3
Transmission Gear Design

3.1 Introduction

Gears are one of the most important components for mechanical systems. Although gears are applied in all machinery, the automotive industry is by far the largest user of gears because of the huge number of vehicle sales. In automotive powertrain systems, gears are indispensable, regardless what type of transmission is used for the vehicle. Different gear designs are applied for different types of transmissions. Lay‐shaft gears are applied mostly for manual transmissions and sometimes also for automatic transmissions, where all gears rotate about fixed axes and multiple transmission ratios are achieved by multiple gear pairs rotating about parallel shafts, as discussed in Chapter 2. Most of the conventional automatic transmissions use planetary gear trains to transmit power and achieve multiple gear ratios, with multiple clutches to control the power flows. Different from lay‐shaft gear systems, not all gears rotate about fixed axes in planetary gear systems. Continuously variable transmissions do not need as many as gears as manual transmissions or conventional automatic transmissions, but they still use gears in the final drive to increase the overall drive train ratio.

The history of gears dates back thousands of years in human civilization [1], but it was not until the eighteenth century that the gear industry started to take shape. The vast majority of gears in the gear industry today are involute gears (i.e. gears with involute tooth profiles), which were originally designed by Leonhard Euler [2]. Gears with cycloid tooth profiles are also used for some applications, such as mechanical clocks and apparatus, which mainly involve the transmission of motion. Many types of gears have been developed for various applications. In vehicle powertrain systems, spur gears, helical gears, and bevel gears (including straight, spiral, and hypoid bevel gears) are mostly applied for the transmission of power. Because of the large volume and wide application, gears are standardized mechanical components in the industry.

Gear design standard and practice is based on the principles and approaches of the theory of gearing. For a comprehensive study of gear design fundamentals, see Analytical Mechanics of Gears by E. Buckingham [3], Gear Geometry and Applied Theory by F.L. Litvin [4], and Handbook of Practical Gear Design by D.W. Dudley [5]. The contents of this chapter will be focused on gear design using existing equations or formulas available in AGMA standards [6,7] rather than on general gear design theory. Following the introduction section, the chapter will first provide a summary of gear design fundamentals on conjugate motions, characteristics of involute gearing, and key definitions and terminologies. The chapter will then present the design of tooth element proportions for spur and helical involute gears according to the related AGMA standards. The design of non‐standard gears, including the long‐short addendum system and the general non‐standard gears will be highlighted in the chapter since the design of these gears is very useful for vehicle transmission applications. The chapter will also cover the calculation of gear forces in the power flows of automotive transmissions. A separate section will be included on the strength design and power rating of gear pairs based on AGMA standard formulation. Note here that the design of hypoid gears used in the final drives of RWD and 4WD vehicles is quite different from cylindrical gears and readers are referred to the related AGMA standard [8] for the design of these gear drives.

The chapter ends with a section on the kinematics of planetary gear trains which are widely applied in automatic transmissions. The section specifically covers the structure and kinematics of three types of planetary gear trains – the simple planetary gear train, dual‐planet planetary gear train, and Ravigneaux planetary gear train – since it is these three types of planetary gear trains that are widely applied in automatic transmissions. The kinematic characteristics and torque relations of these three types of planetary gear trains will be analysed and formulated in this section, which will be applied for the design and analysis of automatic transmissions in Chapter 5.

3.2 Gear Design Fundamentals

3.2.1 Conjugate Motion and Definitions

In planar gearing, a pair of gears transmits rotational motion and power from one axis to another through conjugate motion between the tooth profiles of the two gears. As shown in Figure 3.1, the two gears are centered at point O₁ and O₂, with E as the center distance. The two conjugate tooth profiles, Σ₁ and Σ₂, rotate with the respective gear about the gear axis at point O₁ and O₂ at angular velocities ω₁ and ω₂. According to the Lewis theorem [4], the common normal at the point of contact (point of tangency) between the two conjugate profiles in the process of conjugate motion must intersect the line O₁O₂ connecting the centers of rotation and divide the line O₁O₂ into two segments O₁I and O₂I that are related by the following equation:

(3.1)

where i is the gear ratio. If i is a constant, then the position of point I remains the same in the process of conjugate motion. Thus, to transmit uniform rotary motion from one shaft to another by means of gear teeth, the common normal to the profiles of the gear teeth at all points of contact must pass through a fixed point in the line connecting the centers of the two gears. Point I is in kinematics the instantaneous center of rotation between the two gears and is called the pitch point, denoted as P, in gear design. As shown in Figure 3.1, the two circles centered at point O₁ and O₂ with radius r₁ and r₂ equal to O₁P and O₂P are called the pitch circles. The relative motion between the two gears is equivalent in kinematics to the relative motion between the two circular disks with the pitch radii rolling against each other without sliding. Therefore, the pitch circles are the centrodes of the two gears in kinematics.

Figure 3.1 Definitions of conjugate motion.

During the process of the conjugate motion between a pair of gear tooth profiles, the points of contact at different instants form a line in the fixed frame, which is called the line of action. In other words, the two gear tooth profiles only contact at points along the line of action. The curvilinear nature of the line of action depends on the conjugate profiles. At a point of contact along the line of action, the common normal to the two conjugate profiles passes through the pitch point P and forms angle α with the common tangent to the two pitch circles, as shown in Figure 3.1. This angle α is called the pressure angle in the theory of gearing and is a very important gear design parameter. In kinematics, the pressure angle affects the transmission of force from the input gear to the output gear and the loading condition of the gears, shafts, and bearings.

3.2.2 Property of Involute Curves

As mentioned in the introduction, involute curves are used as the profiles for the vast majority gears in power transmission applications. To understand why this is the case, it is necessary to take a look at the geometry of involute curves and how these curves are generated. A circular involute curve is uniquely defined by its base circle. As shown in Figure 3.2, an involute curve starts from point M_o on the base circle with radius r_b. The straight line AM with end point M rolls on the base circle without sliding and is always in tangency to the base circle. Initially, this line is in tangency to the base circle at point M_o, and as it rolls on the base circle, the end point M will trace out the involute curve. The generation of the involute curve can also be considered as the result of unwinding a line with end point M_o that initially wraps around the base circle. As the line unwinds from the base circle, it is always kept taut so that the portion of the line away from the base circle is always in tangency to the base circle. The end point of the line will then trace out the involute curve as the line unwinds from the base circle.

Figure 3.2 Generation of involute curve.

It is obvious from the generation of involute curves that the normal of an involute curve at any point along it is in tangency to the base circle. Since line AM rolls on the base circle without sliding, the arc length AM_o, r_bϕ, is equal to the length of line AM. As shown in Figure 3.2, the Cartesian coordinates of point M on the involute curve can be determined by the following equations:

(3.2)

(3.3)

where ϕ is the so‐called unwinding angle of the involute curve and must be in radians. The curvilinear equation of the involute curve can also be written in the following form:

(3.4)

(3.5)

(3.6)

where angle α is called the pressure angle that is formed between line OM and line OA and is a variable along the involute curve. For triangle ΔOAM, , therefore, . In gear design, the relation is defined as the involute function denoted as:

(3.7)

3.2.3 Involute Curves as Gear Tooth Profiles

Involute spur gears, i.e. gears with straight teeth, have symmetric involute tooth profiles with the same base circle. The tooth profiles on one side of the tooth can be imagined as the curves formed by evenly spaced knots on a string that is unwinding from the base circle, as shown in Figure 3.3. The distance between two neighboring tooth profiles measured along the normal is constant and is called the normal pitch or the base pitch, denoted as p_b. Apparently, the base pitch is the even distribution of the base circle circumference on the number of gear teeth N:

(3.8)

Figure 3.3 Formation of involute gear teeth.

