2.1 Formation of worm helical surface

The coordinate systems used for the formation of the helical surface of an offsetting normal arc-toothed cylindrical worm are depicted in Figure 1. The vector of the fixed coordinate system is coincident with the worm’s axis. Initially, the arc-shaped blade of lathe tool is situated in the offsetting normal plane of the worm helicoid, formed by vectors and . Here, is the spherical vector function [ 21]. The distance between the offsetting normal plane and the normal plane measured along the vector is the offsetting distance of the arc-shaped blade, as depicted in Figure 1. The point which lies on the vector is the center of the arc-shaped blade of lathe tool, and the length of the vector is the operating distance during worm processing. In this machining process, the worm remains fixed, while the lathe tool rotates around the worm axial along a helical line. When the rotating angle of the lathe tool is , its moving distance is . Here, p is the worm’s helical parameter on its reference cylinder. Accordingly, for the worm helicoid , we can establish the vector equation within as

Figure 1全屏查看

Formation of offsetting normal arc-toothed cylindrical worm helical surface

(1)

where , , and . The symbol is the radius of the arc-shaped blade of lathe tool, and the symbols and are the curvilinear coordinates of the worm helicoid. The symbols and are the circular vector function [ 21].

The first-order partial derivatives of in regard to the two variables and can be respectively determined in as

(2)

(3)

According to Eqs. ( 2) and ( 3), the first fundamental quantity [ 22] of and its unit normal vector can be successively acquired as

(4)

(5)

where , , and .

2.2 Cutting meshing of worm wheel

As described in Figure 2, is utilized to denote the initial location of the worm during the cutting meshing, while is utilized to express its current location. The unit vector is identical with , and the worm angular velocity points to the positive direction of the vector . Likewise, is employed to denote the initial location of the worm wheel during the cutting meshing, while is used to express its current location. The unit vector is identical with . The worm and worm wheel are all right-hand so that the worm wheel angular velocity is toward the positive direction of the vector .

Figure 2全屏查看

Coordinate systems in cutting meshing of worm wheel

The two vectors and are perpendicular to each other since the shaft angle of the worm drive studied here is equal to 90°. The vectors and are coincident and along the communal vertical line of and , respectively. The length of the vector is the center distance a of the worm drive. The rotating angles of the worm and worm gear at the current positions relative to their initial positions are and , respectively. Here, expresses the transmission ratio.

The family of hobbing cutter generating surfaces during machining the worm wheel can be obtained by adding angle into Eq. (1), and represented in as

(6)

where and .

Assuming the angular speed of the worm is during the cutting meshing of the worm gear so that the angular speed of the worm wheel is . Accordingly, the relative angular speed vector and the relative speed vector [ 21] of the cutting meshing of the worm wheel can be respectively determined in as

(7)

(8)

Taking the dot product of Eqs. ( 5) and ( 8), then it can obtain the meshing function [ 21] as

(9)

Next, in , the equation of the worm wheel tooth surface can be determined as

(10)

where and

.

3.1 Method to determine feature point on first-type limit line

The first-type limit line of the worm drive can be determined by connecting some feature points where the first-type limit function equals 0. Generally, in a cylindrical worm gearing, the trend of the first-type limit line aligns roughly with the tooth width of the worm wheel [ 18]. Thereupon, it can be set the coordinate value of the feature point is equal to along the worm wheel axis whose value range can be ascertained according to the tooth width of the worm wheel. Utilizing this information, a nonlinear equation set to determine the feature point can be built from Eqs. ( 9), ( 10), and (15) as

(16)

To solve System (16), we can transform it into a nonlinear equation with one unknown with the help of the eliminating technique and use the geometrical plotting to determine an iterative initial value [ 14]. From the first two expressions in System (16), the trigonometric functions and can be expressed by the two variables and as below:

(17)

Calculating the quadratic sum of and in Eq. (17) leads up to a quadratic equation with one unknown as below:

(18)

The solutions of Eq. (18) are

, (19)

wherein and .

