**7. Appendix**

70 Mechanical Engineering

*torque is applied to the input pinion so that the gear teeth come into contact. When all the gear teeth for both power paths come into contact, then the clocking angle β is, by definition, equal to zero. If the teeth of one power path are not in contact, then the clocking angle β is equal to the angle that the firststage gear would have to be rotated relative to the second-stage pinion to bring all the teeth into* 

The tests show that suitable (47 per cent/53 per cent) load sharing can be achieved merely by taking into account the clocking angle and ensuring proper machining and assembly.

This research into the clocking angle has been followed up by subsequent authors (Parker & Lin, 2004) who have studied how contact between different planetary gears is sequenced.

Choosing the correct assembly for aircraft power transmission is a key factor in the quest for weight reduction. Although technological advances in mechanical components can help achieve weight reduction in gear systems, their influence is much less than that of choosing

Planetary gear or split torque systems are typically used in helicopter gear transmissions. The fundamental advantage of the split torque systems is that less weight is achieved by equal torque transmission and gear transmission ratios. This advantage is based primarily

 In the final transmission stage where the greatest torque is achieved, the use of several paths for the transmission ratio means that, given equal torque and stress levels in the teeth, the ratio between output torque/weight will be better in torque split gear systems

 In the final transmission stage, transmission ratios of around 5:1 or 7:1 are achieved by planetary gearboxes used with a single stage, compared to 10:1 or 14:1 for split torque

 The possibility of achieving higher transmission ratios in split torque gearboxes makes it possible to use a smaller number of gear stages, resulting in lighter gear systems. Split torque gearboxes need fewer gears and bearings that planetary gearboxes, which

 A key factor for aircraft use is that split torque gearboxes improve reliability by using multiple power paths; thus, if one path fails, operation is always assured through

 The main disadvantage of the split torque gearboxes is when torque split between the possible paths is uneven; however, several solutions are available to ensure correct

These arguments would indicate the advisability of using this type of transmission in

*contact"*.

**5. Conclusion** 

the correct gear assembly.

on arguments as follows:

another path.

torque split.

aircraft gear systems.

**6. Nomenclature**  m gear module

than in planetary gear systems.

gears used in the final stage.

means lower transmission losses.

ri radius of the pitch circle of wheel i

The numerical relationships among the teeth number used in the text are listed below. C1, C1', C2, C2', C3 and C3' are functionS of n, a whole number which represents the pitch difference in the curvilinear quadrilateral.

$$a\_1 = \left(z\_1 + z\_3\right)^2 + \left(z\_1 + z\_4\right)^2\tag{14}$$

$$b\_1 = 2 \cdot (z\_1 + z\_3) \cdot (z\_1 + z\_4) \tag{15}$$

$$\mathbf{c}\_1 = \left(\mathbf{z}\_2 + \mathbf{z}\_3\right)^2 + \left(\mathbf{z}\_2 + \mathbf{z}\_4\right)^2\tag{16}$$

$$d\_1 = 2 \cdot \left(z\_2 + z\_3\right) \cdot \left(z\_2 + z\_4\right) \tag{17}$$

$$e\_1 = \left(z\_1 + z\_3\right)^2 + \left(z\_2 + z\_3\right)^2\tag{18}$$

$$f\_1 = 2 \cdot (z\_1 + z\_3) \cdot (z\_2 + z\_3) \tag{19}$$

$$\lg\_1 = \left(z\_1 + z\_4\right)^2 + \left(z\_2 + z\_4\right)^2\tag{20}$$

$$h\_1 = 2 \cdot (z\_1 + z\_4) \cdot (z\_2 + z\_4) \tag{21}$$

$$A\_1 = \frac{z\_1 + z\_4}{z\_3 - z\_4} \tag{22}$$

$$B\_1 = \frac{z\_2 + z\_4}{z\_3 - z\_4} \tag{23}$$

$$C\_1 = 2\pi \cdot \frac{z\_4 + n}{z\_4 - z\_3} \tag{24}$$

$$A\_1 \stackrel{\prime}{=} \frac{z\_1 + z\_3}{z\_4 - z\_3} \tag{25}$$

$$B\_1 "= \frac{\mathbf{z\_2} + \mathbf{z\_3}}{\mathbf{z\_4} - \mathbf{z\_3}} \tag{26}$$

