A new type of toroidal surface enveloping conical worm gearing is proposed in our recent work (Chongfei and Yaping, 2019b). According to its forming principle, the geometrical shape of the generating surface has an important influence on the geometry characteristic of the enveloping worm pair. To explore the reasonable principles for selecting the geometrical parameters of the grinding wheel, some numerical study examples are performed. In this process, the methods for the tooth crest width are developed. Simple strategies for estimating the risk of the worm tooth surface being located in the invalid area and the risk of the curvature interference on the tooth surface are proposed. The numerical result shows that increasing the radius of the toroidal-generating surface and the nominal pressure angle of the grinding wheel are beneficial to improve the engagement behavior of the conical worm pair, but the tooth crest sharpening of the conical worm may happen if they are too large. For the nominal radius of the grinding wheel, it has a negligible effect on the meshing characteristics of this worm set. In addition, the selection principle of the parameters is also suggested.

The classical conical worm drive is composed of an Archimedean conical worm and a conical face gear (Litvin, 1997). In line with its forming mechanism, the screw helicoid of the Archimedean conical worm is a ruled surface and is arduous to grind precisely in line with the forming principle (Bohle, 1955; Nelson, 1961). Apparently, it is not conducive to the hardening of the tooth surface after it is shaped, and the further enhancement of the meshing behavior of the conical worm drive is thus limited (Yaping and Xiangwei, 2018).

Schematic diagram of the toroidal surface enveloping conical worm gearing.

To overcome the shortcoming mentioned above, a new type of toroidal surface enveloping conical worm gearing is suggested in our recent work (Chongfei and Yaping, 2019b). As shown in Fig. 1, the helical surface of this new type of conical worm is ground by a grinding wheel with the toroidal-generating surface. Meanwhile, the coupled worm wheel is enveloped by a taper hob whose working surface is consistent with that of the obtained conical worm.

From the perspective of forming principles, the shape of the generating torus has an important influence on the geometry of the tooth surfaces of this new type of enveloping conical worm drive. As is well known, the shapes of the two tooth surfaces in the mesh usually have a very important influence on the meshing quality of the gear (Litvin, 2004). Besides, there is no relevant experience for this new type of conical worm drive. Therefore, it is indispensable to study the geometry of the grinding wheel for the sake of improving the meshing quality of the toroidal surface enveloping conical worm gearing and providing a reasonable principle of parameter selection for the production.

In this paper, a method to calculate the worm tooth crest width without solving the nonlinear equations is suggested, and new strategies for estimating the risk of the worm tooth surface being located in the invalid area and the risk of the curvature interference on the tooth surface are proposed. Based on this, the effects of geometrical parameters of the abrasion wheel, including the nominal radius of the abrasion wheel, the radius of the toroidal-generating surface, and the pressure angle of the grinding wheel, on the meshing performance of the toroidal surface enveloping conical worm drive are investigated.

For the numerical simulation study, some relevant theoretical background needs to be briefly introduced in this section.

Generating torus of the grinding wheel in

As drawn in Fig. 2, a moveable frame

The generating torus on the abrasion wheel intersects with the shaft section
of the abrasion wheel along the arc. The projection distance of the points

In accordance with the geometry site displayed in Fig. 2, the vector
equation of the toroidal-generating surface,

By definition (Wardle, 2008), the unit normal vector of

As illustrated in the reference (Chongfei and Yaping, 2019b), during the process of
grinding the work blank, the vector equation and the unit normal vector of

The coefficient

Next, the tooth equation of the worm helical surface,

By the literature (Chongfei and Yaping, 2019b), the relative velocity vector of the
abrasion wheel and the worm roughcast in

Based on Eqs. (4) and (6), the meshing function of the abrasion wheel and
the conical worm roughcast in the process of the grinding engagement can be
yielded as

In the process of mesh, the equation of the surface family,

In line with the method elaborated in the literature (Chongfei and Yaping, 2019b), the
relative speed of the worm pair can be obtained in

Based on Eqs. (9) and (10), the mesh function of the toroidal surface
enveloping conical worm gearing can be worked out as

Via the coordinate transformations, the equation of the conical worm wheel
tooth surface can be acquired in

Correspondingly, when

On the basis of the classical differential geometry (Wardle, 2008), the
coefficients of the first and second fundamental forms of

Subsequently, the normal vector,

By definition (Litvin, 2004), the curvature interference limit function
during the grinding mesh of the enveloping conical worm can be written as

By definition (Xuezhu, 1989), the meshing limit function of the worm pair
can be represented as

The geometric representation to display the effects of the radius of
the generating toroidal surface on the worm tooth shape is given in Fig. 3a.
Therein,

Geometric representation of the variation of the grinding wheel geometry parameters.