The arc distance between two neighboring tooth profiles measured on the pitch circle is called the circular pitch or simply pitch, denoted as p_c or just p without the subscript. Similar to the base pitch, the circular pitch is the even distribution of the pitch circle circumference on the number of teeth, i.e. , with r being the pitch circle radius. It can be observed from Eqs (3.6) and (3.8) that the pitch circle radius, the base circle radius, the circular pitch and the base pitch are related to the pressure angle α by the following equations:

(3.9)

3.2.4 Characteristics of Involute Gearing

Because of the geometric properties of involute curves, gears with involute profiles have unique features that are advantageous for applications in power transmission and for manufacturability. The characteristics of involute gears described in the following list warrant that these gears are almost exclusively applied in the gear industry.

1. The line of action for a pair of involute gears is a straight line, as shown in Figure 3.1. As mentioned previously, the normal at any point along an involute curve is always tangent to the base circle. This means that the common normal at any point of contact between the tooth profiles of an involute gear pair must be in tangency to the two base circles. There are only two common tangents to the two base circles in a gear pair, one for each pair of contacting tooth branches. It follows immediately that the tooth profiles only contact each other along a common tangent to the base circles. Thus, the common tangent to the two base circles is the line of action by definition. Furthermore, the line of action, or the common tangent to the two base circles, intersects the gear center line O₁O₂ at the pitch point P.
2. The pressure angle of involute gears is a constant in the process of conjugate motion. This follows directly from (a) since the position and orientation of the common tangent to the two base circles do not change as the gears rotate. This feature is advantageous for the loading conditions on the gear teeth, bearings, and shafts since the contact force is always applied in a constant direction.
3. Involute tooth profiles are mutually conjugative. This means that if the tooth profile of a gear is involute, then the tooth profile of the other gear in the pair is also involute. This feature can be rigorously proven by the theory of gearing [4]. Thanks to this feature, involute gears possess excellent exchangeability and manufacturability.
4. If the pitch diameter of one gear is given and the pitch diameter of the other is infinitely large, then the conjugate motion is between a gear and a rack. If the gear tooth profile is involute, then the rack tooth profile is a straight line. Vice versa, it is also true. In this case, the line of action passes through the pitch point and is tangent to the gear base circle. This feature is very important for gear manufacturing because all involute gears can be cut by a rack cutter – or basic rack as commonly termed in gear theory – with a straight blade. In production, gears are mostly manufactured by a hob in the hobbing process, as shown in Figure 3.4. The hob is a cylindrical worm with multiple straight slots. These slots form the cutting edges for the hob and allow the removal of cutting chips from the gear blank. In the axial section of the hob, the profile of the worm teeth is the same as that of the basic rack. In kinematics, the turning of the hob is equivalent to the translation of a rack with straight teeth. The angular velocities of the hob and the gear blank are controlled at a constant ratio according to the tooth number of the gear being cut and the pitch of the hob. The hob also moves along the gear axis as the feed motion to cut the whole length of the gear teeth.
5. The change of center distance does not change the gear ratio of a pair of involute gears. The gear ratio of a pair of involute gears solely depends on the ratio between the base circle radii. Once the gears are cut, the base circle radii are determined and do not change at whatever center distance the two gears are assembled. As shown in Figure 3.5, the common tangent to the two base circles defines the line of action when the gears are assembled at center distance E′, shown on the right side of Figure 3.5, which is not the same as the standard center distance , with r₁ and r₂ being the pitch radii of the two gears, as shown on the left side of the Figure 3.5. Point I is the instantaneous center of rotation of the two gears. Since the gears are not assembled at the standard center distance, point I is not the point of tangency between the two pitch circles with pitch radius r₁ and r₂ respectively. The two circles with radius and , equal to O₁I and O₂I respectively, intersect at point I and are called the operating pitch circles. The respective radius and are called the operating pitch radius. In kinematics, the motion between the two gears assembled at center distance E^′ is equivalent to the rolling of the two operating pitch circles against each other without sliding.
6. Therefore, the operating pitch circles are actually the centrodes of the two gears. Based on the Lewis theory and Eq. (3.1), the gear ratio for the gear pair is . Since the two triangles, ΔO₁KI and ΔO₂LI, are similar, the ratio between the base circle radii is equal to the ratio between the operating pitch radii, i.e. . It is apparent then that the gear ratio remains the same since . Note that when the two gears are assembled at a non‐standard center distance, the operating pitch radii and , and the operating pressure angle α′ as shown in Figure 3.5 are different from the standard values.

Figure 3.4 Gear generation by rack cutter.

Figure 3.5 Involute gears assembled at non‐standard center distance.

3.3 Design of Tooth Element Proportions of Standard Gears

3.3.1 Gear Dimensional and Geometrical Parameters

The gear pitch circle is the design reference for gear dimension and geometry. The design parameters of the rack cutter and the corresponding parameters of the gear are shown in Figure 3.6. The geometry and dimension of a gear are determined by the design parameters, or the tooth element proportions:

N: number of teeth	P: diametral pitch	αc: cutter pressure angle
a: addendum	b: dedendum	c: clearance ()
r: pitch radius	ra: addendum radius	rd: dedendum radius
rb: base circle radius	pc: circular pitch	pb: base pitch
t: tooth thickness	w: tooth space	pn: cutter normal pitch ()
α: gear pressure angle (for standard gears, )

Figure 3.6 Gear tooth element proportions.

3.3.2 Standardization of Tooth Dimensions

In the AGMA Standard [6,7], the tooth element proportions are designed based on the diametral pitch, which is denoted as P and is defined as:

(3.10)

As defined in Eq. (3.10), the diametral pitch can be considered as the distribution of the number of teeth on the gear pitch diameter, as the term itself implies. The unit of the diametral pitch is the reverse of inch, i.e. . Once the diametral pitch of a gear is specified, the pitch circle diameter and the circular pitch are then determined as:

(3.11)

The diametral pitch is standardized in the gear industry and its values are divided into two groups as listed in the following:

1. Coarse pitch: , , , 2, , , , 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18
2. Fine pitch: 20, 22, 24, 26, 28, 30, 48, 64, 72, 80, 96, 120

The fine pitch gears are mainly used for the transmission of motions in applications with low loads, such as instruments and control mechanisms. For gears used in automotive powertrain applications, the diametral pitches are in the series 6, 7, 8, 9, 10, 12. For passenger car transmission gears, diametral pitches 7, 8, 9, 10, 12 are usually used. For truck transmission gears, the diametral pitches are usually selected from 6, 7, 8, 9, 10.

In SI units, the gear tooth element proportions are designed based on the module, denoted as m and with milimeters (mm) as the unit. The module is the reverse of the diametral pitch numerically, that is, . For passenger vehicle transmission gears, the module is usually in the series 2, 2.5, 2.75, 3, 3.5, and for truck transmission gears, the module is usually selected between 3 and 5.5.

The gear pressure angle α on the pitch circle is the other standardized parameter in the gear industry. For spur gears, the standard pressure angle is 20° or 16°. The standardization of the gear pressure angle is directly related to the cutter pressure angle α_c, as shown in Figure 3.6. For standard gears, the gear pressure angle α is the same as the cutter pressure angle α_c. It is obvious that the diametral pitch and the pressure angle for a pair of gears in mesh must be the same.