Split Torque Gearboxes: Requirements, Performance and Applications 73

3

3

3

Corporation, *U.S. Patent* Number 6,612,195

Corp., *U.S. Patent Number* 6,883,750

2-04-015607-0, Paris, France, pp. 587-662

Corporation. *U.S. Patent* Number 5,117,704

Technologies Corporation. *U.S. Patent* Number 5,113,713

Army Research Laboratory Technical Report 92-C-030

Research Laboratory Technical Report ARL-TR-291

**8. References** 

2004/094093

' *z z <sup>B</sup> z z*

22 3 '*n z zz <sup>C</sup>*

Cocking, H. (1986). The Design of an Advanced Engineering Gearbox. *Vertica.* Vol 10, No. 2, Westland Helicopters and Hovercraft PLC, Yeovil, England, pp. 213-215 Craig, G. A.; Heath, G. F. & Sheth, V. J. (1998). Split Torque Proprotor Transmission,

Gmirya, Y.; Kish, J.G. (2003). Split-Torque Face Gear Transmission, Sikorsky Aircraft

Gmirya, Y. & Vinayak, H. (2004). Load Sharing Gear for High Torque, Split-Path

Gmirya, Y. (2005). Split Torque Gearbox With Pivoted Engine Support, Sikorsky Aircraft

Henriot G. (1979). *Traité théorique et pratique des engrenages (I)*, Ed. Dunod, 6th Edition, ISBN

Isabelle, C.J; Kish & J.G, Stone, R.A. (1992). Elastomeric Load Sharing Device. United

Kish, J.G. & Webb, L.G. (1992). Elastomeric Torsional Isolator. United Technologies

Kish, J.G. (1993a). *Sikorsky Aircraft Advanced Rotorcraft Transmission (ART) Program* – Final Report. NASA CR-191079, NASA Lewis Research Center, Cleveland, OH Kish, J.G. (1993b). Comanche Drive System. *Proceedings of the Rotary Wing Propulsion Specialists Meeting*, American Helicopter Society, Williamsburg, VA, pp. 7 Krantz, T.L.; Rashidi, M. & Kish, J.G. (1992). *Split Torque Transmission Load Sharing*, in:

Krantz, T.L. (1994). *Dynamics of a Split Torque Helicopter Transmission*, in: Technical

Krantz, T.L. (1996). *A Method to Analyze and Optimize the Load Sharing of Split Path* 

Ohio. Army Research Laboratory Technical Report ARL-TR-1067

Technical Memorandum 105,884, NASA Lewis Research Center, Cleveland, Ohio.

Memorandum 106,410, NASA Lewis Research Center, Cleveland, Ohio. Army

*Transmissions*, in: Technical Memorandum 107,201, NASA Lewis Research Center, Cleveland, Ohio. Army Research Laboratory Technical Report ARL-TR-1066 Krantz, T.L. & Delgado, I.R. (1996). *Experimental Study of Split-Path Transmission Load Sharing*,

in: Technical Memorandum 107,202, NASA Lewis Research Center, Cleveland,

Transmissions. Sikorsky Aircraft Corporation. *International Patent PCT*. WO

McDonnell Douglas Helicopter Co., *U.S. Patent* Number 5,823,470

' *z z <sup>A</sup> z z*

1 3

(45)

(46)

(47)

4 3

3 2

4 3

3 4

*z z*

2 34

$$C\_1 "= 2\pi \cdot \frac{z\_3 + n}{z\_3 - z\_4} \tag{27}$$

$$a\_2 = \left(z\_1 + z\_3\right)^2 + \left(z\_1 + z\_4\right)^2\tag{28}$$

$$b\_2 = 2 \cdot \left(z\_1 + z\_3\right) \cdot \left(z\_1 + z\_4\right) \tag{29}$$

$$\mathbf{c}\_2 = \left(\mathbf{z}\_2 - \mathbf{z}\_3\right)^2 + \left(\mathbf{z}\_2 - \mathbf{z}\_4\right)^2\tag{30}$$

$$d\_2 = 2 \cdot (z\_2 - z\_3) \cdot (z\_2 - z\_4) \tag{31}$$

$$\text{i.e.}\_2 = \left(z\_1 + z\_3\right)^2 + \left(z\_2 - z\_3\right)^2\tag{32}$$

$$f\_2 = 2 \cdot (z\_1 + z\_3) \cdot (z\_2 - z\_3) \tag{33}$$

$$\mathcal{g}\_2 = \left(z\_1 + z\_4\right)^2 + \left(z\_2 - z\_4\right)^2\tag{34}$$

$$h\_2 = 2 \cdot (z\_1 + z\_4) \cdot (z\_2 - z\_4) \tag{35}$$

$$A\_2 = \frac{z\_1 + z\_4}{z\_4 - z\_3} \tag{36}$$

$$B\_2 = \frac{z\_2 - z\_4}{z\_4 - z\_3} \tag{37}$$

$$C\_2 = \pi \cdot \frac{\mathbf{2} \cdot \mathbf{n} + \mathbf{z}\_3 + \mathbf{z}\_4}{\mathbf{z}\_3 - \mathbf{z}\_4} \tag{38}$$