Figure 3b is the geometric representation of the effects of the nominal
pressure angle of the abrasion wheel on the worm tooth shape. Therein,

As shown in Fig. 3c, it can be found that the variation of the nominal
radius of the grinding wheel,

On the basis of the presented theoretical background in Sect. 2, four
representative numerical examples labeled (1), (2), (3) and (4),
respectively, are provided to investigate the influence of grinding wheel
parameters on the meshing characteristics of the toroidal surface enveloping
conical worm drive. Among them, Examples (1) and (4) are used to research
the influence of the nominal pressure angle of the abrasion wheel,
Examples (2) and (4) are used to study the effects of the nominal radius
of the grinding wheel, and Examples (3) and (4) are used to research the
influence of the radius of the toroidal-generating surface. The main parameters,
which include the technical parameters of the toroidal surface enveloping
conical worm pairs and the geometrical parameters of the abrasion wheel, are
listed in Tables 1 and 2. In general, coefficients

Parameters of the conical worm drive in each numerical example.

Grinding wheel in the shaft section.

According to the geometric parameters provided in these tables, the skeleton
maps of the grinding wheel in its shaft section are drawn in Fig. 4. In
the picture, the symbol

Generally speaking, the tooth crest width of the worm can be calculated by
solving a series of nonlinear equations (Xuezhu, 1987). But objectively speaking, solving
nonlinear equations is usually a cumbersome process. In this work, a method
is used without solving the nonlinear equations and just according to the geometric
relationship between the grinding wheels corresponding to the two sides of
the conical worm tooth. According to the geometric positional relationship
shown in Fig. 5, the tooth crest width of the toroidal surface enveloping
conical worm,

Geometric parameters of the abrasion wheel in each numerical example.

Geometric representation of the worm tooth width

The values of

Numerical results of tooth profile parameters in the examples by different methods.

In line with the meshing theory (Litvin, 2004; Xuezhu, 1989), the meshing limit line will divide the whole helical surface of the worm into an active zone and an inactive zone if it exists on the tooth surface. For the conical worm pair, the whole worm tooth surface has the risk of being in the inactive zone since the conical worm is offset to one side of the mating worm wheel. Therefore, it is of great significance to discover the influence of process parameters on the distribution of the meshing limit line.

Based on Eqs. (5), (7), (11), and (22), the meshing limit line on the tooth
surface of the conical worm can be determined by the following equations:

By letting

From Eq. (28), it can be found that the necessary condition for

Values of

As we can see, the values of

From the meshing theory (Litvin, 2004; Xuezhu, 1989), it can be known that the curvature interference limit line may cause the undercutting when it exists on the tooth surface.

In this section, the influence of process parameters on the distribution of the curvature interference limit line in the grinding engagement of the worm will be investigated. Since the helical surface of the conical worm is the enveloping surface of the abrasion-wheel-generating surface, the curvature interference (undercutting) may occur on the conical worm helicoid.

By definition, the value of

Value of

From Table 5, it is clear that the curvature interference limit line is
closest to the small end and tooth crest of the conical worm; that is to say,
the risk of tooth undercutting is highest in there. As reflected by the data
in Examples (1) and (4), increasing the nominal pressure angle of the
abrasion wheel can drive the curvature interference limit line away from the
conical worm tooth surface: this means that the risk of undercutting on the
tooth surface of the worm can be reduced. Comparing Examples (2) and (4),
we can find that increasing the nominal radius of the abrasion wheel can
reduce the risk of curvature interference. The values of

The meshing zones and the lines of contact on both the

For Examples (1) and (3), their meshing areas on the

Global meshing performance of the

Global meshing performance of the

To survey the influence of geometric parameters of a grinding wheel on the
local meshing characteristic of conical worm gearing, six points on the
tooth flank of a worm wheel are selected randomly as the inspection points.
After ascertaining these points, the local meshing characteristic
parameters, such as the induced principal curvature

The inspection points selected on the worm gear tooth flank.

Numerical variation of the induced principal curvature at the observation points.

Numerical variation of the sliding angle at the observation points.

In line with the data in these tables, the values of

The mathematical model is constructed for the meshing simulation of the toroidal surface enveloping conical worm gearing, whose conical worm helicoid is ground by an abrasion wheel with the toroidal-generating surface.

The qualitative analysis of the variation of grinding wheel parameters is performed. On the basis of the established mathematical model, the numerical simulations are implemented. In this process, the method to calculate the worm tooth crest width without solving the nonlinear equations is suggested. The simple strategies for estimating the risk of the worm tooth surface being located in the invalid area and the risk of the curvature interference on the tooth surface are proposed.

The results of the numerical examples study disclose the following.

The increase in the nominal pressure angle

Though the variation of the nominal radius of the grinding wheel

All the data used in this article can be made available upon reasonable request. Please contact the corresponding author (Yaping Zhao, zhyp_neu@163.com).

CH conducted theoretical calculation and example studies and wrote the manuscript under the guidance of YZ. YZ verified the results and supervised the whole work.

The authors declare that they have no conflict of interest.

The authors acknowledge the open fund of the key laboratory for metallurgical equipment and control of the Ministry of Education in Wuhan University of Science and Technology (2018B05) and the subsidy from the National Natural Science Foundation of China (51475083).

This research has been supported by the National Natural Science Foundation of China (grant no. 51475083), the Fundamental Research Funds for the Central Universities (grant no. N160304012), and the Chinese National Natural Science Foundation (grant no. U1708254).

This paper was edited by Bahman Azarhoushang and reviewed by two anonymous referees.