The components A and B of Eq. (9) and the trigonometric functions and in Eq. (17) can be successively denoted by the variable via substituting Eq. (19) into them, and the results are

, (20)

, (21)

Likewise, based on Eqs. ( 19)-( 21), the partial derivatives , and of the meshing function in Eqs. ( 12)-( 14) can also be represented by the variable as below:

, , (22)

After substituting Eqs. ( 20)-( 22) into the third expression of System (16), the first-type limit equation can be expressed by the variable as

(23)

where .

Due to the equivalence between the nonlinear equation of Eq. (23) and System (16), solving the equation in Eq. (23) can well determine the feature point. Changing the value of can obtain a group of feature points on the first-type limit line and therefore such a limit line can be acquired by connecting these feature points.

3.2 Method to determine intersection point between first and second-types of limit lines

The second-type limit line is also named the meshing limit line [ 19, 24], which appears on the enveloping surface (the worm helical surface) during the cutting meshing. This limit line represents the envelop line of the contact lines and divides the surface into two sections: the working zone and the non-working zone. On the enveloped surface, the conjugated line of the second-type limit line is not the envelop line of the contact lines but can divide the surface into two sub-zones, which will be tangent smoothly along the conjugated line of the second-type limit line. HU et al [25] mentioned that the first-type limit line and the conjugated line of the second-type limit line had a point of tangency. However, the computing method and the numerical results for such a point were not provided. Furthermore, the correctness of this viewpoint also needs to be verified. In view of this, the method to determine the intersection point between the two types of limit lines will be explained in the following, and its characteristic will be investigated.

The nonlinear equation set to ascertain the intersection point between the two types of limit lines can be built from Eqs. ( 9), ( 14), and (15) as

(24)

The eliminating process of System (24) is the same as that of System (16), and it can also be eliminated as an equivalent nonlinear equation with one unknown as follows:

(25)

The existence of solution of Eq. (25) is corresponding to the existence of the intersection point between the two types of limit lines. The number of the points of intersection between the curve of function and the horizontal axis will be the number of the intersection points between the two types of limit lines.

4.1 Determination of first-type limit line

In this section, we employ an offsetting normal arc-toothed cylindrical worm drive with basic design parameters outlined in Table 1 to investigate the undercutting characteristics of its worm wheel. Table 1 also contains the operating parameters.

Table 1全屏查看

Parameters of worm pair

Description	Symbol	Value
Transmission ratio		29/5
Center distance	a	180 mm
Number of worm threads		5
Modulus of worm		9.5 mm
Tooth number of worm wheel		29
Reference radius of worm		35.5 mm
Helix parameter of worm		23.75 mm
Pressure angle of worm		23°
Modification coefficient of worm wheel		0.7105
Reference lead angle of worm		33.7831°
Tip radius of worm		45 mm
Radius of arc-shaped blade of lathe tool		76 mm
Offsetting distance of lathe tool		-10 mm
Operating distance during machining worm		63.758 mm

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To acquire the first-type limit line for the worm drive, thirteen feature points G ₁- G ₁₃ are set along the axial orientation of the worm wheel. The values of at these points are determined according to the whole face width of the worm wheel and can be found in Table 2. Due to the same determination method of all the feature points, we use the feature point G ₁₃ corresponding to as a case to illustrate the determination of the feature points. For this feature point, the curve of when is drawn in Figure 3, where the value range of is . The null value of in the right part of Figure 3 is caused by in Eq. (19).