3.3.3 Tooth Dimensions of Standard Gears

According to the AGMA standard [6,7], the tooth element proportions, mainly the addendum and the dedendum, are designed by slightly different formulae for coarse pitch and fine pitch, as represented in the following:

(3.12)

(3.13)

For both diametral pitch series, the tooth thickness and the tooth space on the pitch circle are the same, equal to half of the circular pitch:

(3.14)

Following the AGMA standard, other gear dimensional parameters are calculated by the following equations:

(3.15)

(3.16)

(3.17)

(3.18)

(3.19)

(3.20)

(3.21)

3.3.4 Contact Ratio

A gear pair is designed to transmit motion and power from one gear to another continuously. For the gear pair in Figure 3.7, gear 1 (upper gear) is the driving member and gear 2 is the driven member. Line N₁N₂ is the line of action that is tangent to the two base circles. Point B₂ is the intersection between the addendum circle of gear 2 and the line of action, and point B₁ is the intersection of the addendum circle of gear 1 and the line of action. It is obvious that B₂ is the starting point and B₁ is the end point of the mesh cycle. Three cases are shown in Figure 3.7 to demonstrate the effect of the length of line segment B₁B₂ on the continuity of mesh. In case (a), B₁B₂ is just equal to the base pitch p_b. Mesh continuity is barely maintained in this case since when the current tooth pair ends contact at point B₁, the next pair just starts contact at point B₂. In case (b), B₁B₂ is longer than the base pitch p_b. Mesh continuity is well maintained since the next pair of teeth is already in contact before the current pair comes out of contact. This guarantees that at least one pair of teeth will be in contact during the mesh process. In case (c), B₁B₂ is shorter than the base pitch p_b. In this case, mesh continuity cannot be maintained since when the current pair of teeth comes out contact, the next pair is not yet at contact. In gear design, mesh continuity is quantified by the contact ratio, defined as the ratio between the length of line segment B₁B₂ and the base pitch: .

Figure 3.7 Gear mesh process and definition of contact ratio.

The contact ratio of a pair of involute gears can be determined using Figure 3.8. By considering gear 1 as the driving member and gear 2 as the driven member, the starting point and end point of mesh are at B₁ and B₂, which are respectively the intersection between the line of action and the addendum circles of gear 2 and gear 1. Points K and L are the points of tangency between the line of action and the two base circles. For ΔO₁KB₂ and ΔO₂LB₁, the lengths of line segments KB₂ and B₁L can be determined as:

(3.22)

where r_a1 and r_a2 are the respective addendum radii of gear 1 and gear1, r_b1 and r_b2 are the respective base circle radii. Length KL is related to the center distance E and is determined as:

(3.23)

Figure 3.8 Determination of contact ratio.

It is apparent that . Therefore the contact ratio is determined by the following equation:

(3.24)

In gear design, the contact ratio must be higher than 1.0. Higher contact ratio is beneficial for the sharing of contact load and also for the reduction of gear noise. It is observed from the equation above that larger addendum radii yield higher contact ratio. In gear design practice, non‐standard gears with modified addendum radii are sometimes used to increase the contact ratio, as discussed later in the design of non‐standard gears.

3.3.5 Tooth Thickness and Space along the Tooth Height

The tooth thickness is very important for gear design and manufacturing. If the tooth thickness is not controlled with high accuracy in gear manufacturing, the gear teeth may not be properly meshed. There will be either interference or backlash between the teeth of the two gears. The tooth thickness calculated by Eq. (3.20) is measured on the pitch circle. Along the tooth height, the tooth thickness varies from the largest at the base circle to the smallest at the addendum. The tooth thickness at different radii from the gear center is crucial to the mesh condition of non‐standard gears, as discussed in the next section. In the following, a set of equations formulated in reference [4] are represented for the calculation of tooth thickness of involute gears at any radius from the gear center.

As shown in Figure 3.9, arc length is the standard tooth thickness t on the pitch circle with the pitch radius r, and arc length is the tooth thickness t_x on the circle with radius r_x. Angles β and β_x are respectively the angle between line OA and the middle line of the tooth and between line OA_o and the middle line of the tooth. The tooth thickness on the pitch circle is equal to 2βr. The angle between line OA_o and line OA is determined by Eq. 3.7 and is equal to Inv(α_c), and the angle between line OA_o and line OB is equal to Inv(α_x). The following equations can be obtained by observing Figure 3.9:

(3.25)

(3.26)

(3.27)

where α_c is the cutter pressure angle and is the same as the pressure angle α for standard gears. The tooth thickness at radius r_x and the tooth thickness at pitch radius are equal to 2β_xr_x and 2βr respectively. Therefore:

(3.28)

Figure 3.9 Tooth thickness at different radius.

The tooth space w_x at radius r_x is equal to the circular pitch on the circle with radius r_x minus the tooth thickness t_x, and is determined by the following equation:

(3.29)

Summarizing these derivations, the tooth thickness and the tooth space at a given radius r_x for an involute gear are determined by the following set of equations:

(3.30)

(3.31)

(3.32)

(3.33)

The tooth thickness and tooth space change along the tooth height and are critical for the proper mesh of general non‐standard gears discussed in the next section. For the derivation of mesh condition for general standard gears, Eqs (3.32) and (3.33) can be rewritten as:

(3.34)

3.4 Design of Non‐Standard Gears

3.4.1 Standard and Non‐Standard Cutter Settings

Non‐standard gears differ from standard gears in the cutter settings that are used to cut them. As mentioned previously, gears can be cut by a rack cutter or “basic rack” as shown in Figures 3.4 and 3.6. In the cutting process, the cutter pitch line is in tangency to the pitch circle of the gear being cut. When a standard gear is cut, the middle line of the rack cutter coincides with the pitch line and is in tangency with the gear pitch circle, as shown in Figure 3.10. In this standard cutter setting, the distance between the middle line of the rack cutter and the gear center is equal to the gear pitch radius. However, the rack cutter can be displaced either toward or away from the gear center. If displaced away from the gear center, the cutter displacement, denoted as e in Figure 3.10, is defined to be positive, otherwise, the displacement is negative. Gears that are cut when the cutter displacement is not zero are non‐standard gears. Obviously, the cutter displacement for standard gears is equal to zero.

Figure 3.10 Standard and non‐standard cutter settings.

The tooth thickness t_c and tooth space w_c of the rack cutter on the middle line are designed as the same and are both equal to half a pitch, i.e. as shown in Figure 3.10. In the cutting process, the rack cutter travels a distance equal to the pitch when the gear rotates through a circular pitch, which is also equal to . Therefore, for standard gears, the tooth thickness and tooth space of the gear are respectively equal to the tooth space and tooth thickness of the rack cutter on its middle line:

(3.35)

For non‐standard gears, the cutter middle line does not coincide with cutter pitch line, that is in tangency with the gear pitch circle. For the positive cutter displacement shown in Figure 3.10, the tooth space on its pitch line becomes wider, while the tooth thickness becomes narrower, resulting in an increase in the tooth thickness and a decrease in the tooth space on the gear pitch circle, as determined by the following equations:

(3.36)

where α_c is the cutter pressure angle and e is the cutter displacement, shown as positive in Figure 3.10. It is obvious from Eq. (3.36) that when the cutter displacement is positive, the gear tooth thickness on the pitch circle is increased and the tooth space is decreased. It is also obvious that the gear dedendum and dedendum radius will be changed by the cutter displacement as follows:

(3.37)

where r and b are the gear pitch radius and the standard gear dedendum respectively. For positive cutter displacement, the dedendum will be decreased by an amount equal to the cutter displacement, while the dedendum circle radius will be increased by the cutter displacement.

