generation process
- \({\varphi }\) :
-
Tool rotation angle in the face-hobbing process
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Astoul, J., Geneix, J., Mermoz, E. et al. A simple and robust method for spiral bevel gear generation and tooth contact analysis. Int J Interact Des Manuf 7, 37–49 (2013). https://doi.org/10.1007/s12008-012-0163-y
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DOI: https://doi.org/10.1007/s12008-012-0163-y