Design Method for N-Lobed Noncircular Bevel Gears

As a type of spatial transmission mechanism, noncircular bevel gears (NBGs) can transfer power and motion between two intersecting axes with variable transmission following a suitable program of motion. Utilizing the spherical triangle theorem and meshing principle, parametric equations are established in the spherical polar coordinate system for the driving and driven gears, for the pitch curves, and for the addendum and dedendum curves of a NBG for a given transmission ratio and axis angle. A formulation of the tooth profile of a NBG is deduced using an analytic method. Three-dimensional models of the 3- and 4-lobed NBGs are derived in verifying this method.


Introduction
With noncircular bevel gears (NBGs), power and motion can be transferred between two intersecting axes with a variable transmission executed by a suitable program of motion. NBGs have many advantages such as smooth motion, compact structure, and accurate transmission. Because of their performance, they have also been used in highly specialized applications such as limited-slip differentials. Compared with cylindrical gears, the pitch curves of NBGs are spatial curves. Therefore, designing them is more difficult than either noncircular or bevel gears. Their theory of transmission is incomplete, the research into applications is not mature, and analyses and calculations need to be improved if NBGs are to be developed further 1 .
On the basis of the research method for planar noncircular gears, Ollson 2 proposed a design and manufacture method for NBGs employing the spherical polar coordinate system. The pitch surface of a N-lobed elliptical bevel gear was analyzed by Figliolini and Angeles 3 . Many scholars have studied the tooth shape, pitch curve, and machining method of NBGs. Because of its variable transmission ratio, Wang and collaborators 4,5 applied noncircular gears to a limited-slip differential and were granted patents for the device. Jia and collaborators 6,7 applied NBGs to a limited-slip differential with a variable transmission ratio and studied methods for meshing NBGs. Jia and collaborators also put forward a method of machining a tooth surface by wire cutting on a NC machine tool and thereby solved the problem of requiring a small taper for the domestic machine tool 8 . Zhao and collaborators 9 also applied this machining method for their NBGs. Lin and collaborators 10 studied the method of calculation and transmission performance of ellipse bevel gears. Using screw theory, Lin and collaborators 11 studied the compound transmission mechanism of a curved-face gear. Xia and collaborators 12 studied the geometric parametric equation and CAD modeling of NBGs. Lv and collaborators 13 proposed a new kind of shaping method for the pitch surface that solves the problems caused by convex and concave tips. Zheng and collaborators 14 proposed a universal method that is applicable to tooth profiles. They also analyzed the generation concept of a crown tooth to generate a tooth surface 15 . Shi and collaborators 16 analyzed the minimum teeth number to avoid undercutting. Additionally, the varying-coefficient-profile-shift-modification method is used to avoid undercutting, thus ensuring the root part of the tooth face does not participate during meshing. They further presented a design method for NBGs having a concave pitch curve described in the spherical polar coordinate system 17 and proposed a method to determine whether a gear is continuously driven based on the coincidence degree defined by the engagement angle 18 .
The above research still must surmount production difficulties and high manufacturing costs. The equivalent design method [5][6][7][8][9][10][11] had been adopted and approximate solutions were obtained using the results of previous studies. To improve on these results, the spherical triangle theorem is implemented and the meshing principle adopted. The parametric equations for the driving and driven gears, the pitch surface, and the addendum and dedendum surfaces of NBGs for a given transmission ratio and axis angle are established in the spherical polar coordinate system. Using the analytic method, the formulation of tooth profile of NBGs are deduced. The three-dimensional models of the 3-lobed and 4-lobed NBGs are presented in a verification of this research method and thereby provide a theoretical basis for the manufacture of the NBGs.

