1 Introduction

Gears are essential transmission components in many areas such as aviation, automobile, engineering machinery, and so on [1,2,3]. In recent years, for face-milled spiral bevel and hypoid gears, the completing process method is widely used for increased efficiency, cost reduction, machining accuracy improvement, and tooth strength enhancement. The “completing process method” is an advanced manufacturing method, also named the “duplex helical method”, “duplex spread-blade method” and “double-cut method”. It includes only two processes: (a) finish machining of the wheel and (b) finish machining of the pinion. Compared with the traditional “five-cut method”, the “completing process method” is characterized by only one cutter for machining both the concave and convex flank of the pinion simultaneously [4,5,6].

For the standard-depth-taper face-milled gear pair, the tooth space width and the tooth thickness width change in proportion to the cone distance at any particular section along the face width. As Fig. 1 shows, the spiral angle \(\beta_{mV}\) at the concave mean point \(P_{V}\) along the pitch cone and the spiral angle \(\beta_{mX}\) at the convex mean point \(P_{X}\) along the pitch cone are equal, and they are both equal to the designed mean spiral angle. Now the concave tooth line and the convex tooth line tilt to each other [7]. But with the completing process method, the cutter cuts the concave flank and the convex flank simultaneously. Assuming that the concave tooth line \(mm\) stays the same, then the convex tooth line turns into \(n^{\prime}n^{\prime}\) from \(nn\). If by the formate method, the concave tooth line \(mm\) and the convex tooth line \(n^{\prime}n^{\prime}\) are approximately two concentric circular arcs, and do not tilt to each other any more [8]. Therefore, the completing process method leads to less material removed from the heel and more material removed from the toe, resulting in thicker tooth thickness width at the heel and thinner tooth thickness width at the toe than normal. In essence, the spiral angle \(\beta_{mV}\) and \(\beta_{mX}\) become unequal, so the tooth space width and the tooth thickness width cannot shrink in proportion to the cone distance, which affects proper meshing of gear pair. Hence, the gear blank dimensions need to be redesigned. Since the values of the cutter diameter, the pressure angle and the spiral angle are determined, the root angle is the only parameter that can be modified.

Fig. 1
figure 1

Pitch plane of face-milled spiral bevel gear

Full size image

The Gleason Company proposed the completing process method in the 1930s and disclosed the calculating instructions for the generated small face-milled spiral bevel gears duplex helical method (SGDH) in 1965 [9]. The improved method was applicable to large-module face-milled spiral bevel gears up to 1978 [10]. But the manufacturing principles are not publicly disclosed.

In recent years, scholars have continued research into the completing process method. Shtipelman pointed out that when both gear and pinion were generated by the duplex method, the gear and pinion dedendum angles had to be computed so that the spiral angles at the opposite sides of the gear and pinion teeth would have initial values, respectively [11]. In [7], the abnormal contraction of the tooth thickness width and tooth space width caused by the duplex helical method was introduced, and the formulas of the duplex taper were derived. In [8], the duplex contraction of the exact duplex helical method by the Gleason Works was introduced, and the calculating formula of the sum of the root angle of the gear and pinion was deduced, and the sum was distributed according to tooth depth ratio of inclined point. In [12], the basic machine settings of the spiral bevel and hypoid gears generated by the duplex helical method were determined, and the hypoid gear dimensions were modified based on the root angle of the pinion. In [13], three reference points were used to calculate the basic machine-tool settings for spiral bevel and hypoid gears manufactured by the duplex helical method, and the resulting new mean dedendum of the pinion was different from the mean dedendum of the hypoid gears’ blank dimensions, and the modified mean dedendum was used. For the completing process method, some other studies were also carried out, such as meshing performance analysis [14], tooth surface reconstructing method [15], tooth surface modification [16], and flank deviation correction [17]. These studies lay a foundation for the development of the completing process method. But the machining principle of the completing process method is not taken into account in the existing gear blank modification method.

This paper proposes a novel taper design method for face-milled spiral bevel and hypoid gears by the completing process method based on the machining principle. The root angles of the wheel and the pinion are redesigned after the computation of blank dimensions of a gear pair according to ISO standard. The wheel is cut by the formate method, an appropriate root angle of the wheel is searched by iteration so that the spiral angle \(\beta_{mWV}\) at the wheel concave mean point and \(\beta_{mWX}\) at the wheel convex mean point along the pitch cone are both equal to the original designed wheel mean spiral angle. The pinion is cut by the generating method, and in the same way, an appropriate root angle of the pinion is searched by iteration so that the spiral angle \(\beta_{mPV}\) at the pinion concave mean point and \(\beta_{mPX}\) at the pinion convex mean point along the pitch cone are both equal to the original designed pinion mean spiral angle. In this way, the ratios of the wheel tooth thickness and tooth space width to the wheel cone distance are more stable, and the ratios of the pinion tooth thickness and tooth space width to the pinion cone distance are more stable, too.

