Study on the toroidal helix and manufacturing method of enveloping hourglass worm tooth surface

As a complex space enveloped surface, the tooth surface of dual-lead helical gear enveloping hourglass worm is characteristic with difference tooth profile, unequal pitch and variable tooth thickness. To grasp the geometry characteristics and precision manufacturing method of it, the analytical method for toroidal helix is presented. The mathematical model of toroidal helix is developed, and the geometry characteristics of helix angle, axial tooth thickness, and tooth surface undercutting are analyzed. The helix angle of toroidal helix presents the convex shape, and the undercutting characteristic is affected by the pressure angle, helix angle, transmission ratio, and normal modulus. Based on the toroidal helix of tooth surface, the precision manufacturing method is proposed, and the measuring method is developed. The hourglass worm sample was manufactured and measured, and the transmission accuracy of hourglass worm drive is tested. The results show that the manufacturing and measuring method of toroidal helix is effective and feasible.


Introduction
The tooth surface of dual-lead helical gear enveloping hourglass worm is a complex space enveloped surface, and it with difference tooth profile, unequal pitch, and variable tooth thickness. 1 The toroidal helix is completely different from the usual helix, which is characterized with constant pitch and constant tooth profile. Though studying the geometric characteristics of toroidal helix and understanding the sharpening and undercutting, it can provide theoretical guidance and reference for the parameter design, manufacturing and accuracy testing of this novel dual-lead helical gear enveloping hourglass worm drive.
Scholars have discussed the tooth surface geometric characteristics of various worm gearing. Mohan and Shunmugan 2 analyzed the geometrical performances of toroidal worm drive, and simulated the worm wheel tooth profile generated by different axial sections. Dudas et al. 3 developed the arched profile spiroid worm gear drive, and proposed a new manufacturing technology for the spiroid worm with lathe. Zhao et al. [4][5][6][7] proposed the novel double anchor ring secondary enveloping toroidal worm gearing, the tooth flank modification theory, tooth contact, and mismatched performance are investigated. Deng et al. [8][9][10][11][12] proposed a novel non-backlash rolling body enveloping toroidal worm gearing, as well as studied the optimal process of machining complex surfaces and the meshing characteristics. Yang et al. 13 proposed a method to generate the spiral groove of double enveloping worm gear hob, and established the geometric model of its rake face. Rui et al. 14 presented a method to investigate the grinding interference of hourglass worm wheel hob relief surfaces, and the design method are proposed to avoid grinding interference. Simon 15 studied the LTCA of cylindrical worm gearing with the tooth surface manufacture errors and the assembly errors of shaft misalignments. Falah and Elkholy 16,17 analyzed the performance of TCA on single-enveloping toroidal worm gearing with assembly errors, and developed the stress analysis as well as load sharing with transmission errors. Polowniak et al. 18 presented a novel construction method to the hourglass worm wheel tooth profile, and it is enveloped by the double enveloping worm gear hob with the rectilinear and concave-convex. Seol and Chumg 19 and Seol and Park 20,21 investigated the geometric interference of worm drive with the oversized hob and localized contacts. Kacalak 22 proposed a novel structure of non-backlash worm gearing, which backlash can be adjusted or eliminated though the specially structure designed Moeso proposed a whirl-machining process to manufacture the toroidal worm, and the whirl-machining model of toroidal worm is developed through CNC machining machine, as well as the benefits of whirl-machining process in toroidal worm manufacturing are verified. 23 Chen et al. 1,24 and Zheng et al. 25 proposed a novel worm drive, which is consisted of a dual-lead helical gear and an hourglass worm, as well as the meshing characteristics of backlash-adjustable and wear-compensable are developed. However, as a novel hourglass worm, we have an insufficient understanding of the geometry characteristics and precision manufacturing method.
This paper presented a tooth surface analytical method for the dual-lead helical gear enveloping toroidal worm, and the mathematical model of toroidal helix is developed, as well as the geometry characteristics of helix angle, axial tooth thickness, and tooth surface undercutting are analyzed. The precision manufacturing method and measuring method is proposed based on the toroidal helix, and the toroidal worm sample was machined and measured, as well as the transmission errors is tested.