Pitch Curve
The two pitch surfaces in the mesh (Figure 1) can be formally expressed as u 1 = u 1 (u 1 ) and u 2 = u 2 (u 2 ). Here, u denotes the pitch curve polar angle, u the perigon, and v the instantaneous angular velocity. The transmission ratio of the NBGs is 16 At point P, n P1 ! = n P2 ! , vp ! represents the velocity in the common normal direction, and u 0 the original angle; therefore, the transmission ratio is given by The last identity is obtained using a basic trignometric identity. The pitch curve equation is and from equation (1) we have Using Figure 2, the sum of the angles from P to C and D is a constant l, and the number of cycles of u 2 is n 2 . Therefore, the equation for the pitch curve of an elliptical bevel gear is then Where and with A 1 = sin u 0 Á sin l + cos u 0 Á cos l 1 À cos u 0 Á cos f + sin u 0 Á sin f, B 1 = sin u 0 Á sin l + cos u 0 Á cos l 1 À cos u 0 Á cos f À sin u 0 Á sin f:

Dedendum and Addendum Curves
The equations of the dedendum and addendum curves ( Figure 3) are 16 cos g = cos u Á cos a a + sin u Á sin a a Á cos z cosDu g = cos a a À cos u Á cos g sin u Á sin g 8 < : , ð7Þ cos b = cos u Á cos a f + sin u Á sin a f Á cos j where & :

Tooth Profile
The included angle, denoted by l, is defined as the angle between the normal arc of a tooth profile and the spherical orthodrome and is the polar angle between the normal arc of the left and right tooth profiles of a NBG. With d denoting the azimuth angle of the tangent of the intersection point of the pitch curves, the difference l2d is then a constant value. The tooth profiles can be derived from the Willis theorem. 19 The concave pitch curve of NGBs can be processed by the bevel gear milling cutter, so the deduced tooth profile equation is universal. The tooth profile of the bevel gear milling cutter is shown in Figure 4. The initial meshing point A 0 is on the pitch circle of the cutter. When the cutter rotates to A 1 , they are meshed in N 1 , and where u B denotes the polar angle of the base circle, u the polar angle of the pitch circle, and a n the profile angle of the bevel gear cutter. With A n N n denoting the length between A n on the pitch circle and N n in the tooth profile, we deduce from Eq. (9) A n N n = S n cos a n , ð10Þ where S n denotes the pitch circle length between the two intersection points, and n = 1, 2:::. By the principle of gear engagement, the arc length of the NBGs and the helical curve arc of the cutter are the same. The tooth profile of any point A 0 on the pitch curve of the NBGs is as follows: make a right spherical triangle MA 1 N 1 T 1 at any point A 1 on the pitch circle; find the tangent _ where \T 1 A 1 N 1 = a n and _ A 0 A 1 denotes the arc from the intersection point of the pitch curve of the NBGs and the tooth profile to the other intersection point; hence, point N 1 is the trail of the tooth profile, which is above the pitch curve (see Figure 5).
Similarly, make a right spherical triangle MA 2 N 2 T 2 at any point A 2 on the pitch circle, a tangent _ A 2 T 2 to the pitch curve at point with \T 2 A 2 N 2 = a n . The point N 2 is the trail of the tooth profile, which is below the pitch curve.

Design Examples
The driving gear is a 3-order NBG whereas the driven gear is of 4-order for the given parameter settings l = 748 and u 0 = 908. The equations for the pitch curve, and the addendum and dedendum curves of the driving gear were obtained along with those of the driven gear. The curvature of the pitch curve was deduced, with the minimum number of teeth of the NBGs being 38.
To ensure a completed tooth shape, the number of teeth should satisfy condition where S denotes the length of the pitch curve of the shaper cutter and S = 2p Á sin v, S i the length of the pitch curve of the NBGs, obtained from du, Z the number of teeth for the shaper cutter, Z i the number of teeth for driving gear and driven gear, and v the cone angle of the shaper cutter. Three-dimensional models of the 3-order and 4-order NBGs are illustrated in Figure 6.

Conclusions
By applying the spherical triangle theorem and adopting the meshing principle, a general design method for NBGs has been proposed. The following summarizes the results obtained: (1) The equations that determine the pitch curve of the NBGs were obtained for any order and in any configuration during their pure rolling motion for a given transmission ratio and axis  angle; the equations are expressed in the spherical polar coordinate system. (2) The equations of the addendum and dedendum curves for the driving and driven gear were derived. (3) Using an analytic method, a formulation of the tooth profile for the NBGs was deduced and three-dimensional models of a pair of conjugate NBGs were developed in a verification of the correctness and reliability of this modification method.

Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.