In order to ensure that the change of the whole tooth depth along the face width is relatively even, the root line is tilted about the root mean point. Thus, the mean addendum and the mean dedendum stay the same before and after the redesign.

The goal is blank modification for the completing process method, and the machine-tool settings in this paper are calculated based on the conventional primary cradle-type generator. Gear blank design is the first step of gear development. After the step of blank modification, the machining principle of the completing process method will be studied, and it will be based on the CNC free-form type generator. Then, the machine-tool settings of the completing process method on the CNC free-form type generator will be determined based on the contact characteristics [18] for good performance. The method is applicable to face-milled spiral bevel and hypoid gear by the completing process method. After redesign by this method, a duplex taper will be developed.

2 Redesign of Wheel Root Angle

The original design of the blank dimensions is accomplished according to ISO standard [19]. The redesign process of the wheel root angle is as follows.

2.1 Determination of Wheel Cutter Parameters

The wheel can be cut by the formate method or by the generating method [20]. Since the wheel is usually cut by the formate method to increase efficiency in actual production [21] when the pitch angle is larger than 70°, the wheel is cut by the formate method here. If this method is to be applied to the wheel by the generating method, then the formulas in Sect. 2.2 and Sect. 2.3 are replaced by the formulas of the generating method.

The inner blade angle and the outer blade angle of the wheel cutter are equal to the pressure angle at the concave root mean point and the pressure angle at the convex root mean point, respectively. The inner blade angle of the wheel cutter is positive, and the outer blade angle is negative. The cutter radius of the wheel \(r_{cW}\) is selected from standard specifications. The point width of the wheel cutter can be determined by the following equations.

If the backlash is not considered, the tooth space width of the wheel should be equal to the tooth thickness width of the pinion [22]. Figure 2 is the pitch plane of the wheel. The outer circular tooth thickness of the wheel \(S_{eW}\) and the outer circular tooth thickness of pinion \(S_{eP}\) are already determined during the original computation of blank dimensions. The mean circular tooth space width of the wheel \(cd\) can be calculated as follows:

$$cd \approx \frac{{R_{mW} }}{{R_{eW} }}S_{eP}$$
(1)

where \(R_{mW}\) is the mean cone distance of the wheel, and \(R_{eW}\) is the outer cone distance of the wheel.

Fig. 2
figure 2

Point width of wheel cutter

Full size image

The mean pitch normal chordal tooth space width of the wheel \(\overline{ab}\) can be obtained by the following equation:

$$\overline{ab} = cd\cos \beta_{mW}$$
(2)

where \(\beta_{mW}\) is the mean spiral angle of the wheel.

The point width of the wheel cutter can be determined as follows:

$$W_{W} = \overline{ab} - h_{fmW} \left( {\tan \alpha_{cWX} + \tan \alpha_{cWV} } \right)$$
(3)

where \(h_{fmW}\) is the mean dedendum of the wheel, and can be calculated by the original designed parameters because it does not change.

2.2 Determination of Wheel Machine Settings

Since the blade angles of the wheel cutter are equal to the corresponding pressure angle at the root mean point, the axis of the wheel cutter is perpendicular to the root cone of the wheel, which means that there is no tilt angle [23]. The wheel machine settings (shown in Fig. 3) can be determined as follows:

Fig. 3
figure 3

Wheel machining principle

Full size image

Vertical:

$$V_{W} = r_{cW} \cos \beta_{mRW}$$
(4)

Horizontal:

$$H_{W} = L_{mW} - r_{cW} \sin \beta_{mRW}$$
(5)

where \(L_{mW} = R_{mW} \cos \left( {\delta_{W} - \delta_{fW} } \right) + \left( {t_{zRW} - t_{zW} } \right)\cos \delta_{fW}\).

Cradle angle:

$$q_{W} = \arctan \left( {\frac{{V_{W} }}{{H_{W} }}} \right)$$
(6)

Radial distance:

$$s_{W} = \sqrt {V_{W}^{2} + H_{W}^{2} }$$
(7)

Machine root angle:

$$\Gamma_{MW} = \delta_{fW}$$
(8)

Machine center to crossing point:

$$\Delta X_{W} = t_{zRW}$$
(9)

where \(\delta_{fW}\) is the root angle of the wheel, \(t_{zRW}\) is the root apex beyond crossing point of the wheel, \(t_{zW}\) is the pitch apex beyond crossing point of the wheel.

