**3. Feasible geometric configurations**

To ensure simultaneous meshing of four gears (Fig. 1), configuration must comply with certain geometric constraints. A number of studies describe the complexity of simultaneous gearing in split torque gearboxes (Kish, 1993a) and in planetary gear systems (Henriot, 1979, Parker & Lin, 2004); other studies approach the problem generically (Vilán-Vilán et al., 2010), describing possible solutions that ensure the simultaneous meshing of four gears.

For four gears to mesh perfectly, the teeth need to mesh simultaneously at the contact points. The curvilinear quadrilateral and the pitch difference are defined below in order to express the meshing condition. From now on we will use this nomenclature of our own devising -that is, curvilinear quadrilateral - to indicate the zone defined by portions of pitch circles in the meshing area (Fig. 5). The pitch difference is the sum of pitches in the input and output gears minus the sum of pitches in the idler gears at the curvilinear quadrilateral. For perfect engagement between the four gears, the pitch difference must coincide with a whole number of pitches.

Split Torque Gearboxes: Requirements, Performance and Applications 61

where n is the pitch difference in the curvilinear quadrilateral. As previously mentioned, n

We thus obtain an equation with four unknowns (α, β, γ, δ). The three remaining relationships can be obtained from the quadrilateral that joins the centres of the pitch circles (this quadrilateral will be denoted the rectilinear quadrilateral). Finally, we come to a transcendental equation (2) from which α can be obtained according to the number of teeth

1 11 1 1

 

*cab ef A B C g <sup>d</sup>*

1 11 1

*cab h AB <sup>C</sup>*

arccos

.cos cos ' ' arccos '

111

111

111 1

*cab d*

*ABC* 11 1

11 1

a1, b1, c1, d1, e1, f1, g1, h1, A1, B1, C1, A1', B1' and C1' are numerical relationships among the teeth number from each wheel that must mesh simultaneously. The value of each coefficient

The transcendental equation for obtaining α has several solutions, all representing possible assemblies for the starting gears. For example, for four-gear meshing with the next teeth numbers: z1=30, z2=50, z3=20 and z4=12 (see Nomenclature), all the possible solutions for the gear can be encountered. In this case solutions are n = -12, -11, -3, -2, -1, 0, 1, 2, 3, 4, 7, 29 and 30, where n is the pitch difference between the two sides of the curvilinear quadrilateral (a whole number that ensures suitable meshing). Fig. 7 shows some of the possible meshing solutions.

 

  1

cos

(4)

*ABC* ''' (5)

cos

(2)

(3)

1

*d*

must be a whole number to ensure suitable meshing between gears.

cos arccos

Once the angle α has been determined, we can calculate:

in the gears.

is listed in the Appendix.

Fig. 5. General conditions for simultaneous meshing of four gears

A relationship is thus established between the position of the gears, as defined by the relative distance between centres, and the number of teeth in each of the gears. Below we explore two possible cases of over-constrained gears:


#### **3.1 Case 1. Four outside gears**

For a gearbox with the geometry illustrated in Fig. 6, it is possible to locate the different positions that will produce suitable meshing between gears, in function of the number of teeth in each gear, by defining the value of the angles α, β, δ and γ.

Fig. 6. Nomenclature for the four-gear case

The condition described in the previous section can be mathematically expressed as follows (see Nomenclature):

$$r\_1 \cdot \alpha + r\_2 \cdot \beta - r\_3 \cdot \gamma - r\_4 \cdot \delta = n \cdot (m\pi) \quad n \in Z \tag{1}$$

A relationship is thus established between the position of the gears, as defined by the relative distance between centres, and the number of teeth in each of the gears. Below we

For a gearbox with the geometry illustrated in Fig. 6, it is possible to locate the different positions that will produce suitable meshing between gears, in function of the number of

The condition described in the previous section can be mathematically expressed as follows

*r r r r nm n Z* 1 2 34

 

(1)

Fig. 5. General conditions for simultaneous meshing of four gears

teeth in each gear, by defining the value of the angles α, β, δ and γ.

explore two possible cases of over-constrained gears:

CASE 2. Three outside gears and one ring gear.