Table 2全屏查看

Numerical results of feature points

Point	/mm	/rad	/rad	/rad	/mm	/mm	z 1/mm	/mm
G 1	-42.0000	0.3095	-1.2127	-5.0424	-42.0000	134.6119	31.3630	64.6072
G 2	-35.0000	0.4493	-1.5581	-4.6338	-35.0000	136.3953	19.8920	57.0586
G 3	-28.0000	0.5694	-1.9497	-4.2154	-28.0000	137.5831	6.8962	50.9695
G 4	-21.0000	0.6401	-2.4544	-3.7678	-21.0000	137.5184	-7.6325	47.5788
G 5	-14.0000	0.6504	-2.8553	-3.5123	-14.0000	136.1616	-17.5412	47.1017
G s	—	0.6455	-2.9636	-3.4717	-11.2013	135.4889	-19.9266	47.3290
G 6	-7.0000	0.6334	-3.0887	-3.4505	-7.0000	134.5071	-22.4446	47.8930
G 7	0	0.6055	-3.2377	-3.4751	0	133.1406	-24.9572	49.2194
G 8	7.0000	0.5706	-3.3474	-3.5360	7.0000	132.2271	-26.3432	50.9072
G 9	14.0000	0.5300	-3.4354	-3.6147	14.0000	131.7799	-27.0911	52.9202
G 10	21.0000	0.4839	-3.5067	-3.7059	21.0000	131.7587	-27.3715	55.2613
G 11	28.0000	0.4324	-3.5606	-3.8100	28.0000	132.0983	-27.2107	57.9483
G 12	35.0000	0.3752	-3.5931	-3.9304	35.0000	132.7178	-26.5631	61.0063
G 13	42.0000	0.3121	-3.5986	-4.0719	42.0000	133.5255	-25.3523	64.4635

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Figure 3全屏查看

Function curve when

Distinctly, an intersection point between the horizontal axis and curve nears the point . This allows us to use as the initial value for solving iteratively. Then the values of the two angles and can be orderly calculated using Eqs. ( 19) and ( 17). The numerical results for this feature point are recorded in Table 2. Likewise, other feature points are determined by means of this method and their numerical outcomes are also provided in Table 2.

In addition, the curve is drawn in Figure 4, and the value range of is as well. Figure 4 reflects that there is only one intersection point between the horizontal axis and curve, and this means that the first- and second-type of limit lines only have one intersection point, marked as G _s. The numerical results for this intersection point between the two types of limit lines are listed in Table 2 as well.

Figure 4全屏查看

Function curve

Pursuant to the data listed in Table 2, the first-type limit line L ₁ and its conjugated line are plotted in the worm wheel axial section and worm axial section, as seen in Figures 5 and 6.

Figure 5全屏查看

First-type limit line in worm wheel axial section

Figure 6全屏查看

Conjugated line of first-type limit line in worm axial section

In Figure 5, line H ₁ H ₂ is the conjugated line of the worm tooth crest, forming the boundary of the meshing zone on the worm wheel tooth surface. In Figure 6, line L ₂ signifies the second-type limit line. The working zone and the non-working zone are respectively on the right flank and left flank of line L ₂. The conjugated line of line L ₂ in Figure 5 divides the worm wheel tooth surface into two sub-zones (the left flank of line L ₂ is called sub-zone and the right flank of line L ₂ is called sub-zone ) which are respectively determined by the two solutions of in the meshing equation [ 13]. The sub-meshing zones H ₁ K ₁ K ₂ H ₁ and H ₂ K ₁ K ₂ H ₂ are located in and , respectively. From the meshing equation in Eq. (9), the two solutions of in the sub-zones and can be respectively denoted by the angles and as

(26)

where and .

The point G _s in Figure 5 is the intersection point between L ₁ and L ₂, and it can be discovered that the first-type limit line L ₁ is located below the conjugated line of the second-type limit line L ₂. At the point G _s, the dot product is equal to 0 while the norm of the vector is equal to 0.1924 mm, and therefore the vector is perpendicular to the vector . This means that the sliding angle [ 20] at the point G _s is equal to 0, and therefore this point is the lubrication weak point of the worm drive. Moreover, Figures 5 and 6 also reflect that the first-type limit line L ₁ on is located outside the meshing zone of the worm drive and its conjugated line is also located outside the practical tooth surface of the worm.