2.3 Calculation of Wheel Mean Point Parameters

With the formate method, the tooth surface of the wheel is a complete copy of the conical surface of the wheel cutter [24]. Any point on the cutter surface can be expressed by a set of parameters \(\left( {h_{c} ,\theta_{c} } \right)\), and \(h_{c}\) represents the height from the cutter top plane to this point along the cutter axis, and \(\theta_{c}\) represents the rotation angle in the cutter transverse plane. With the coordinate of the wheel pitch mean point on the rotation projection plane given, the corresponding parameters \(\left( {h_{c} ,\theta_{c} } \right)\) on the cutter surface can be obtained by iteration, with which the position vector \(r_{c}\) and the normal vector \(n_{c}\) of this point in the cutter coordinate system can be expressed. Then the position vector \(r_{W}\) and the normal vector \(n_{W}\) of the wheel pitch mean point in the wheel coordinate system can be obtained through coordinate transformation. The position vector and the normal vector of the wheel concave mean point \(P_{WV}\) and convex mean point \(P_{WX}\) can be calculated by the following equations.

$$r_{W} = M_{WM} M_{Mc} r_{c}$$
(10)
$$n_{W} = m_{WM} m_{Mc} n_{c}$$
(11)

Here, the transfer matrix from the cutter coordinate system to the machine coordinate system is:

$$M_{Mc} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & {H_{W} } \\ 0 & { - 1} & 0 & {V_{W} } \\ 0 & 0 & { - 1} & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right]$$
(12)
$$m_{Mc} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & { - 1} & 0 \\ 0 & 0 & { - 1} \\ \end{array} } \right]$$
(13)

The transfer matrix from the machine coordinate system to the wheel coordinate system:

$$M_{WM} = \left[ {\begin{array}{*{20}c} {\cos \Gamma_{MW} } & 0 & { - \sin \Gamma_{MW} } & { - \Delta X_{W} } \\ { - \sin \Gamma_{MW} } & 0 & {\cos \Gamma_{MW} } & 0 \\ 0 & { - 1} & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right]$$
(14)
$$m_{WM} = \left[ {\begin{array}{*{20}c} {\cos \Gamma_{MW} } & 0 & { - \sin \Gamma_{MW} } \\ { - \sin \Gamma_{MW} } & 0 & {\cos \Gamma_{MW} } \\ 0 & { - 1} & 0 \\ \end{array} } \right]$$
(15)

2.4 Calculation of Spiral Angle

With the position vector and the normal vector of a point on a gear tooth flank given, the spiral angle at this point along the pitch tooth line can be obtained. As Fig. 4 shows, \(x\) is the gear axis, and \(\theta\) is the directed angle in \(yz\) plane, which can be determined by the following piecewise function.

$$\left\{ \begin{gathered} y > 0,z > 0: \, \theta = \arctan \left( {z/y} \right) \\ y > 0,z < 0: \, \theta = \arctan \left( {z/y} \right) + 2\pi \\ y < 0: \, \theta = \arctan \left( {z/y} \right) + \pi \\ y = 0,z > 0: \, \theta = \pi /2 \\ y = 0,z < 0: \, \theta = 3\pi /2 \\ y > 0,z = 0: \, \theta = 0 \\ y < 0,z = 0: \, \theta = - \pi \\ \end{gathered} \right.$$
(16)
Fig. 4
figure 4

Spiral angle at any point

Full size image

In the axial section plane \(xr\), \(N\) is the unit normal vector at that point perpendicular to the pitch cone, pointing to the outside of the pitch cone. \(L\) is the normal vector along the pitch cone generatrix, pointing to the gear heel. \(T\) is the unit tangent vector at that point along the pitch tooth line. These vectors can be obtained as follows:

$$N = \left\{ {\begin{array}{*{20}c} { - \sin \delta } & {\cos \delta \cos \theta } & {\cos \delta \sin \theta } \\ \end{array} } \right\}$$
(17)
$$T = N \times n/\left| {N \times n} \right|$$
(18)
$$L = \left\{ {\begin{array}{*{20}c} {\cos \delta } & {\sin \delta \cos \theta } & {\sin \delta \sin \theta } \\ \end{array} } \right\}$$
(19)

where \(n\) is the calculated normal vector at this point, and \(\delta\) is the pitch angle.

Then the spiral angle at this point along the pitch cone can be obtained by this:

$$\beta = a\cos \left( {L \cdot T} \right)$$
(20)

If the calculated \(\beta > \pi /2\), then \(\beta = \pi - \beta\).

