Tooth Surface Modeling and Stress Analysis of 3-DOF Spherical Gear

A new type of spherical gear is proposed, which is formed by combining the tooth profile of the plane involute annular tooth surface and spherical involute spherical cone tooth surface. 3-DOF of pitch, deflection and rotation can be realized without principle error transmission. Based on the forming principle of plane involute and spherical involute, the equations of annular tooth surface and spherical cone tooth surface are derived by coordinate transformation theory; Combined with MATLAB and Pro/E, the 3D accurate model of 3-DOF spherical gear was obtained, and the contact stress of tooth surface was analyzed based on the finite element method. The simulation results show that the stress concentration always occurs at the tooth root and the tooth top of the spherical and conical tooth surface under the condition of constant spherical distance and the same torque but different coaxial intersection angle, which can provide reference for the tooth surface modification and optimization of spherical gear.


Introduction
Traditional gear has only one degree of freedom, which cannot meet the requirements of multi-drive degrees of freedom in emerging fields. The development of modern transmission field urgently needs the new type gear with multi-degree of freedom.
In the early 1980s, Ole. Monlang [1] invented a Trallfa flexible wrist based on spherical crown gear. However, its disadvantages include the inability to achieve accurate spherical motion with fixed transmission ratio, and its difficulty in processing and manufacturing and high cost. Shyue-cheng Yang et al [2] proposed to change the discrete conical teeth of Trallfa spherical gear into arc-shaped teeth, and transfer the motion through the engagement of arc-shaped convex teeth and arc-shaped concave teeth. Then they proposed a spherical gear with a discrete ring involute tooth profile [3] . Yang, Hsueh Cheng et al [4] designed a spherical gear pair with continuous involute tooth profile, which was shaped as convex and concave drum. Pan cunyun [5] proposed the involute ring spherical gear, which is formed by the rotation of the plane involute around the axis. This kind of continuous gear and tooth mechanism fundamentally overcomes the transmission error defect of discrete Trallfa gear and realizes continuous meshing transmission and is easy to manufacture.
In this paper, a new spherical gear structure with 3-DOF is proposed, which combines a planar involute ring tooth surface with a spherical involute cone tooth surface. Based on MATLAB and Pro/E, the precise 3d model of spherical gear was established. Besides, the Workbench was used for static analysis of spherical gear to obtain the contact stress of tooth surface, which provided basis and reference for the optimization design of spherical gear.

Analysis of transmission principle of spherical gear
As shown in figure 1, the meshing motion of the spherical gear can be regarded as pure rolling between two spheres, which always maintain tangential contact, and the transmission ratio is 1:1. The rotation of the spherical gear 1 around the 1 x 、 1 y and 1 z axis of the coordinate system with the center of the spherical as the origin and the axis can respectively realize the pitch, deflection and rotation motion of the gear pair. The weft of the pitch sphere is the pitch line of the partial pendulum meshing motion, and the longitude of the pitch sphere is the pitch line of the pitch meshing motion.

3.2Equation of annular tooth surface
where v  is the included angle between left and right involute symmetric points,  p is the node pressure angle, and z is the number of teeth.
As shown in figure 3, for the middle convex tooth, the left and right involutes are symmetric about the z axis, and the rotation angle 1 ( ) For the middle concave tooth, the left involute rotation angle 2 ( )

3.3Equation of spherical cone tooth surface
In figure where s  is the center angle corresponding to the tooth thickness, Then the rotation angle of the involute with variable cone angle is ( ) The    As can be seen from the above, with the increase of the intersection angle of the two spherical gear, the contact area of gear teeth increases first and then decreases, while the maximum stress value of gear teeth decreases first and then increases. It meets the law that spherical gear teeth from low latitude to high latitude gradually reduced. When the axial intersection angle is 90  , the maximum stress of gear tooth reaches the minimum of 24MPa. When the axial intersection angle is180  , the maximum stress of gear teeth is 61MPa. The stress concentration occurs at the tooth root and the tooth top of the spherical gear under different meshing conditions, which can provide guidance for the tooth surface modification and optimization design of spherical gear.

Conclusion
According to the generating principle of involute, the equation of spherical and ring tooth surface in the direction of longitude and weft is deduced. Based on the finite element method, the contact stress of the spherical gear pair was analyzed statically, and the contact stress of the contact position of the tooth surface was extracted. It was found that the stress concentration always occurred at the tooth root and the tooth top of the spherical and ring tooth surface of the gear, and the contact stress decreased first and then increased with the increase of the axial intersection angle.