On the other hand, the parameters of point H ₁ can be obtained according to Ref. [ 13]. Based on this, from Eq. (15), the value of at the point H ₁ can be calculated out as 1.5716 and this indicates that such a point is located on the side of non-undercutting because the direction of determined in Eq. (5) is from the space to the inside of the worm. This also manifests that there is no undercutting on the whole meshing zone H ₁ K ₁ H ₂ K ₂ H ₁.

4.2 Discussion on first-type limit line

The location of the first-type limit line for the worm drive is influenced by various factors, including the offsetting distance of the lathe tool, the number of worm threads , the tooth number of the worm wheel, and the modification coefficient of the worm wheel. Figures 7- 10 demonstrate how these parameters affect the position of the first-type limit line.

Figure 7全屏查看

Influence of offsetting distance of lathe tool on first-type limit line

Figure 8全屏查看

Influence of number of worm threads on first-type limit line

Figure 9全屏查看

Influence of tooth number of worm wheel on first-type limit line

Figure 10全屏查看

Influence of modification coefficient of worm wheel on first-type limit line

From the obtained results shown in Figures 7- 10, it can be found that when increasing and decreasing , and , the line L ₁ will move toward the worm wheel dedendum. In consequence, the area where the undercutting is most easily produced on the worm wheel tooth surface is in the middle of the worm wheel dedendum and near the conjugated line H ₁ H ₂ of the worm tooth crest.

Moreover, in order to explain the changing law of -values near the line L ₁, the -values of some meshing points on the conjugated line of the worm tooth crest are obtained by calculation and marked in Figure 10. These results reflect that the -values are positive and negative on the non-undercutting and undercutting areas, respectively. Thus, the value of the meshing points on the worm tooth crest can be utilized to determine whether undercutting exists in the meshing zone. For details, see Section 4.4.

4.3 Undercutting mechanism

As recounted in the introduction, the first-type limit line is the spine curve of the worm wheel tooth surface, dividing it into non-undercutting and undercutting areas. During the cutting meshing of the worm wheel, the undercutting area is located in the interior of the hob teeth and will be cut off while the non-undercutting area will form the practical worm wheel tooth surface. The undercutting mechanism will be elucidated through an example.

The acquired results of the worm drive with are drawn in Figures 11 and 12.

Figure 11全屏查看

Limit lines in worm axial section

Figure 12全屏查看

Limit lines in worm wheel axial section

The points G _w1 and G _w2 represent the intersection points between the first-type limit line and the worm tooth crest. Notably, the conjugated line of worm tooth crest between the two points G _w1 and G _w2 does not exist on the practical worm wheel tooth surface although Figure 12 shows that such a line is located above the first-type limit line. This phenomenon is caused by the projection method and Figure 12 cannot reflect the actual situation of the limit lines and the meshing zone. To address this, we create a three-dimensional schematic diagram of the limit lines and meshing zone on the worm wheel tooth surface, as seen in Figure 13.

Figure 13全屏查看

Schematic diagram of limit lines and meshing zone on worm wheel tooth surface

In Figure 13, the surface ( and ) is the non-undercutting area while the surface ( and ) is the undercutting area. The two surfaces and have a common curve, i.e., the first-type limit line L ₁. Moreover, although the equations of the surfaces and are identical, the value ranges of their parameters are different. The points K ₁, K ₂ and K ₃ correspond to the intersection points between the second-type limit line and the worm gear tooth crest, the worm tooth crest, and a supplementary point on the second-type limit line, respectively. On the conjugate line of L ₂, the point G _s is located between the two points K ₁ and K ₂, and the conjugated line of the second-type limit line G _s K ₂ is on the undercutting area . The conjugated line of the second-type limit line G _s K ₃ divides the surface into two parts of and which are also corresponding to the angles and respectively. Obviously, a part of the meshing zone G _w1 K ₂ G _w2 G _s G _w1 is located on the undercutting area and this can lead the undercutting to appear on the worm wheel tooth surface. On the other hand, the transversals and of the two surfaces and when (see Figure 12) are respectively obtained and drawn in Figure 14. This figure reflects that the intersection points between the conjugated lines of the worm tooth crest and the second-type limit line are all on the transversal which is located on the undercutting area, and this result is the same as that of Figure 13. As a result, the undercutting mechanism for the worm wheel is essentially that a part of the conjugated line of worm tooth crest and a part of the meshing zone will come into the undercutting area and will be cut off during the cutting meshing of the worm wheel. At this time, the first-type limit line L ₁ can form the boundary of the meshing zone on .