The spiral angle \(\beta_{mWV}\) at the wheel concave mean point and \(\beta_{mWX}\) at the wheel convex mean point along the pitch line can be determined by the above method.

This method can be applied to any type of bevel gear, if \(\delta\) is replaced by other cone angle, then the calculated spiral angle is along that cone angle.

2.5 Redesign of Wheel Root Angle

The demand for the wheel blank by the completing process method is: the redesigned wheel concave mean spiral angle \(\beta_{mWV}\) and the redesigned wheel convex mean spiral angle \(\beta_{mWX}\) along the pitch cone are equal, and both equal to the original designed mean spiral angle \(\beta_{mW}\). The constrain can be expressed by the following equation:

$$\beta_{mWX} = \beta_{mWV} = \beta_{mW}$$
(21)

This equation is equivalent to two independent constrains. When the two independent variables, that is the wheel root angle \(\delta_{fW}\) and the wheel root mean spiral angle \(\beta_{mRW}\), are assigned new values, the wheel machine settings are recalculated, then new \(\beta_{mWV}\) and \(\beta_{mWX}\) are obtained. Through iteration like this, the appropriate new wheel root angle \(\delta_{fW}^{\prime }\) and new wheel root mean spiral angle \(\beta_{mRW}^{\prime }\) can be searched finally. The iteration is based on the secant method.

During iteration, the initial value of the wheel root angle can be given the original designed parameter, and the initial value of the wheel root mean spiral angle can be calculated by the original designed parameters.

3 Redesign of Pinion Root Angle

3.1 Determination of Pinion Cutter Parameters

The cutter radius of the pinion \(r_{cP}\) is selected from standard specifications.

The inner blade angle \(\alpha_{cPX}\) and the outer blade angle \(\alpha_{cPV}\) of the pinion cutter are equal to the pinion convex and concave mean root pressure angle, respectively. The inner blade angle is negative, and the outer blade angle is positive.

After the wheel root angle is redesigned, the coordinate of the redesigned wheel concave mean point and the redesigned wheel convex mean point are determined. But these two points are two endpoints of the mean tooth space circular arc of the wheel. Then the convex mean point is turned an angular pitch about the wheel axis so that these two points become two endpoints of the wheel mean tooth thickness circular arc. Thus, the circular tooth thickness \(gh\) (shown in Fig. 5) can be determined.

Fig. 5
figure 5

Point width of pinion cutter

Full size image

The mean normal chordal tooth thickness of the wheel can be calculated as follows:

$$\overline{ef} \approx gh\cos \beta_{mW}$$
(22)

If the backlash is not taken into account, the pinion tooth space width should be equal to the wheel tooth thickness width at the pitch plane. The pinion cutter pitch width \(ef\) should be equal to the pinion mean normal chordal tooth space width. So the point width of the pinion cutter can be estimated by the following equation.

$$W_{P} = \overline{ef} - h_{fmP} \left( {\tan \left| {\alpha_{cPX} } \right| + \tan \left| {\alpha_{cPV} } \right|} \right)$$
(23)

where \(h_{fmP}\) is the pinion mean dedendum.

3.2 Determination of Pinion Machine Settings

The axis of the pinion cutter is perpendicular to the pinion root cone, so there is no tilt. The pinion machine settings (shown in Fig. 6) can be determined as follows:

Fig. 6
figure 6

Machining principle of pinion

Full size image

Vertical:

$$V_{P} = r_{cP} \cos \beta_{mRP}$$
(24)

Horizontal:

$$H_{P} = L_{mP} - r_{cP} \sin \beta_{mRP}$$
(25)

where \(L_{mP} = R_{mP} \cos \left( {\delta_{P} - \delta_{fP} } \right)\).

Cradle angle:

$$q_{P} = \arctan \left( {\frac{{V_{P} }}{{H_{P} }}} \right)$$
(26)

Radial distance:

$$s_{P} = \sqrt {V_{P}^{2} + H_{P}^{2} }$$
(27)

Roll ratio:

$$R_{aP} = \frac{{L_{mP} }}{{R_{mP} \sin \delta_{P} }}$$
(28)

Machine root angle:

$$\Gamma_{MP} = \delta_{fP}$$
(29)

Here, \(\beta_{mRP}\) is the pinion mean root spiral angle, and \(R_{mP}\) is the pinion mean cone distance, and \(\delta_{P}\) is the pinion pitch angle, and \(\delta_{fP}\) is the pinion root angle.

