In comparison with traditional gear design based on pre-selected, typically standard generating rack parameters and its addendum modification also known as the X-shift, the alternative Direct Gear Design^{®} method [1, 2] provides certain advantages for custom high-performance gear drives that include increased load capacity, efficiency, and lifetime and reduced size, weight, noise, vibrations, and cost.

Pitch factor analysis is one of Direct Gear Design’s methodical approaches to describe the involute gear mesh geometry and explore its characteristics. It divides the operating circular pitch of the involute gear mesh into three segments: the driving or load (or motion) transmitting segment related to the drive tooth flanks, the coast segment that may transmit load (or motion) in reverse related to the coast tooth flanks, and the noncontact segment that is excluded from load (or motion) transmissions related to the tooth tip lands and radii. Ratios of these segments to the operating circular pitch are called the pitch factors. Combinations of these factors greatly affect involute gear mesh parameters that define gear drive performance.

This paper introduces an analytical approach that describes main gear mesh characteristics such as operating pressure angles and contact ratios as functions of the pitch factors. It also considers areas of existence of involute gear pairs with the given constant values of the pitch factors.

**DEFINITION OF GEAR TOOTH PROFILES**

The Direct Gear Design method does not use pre-selected basic or generating rack to define the gear geometry. Two involute curves unwound from the base circle, the arc distance between them, and tooth tip circle describe a gear tooth profile (see Figure 1). The equally spaced teeth form the gear. The root fillet profile connecting neighboring tooth flanks is not in contact with the mating gear teeth. However, this portion of the tooth profile is critical because this is the area of the bending stress concentration. It is designed to exclude any kind of interference with the mating tooth tip and minimize bending stress.

Similarly, Figure 2 describes an asymmetric gear tooth profile. Asymmetric teeth are beneficial for mostly unidirectional gear transmissions where one (drive) tooth flank carries higher load and a longer period of time than the opposite (coast) tooth [2, 3]. Asymmetric tooth flanks are formed by two involute curves unwound from two different base circles for drive and coast flanks. The design intent of asymmetric tooth gear design is to improve drive tooth flank performance on account of less-loaded coast tooth flank.

**DEFINITION OF PITCH FACTORS**

In order to maximize gear drive performance, Direct Gear Design defines the gear tooth geometry elements (flanks, tooth tips, and root fillet) independently. Such approach allows the gear parameters’ range to expand beyond the standard gear design limitations. Pitch factor analysis helps to understand relations and limits between such critical gear mesh parameters as operating pressure angle and contact ratio that define the gear tooth flank durability and bending strength.

The gear mesh operating circular pitch can be presented as:

Equation 1

where:

indexes 1 and 2 are for mating pinion and gear accordingly;

z* _{1}* and z

_{2}are numbers of teeth;

d_{w1} and d_{w2} are operating (rolling) pitch diameters equal d_{w1} = 2a_{w}/(1+u), d_{w2} = ud_{w1};

a_{w} is a center distance;

u = z_{2}/z_{1} is gear ratio;

S_{w1} and S_{w2} are the pinion and gear tooth thicknesses at the operating pitch diameter;

S_{bl} is arc backlash.

Figure 3 and Figure 4 show the symmetric and asymmetric tooth gear mesh and operating pitch components.

The following equations describe the most general case of asymmetric tooth gears. For symmetric tooth gears, these equations are simplified because the drive and coast gear flank parameters are identical.

The tooth thicknesses S_{w1} and S_{w2} from Figure 2 are:

Equation 2

where:

S_{d1,2} are projections of the addendum portion of the drive involute flank on the pitch circle:

Equation 3

a_{wd} – drive flank operating pressure angle equal a_{wd} = arccos d_{bd1,2}/d_{w1,2};

a_{ad1,2} – drive flank involute angles at the tooth tips that are equal a_{ad1,2} = arccos d_{bd1,2}/d_{a1,2}.

inv(x)=tan(x)-x – involute function of the x (in radians),

signs ∓ and ± denote – the top sign is for external gear mesh and the bottom sign is for internal gear mesh;