$$A\_2 = \frac{z\_1 + z\_3}{z\_4 - z\_3} \tag{39}$$

$$B\_2 \, ^\prime = \frac{z\_2 - z\_3}{z\_4 - z\_3} \tag{40}$$

$$C\_2 \, != \pi \cdot \frac{2 \cdot n + z\_3 + z\_4}{z\_3 - z\_4} \tag{41}$$

$$A\_3 = \frac{z\_1 + z\_4}{z\_3 - z\_4} \tag{42}$$

$$B\_3 = \frac{z\_4 - z\_2}{z\_3 - z\_4} \tag{43}$$

$$C\_3 = \pi \cdot \frac{2 \cdot n - 2 \cdot z\_2 + z\_3 + 3 \cdot z\_4}{z\_4 - z\_3} \tag{44}$$

$$A\_3 "= \frac{z\_1 + z\_3}{z\_4 - z\_3} \tag{45}$$

$$B\_3 = \frac{z\_3 - z\_2}{z\_4 - z\_3} \tag{46}$$

$$\mathbf{C}\_3 \text{ '} = \pi \cdot \frac{\mathbf{2} \cdot \mathbf{n} - \mathbf{2} \cdot \mathbf{z}\_2 + \mathbf{3} \cdot \mathbf{z}\_3 + \mathbf{z}\_4}{\mathbf{z}\_3 - \mathbf{z}\_4} \tag{47}$$

#### **8. References**

72 Mechanical Engineering

' 2 *z n <sup>C</sup>*

1

3

<sup>2</sup> <sup>2</sup>

<sup>2</sup> <sup>2</sup>

2 2

2 2

1 4

*z z <sup>A</sup> z z*

<sup>2</sup> *nz z <sup>C</sup> z z*

4 3

2 4

4 3 *z z <sup>B</sup> z z*

3 4

1 3

4 3

2 3

4 3

3 4

*z z*

1 4

*z z <sup>A</sup> z z*

22 3 *n zz z <sup>C</sup> z z*

3 4

4 2

3 4 *z z <sup>B</sup> z z*

4 3

3 4

23 4

3 4

2

2

2

2

2 '*nz z <sup>C</sup>*

3

3

' *z z <sup>B</sup> z z*

' *z z <sup>A</sup> z z*

2

2

3

3 4

(27)

2 13 14 *a zz zz* (28)

2 13 14 *b zz zz* 2 (29)

2 23 24 *c zz zz* (30)

*d zz zz* 2 23 24 2 (31)

2 13 23 *e zz zz* (32)

2 13 23 *f* 2 *zz zz* (33)

2 14 24 *g zz zz* (34)

*h zz zz* 2 14 24 2 (35)

(36)

(37)

(39)

(40)

(42)

(43)

(44)

(41)

(38)

*z z*


**1. Introduction** 

**2. Nomenclature** 

*L t* ,

 1 *<sup>m</sup> kt k t* 

,

**q**

*L t*

stiffness matrix

0 0 *<sup>T</sup>* **G VV**

 **<sup>e</sup> q**

, *C C m r* : pinion, gear torque

*t k M e M dM* 

with tooth shape modifications and errors

<sup>0</sup> *k* : mesh stiffness per unit of contact length

*<sup>p</sup> k* : modal stiffness associated with (

**<sup>G</sup>** *t kM dM*

*k M* , mesh stiffness per unit of contact length at *M*

*b* : face width

**4** 

Velex Philippe

*France* 

**On the Modelling of Spur and** 

*University of Lyon, INSA Lyon, LaMCoS UMR CNRS,* 

**Helical Gear Dynamic Behaviour** 

This chapter is aimed at introducing the fundamentals of spur and helical gear dynamics. Using three-dimensional lumped models and a thin-slice approach for mesh elasticity, the general equations of motion for single-stage spur or helical gears are presented. Some particular cases including the classic one degree-of-freedom model are examined in order to introduce and illustrate the basic phenomena. The interest of the concept of transmission errors is analysed and a number of practical considerations are deduced. Emphasis is deliberately placed on analytical results which, although approximate, allow a clearer understanding of gear dynamics than that provided by extensive numerical simulations.

*e M* , *E t MAX* ( ) : composite normal deviation at *M* , maximum of *e M* at time *t* .

*H x* : unit Heaviside step function ( *Hx i* 1 1; 0 *f x H x otherwise* )

*<sup>p</sup>* , **Φ<sup>p</sup>** )

**F V M** : time-varying, possibly non-linear forcing term associated

**K V M V M** : time-varying, possibly non-linear gear mesh

Some extensions towards continuous models are presented.

*E E*, \*: actual and normalized depth of modification at tooth tips

*mk* , *k t* ,**q** : average and time-varying, non-linear mesh stiffness

, linear time-varying mesh stiffness

*T*