3.3 Calculation of Pinion Mean Point Parameters

The pinion is cut by the generating method, and the cradle rotates with the cutter on it, which can be imaged as a generating gear that meshes with the pinion with line contact at the roll ratio. Any point on the cutter surface can be expressed with a set of \(\left( {h_{c} ,\theta_{c} } \right)\). As the same with the wheel, \(h_{c}\) represents the height along the cutter axis from the cutter top plane to this point, and \(\theta_{c}\) represents the directed angle on the cutter transverse plane. When the coordinate of the pinion mean point on the rotation projection plane is given, the corresponding parameters \(\left( {h_{c} ,\theta_{c} } \right)\) on the cutter surface can be solved by iteration. The rotation angle of the generating gear can be determined through meshing equation. The position vector \(r_{c}\) and the normal vector \(n_{c}\) of this point on the cutter in the cutter coordinate system can be expressed in the pinion coordinate system by transfer matrix, and becomes the position vector \({\mathbf{r}}_{P}\) and the normal vector \(n_{P}\) on the pinion flank, which can be calculated as follows:

$$r_{P} = M_{PPd} M_{PdM} M_{MG} M_{Gc} r_{c}$$
(30)
$$n_{P} = m_{PPd} m_{PdM} m_{MG} m_{Gc} r_{c}$$
(31)

Here, the transfer matrix from the cutter coordinate system to the generating gear coordinate system is:

$$M_{Gc} = \left[ {\begin{array}{*{20}c} {\sin \beta_{mRP} } & {\cos \beta_{mRP} } & 0 & {s_{P} \cos q_{P} } \\ {\cos \beta_{mRP} } & { - \sin \beta_{mRP} } & 0 & { - s_{P} \sin q_{P} } \\ 0 & 0 & { - 1} & {\left( {t_{zP} - t_{zRP} } \right)\sin \delta_{fP} } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right]$$
(32)
$$m_{Gc} = \left[ {\begin{array}{*{20}c} {\sin \beta_{mRP} } & {\cos \beta_{mRP} } & 0 \\ {\cos \beta_{mRP} } & { - \sin \beta_{mRP} } & 0 \\ 0 & 0 & { - 1} \\ \end{array} } \right]$$
(33)

The transfer matrix from the generating gear coordinate system to the machine coordinate system is:

$$M_{MG} = \left[ {\begin{array}{*{20}c} {\cos \varphi_{GP} } & { - \sin \varphi_{GP} } & 0 & 0 \\ {\sin \varphi_{GP} } & {\cos \varphi_{GP} } & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right]$$
(34)
$$m_{MG} = \left[ {\begin{array}{*{20}c} {\cos \varphi_{GP} } & { - \sin \varphi_{GP} } & 0 \\ {\sin \varphi_{GP} } & {\cos \varphi_{GP} } & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]$$
(35)

where \(\varphi_{GP}\) represents the rotation angle of the generating gear about the axis \(z_{G}\), and its value is positive if the rotation direction is in accord with the right-hand rule, otherwise negative.

The transfer matrix from the machine coordinate system to the fixed pinion coordinate system is:

$$M_{PdM} = \left[ {\begin{array}{*{20}c} {\cos \delta_{fP} } & 0 & {\sin \delta_{fP} } & { - t_{zP} } \\ {\sin \delta_{fP} } & 0 & { - \cos \delta_{fP} } & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right]$$
(36)
$$m_{PdM} = \left[ {\begin{array}{*{20}c} {\cos \delta_{fP} } & 0 & {\sin \delta_{fP} } \\ {\sin \delta_{fP} } & 0 & { - \cos \delta_{fP} } \\ 0 & 1 & 0 \\ \end{array} } \right]$$
(37)

The transfer matrix from the fixed pinion coordinate system to the moving pinion coordinate system is:

$$M_{PPd} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & {\cos \varphi_{P} } & {\sin \varphi_{P} } & 0 \\ 0 & { - \sin \varphi_{P} } & {\cos \varphi_{P} } & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right]$$
(38)
$$m_{PPd} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & {\cos \varphi_{P} } & {\sin \varphi_{P} } \\ 0 & { - \sin \varphi_{P} } & {\cos \varphi_{P} } \\ \end{array} } \right]$$
(39)

where \(\varphi_{P}\) represents the rotation angle of the pinion about the axis \(x_{P}\), and its value is positive if the rotation direction is in accord with the right-hand rule, otherwise negative.

In the machining process of the pinion, the generating gear and the pinion accord with meshing equation [25], through which the rotation angle of the generating gear \(\varphi_{GP}\) and that of the pinion \(\varphi_{P}\) can be obtained.