3.4.2 Avoidance of Tooth Undercutting and Minimum Number of Teeth

Tooth undercutting is a phenomenon that occurs in the gear cutting process. When it occurs in the cutting process of involute gears, the tooth profile is not the involute curve over the whole tooth, but has an undercut portion in the root area. Figure 3.11 shows the comparison between a gear tooth with normal involute profile and an undercut gear tooth. When there is no undercutting, the envelope of the cutter traces at different positions during the cutting process forms the involute profile for the whole tooth. However, when undercutting occurs, the profile on the tooth root is not the involute profile required for the proper mesh of gears. It is obvious that undercutting must be avoided in gear design since an undercut gear cannot be meshed properly and the undercut root makes the gear strength much lower. The problem of undercutting can be solved in general by methods in the theory of gearing; for the case of involute gears, it can be solved based on involute curve geometry and the cutter setup [4], as shown in Figure 3.12.

Figure 3.11 Undercutting of involute gears.

Figure 3.12 Undercutting avoidance for involute gears.

The regular tooth profile of an involute gear must be above the base circle since involute curves start from the base circle. As shown in Figure 3.12, the rack cutter is displaced with an offset e. The cutter addendum is a and the cutter clearance is c. The cutter pressure angle and pitch are denoted as α_c and p_c. The cutter and the gear tooth profiles are denoted as Σ₁ and Σ₂ respectively. Point G is the starting point of the gear tooth profile and α_G is the pressure angle at point G. Clearly, and . Line PL is the line of action between the rack cutter and the gear and is in tangency to the gear base circle at point L. Point G is on the line of action since it is a normal contact point between the rack cutter profile and the gear tooth profile. For triangle ΔGOL, the following inequality can be derived:

(3.38)

For standard gears, the cutter displacement e is zero, therefore,

(3.39)

Equations (3.9) and (3.12) are used in the derivation of the inequality just given. It follows directly that to avoid undercutting of standard involute gears, the number of teeth of the gear must be larger than . This quantity is rounded up to the integer and is called the minimum number of teeth, denoted as N_min, for the avoidance of undercutting.

For non‐standard gears, the cutter displacement e is not zero and the inequality in Eq. (3.38) can be transformed as follows:

(3.40)

(3.41)

(3.42)

where P is the diametral pitch, Pe is unitless and is denoted as ζ, which is called the modification coefficient for non‐standard gears. The condition for non‐undercutting can then be represented as . In gear design practice, there are two cases involving the use of modification coefficient ζ. In the first case, and , i.e. the number of teeth of the gear is smaller than the minimum number of teeth for the avoidance of undercutting. In this case, the cutter must be displaced away from the gear center and the minimum cutter displacement is determined by:

(3.43)

In the second case, and . The cutter setting can be either standard or the cutter can be displaced toward the gear center to change the tooth thickness for non‐standard gears.

3.4.3 Systems of Non‐standard Gears

Non‐standard gears are designed for several purposes:

• To avoid undercutting of a gear with fewer teeth than the minimum number of teeth, N_min.
• To enlarge the tooth thickness of the pinion, the smaller gear, for the balance of strength in a gear pair. This is possible when the sum of the numbers of teeth in the gear pair is larger than 2N_min, as will be shown later.
• To accommodate the assembly of gear pairs at non‐standard center distance. This is often useful for gear designs that involve multiple gear pairs on the parallel shafts, such as in manual transmissions.
• To increase the contact ratio by increasing the tooth height.
• To optimize the tooth element proportions for specific applications, such as gear pumps and meters.

There are two systems for the design of non‐standard gears in AGMA Standard [6,7], as categorized by the sum of cutter displacements for the pinion and for the gear, i.e. :

1. Long‐short addendum gear system:
2. General non‐standard gear system:

3.4.4 Design of Long‐Short Addendum Gear System

There must be no tooth undercutting for each of the two gears in the pair, so the cutter displacement, e_p for the pinion and e_g for the gear, must satisfy the inequality expressed by Eq. (3.42):

(3.44)

(3.45)

Since for the long‐short (L‐S) addendum system, it follows directly that for the design of non‐standard gears in this system, the following condition must be satisfied:

(3.46)

In the design of L‐S addendum gears, the first step is to check whether the inequality in Eq. (3.46) is satisfied. The cutter displacements must then be chosen according to the design priority. Usually, the pinion cutter displacement e_p is chosen to be positive for an increase in the pinion tooth thickness. The gear cutter displacement e_g has the same magnitude as e_p but is opposite in sign. Obviously, both e_p and e_g must satisfy Eqs (3.44) and (3.45). Once the cutter displacements are chosen, the tooth element proportions are then calculated by the following standard equations:

(3.47)

(3.48)

(3.49)

(3.50)

(3.51)

It can be seen from Eq. (3.49) that the addendum of the pinion becomes longer and the addendum of the gear becomes shorter in the so‐called L‐S addendum system. It should be emphasized here that non‐standard gears in the L‐S addendum system are assembled at the standard center distance and the pressure angle is the same as the standard pressure angle. In summary, the characteristics of non‐standard gears in the L‐S addendum system are:

• The sum of cutter displacements is zero. Usually, the pinion cutter displacement is positive and the gear displacement is negative.
• The center distance is the same as the standard center distance E.
• The operation pressure angle of the non‐standard gears in the L‐S addendum system at assembly is the same as the standard pressure angle.
• The tooth thickness differs from that of the standard gears. Usually, the pinion tooth thickness is increased due to the positive cutter displacement for strength balance in the gear pair. The gear tooth thickness is reduced by the same amount.
• The addendum and dedendum differ from the standard gears.
• The operation pitch circles are the same as those in standard gears.

3.4.5 Design of General Non‐Standard Gear System

As mentioned in Section 3.2, the change of center distance does not change the gear ratio of a pair of involute gears. However, if a pair of involute gears is assembled at a non‐standard center distance, the operating pitch circles, i.e. the centrodes in kinematics, and the operating pressure angle are different from the standard values, as shown in Figure 3.13. It is obvious that the non‐standard center distance E′ is the sum of the operating pitch radius of the pinion and the gear, as expressed by the following equations:

(3.52)

(3.53)

Figure 3.13 Operating pitch circles and operating pressure angle.

In the meshing process, the two operating pitch circles roll against each other without sliding. There cannot be either backlash or interference between the meshing teeth on the operating pitch circle. Using Eq. (3.36), the pinion tooth thickness and the gear tooth space on the respective pitch circles for the non‐standard gear pair can be determined as follows:

(3.54)

(3.55)

The pinion tooth thickness and the gear tooth space on the respective operating pitch circles, and , can be found by Eq. (3.34) with the substitution of r_x by the operating pitch radius, and respectively and the substitution of α_x by α^′, i.e.

(3.56)

(3.57)

For the proper mesh of the non‐standard gears, the following equation must be observed:

(3.58)

On the left side of the equation above, the numerator is the pinion angle of rotation corresponding to the tooth thickness and the denominator is the gear angle of rotation corresponding to the tooth space on the respective operation pitch circle. It follows directly from Eqs (3.56–3.58) that the mesh condition for non‐standard gears is represented by the following equation:

(3.58a)

This equation is the mesh condition of general non‐standard gears and must be satisfied in the design of general non‐standard involute gears. For production gears, there is always a certain amount of tooth backlash which is controlled by tooth thickness tolerance specifications [9,10]. In manual transmissions, there are multiple pairs of gears on parallel shafts. In other words, multiple gear pairs are assembled at the same center distance. Since the standard center distance depends on the sum of teeth, which are integers and must satisfy the transmission ratio requirement, and the diametral pitch, which is selected in the standardized series, it is often impossible to have all the gear pairs with the same standard distance. In such applications, general non‐standard gears can be designed to accommodate the center distance constraints, as described in the following steps:

Step 1: The operating pressure angle α^′ is determined from Eq. (3.53) as:

(3.59)

where E′ is the non‐standard distance at which the gears are to be assembled, i.e. the distance between the two gear shafts; α_c is the cutter pressure angle or the standard gear pressure angle; is the sum of teeth for the gear pair that must provide the specified gear ratio; and P is the diametral pitch selected in the standard series.