S_{c1,2} are projections of the addendum portion of the coast involute flank on the pitch circle:

Equation 4

a_{wc} – coast flank operating pressure angle equal a_{wc} = arccos d_{bc1,2}/d_{w1,2};

a_{ac1,2} – coast flank involute angles at the tooth tips that are equal a_{ac1,2} = arccos d_{bc1,2}/d_{a1,2};

S_{v1,2} are projections of the tooth tip lands on the pitch circle:

Equation 5

Then, the gear mesh operating circular pitch for asymmetric tooth gears from Equation 1 is:

Equation 6

A pitch factor equation is a result of the division of Equation 6 by operating circular pitch p_{w}:

Equation 7

where:

q_{d} is the drive pitch factor defined as:

Equation 8

q_{c} is the coast pitch factor defined as:

Equation 9

and the q_{v} is the non-contact pitch factor defined as:

Equation 10

The drive pitch factor is:

for the external gear mesh:

Equation 11

for the internal gear mesh:

Equation 12

The coast pitch factor is:

for the external gear mesh:

Equation 13

for the internal gear mesh:

Equation 14

Assuming the arc backlash Sbl = 0, the non-contact pitch factor can be presented as:

Equation 15

where:

m_{a1,2} = S_{a1,2}/m_{w} is relative tooth tip thicknesses;

m_{w} is operating module.

Then, the drive and coast pressure angles can be defined by the following equations:

for the external gear mesh:

Equation 16

Equation 17

for internal gear mesh:

Equation 18

Equation 19

The drive and coast contact ratios are:

for the external gear mesh:

Equation 20

Equation 21

for internal gear mesh:

Equation 22

Equation 23

For symmetric tooth gears, the pitch factor θ from Equation 7 is:

Equation 24

This equation shows that for symmetric gears, the pitch factor is always ≤ 0.5. For the standard 20º pressure angle gears, θ = 0.25–0.30, and for the 25º pressure angle gears, θ = 0.30–0.35. In custom symmetric gears, the pitch factor θ can reach values of 0.40–0.45. The pitch factor q = 0.5 is practically not possible if the mating gears do not have the pointed tooth tips.

For gears with asymmetric teeth, the drive pitch factor q_{d} from Equation 7 is:

Equation 25

Reduction of the coast pitch factor q_{c} and the non-contact pitch factor q_{v} allows a significant increase of the drive pitch factor q_{d}. A practical range of the drive pitch factor q_{d} varies between 0.40 and 0.6. Although, in theory, it could be close to 1.0 for irreversible asymmetric gears with extremely low numbers of teeth. Examples of such gear profiles and data are shown in Figure 5 and Table 1.

Figure 6 presents a sample of the drive pressure angle versus the drive contact ratio a_{wd} – e_{ad} chart at different values of q_{d} and the mesh images for gear pairs with numbers of teeth z* _{1}* = 21 for the pinion and z

_{2}= 37 for the gear. This chart shows that the symmetric gear solutions lay below the curve q

_{d}= 0.5, and the asymmetric gear meshes are located below and above this curve.

**AREA OF EXISTENCE AND PITCH FACTORS**

Direct Gear Design presents the area of existence of asymmetric tooth gears with the given numbers of teeth z_{1} and z_{2}, constant asymmetry factor K = cos a_{wc}/cos a_{wd}, and relative tooth tip thicknesses m_{a1} and m_{a2}. [2, 4]. The pitch factors q_{d}, q_{c}, and q_{v} in such areas of existence are varying. Figure 7 presents the overlaid areas of existence of spur external gears with the different constant drive flank pitch factors q_{d}. This type of area of existence of involute gears defines only the drive flank gear meshes. If q_{d} ≤ 0.5, the gears can have symmetric or asymmetric teeth. If q_{d} > 0.5, the gears can have only asymmetric teeth. The gears with symmetric teeth are always reversible. The gears with asymmetric teeth can be reversible or irreversible depending on the coast flank pitch factor q_{c} selection.