The meshing equation can be simplified as follows:

$$U\sin \varphi_{GP} + V\cos \varphi_{GP} = W$$
(40)

Then the rotation angle of the generating gear can be calculated using the following equation:

$$\tan \frac{{\varphi_{GP} }}{2} = \frac{{U - \sqrt {U^{2} + V^{2} - W^{2} } }}{W + V}$$
(41)

Then the rotation angle of the pinion:

$$\varphi_{P} = R_{aP} \cdot \varphi_{GP}$$
(42)

where

$$\left\{ \begin{gathered} U = T_{y} \cos \delta_{fP} \\ V = - T_{x} \cos \delta_{fP} \\ W = - T_{z} \left( {1/R_{aP} - \sin \delta_{fP} } \right) \\ \end{gathered} \right.$$
(43)
$$\left\{ \begin{gathered} T_{x} = y_{G}^{G} nz_{G}^{G} - z_{G}^{G} ny_{G}^{G} \hfill \\ T_{y} = z_{G}^{G} nx_{G}^{G} - x_{G}^{G} nz_{G}^{G} \hfill \\ T_{z} = c_{G}^{G} ny_{G}^{G} - y_{G}^{G} nx_{G}^{G} \hfill \\ \end{gathered} \right.$$
(44)

\(\left( {x_{G}^{G} ,y_{G}^{G} ,z_{G}^{G} } \right)\) is the position vector in the generating gear coordinate system that \(r_{c}\) is turned into by the transfer matrix \(M_{MG}\) and \(M_{Gc}\), and \(\left( {nx_{G}^{G} ,ny_{G}^{G} ,nz_{G}^{G} } \right)\) is the normal vector in the generating gear coordinate system that \(n_{c}\) is turned into by the transfer matrix \(m_{MG}\) and \(m_{Gc}\).

3.4 Redesign of Pinion Root Angle

With the coordinate on the rotation projection plane given, the position vectors and the normal vectors of the pinion concave mean point and the pinion convex mean point can be obtained by the above equations. According to the calculation method of the spiral angle in Sect. 2.4, the pinion concave mean spiral angle \(\beta_{mPV}\) and the pinion convex mean spiral angle \(\beta_{mPX}\) along the pinion pitch line can be calculated.

The demand for the pinion blank by the completing process method is: the redesigned spiral angle \(\beta_{mPV}\) and \(\beta_{mPX}\) are both equal to the original designed pinion mean spiral angle \(\beta_{mP}\), which can be expressed as:

$$\beta_{mPX} = \beta_{mPV} = \beta_{mP}$$
(45)

This equation contains two independent constrains. When the pinion root angle \(\delta_{fP}\) and the pinion root mean spiral angle \(\beta_{mRP}\) are changed, the pinion machine settings are recalculated as well as the spiral angle \(\beta_{mPV}\) and \(\beta_{mPX}\). This iteration is executed till the above constrain is satisfied, then the redesigned pinion root angle \(\delta_{fP}^{\prime }\) and the redesigned pinion root mean spiral angle \(\beta_{mRP}^{\prime }\) are finally determined. The iteration is based on the secant method.

During the iteration, the initial value of the pinion root angle can be given the original designed value, and the initial value of pinion root mean spiral angle can be calculated with the original designed parameters.

4 Redesign of Other Relevant Blank Dimensions

With the redesigned wheel root angle \(\delta_{fW}^{\prime }\) determined, the face cone of the pinion needs to be parallel to the root cone of the wheel; in the same way, with the redesigned pinion root angle \(\delta_{fP}^{\prime }\) determined, the face cone of the wheel needs to be parallel to the root cone of the pinion. Then the wheel face angle and the pinion face angle are redesigned as follows:

$$\left\{ \begin{gathered} \delta_{aP}^{\prime } = \Sigma - \delta_{fW}^{\prime } \hfill \\ \delta_{aW}^{\prime } = \Sigma - \delta_{fP}^{\prime } \hfill \\ \end{gathered} \right.$$
(46)

where \(\delta_{aP}^{\prime }\) is the redesigned pinion face angle, and \(\delta_{aW}^{\prime }\) is the redesigned wheel face angle, and \(\Sigma\) is the shaft angle.

With the mean addendum and dedendum of the gear pair staying the same, after the wheel root angle and the pinion root angle are redesigned, some other relevant blank dimensions change subsequently. The following equations apply to both the wheel and the pinion.