Step 2: The sum of the cutter displacement is then calculated based on the mesh condition for general non‐standard gears represented by Eq. (3.58a), as follows,

(3.61)

The sum of cutter displacements determined above is then distributed between the pinion and the gear in the pair based on considerations of strength and of avoidance of undercutting and interference [7,11,12].

Step 3: The tooth element proportions of the general non‐standard gears are then calculated by the following equations:

(3.61)

(3.62)

(3.63)

(3.64)

(3.65)

(3.66)

(3.67)

where h_o is the standard tooth height, which is equal to the sum of addendum and dedendum, i.e. ; ΔE is the difference between the center distance at which the gears are assembled and the standard center distance, and is equal to ). The addendum radii determined by Eq. (3.66) are designed to provide the standard clearance that is equal to . Obviously, the tooth height of general standard gears determined by Eq. (3.67) is different from the standard value and is shortened by . After all tooth element proportions are calculated, Eq. (3.24) is then used to check the contact ratio of the designed general non‐standard gears. Note here that when Eq. (3.24) is used for the contact ratio of general non‐standard gears, the operating pressure angle α′ and the non‐standard center distance E′ must be used in the equation instead of the standard pressure angle α and the standard center distance E.

3.5 Involute Helical Gears

Involute helical gears have an involute tooth profile on the transverse section, which is the cross‐section perpendicular to the gear axis. The surface of an involute helical gear is a helicoid which is formed by an involute curve on the transverse section in a screw motion along the gear axis, as shown in Figure 3.14. Intuitively, an involute helical gear can be thought as the result of a spur involute gear having undergone a uniform twist. The base circles for all the involute tooth profiles on the transverse sections form the so‐called base cylinder with base radius r_b. Similarly, the pitch circles on the transverse sections form the pitch cylinder with pitch radius r. The tooth surface of an involute helical gear is between the base cylinder and the addendum cylinder with radius r_a. The intersection between the tooth surface and any cylinder with radius between r_b and r_a is a helical line or helix. The angle between the tangent to the helix on the pitch cylinder and the gear axis is called the helical angle, denoted as β. This helical angle can also be defined by spreading the helix on the pitch cylinder on a plane as shown in Figure 3.14. A helical gear can be right handed or left handed, depending the direction of the helix. The gear shown in Figure 3.14 is right handed. For helical gears with parallel axes, the two gears in mesh have the same magnitude for the helical angle, but opposite hands of the helix in external gearing. For internal gearing, the helical angle and helix direction are the same.

Figure 3.14 Basic geometry of involute helical gears.

3.5.1 Characteristics of Involute Helical Gearing

Involute helical gears have all the characteristics of involute gears, plus additional features highlighted in the following:

1. On the transverse section, involute helical gears just behave as spur involute gears. Because of this, all of the equations and conclusions for the design and analysis of spur involute gears are valid if the mesh for helical involute gears is considered on the transverse section.
2. The involute tooth profiles along the gear axes come in and out of mesh at different time, i.e. at different rotational positions. This provides an extra contact ratio, called lengthwise contact ratio, in addition to the contact ratio on the transverse section. Because of the enhanced contact ratio, involute helical gears are stronger and quieter than spur involute gears.
3. Helical and spur involute gears are cut by the same machine tools and cutters. The manufacturing costs for helical and spur gears are about the same.
4. The tooth geometry design of helical gears is based on the tooth form on the normal section, which is the intersection between the tooth surface and the plane normal to the helix on the pitch cylinder. The normal diametral pitch P_n and the normal pressure angle α_n are standardized for helical gears. For the proper mesh of a pair of helical gears, the normal diametral pitch and the normal pressure angle must be the same respectively, i.e. and .
5. Helical gears have axial loads, or thrusts. This is the disadvantage of helical gears.

Because of the advantages in strength and quietness as compared with spur gears, involute helical gears are widely applied for passenger vehicle transmissions. Having an axial load is a shortcoming of involute helical gears. When there are two gears on the same shaft in the power flow of a gear box, one driven for a gear pair and one driving for another gear pair, the axial load can be partially cancelled in gear design so as to minimize the thrust on the bearing.

3.5.2 Design Parameters on the Normal and Transverse Sections

As mentioned in the design of spur involute gears, the tooth geometry of involute spur gears is based on the rack cutter with a straight tooth, or the so‐called basic rack shown in Figure 3.15a. Similarly, the geometry of involute helical gears is based a basic rack shown in Figure 3.15b. The tooth surface of the rack cutter, or basic rack, is an inclined plane for involute helical gears. On the pitch plane, that is tangent to the pitch cylinder, the rack tooth profile is a straight line that forms the helical angle β with the lengthwise direction. The distance between two neighboring teeth measured on the transverse section is the transverse pitch p_t. The normal pitch p_n is measured perpendicular to the tooth profile on the pitch plane, as shown in Figure 3.15c. The transverse pressure angle α_t and the normal pressure angle α_n are respectively defined on the transverse section and the normal section of the basic rack, as shown in Figure 3.15d.

Figure 3.15 Transverse and normal sections of the rack cutter for involute helical gears.

It can be observed from Figure 3.15c that the transverse pitch and the normal pitch are related to the helical angle by . Reversing both sides of this relation leads to the relation between the transverse diametral pitch P_t and the normal diametral pitch P_n. From Figure 3.15d, the relation between the transverse pressure angle α_t and the normal pressure angle α_n can be derived since the tooth height is the same whether it is measured on the transverse section or the normal section. In summary, the normal diametral pitch and the normal pressure angle are converted to the transverse diametral pitch and the transverse pressure angle by the following equation:

(3.68)

(3.69)

3.5.3 Tooth Dimensions of Standard Involute Helical Gears

As mentioned previously, the normal pressure angle α_n and the normal diametral pitch P_n for involute helical gears are standardized in the gear industry. The tooth element proportions of involute helical gears are based on the tooth form on the normal section. According to the AGMA standard [7,12], the addendum and the dedendum are designed by the following equations:

(3.70)

The basic design parameters of an involute helical gear are its normal pressure angle α_n, normal diametral pitch P_n, helical angle β, and number of teeth N. The dimensional parameters are determined by the following equations:

(3.71)

(3.72)

(3.73)

(3.74)

(3.75)

(3.76)

(3.77)

(3.78)

3.5.4 Minimum Number of Teeth for Involute Helical Gears

Since involute helical gears have the same characteristics as spur involute gears on the transverse section, the condition for the avoidance of undercutting represented by Eq. (3.39) can be applied directly if the parameters on the transverse section are used. Therefore, the following inequality must be observed to avoid undercutting of standard involute helical gears,

(3.79)

As can be observed from Eq. (3.79), involute helical gears can be designed with fewer teeth than spur gears without the issue of undercutting. The higher the helical angle, the fewer the number of teeth the gear can have without undercutting. For example, if the normal pressure angle is 20° and the helical angle is 45°, the minimum number of teeth for an involute helical gear for the avoidance of undercutting is as few as seven.

