**3.2 Case 2. Three outside gears and one ring gear**

In this case torque is transmitted from a driving pinion (1) to a ring gear (2) through two idler pinions (3) and (4). Two solutions are available depending on the geometry of the rectilinear quadrilateral that joins the centres of the pitch circles, either crossed (Fig. 8 (a)) or non-crossed (Fig. 8 (b)). The starting equation is different for each of these cases.

Fig. 8. Solutions for three outside gears and one ring gear: (a) crossed quadrilateral (b) noncrossed quadrilateral

In this case torque is transmitted from a driving pinion (1) to a ring gear (2) through two idler pinions (3) and (4). Two solutions are available depending on the geometry of the rectilinear quadrilateral that joins the centres of the pitch circles, either crossed (Fig. 8 (a)) or

(a) (b)

Fig. 8. Solutions for three outside gears and one ring gear: (a) crossed quadrilateral (b) non-

non-crossed (Fig. 8 (b)). The starting equation is different for each of these cases.

Fig. 7. Feasible solutions for given numbers of teeth

**3.2 Case 2. Three outside gears and one ring gear** 

crossed quadrilateral

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

$$\pi \, z\_1 \cdot \alpha + z\_2 \cdot \beta + z\_3 \cdot \gamma - z\_4 \cdot \delta = \pi \cdot \left(2 \cdot n + z\_3 + z\_4\right) \tag{6}$$

Finally, we come to the same transcendental equation (2), where the coefficients are a2, b2, c2, d2, e2, f2, g2, h2, A2, B2, C2, A2', B2' and C2', whose values are listed in the Appendix.

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

$$z\_1 \cdot \alpha - z\_2 \cdot \beta - z\_3 \cdot \gamma - z\_4 \cdot \delta = \pi \cdot \left(2 \cdot n - 2 \cdot z\_2 + z\_3 + z\_4\right) \tag{7}$$

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.

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

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

Fig. 9. Idler pinions in an outside gear

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 curvilinear quadrilateral:

$$z\_1 \cdot \alpha + z\_2 \cdot \beta - 2 \cdot z\_3 \cdot \gamma = n \cdot 2\pi \tag{8}$$

$$
\alpha + \beta + 2 \cdot \gamma = 2\pi \tag{9}
$$

$$\left( (z\_1 + z\_3) \cdot \sin \left( \frac{a}{2} \right) = \left( z\_2 + z\_3 \right) \cdot \sin \left( \frac{\beta}{2} \right) \tag{10}$$

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

Split Torque Gearboxes: Requirements, Performance and Applications 65

One type of pivoted systems is described in detail in a patent (Gmirya, 2005) for split torque reduction applied to an aerial vehicle propulsion system (Fig. 11). *"The input pinion (64) engages with gears (66) and (68). The input pinion is defined along the gear shaft AG, the first gear* 

Fig. 11. Perspective view of the split torque gearbox with pivoted engine support (Gmirya, 2005)

Fig. 10. Schematic view of split torque main transmission

**4.2 Pivoted systems** 

$$\frac{z\_1 + z\_3}{z\_2 + z\_3} \cdot \sin\left(\frac{a}{2}\right) = \sin\left[\frac{z\_3 + n}{z\_2 + z\_3} \cdot \pi - \frac{z\_1 + z\_3}{z\_2 + z\_3} \cdot \left(\frac{a}{2}\right)\right] \tag{11}$$

The solutions for the other angles can now be obtained:

$$\beta = 2 \cdot \arcsin\left[\frac{z\_1 + z\_3}{z\_2 + z\_3} \cdot \sin\left(\frac{a}{2}\right)\right] \tag{12}$$

$$
\gamma = \pi - \frac{\alpha}{2} - \frac{\beta}{2} \tag{13}
$$

#### **4. Load sharing**

The main problem in the design of split torque gearboxes is to ensure that torque is equally split between different paths. Small deviations in machining can result in one of the paths with 100% of torque and the other path operating entirely freely (Kish & Webb, 1992). This situation causes excessive wear in one of the paths and renders the torque split system ineffective.

Below we describe approaches to ensuring equal torque split between different paths in split torque gear arrangements. The main types are:


The use of any of these systems to ensure correct torque split makes the gearbox heavier and assembly and maintenance more complex, which is why a number of authors do not support the use of systems that ensure torque split. Described below are the main systems that ensure correct torque split and discussed also are the proposals of authors who advocate for not using special systems.

#### **4.1 Geared differential**

One way to ensure correct torque split between two branches is to use a differential system. The great disadvantage of this system, however, is that resistive torque lost in one branch leads to loss of the full engine torque. Different differential mechanisms can be used, with assemblies very similar to those in vehicles or to the system depicted in Fig. 10. Assembled at the entry point to the gearbox is an input planetary system that acts as a differential that ensures load sharing. This transmission accepts power from three input engines, each of which has a differential system that ensures balanced torque splitting. Power is input from each engine to the sun gear of the differential planetary system. The carrier is the output to a bevel pinion that drives one torque splitting branch while the ring gear drives the other torque splitting branch. As the carrier and the ring gear rotate in opposite directions, the bevel pinions are arranged on opposite sides to ensure correct rotation direction. Each output bevel gear drives one pinion which then combines power into the output gear.