Mathematical model of toroidal helix
The toroidal helix is the intersecting line of two surface, which are the tooth surface of dual-lead helical gear enveloping hourglass worm and surface of annular rotating, as shown in Figure 1. Where, R 2 is the radial value of tooth surface, r j is the generatrix torus arc radius of toroidal rotary surface, and r f2 ł r j ł r a2 , r f2 , and r a2 are the torus arc radius of the addendum and dedendum of tooth surface, respectively. And a is the generating central distance.
The annular revolving surface in Figure 1 can be represented as: Where, f 1 and f 2 are the parameters of annular surface, and should satisfy the following boundary conditions: Combining the equation (1) with the equation of hourglass worm tooth surface, which has deduced in document 1,24 , the toroidal helix equation can be obtained as follows: Geometry characteristics of toroidal helix

Helix angle
The geometric relationship of the tangent vector t p at any point P on the toroidal helix is shown in Figure 2.
Here, l is the spiral rise angle of tooth surface at this point on the toroidal helix. According to the differential geometry, the tangent vector of point P on toroidal helix can be represented as follows: Where, n 1p is the normal vectors of tooth surface on point P, while n 2p is the normal vectors of rotating surface on point P. As shown in Figure 2 and the definition of helix angle, the calculation formula of helix angle on the toroidal helix can be obtained as follows: A dual-lead helical gear enveloping hourglass worm drive for the solar power tracking system is selected as an example, and the geometric parameters is shown in Table 1.
All the subsequent calculation parameters are same as above. The change curves of the left-hand and righthand helix angles on the pitch circle, the dedendum circle and the addendum circle of this novel hourglass worm can be obtained through the analysis and calculation, and it is shown in Figure 3.
From Figure 3, the both helix angle of this tooth surface is a convex shape, it first increases and then decreases, the change trend of the dedendum circle is obvious, the pitch circle is the second, while the changes of the addendum circle are relatively small. The maximum differences of the toroidal helix angle changes on the left-hand tooth surface of addendum circle, pitch circle and dedendum circle is 3°, 2.47°, and 2.12°, respectively. While the maximum differences of the helix toroidal angle change on the right-hand tooth surface of addendum circle, pitch circle, and dedendum circle is 2.84°, 2.33°, and 2.29°, respectively.

Tooth thickness
The thickness of this hourglass worm can be controlled by fixing one tooth surface and rotating a certain angle of the other tooth surface. The relationship of conversion principle and tooth thickness control are shown in Figure 4. Curve L 1 is the toroidal helix on the righthand tooth surface. Curve L 2 is the initial toroidal helix on the left-hand tooth surface, the curve L 2 is rotate around the axis by an angle to obtain the curve L 3 , and the tooth thickness relationship between L 1 and L 3 is correct, P 1 , P 2 , and P 3 are the intersections of curve L 1 , curve L 2 , and curve L 3 with the throat plane respectively, and r c is the toroidal helix radius on the Plane Bringing the r j = r c and z 2 = 0 into equation (3), combined with the corresponding left-hand and right-hand  tooth surface equations, the values of P 1 and P 2 can be obtained as x 2 ð Þ P1 , y 2 ð Þ P1 , 0 À Á and x 2 ð Þ P2 , À y 2 ð Þ P2 , 0Þ, respectively.
Curve L 3 is obtained by turning curve L 2 by an angle f around axis Z 2 . The rotation angle f can be expressed as: In formula (6), s n is the normal tooth thickness of the middle tooth.
The hourglass worm can be intercepted by plane P passing through axis z 2 , as shown in Figure 5. The intersection of plane P and hourglass worm is the axial tooth profile of dual-lead helical gear enveloping hourglass worm. u P is the included angle between plane P and x 2 axes, u P 2 0, 2p ½ .
In Figure 5, the axial tooth thickness of hourglass worm should satisfy the follows: Combining the worm tooth surface equation with equation (7), the axial tooth profile of tooth surface can be confirmed. By intersecting the circular arc r j with the axial tooth profile, the coordinate values of P L and P R can be obtained as where P L and P R are the intersection points of the circular arc and the both sides tooth surfaces of a certain tooth, respectively. The arc length between P L and P R is defined as the axial tooth thickness of the tooth on the arc.
As shown in Figure 6, the axial tooth thickness of dual-lead helical gear enveloping hourglass worm tooth surface is: The variation of axial tooth thickness on the addendum circle, the pitch circle and the dedendum circle of dual-lead helical gear enveloping toroidal worm can be obtained, as shown in Figure 7. The axial tooth thickness is distributed in sinusoidal wave shape, and the change trend of the addendum circle is significant, the change of pitch circle is gentle, as well as the change of tooth dedendum circle is extremely small. The maximum difference of axial tooth thickness change on the addendum circle, the pitch circle, and the dedendum circle are 0.332, 0.108, and 0.020 mm, respectively. Therefore, the dual-lead helical gear enveloping hourglass worm is not prone to the phenomenon that the tooth tip becomes sharp.