3.5.5 Contact Ratio of Involute Helical Gears

The contact ratio of involute helical gears has two components: a transverse contact ratio and a lengthwise contact ratio. The transverse contact ratio is also termed the involute contact ratio, which is determined by Eq. (3.24). For involute helical gears, the transverse diametral pitch P_t and the transverse pressure angle α_t are used in Eq. (3.24), since these gears behave as spur involute gears on the transverse section. The lengthwise contact ratio comes from the fact that the tooth profiles on the two ends of the tooth length do not come in and out of mesh at the same time. Figure 3.16 shows the planar spread of the pitch cylinder and the tooth helixes on it, with β and L as the helical angle and the tooth length. Line AB shows the position of a tooth that just enters mesh. For spur gears, the tooth lines on the planar spread are straight lines parallel to the gear axis and separated by the circular pitch. When a spur gear rotates through a circular pitch, points A and B on tooth ends move to points A′ and B′, which indicate the end of the mesh for the tooth. However, in the case of helical gears, only the mesh on the end section with point A is finished. When the gear rotates through a circular pitch, the same tooth is still in mesh elsewhere along the tooth length. For the whole tooth to be out of mesh, the gear has to rotate additionally until the other end of the tooth helix, point B, moves through on the pitch circle an arc length that is equal to L tan β. The lengthwise contact ratio m_l is then defined as the ratio between the length of line segment B′B′′ and the circular pitch on the transverse section. The total contact ratio of helical gears is the sum of the transverse contact ratio and the lengthwise contact ratio as follows:

(3.80)

(3.81)

Figure 3.16 Lengthwise contact ratio for involute helical gears.

3.5.6 Design of Non‐standard Involute Helical Gears

The design of non‐standard involute helical gears is similar to the design of spur gears, as described in Section 3.4. Firstly, the transverse diametral pitch P_t and the transverse pressure angle α_t are converted from the normal diametral pitch P_n and normal pressure angle α_n by Eqs (3.68) and (3.69). Then the non‐standard center distance E^′ and the mesh condition on the transverse section are used to determine the operating transverse pressure angle and the sum of the cutter displacements by the following equations:

(3.82)

(3.83)

The equations for the design of non‐standard spur gears, Eqs (3.61–3.67), are also applicable for the design of non‐standard involute helical gears if the transverse pressure angle α_t is used. It is noted here that the normal diametral pitch P_n is used for the calculation of addendum and dedendum in these equations. The contact ratio of non‐standard involute helical gears is calculated by Eq. (3.81), where the standard center distance E and the standard transverse pressure angle α_t are replaced by the non‐standard center distance E′ and non‐standard transverse pressure angle .

3.6 Gear Tooth Strength and Pitting Resistance

3.6.1 Determination of Gear Forces

Figure 3.17 shows a pair of helical gears with the pinion as the driving member. The driving torque and the pinion angular velocity are in the same direction. Since the pinion is the driving member, the pinion tooth side that contacts the gear tooth is indicated as shown in the figure. The contact force between the pinion and the gear has three components: radial, tangential, and axial. The directions of the three force components applied to the pinion are shown in Figure 3.17. The radial component is always toward the center, the tangential component always forms a torque about the center to resist the driving torque, and the direction of the axial component or thrust depends on the hand of the helix and can be readily determined once the contact side of the driving member is indicated. The three force components applied by the pinion to the gear member are just in the opposite directions.

Figure 3.17 Directions of gear force components.

The magnitudes of the three force components shown in Figure 3.17 are related to the normal pressure angle α_n and the helical angle β. As shown in Figure 3.18, the contact force N acts on the normal section that is perpendicular to the helix on the pitch cylinder. The angle between the contact force and the intersection between the normal section and the pitch cylinder is the normal pressure angle α_n. The contact force is decomposed to two components on the normal section: the radial component F_r which is always toward the gear center and the tangential component on the normal section F_nt. F_nt is further decomposed into the tangential component F_t on the transverse section and axial component F_a. It is obvious that the driving torque T_d is equal to the tangential component F_t times the pitch radius r_p of the driving member, which is the pinion as shown in Figure 3.17, i.e. . By observing Figure 3.18, the gear contact force and the three gear force components are determined by the following equations:

(3.84)

(3.85)

(3.86)

(3.87)

Figure 3.18 Determination of gear force components.

Note that Eqs (3.84–3.87) are also valid for spur gears. To determine the gear force components of spur gears, the normal pressure angle α_n is replaced by the pressure angle α of the spur gears and the helical angle β is zero. Obviously, the axial load for spur gears is equal to zero.

3.6.2 AGMA Standard on Bending Strength and Pitting Resistance

Gear strength and durability are rated on bending stress and contact stress in the AGMA standard [11,13]. For gears used in automotive transmissions, durability, which is related to pitting resistance, and tooth strength, which is related to fracture and fatigue resistance, are the most important design considerations. The AGMA standard Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth provides the formula for bending stress calculations based on the modification of the Lewis method for gear bending stress, and the formula for contact stress based on the modification of the Hertzian contact theory. Although these formulations are highly empirical by nature and may yield very different results since a large number of modifiers or factors are used to account for variations in materials, manufacturing, and applications, they are based on experiences, expertise, and real data from the whole gear industry and they have been proved to be valuable and practical in the design and application of gear transmission systems. The following text highlights the AGMA formulations on gear tooth bending strength and pitting resistance and their applications in gear design practice.

3.6.3 Pitting Resistance

In the AGMA fundamental pitting resistance formulation for involute spur and helical gears, the contact stress is calculated by the following equation:

(3.88)

In this formulation five quantities carry a unit:

• s_c: This is the calculated contact stress in lb/in² or MPa.
• C_p: Elastic coefficient in or , C_p is tabulated in the AGMA standard in terms of materials.
• F_t: Tangent component of the contact force in Eq. (3.85) in lb or N.
• d: Operating pinion pitch diameter, in in. or mm.
• F: Net face width of the narrowest member, in in. or mm, the same as tooth length L in Eq. (3.82).

The other six quantities are unitless factors named as follows:

• C_a: Application factor for pitting resistance.
• C_s: Size factor for pitting resistance.
• C_m: Load distribution factor for pitting resistance.
• C_f: Surface condition factor for pitting resistance.
• I: Geometry factor for pitting resistance.

The contact stress calculated by Eq. (3.88) must be lower than the allowable contact stress, which is modified by a another set of factors, as shown below:

(3.89)

The quantities in this formulation are as follows:

• s_ac: Allowable contact stress in lb/in² or MPa.
• C_L: Life factor for pitting resistance.
• C_H: Tooth hardness ratio for pitting resistance.
• C_T: Temperature factor for pitting resistance.
• C_R: Reliability factor for pitting resistance.

3.6.4 Bending Strength

In the AGMA’s fundamental bending strength formulation for involute spur and helical gears, the bending stress is calculated by the following equation:

(3.90)

In this formula, s_t is the calculated bending stress in lb/in² or MPa. F_t and F are the same as in the formulation for pitting resistance. P_t is the transverse diametral pitch. The six unitless modification factors are as follows:

• K_a: Application factor for bending strength.
• K_B: Rim thickness factor for bending strength.
• K_s: Size factor for bending strength.
• K_m: Load distribution factor for bending strength.
• K_v: Dynamic factor for bending strength.
• J: Geometry factor for bending strength.

The bending stress calculated by Eq. (3.90) must be lower than the allowable bending stress modified as follows:

(3.91)

In this formulation, the allowable bending stress s_at is modified by three factors:

• K_L: Life factor for bending strength.
• K_T: Temperature factor for bending strength.
• K_R: Reliability factor for bending strength.