  1. (1)

    The unchanged mean addendum \(h_{am}\) and dedendum \(h_{fm}\):

    $$\left\{ \begin{gathered} h_{am} = h_{ae} - \frac{b}{2}\tan \left( {\delta_{a} - \delta } \right) \hfill \\ h_{fm} = h_{fe} - \frac{b}{2}\tan \left( {\delta - \delta_{f} } \right) \hfill \\ \end{gathered} \right.$$
    (47)

where \(b\) represents face width, and \(h_{ae}\) represents outer addendum, and \(h_{fe}\) represents outer dedendum, and \(\delta_{a}\) represents face angle, and \(\delta\) represents pitch angle, and \(\delta_{f}\) represents root angle, and the values of these parameters before redesign are used.

  1. (2)

    After the redesigned root angle \(\delta_{f}^{\prime }\) is determined, the corresponding changed parameters are:

Dedendum angle:

$$\theta_{f}^{\prime } = \delta - \delta_{f}^{\prime }$$
(48)

Outer dedendum:

$$h_{fe}^{\prime } = h_{fm} + \frac{b}{2}\tan \theta_{f}^{\prime }$$
(49)

Inner dedendum:

$$h_{fi}^{\prime } = h_{fm} - \frac{b}{2}\tan \theta_{f}^{\prime }$$
(50)

Root apex beyond crossing point:

$$\begin{gathered} t_{zR}^{\prime } = t_{z} + \left( {R_{e} \sin \delta - h_{fe}^{\prime } \cos \delta } \right)/\tan \delta_{f}^{\prime } \\ - R_{e} \cos \delta - h_{fe}^{\prime } \sin \delta \\ \end{gathered}$$
(51)
  1. (3)

    After the redesigned face angle \(\delta_{a}^{\prime }\) is determined, the corresponding changed parameters are:

Addendum angle:

$$\theta_{a}^{\prime } = \delta_{a}^{\prime } - \delta$$
(52)

Outer addendum:

$$h_{ae}^{\prime } = h_{am} + \frac{b}{2}\tan \theta_{a}^{\prime }$$
(53)

Inner addendum:

$$h_{ai}^{\prime } = h_{am} - \frac{b}{2}\tan \theta_{a}^{\prime }$$
(54)

Face apex beyond crossing point:

$$\begin{gathered} t_{zF}^{\prime } = t_{z} + \left( {R_{e} \sin \delta + h_{ae}^{\prime } \cos \delta } \right)/\tan \delta_{a}^{\prime } \\ + h_{ae}^{\prime } \sin \delta - R_{e} \cos \delta \\ \end{gathered}$$
(55)

Outer diameter:

$$d_{ae}^{\prime } = d_{e} + 2h_{ae}^{\prime } \cos \delta$$
(56)

Crown to crossing point:

$$t_{xo}^{\prime } = \frac{{d_{ae}^{\prime } }}{{2\tan \delta_{a}^{\prime } }} - t_{zF}^{\prime }$$
(57)

Front crown to crossing point:

$$t_{xi}^{\prime } = t_{xo}^{\prime } - b\frac{{\cos \delta_{a}^{\prime } }}{{\cos \theta_{a}^{\prime } }}$$
(58)

Here, the symbol “\(^{\prime }\)” means the redesigned value, and if without this symbol, it means using the original designed value.

5 Numerical Examples and Discussion

A spiral hypoid gear pair is redesigned by this new method after blank design of standard depth taper according to ISO standard. And the results are compared with those of the duplex taper modified by ISO standard. The basic parameters of this gear pair is shown in Table 1.

Table 1 Basic parameters of gear pair
Full size table

Through iteration, the redesigned wheel concave mean spiral angle \(\beta_{mWV}\) and the redesigned convex mean spiral angle \(\beta_{mWX}\) are both equated to the original designed wheel mean spiral angle. The original designed wheel mean spiral angle \(\beta_{mW}\) is 35.836°, and the redesigned wheel root angle by iteration \(\delta_{fW}^{\prime }\) is 75.3628°, and the redesigned wheel root mean spiral angle \(\beta_{mRW}^{\prime }\) is 35.9427°. The wheel mean point parameters are compared between redesign by the new method and modification by ISO standard in Tables 2 and 3.

Table 2 Comparison of parameters of wheel concave mean point
Full size table
Table 3 Comparison of parameters of wheel convex mean point
Full size table

After the redesign of the wheel, the point width of the pinion cutter is estimated at 4.709 mm. Through iteration, the redesigned pinion concave and convex mean spiral angle are both equated to the original designed pinion mean spiral angle. The original designed pinion mean spiral angle is 43.850000°, and the redesigned pinion root angle \(\delta_{fP}\) is 10.9948°, the redesigned pinion root mean spiral angle \(\beta_{mRP}\) is 43.8498°. The pinion mean point parameters are compared between redesign by the new method and modification by ISO standard in Tables 4 and 5. It can be observed that for the duplex taper modified by ISO standard, the wheel concave and convex mean spiral angle are not equal to the original designed wheel mean spiral angle; and after redesign by the new method, the wheel concave and the convex mean spiral angle are both equal to the original designed wheel mean spiral angle. Likewise, both the pinion concave and convex mean spiral angle become equal to the original designed pinion mean spiral angle after redesign, though they are not for the duplex taper modified by ISO standard.