Undercutting curve
The enveloping curve of the contact lines in the developing process is the undercut curve. If the root tangent    enters the effective area of tooth surface, the undercut phenomenon will occur on and resulting in poor meshing quality, and reducing the carrying capacity and shorting the service life.
Taking the parameters Table 1 as an example, it can be calculated that the root tangent of the enveloping toroidal worm right-hand tooth surface, and it can be shown in Figure 8.
In Figure 8, P C is the intersection of the root tangent and the end face, and its coordinate value is (x c , y c , z c ), r f2 is the radius of tooth root arc of enveloping hourglass worm with variable tooth thickness involute gear, l is the effective length of worm, g 1 is the distance between the root tangent and the tooth root circle at the outlet end of the hourglass worm.
If g 1 \0, the tooth surface will be undercutting. If g 1 .0, the tooth surface will not be undercutting. Therefore, g 1 is taken as the judgment function to judge whether the hourglass worm will be undercut, which can be described as following: Taking the aforementioned parameters in Table 1 as an example, the influence laws of main geometric parameters such as helix angle, normal modulus, transmission ratio, and pressure angle on the undercutting are analyzed, as shown in Figure 9.
The pressure angle and helix angle have very little influence on the tooth undercutting. The change of transmission ratio and normal modulus will lead to the change of dedendum arc radius of hourglass worm, so it has a slight influence on tooth surface undercutting. Therefore, the tooth surface of dual-lead helical gear enveloping hourglass worm is less prone to undercutting.

3D model of toroidal helix tooth surface
Based on the toroidal helix model in equation (3), the toroidal helix of hourglass worm in Table 1 can be calculated by the MATLAB, and the two sides of toroidal helix is shown in Figure 10(a). Based on the toroidal helixes, the tooth surface can be developed by the surface fitting in Siemens Unigraphics NX, and the hourglass tooth surface is shown in Figure 10

Processing method of toroidal helix tooth surface
Though the toroidal helix, the tooth surface can be milled by the five-axis linkage vertical machining center, as shown in Figure 11. Figure 11(a) show the toroidal helix machining path on the tooth surface, and it is simulated in the HyperMill as shown in Figure 11(b).

Manufacturing of toroidal helix tooth surface
The tooth surface was manufactured by the highprecision five-axis machining center DMU 60 mono BLOCK, and the machining processing site is shown in Figure 12, as well as the roughness is tested by the Beijing Time-top SF200, and the tooth surface roughness is R a 0.653 mm.

Profile errors of tooth surface
The free scanning function is used to obtain the tooth shape data points of hourglass worm, and it can be shown in Figure 13.
In Figure 13, the position ffi is the initial reference position of the probe, the position ffl is the moving schematic diagram of the scanning process, and the  axial tooth profile of different circumferential angles can be measured by rotating the C-axis.
The tooth surface is measured by the Klingelnberg P26 in Figure 14(a), and the measuring points is shown in Figure 14(b). The measured data points need to be converted to the actual data through probe radius compensating. Through the data processing, 26-28 the tooth surface deviations can be obtained in Table 2.  In Table 2, Df max is the maximum deviation of tooth surface, while Df S is the average deviation. The maximum deviation is 24.7 mm on the left-hand side, while the average deviation is 12 mm. The maximum deviation is 17.2 mm on the right-hand side, while the average deviation is 9.3 mm.

Transmission errors of hourglass worm drive
The reducer prototype was assembled and the transmission accuracy was tested, as shown in Figure 15. [29][30][31] The rotation angle of worm was collected though angle encoder I, while the rotation angle of worm wheel was collected though angle encoder II. Based on the angle encoder, the input angle u 1 and the output angle u 2 can be obtained, and the transmission error is e = u 2 2 u 1 /i 21 , where i 21 is the transmission ratio.     The transmission errors of this prototype are measured as shown in Figure 16. In Figure 16, the clockwise rotation refers to the right-hand tooth surface of hourglass worm meshing with the left-hand tooth surface of dual-lead helical gear, and it is abbreviated as ''cw,'' while the counter-clockwise rotation is abbreviated as ''ccw.'' In the clockwise rotation, the short-cycle transmission error is 22$, while the longcycle is 59$. And in the counter-clockwise rotation, the short-cycle transmission error is 53$, while the longcycle is 81$.
The long-cycle transmission error takes the shape of sinusoidal wave, which is caused by the eccentricity error of dual-lead helical gear. And the short-cycle transmission error is 22$ in the clockwise, while that is 53$ in the counter-clockwise. That is because the maximum deviation is 24.7 mm on the left-hand tooth surface of hourglass worm, which is the meshing tooth surface on the counter-clockwise rotation, while the maximum deviation is 17.2 mm on the clockwise meshing tooth surface, as shown in Table 2.

Conclusions
(1) The helix angle of the hourglass worm toroidal helix shows the convex trend, and the change trend of root circle is obvious, while the addendum circle is relatively small. (2) The axial tooth thickness is distributed in sinusoidal wave shape, and the change trend of addendum circle is significant, while the root circle is extremely small. (3) The pressure angle and helix angle have little effect on undercutting of tooth surface, while the transmission ratio and normal modulus have a slight influence on the undercutting. (4) The tooth surface can be precision manufactured though the toroidal helix, and the hourglass worm sample show that the maximum deviation of tooth surface is 24.7 mm, while the maximum transmission error of hourglass worm drive is 81$.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by the Science and Technology