In the AGMA standard for gear pitting resistance and bending strength, the allowable contact stress s_ac and the allowable bending stress s_at are tabulated in terms of gear materials, grades, heat treatment, and hardness. These allowable stress values are based on 10⁷ cycles with 99% reliability. There are a large number of factors in the AGMA formulation, as shown in Eqs (3.88–3.91). The values of these factors have to be chosen carefully from plots or formulations recommended by the AGMA standard. The modification factors that share the same name (i.e. the same subscripts, for example, C_a and K_a) in the formulation have the same values and share the same plot or formula as provided in the standard. A separate AGMA standard [13,14] needs to be used for the values of the two geometry factors I and J.

3.7 Design of Automotive Transmission Gears

Involute helical gears are widely applied for automotive transmission gears. The design of transmission gears is based on the same principle and theory as for other applications and should take into considerations the vehicle operation conditions and requirements. Generally, durability, compactness, cost, and quietness are the design priorities for automotive gears, especially so for passenger vehicles. Because of the maturity of the automotive industry, the gear design for new transmission development can rely on the experience and data gained from existing products that have been proved in the industry. As the crystallization of these experiences and data, the AGMA standards on gear design provide guidelines in the highly empirical transmission gear design and development process. For the design of transmission gears, a series of AGMA gear design standards should be referenced or followed, as listed below:

• AGMA 933‐B03: Basic Gear Geometry
• AGMA 913‐A98: Method for Specifying the Geometry of Spur and Helical Gears
• AGMA 2000‐A88: Gear Classification and Inspection Handbook
• AGMA 2002‐B88: Tooth Thickness Specification and Measurement
• AGMA 2001‐D04: Fundamental Rating factors and Calculation Methods for Involute Spur and Helical Gear Teeth
• AGMA 6002‐B93: Design Guide for Vehicle Spur and Helical Gear Drives
• AGMA 908‐B89: Information Sheet – Geometry Factors for Determining the Pitting Resistance and Bending Strength for Spur, Helical and Herringbone Gear Teeth, Parts 1 & 2
• AGMA 918‐A93: Summary of Numerical Examples Demonstrating the Procedures for Calculating Geometry Factors for Spur and Helical Gears

For the design of transmission gears, the following data are usually known or specified:

• Maximum engine torque and power
• Vehicle weight and dimensional data
• Transmission ratios
• Space envelope for the transmission assembly
• Expected service life

The design of transmission gearing is a process that does not have a direct closed‐form solution. Experiences and data on existing products in the industry play an important role in the selections of initial design parameters. The following summarizes the steps highlighting the gear design procedure for two example manual transmissions, one FWD and the other RWD, shown in Figures 2.4 and 2.5. These manual transmissions are typical for FWD passenger cars and for RWD commercial vehicles such as pickup truck and cargo vans. The stick diagrams are shown again in Figure 3.19 for convenience.

Figure 3.19 Transmission gear design examples.

Step 1: Selection of normal pressure angle α_n, helical angle β and normal diametral pitch P_n

The normal pressure angle α_n is usually chosen as the standard value of 16° or 20°. The helical angle β is a very important design parameter that affects center distance, axial load, and gear contact ratio. For passenger car transmission gears, β can be initially chosen in the range . For commercial vehicles, the helical angle should be chosen at smaller values to reduce the axial load and can be chosen initially in the range . The final value for the helical angle depends on center distance and load constraints. Note that the helical gears on the output shaft in the MT shown in Figure 3.19a, and the helical gears on the counter shaft in the MT shown in Figure 3.19b, should have the same hands of helix so that the axial loads at the two gear meshes on these two shafts will partially cancel each other out. The gears on the input shaft and the final drive ring gear N_1o are right handed so that the axial load on the final drive ring gear will push the gear box against the engine. The normal diametral pitch P_n directly affects the dimension of the gears and the whole transmission size. It is recommended that the normal diametral pitch be selected according to the engine output power P_e as listed below:

• 

If the SI unit is used, then these diametral pitch values are converted to the module values by . This value is then rounded to the closest module value in the standard series. For example, a normal diametral pitch is converted to a normal module  mm. In many cases, a vehicle may be powered by optional engines with different output powers. Then the selection of the normal diametral pitch should fit the engine option with the highest power. For the FWD MT shown in Figure 3.19a, the final drive gears are subjected to the highest torque, and if necessary the normal diametral pitch for the final drive may be chosen as the coarser value next to the diametral pitch selected for all other gears.

Note that if strength and endurance requirements are satisfied, finer diametral pitch or smaller normal module are preferred. With the same size, gears can be designed with more teeth if a finer pitch is used. Gear pairs with more teeth and finer pitch in general run smoother and more quietly. For compact cars, the transmission gears are often designed with a normal module of 2 mm or a normal diametral pitch of 12.

Step 2: Selection of the numbers of teeth for different gears

The numbers of teeth must firstly guarantee that the specified gear ratios be achieved accurately, usually up to the second digit after the decimal point of the gear ratio value. Secondly, the numbers of teeth must also satisfy the constraints on the center distance. For the FWD MT in Figure 3.19a, the numbers of teeth and the center distances of gear pairs on the input and output shafts are constrained by the following equations:

(3.92)

(3.93)

where the subscript j is 1, 2, 3, 4, 5, and r respectively. Apparently, the shaft distance or center distance, E_io is the same for all the six gear pairs. The value of E_io is usually specified within a narrow range due to assembly constraints. With the normal diametral pitch P_n and the helical angle β_j chosen in Step 1, the numbers of teeth for each gear pair, N_ji and N_jo, can be determined by combining Eqs (3.92) and (3.93). But the solutions for N_ji and N_jo are generally not integers and must be rounded to the closest integers. These rounded integer numbers of teeth are then plugged back into Eq. (3.92) to solve for the helical angles. It is possible to design standard helical gears for all forward gears in this way. The reverse gears in the FWD MT shown in Figure 3.19a can be designed as a non‐standard spur gear since the center distance of the spur reverse gears only depends on the numbers of teeth once the diametral pitch is chosen.

For the FWD MT shown in Figure 3.19a, the first gear N_1i, the second gear N_2i, and reverse gear N_ri, are machined on the input shaft. The numbers of teeth of these three gears can be chosen as: , , . The numbers of teeth for gears N_1o, N_2o, and N_ro are then determined by the specified ratios. The numbers of teeth for the first, second, and reverse gears and the selected normal diametral pitch are then plugged in Eq. (3.92) to solve the helical angles.

For the RWD MT in Figure 3.19b, the numbers of teeth and the center distances between the output shaft that is coaxial with the input shaft and the counter shaft must satisfy the following equations:

(3.94)

(3.95)

where the subscripts j is 1, 2, 3, 4, 5, and r respectively. In the design, there is an input gear pair, N_i and N_c, which is in the power flows of all gears except the 4th gear. Except for the direct 4th gear, the transmission ratios are the multiplication of the common input gear ratio and the other ratio designated for each gear. The input gear ratio, , is usually designed to be around , with and . This allows for a near balanced strength for the input gears which undergo the longest service life under load. The numbers of teeth for other gear pairs are then chosen based on the gear ratio requirements and center distance constraints. The selection for the number of teeth of the first gear on the counter shaft, N_1c, is very important because the first gear pair determines the radial dimension of the transmission housing and has the largest strength imbalance between the pinion and the gear. In general, N_1c is chosen to be . After the numbers of teeth for the first gears are chosen, the numbers of teeth for all other gears can then be chosen by using Eqs (3.94) and (3.95). As a rule of thumb, the counter shaft gear of the next higher gear is designed with more teeth. For example, N_2c is chosen to be . The center distance constraints can be satisfied by choosing different helical angles for each gear pair. If the helical angle determined based on the center distance constraint is way off the range recommended in Step 1, then general non‐standard gears can be designed to accommodate the center distance with the helical angle in the recommended range.