CASE 1. Four outside gears.

**3.1 Case 1. Four outside gears** 

Fig. 6. Nomenclature for the four-gear case

(see Nomenclature):

where n is the pitch difference in the curvilinear quadrilateral. As previously mentioned, n must be a whole number to ensure suitable meshing between gears.

We thus obtain an equation with four unknowns (α, β, γ, δ). The three remaining relationships can be obtained from the quadrilateral that joins the centres of the pitch circles (this quadrilateral will be denoted the rectilinear quadrilateral). Finally, we come to a transcendental equation (2) from which α can be obtained according to the number of teeth in the gears.

$$\begin{aligned} &e\_1 - f \cdot \cos\left[A\_1 \cdot \mathfrak{a} + B\_1 \cdot \arccos\left(\frac{c\_1 - a\_1 + b\_1 \cdot \cos\mathfrak{a}}{d\_1}\right) + C\_1\right] = g\_1 - \\ &- h\_1 \cdot \cos\left[A\_1' \cdot \mathfrak{a} + B\_1' \cdot \arccos\left(\frac{c\_1 - a\_1 + b\_1 \cdot \cos\mathfrak{a}}{d\_1}\right) + C\_1'\right] \end{aligned} \tag{2}$$

Once the angle α has been determined, we can calculate:

$$\beta = \arccos\left(\frac{c\_1 - a\_1 + b\_1 \cdot \cos a}{d\_1}\right) \tag{3}$$

$$
\gamma = A\_1 \cdot \alpha + B\_1 \cdot \beta + C\_1 \tag{4}
$$

$$
\delta = A'\_1 \cdot \mathcal{a} + B'\_1 \cdot \mathcal{B} + \mathcal{C}'\_1 \tag{5}
$$

a1, b1, c1, d1, e1, f1, g1, h1, A1, B1, C1, A1', B1' and C1' are numerical relationships among the teeth number from each wheel that must mesh simultaneously. The value of each coefficient is listed in the Appendix.

The transcendental equation for obtaining α has several solutions, all representing possible assemblies for the starting gears. For example, for four-gear meshing with the next teeth numbers: z1=30, z2=50, z3=20 and z4=12 (see Nomenclature), all the possible solutions for the gear can be encountered. In this case solutions are n = -12, -11, -3, -2, -1, 0, 1, 2, 3, 4, 7, 29 and 30, where n is the pitch difference between the two sides of the curvilinear quadrilateral (a whole number that ensures suitable meshing). Fig. 7 shows some of the possible meshing solutions.

Split Torque Gearboxes: Requirements, Performance and Applications 63

1 2 34 3 4 *z z z z nz z*

Finally, we come to the same transcendental equation (2), where the coefficients are a2, b2, c2,

1 2 34 <sup>234</sup> *z z z z n zzz*

Finally, we come to the same transcendental equation (2), where the coefficients become a2, b2, c2, d2, e2, f2, g2, h2, A3, B3, C3, A3', B3' and C3', whose values are listed in the Appendix.

 

A common split torque gear assembly is one with two equally sized idler pinions (Fig. 9).

The solution is obtained by particularizing the general solution for four outside wheels and imposing the condition z3= z4, or = . The following equations are defined for the

12 3 *zz zn*

1 3 2 3 sin sin

 

 

2 2

 

 (10)

2 2 (8)

2 2 (9)

*zz zz* 

Resolving the system, the following transcendental function in is obtained:

 

2 (6)

2 2 (7)

For the crossed quadrilateral configuration, the starting equation is (see Nomenclature):

 

d2, e2, f2, g2, h2, A2, B2, C2, A2', B2' and C2', whose values are listed in the Appendix.

Fig. 9. Idler pinions in an outside gear

curvilinear quadrilateral:

 

For the non-crossed quadrilateral configuration, the starting equation is:

 

**3.3 A particular case: Outside meshing with equal intermediate pinions** 

 

Fig. 7. Feasible solutions for given numbers of teeth