The areas of existence in Figure 7 is limited by the interference isograms a_{pd1} = 0º and a_{pd2} = 0º that describe a beginning of undercut of the involute drive flank near the root fillet by the mating tooth tip and the isogram e_{ad} = 1.0, the minimal value of the drive flank transverse contact ratio for spur gears.

The interference isograms a_{pd1} = 0º and a_{pd2} = 0º are defined by Equation 16 and the following equations [1]:

Equation 26

and

Equation 27

respectively.

The minimal for spur gears’ contact ratio isogram e_{ad} = 1.0 is defined by Equations 16 and 20.

In point A of the area of existence where the drive flank pressure angle a_{wd} is maximum and the contact ratio ead = 1.0, the pressure angle and contact ratio isograms have a common tangent point, and the first derivatives of these isogram functions should be equal:

Equation 28

or a_{ad1} = a_{ad2}.

This means that the points A of the areas of existence lay on the straight line a_{ad1} = a_{ad2}. The pressure angle equation at point A is defined as a solution of Equations 16 and 20.

Equation 29

Its solution is [2]:

Equation 30

In point B at the intersection of the interference isograms a_{pd1} = 0º and a_{pd2} = 0º, the pressure angle is minimum and the contact ratio ead is maximum. This maximum contact ratio is defined as a solution of Equations 16, 20, 26, and 27:

Equation 31

The drive flank pressure angle awd at point B is [1]:

Equation 32

The pressure angles awd and contact ratios ead at points A and B of the areas of existence from Figure 7 are presented in Table 2.

Points of the area of existence with the constant drive flank pitch factor do not define complete mating gear teeth, but just their drive flanks. This allows independently selecting the tooth tip thicknesses and the coast tooth flank parameters of asymmetric gears.

When some point of the area of existence with the coordinates a_{ad1} and a_{ad2} is chosen, the pressure angle a_{wd} is calculated by Equation 16. Then, after selection of the relative tooth tip thicknesses m_{a1} and m_{a2}, the non-contact pitch factor q_{v} is calculated by Equation 15. This allows the coast flank pitch factor to be defined from Equation 7:

Equation 33

If the tooth tip radii are equal to zero, the asymmetry factor K can be defined as a solution of equations K = cos a_{wc}/cos a_{wd} = cos a_{ac1,2}/ cos a_{ad1,2} and Equation 16:

Equation 34

Then, the coast flank pressure angle can be defined:

Equation 35

The coast flank contact ratio is defined by Equation 21.

**EXAMPLE OF APPLICATION**

Figure 8 presents an experimental asymmetric tooth spur gear set of an electric generator driven by the 9I56 gas turbine engine. Table 3 show the main tooth geometry data of this gear set.

This gear set has a high drive pitch factor q_{d} = 0.58 that is not achievable for gears with symmetric teeth. In comparison testing with the baseline helical gear set designed by standards, the experimental asymmetric tooth spur gear set demonstrated significant stress and a vibration level reduction [3].

**SUMMARY**

A simultaneous increase of the drive pressure angle and the drive contact ratio maximizes gear drive performance. It allows reducing the contact and bending stress and increasing load capacity and power transmission density. This indicates potential advantages of the directly designed asymmetric tooth gears over the symmetric ones for gear drives that transmit load mostly in one direction.

The pitch factor analysis is the additional Direct Gear Design analytical tool that can be used for comparison of different gear geometry solutions, helping the designer better understand the available options and choose the optimal one.

**REFERENCES**

- Vulgakov, E.B. (1974) Gears with improved characteristics. Mashinostroenie, 264 p.
- Kapelevich, A.L. (2013) Direct Gear Design. CRC Press, 324 p.
- Kapelevich, A.L. (2000) Geometry and design of involute spur gears with asymmetric teeth. Mechanism and Machine Theory, 35, 117–130.
- Kapelevich, A.L. & Shekhtman, Y.V. (2010) Area of existence of involute gears. Gear Technology. January/February, 64–69.