Figure 14全屏查看

Transversals of two surfaces and when

Furthermore, the machining simulation is carried out by VERICUT software to verify the correctness of the undercutting mechanism of the worm drive. A virtual five-axis machine tool system is built in VERICUT as shown in Figure 15, and the generating surface of the hobbing cutter is established according to the vector equation of the worm helicoid. Figure 16 shows the virtual model of the worm wheel via generating motion. We can see that the undercutting area acquired by VERICUT software is located at the worm wheel tooth root and near the negative side of the axis , and this result is consistent with Figure 11. Based on this, the results obtained above indicate that the proposed computing method of the undercutting area is correct. On the other hand, the transition curved surface shown in Figure 16 is formed in the process of machining worm wheel. This area is outside the meshing zone of the worm drive, and therefore it will not be discussed here.

Figure 15全屏查看

Simulation model of worm wheel generated by cylindrical hob in VERICUT

Figure 16全屏查看

Virtual model of the worm wheel

4.4 Characteristic quantity for judging curvature interference

According to the above analyses, the values of at the meshing points on the worm tooth crest can be utilized as the characteristic quantity for judging whether the curvature interference occurs in the meshing zone since the undercutting appears on the worm tooth crest firstly. If the values of at the meshing points of the worm tooth crest are all greater than 0, there will be no undercutting in the meshing zone. On the contrary, if the values of at the meshing points of the worm tooth crest are not all greater than 0, the undercutting will appear in the meshing zone. Therefore, the curve on the worm tooth crest can be used to reflect the undercutting characteristic of the worm wheel visually. This method is simpler than the above method because the computation of the values of at the meshing points of the worm tooth crest is relatively easy.

The values of the angle along the worm tooth crest are identical, which can be easily computed out from the equation of worm tooth crest via iteration and can be denoted as . After setting the value of according to the worm thread length, the -value can be figured out as , and then the value of the angle can be worked out from Eq. (26). In the end, the value of can be obtained from Eq. (15) based on the preceding results and the curve can be determined.

The curves corresponding to the results in Figure 10 are drawn in Figure 17. The value range of is set within , accounting for the presence of the non-working area on the worm helical surface. The solid lines and dotted lines in Figure 17 are respectively obtained by using the angles and of Eq. (26).

Figure 17全屏查看

Curve of

Figure 17 shows that the minimum value of is generally on the solid line, and therefore this minimum value is calculated by the angles . This indicates that the undercutting occurs first in sub-meshing zone K ₁ K ₂ H ₁ K ₁ and then in sub-meshing zone K ₁ K ₂ H ₂ K ₁. With the decrease of , the curve will move down and intersect with the abscissa axis. This reflects that the risk of producing undercutting is increased and these results are coincident with the results shown in Figure 10. When , some values of are less than 0 and this manifests that the undercutting appears in the meshing zone. According to Figure 17, it can be found that the axial position of the undercutting area on the worm helicoid is roughly located in the range when . Thus it can be seen that the results reflected by the curve in Figure 17 and the results displayed in Figure 11 are coincident. As a result, using the characteristic quantity proposed in this chapter to judge the undercutting for the worm wheel is effective and feasible. This approach can be extended to investigate undercutting characteristics for the worm wheels in other types of cylindrical worm drives, contributing to the preliminary determination of design and technological parameters for worm drives.