Fig. 10. Schematic view of split torque main transmission

#### **4.2 Pivoted systems**

64 Mechanical Engineering

1 3 2 3 2 arcsin sin

> 2 2

*z z z z*

The main problem in the design of split torque gearboxes is to ensure that torque is equally split between different paths. Small deviations in machining can result in one of the paths with 100% of torque and the other path operating entirely freely (Kish & Webb, 1992). This situation causes excessive wear in one of the paths and renders the torque split system ineffective.

Below we describe approaches to ensuring equal torque split between different paths in split

1. Geared differential. This differential mechanism, frequently used in the automotive

2. Pivoted systems. These use a semi-floating pinion constrained both to pivot normal to

3. Quill shafts. A torsion divider with a separate gear and pinion, each supported on its own bearings, are connected through the quill shaft, which allows torsional flexibility. The use of any of these systems to ensure correct torque split makes the gearbox heavier and assembly and maintenance more complex, which is why a number of authors do not support the use of systems that ensure torque split. Described below are the main systems that ensure correct torque split and discussed also are the proposals of authors who

One way to ensure correct torque split between two branches is to use a differential system. The great disadvantage of this system, however, is that resistive torque lost in one branch leads to loss of the full engine torque. Different differential mechanisms can be used, with assemblies very similar to those in vehicles or to the system depicted in Fig. 10. Assembled at the entry point to the gearbox is an input planetary system that acts as a differential that ensures load sharing. This transmission accepts power from three input engines, each of which has a differential system that ensures balanced torque splitting. Power is input from each engine to the sun gear of the differential planetary system. The carrier is the output to a bevel pinion that drives one torque splitting branch while the ring gear drives the other torque splitting branch. As the carrier and the ring gear rotate in opposite directions, the bevel pinions are arranged on opposite sides to ensure correct rotation direction. Each output bevel gear

2 2

2

  

(13)

(11)

(12)

1 3 3 13 2 3 23 23

*z z zn zz z z zz zz*

> 

sin sin

The solutions for the other angles can now be obtained:

torque gear arrangements. The main types are:

advocate for not using special systems.

**4.1 Geared differential** 

**4. Load sharing** 

sector, delivers equal torques to the drive gears of a vehicle.

drives one pinion which then combines power into the output gear.

the line of action and to seek a position where tooth loads are equal.

One type of pivoted systems is described in detail in a patent (Gmirya, 2005) for split torque reduction applied to an aerial vehicle propulsion system (Fig. 11). *"The input pinion (64) engages with gears (66) and (68). The input pinion is defined along the gear shaft AG, the first gear* 

Fig. 11. Perspective view of the split torque gearbox with pivoted engine support (Gmirya, 2005)

Split Torque Gearboxes: Requirements, Performance and Applications 67

input shaft (1) is assembled with two separate bearings (2) and the input gear (3).The output shaft (4) is assembled with two separate bearings (5) and, in this case, two output pinions (6). The quill shaft is a third shaft (7) that connects the other two shafts. Due to a lower polar moment of inertia, it admits torsional flexibility, resulting in a small angular deviation between the input and output shafts. The value of the angular deviation is proportional to the transmitted torque; thus, if one path transmits more torque than the other, the angular

Elastomeric elements are frequently used in quill shafts given their low elastic modulus. For example, one system (Isabelle et al., 1992), based on using elastomers (Fig. 13), consists of *"an annular cylindrical elastomeric bearing (14) and several rectangular elastomeric bearing pads (16). The elastomeric bearing (14) and bearing pads (16) have one or more layers (60); each layer (60) has an elastomer (62) with a metal backing strip (64) secured by conventional means such as* 

The annular cylindrical elastomeric bearing (14) absorbs possible misalignments between shafts resulting from defects in assembly. The rectangular elastomeric bearing pads (16) are responsible for providing torsional flexibility to the shafts of the possible gear paths in order

Another elastomer-based system (Kish & Webb, 1992) (Fig. 14) consists of an assembly with *"a central shaft (21) and a pair of bull pinions (22) and (23). The shaft (21) is supported by the bearings (24) and (25); a gear flange (26) at the end of the shaft has bolt holes (27) and teeth (28) on the outer circumference. A spur gear (29) is held to the flange (26) using upper and lower rims (30) and (31), consisting of flat circular disks (32) with bolt holes (33) and an angled outer wall (34). Gussets (35) between the wall and the disk increase rim stiffness to minimize deflection. One or more elastomer layers (36), bonded to the outer surface (37) of the wall (34), act as an elastomeric torsional* 

deviation is greater, allowing the shaft that transmits less torque to increase its load.

**4.3.2 Quill shafts based on elastomeric elements** 

Fig. 13. Elastomeric load sharing device (Isabelle et al., 1991)

*vulcanization, bonding or lamination"*.

to ensure equal torque transmission.

*isolator".*

(66) defines a first gear rotation shaft A1 and the second gear (68) defines a second gear rotation shaft A2. The axes AG, A1 and A2 are preferably located transversally to the pivot axis Ap. The first gear (66) and the second gear (68) engage an output gear (70). The output gear (70) defines an output rotation shaft A0 and is rotationally connected to the translational driveshaft (44) and the rotor driveshaft (46) to power, respectively, the translational propulsion system and the rotor system".

The assembly transmits torque from the pinion (64), which operates at very high revolutions, to the output shaft (44 -46) via two paths. The pivot system works as follows: since the input pinion (64) meshes with two gears (66) and (68), the pivoted engine arrangement permits the input pinion (64) to float until gear loads between the input gear (64), the first gear (66) and the second gear (68) are balanced. Irrespective of gear teeth errors or gearbox shaft misalignments, the input pinion will float and split torque between the two gears.