Table 4 Comparison of parameters of pinion concave mean point
Full size table
Table 5 Comparison of parameters of pinion convex mean point
Full size table

After the wheel and pinion root angle are redesigned, the other relevant blank dimensions are changed as Table 6 shows.

Table 6 Comparison of parameters of gear pair
Full size table

The wheel chordal thicknesses and chordal space widths along the wheel pitch cone are shown in Table 7. The pinion chordal thicknesses and chordal space widths along the pinion pitch cone are shown in Table 8.

Table 7 Comparison of wheel chordal thickness and chordal space width along pitch cone
Full size table
Table 8 Comparison of pinion chordal thickness and chordal space width along pitch cone
Full size table

The ratios of the wheel chordal thickness to the wheel cone distance at a series of sections along the wheel pitch cone are plotted in Fig. 7a. For the duplex taper modified by ISO standard, the range of the ratios of the wheel chordal thickness is from 0.047143 to 0.050981; and for the redesign by the new method, the range is from 0.047149 to 0.049451. The ratios of the wheel chordal space width to the wheel cone distance are plotted in Fig. 7b. For the duplex taper modified by ISO standard, the range of the ratios of the wheel chordal space width is from 0.106546 to 0.110378; and for the redesign by the new method, the range is from 0.108073 to 0.110373. The range of the ratios of the wheel chordal thickness to the wheel cone distance is reduced 40.02% after redesign by the new method, and it means that the ratios of the wheel chordal thickness to the wheel cone distance are more stable after redesign. Likewise, the range of the ratios of the wheel chordal space width to the wheel cone distance is reduced 39.98% after redesign by the new method, and it means that the ratios of wheel chordal space width to the wheel cone distance are more stable after redesign, too.

Fig. 7
figure 7

The ratio of the wheel chordal thickness and chordal space width to the wheel cone distance. a The ratio of the wheel chordal thickness. b The ratio of the wheel chordal space width

Full size image

The ratios of the pinion chordal thickness to the pinion cone distance at a series of sections along the pinion pitch cone are plotted in Fig. 8a. For the duplex taper modified by ISO standard, the range of the ratio of the pinion chordal thickness is from 0.129003 to 0.132872; and for the redesign by the new method, the range is from 0.130454 to 0.132581. The ratios of the pinion chordal space width to the pinion cone distance are plotted in Fig. 8b. For the duplex taper modified by ISO standard, the range of the ratio of the pinion chordal space width is from 0.04951 to 0.053559; and for the redesign by the new method, the range is from 0.049816 to 0.052042. The range of the ratios of the pinion chordal thickness to the pinion cone distance is reduced 45.02% after redesign by the new method, and it means that the ratios of the pinion chordal thickness to the pinion cone distance are more stable after redesign. Likewise, the range of the ratios of the pinion chordal space width to the pinion cone distance is reduced 45.02% after redesign by the new method, and it means that the ratios of the pinion chordal space width to the pinon cone distance are more stable after redesign, too.

Fig. 8
figure 8

The ratio of the pinion chordal thickness and chordal space width to the pinion cone distance. a The ratio of the pinion chordal thickness. b The ratio of the pinion chordal space width

Full size image

The gear pair in this paper is with offset, this method also applies to face-milled spiral bevel gear pair without offset.

6 Conclusion

This paper proposes an novel calculation method of taper design based on machining theory aiming at spiral bevel and hypoid gears by the completing process method. With which, for both the wheel and the pinion, the redesigned concave and convex mean spiral angle can be both equal to its respective original designed mean spiral angle. Thus, the ratios of the wheel tooth thickness and tooth space width to the wheel cone distance are more stable, and the ratios of the pinion tooth thickness and tooth space width to the pinion cone distance are more stable, too. Finally, this method is applied to a spiral hypoid gear pair, and it can be drawn that for the wheel, the ranges of the ratios of the chordal thickness and the chordal space width to the cone distance are reduced more than 39% compared with those modified by ISO standard; for the pinion, the ranges of the ratios of the chordal thickness and the chordal space width to the cone distance are reduced more than 45% compared with those modified by ISO standard.