Step 3: Selection of the tooth length (face width) for gear pair

The tooth length L affects the contact area, and therefore the load capacity of transmission gears. The ratio between the tooth length and the normal diametral pitch is usually in the range: . For transmission gears with a normal diametral pitch of 12, the tooth length can be designed as inch.

Step 4: Tooth element proportions and tolerance specifications

After Step 3, the tooth element proportions can then be calculated by equations in Sections 3.3, 3.4, and 3.5. The contact ratio for each gear pair is calculated using Eqs (3.24), (3.80), and (3.81). Gears for automotive transmissions usually fall in class 9, 10, or 11 in the AGMA standard, depending on the category of the vehicle using the transmission. Gear tooth tolerances are specified in AGMA standards [9,10] based on the selected gear class.

Step 5: Gear strength and pitting resistance check

The gear tooth strength and pitting resistance are rated respectively using Eqs (3.90) and (3.91), and Eqs (3.90) and (3.91). Firstly, the gear force components are calculated by Eqs (3.84–3.87). The driving torque in Eq. (3.87) is the maximum engine output torque of the engine. For the FWD MT in Figure 3.19a, the first gears, 5th gears and the final drive gears should be the focus of strength and pitting resistance rating since these gears are most likely to be subjected to high loads or long service cycles. For the RDW MT in Figure 3.19b, the input gears, the first gears and the 5th gears should be focused on for the same reason. The allowable contact stress s_ac and the allowable bending stress s_at are related to gear materials, hardness, and heat treatment, and are tabulated in the AGMA standard [15] accordingly. As shown in the equations, the tooth bending strength and pitting resistance formulation largely depends on the various modification factors. The selection of these factors should follow the guidelines in the AGMA standard [11,14,15] and also account for the operation conditions characteristic of the automotive transmissions.

Dynamic factorsC_v and K_v: The dynamic factors account for extra gear load in addition to the normal gear load that is generated by the non‐perfect conjugate motions in a gear pair. The root cause for this extra load is the gear transmission errors defined as the deviation from the theoretical uniform gear rotations. The values of C_v and K_v are mainly related to the accuracy of the gears and the pitch line speed. In the AGMA standard [11], the dynamic factors, C_v and K_v, are plotted against the pitch line speed for different gear classes. The pitch line speed of transmission gears is approximately in the range . For transmission gears in AGMA Class 10, the values of C_v and K_v are found to be in the range of based on the AGMA standard.

Application factorsC_a and K_a: The application factors account for the momentary peak load which is appreciably larger than the nominal or design load. For automotive transmission gears, there are rarely such operational conditions that would create momentary load spikes. If the maximum engine torque is used to calculate the gear forces, as mentioned previously, the values of C_a and K_a can be taken as one.

Size factorsC_s and K_s: The size factor accounts for the non‐uniformity of material property in gears that have large dimensions. Transmission gears are normally sized and the size factor is set to be one.

Load distribution factorsC_m and K_m: The load distribution factor reflects the variations of load distribution over the tooth length caused by manufacturing errors, bearing and shaft misalignments, and deflections. Following the AGMA guidelines, the values of the load distribution factors, C_m and K_m, are selected to be .

Life factorsC_L and K_L: The allowable contact and bending stresses, s_ac and s_at, are established in the AGMA standard for 10⁷ tooth load cycles with 99% reliability. The life factor is used to modify the allowable stresses if the designed life of gears differs from 10⁷ tooth load cycles. The load cycles of the final drive ring gear are in the order of 10⁸, corresponding to roughly 120,000 miles or 190,000 kilometers. The load cycles of the final drive pinion can then be approximated as (10⁸). For the gears on the input shaft in the FWD MT shown in Figure 3.19a, the load cycles can be calculated based on the percentage of total mileage each gear contributes. This contribution percentage can be estimated as: 1st gear (1%), 2nd gear (3%), 3rd gear (8%), 4th gear (28%), and 5th gear (60%). The load cycles of the gears on the output shaft are the multiplication of the contribution percentage and the final drive pinion load cycles. The load cycles of the gears on the input shaft are then determined by multiplying the respective gear ratio and the load cycles of the gear on the output shaft. For the RWD MT in Figure 3.19b, the load cycles of the gears on the counter shaft for each gear can be determined in similar fashion, i.e. by the multiplication of the respective contribution percentage and the final drive pinion load cycle. Note that the load cycles for the input gear N_i in Figure 3.19b should be calculated as the multiplication of the ratio and the sum of load cycles of the 1st, 2nd, 3rd, and 5th gears. After the load cycles are determined, the life factors can then be found from the AGMA standard plots to modify the allowable stresses.

Temperature factorsC_T and K_T: In general, automotive transmission gears operate at temperatures below 250 °F. The temperature factors are taken as unity.

Reliability factorsC_R and K_R: As mentioned previously, the allowable stresses are established for 10⁷ tooth load cycles as 99% reliability. This means that statistically there is one failure out of 100 after 10⁷ tooth load cycles. This reliability is not sufficient for automotive transmission gears. As recommended by the ASMG standard, the values of C_R and K_R can be taken as 1.25, or 1.5 for higher reliability, corresponding to one failure in a thousand or one failure in ten thousand after 10⁷ tooth load cycles.

The factors described above are common to both the pitting resistance and bending resistance. Two additional modification factors – the surface condition factor C_f and the hardness ratio factor C_H – are used for the formulation of pitting resistance. One additional factor, the rim thickness factor K_B, is used for the formulation of bending strength. These factors are explained in the following specifically for transmission gears.

Surface condition factorC_f and hardness ratio factorC_H: The surface condition factor reflects the influence of surface finish and residual stress of the tooth surface on pitting resistance. For automotive transmission gears, C_f can be taken as 1.0 because these gears are normally sized and have sufficient accuracy in tooth surface finish. The hardness ratio factor C_H depends on the gear ratio and the hardness of the pinion and the gear. In the AGMA standard [11], this factor is plotted against the gear ratio according to the pinion and gear tooth hardness ratios. For transmission gears, the hardness ratio factor C_H can also be taken as unity since the gear ratio is usually smaller than 4.0 and there is no significant difference in the tooth hardness between the pinion and the gear.

Rim thickness factorK_B: This factor applies to ring gears where gear teeth are manufactured on a thin ring, such as the ring gear of the final drive in the FDW MT in Figure 3.19a. The ring gear of the final drive is bolted to the differential carrier and must be designed with sufficient strength and rigidity to support the gear teeth on it. The rim thickness factor K_B is therefore set as unity for transmission gears.

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Example

A manual gear box for a RWD vehicle is shown in the figure. The normal diametral pitch, the normal pressure angle and the helical angle of the gears labeled as Ni, Nc, N1c, N1o, N2c, and N2o are given as: 8, 16, and 20° respectively. Gears Ni and Nc are standard helical gears. The numbers of teeth of gears for the first and second speeds are given with the drawing. All gears on the counter shaft are right‐handed. Calculate the following for the gear design:

1. The shaft distance between the input shaft and the counter shaft.
2. The gears N10 and N1c have a tooth length of 0.95 inch. The two gears are designed as non‐standard gears using the long‐short addendum system. It is required that the transverse tooth thickness of gear N1c be increased by 10% compared with the standard tooth thickness. Determine the cutter displacements for the two gears and the contact ratio.
3. Determine the operating pressure angle and the sum of the cutter displacements for gears N2c and N2o.
4. If the engine torque is 200 ft.lb, determine the gear forces on the counter shaft in first gear. Show the directions of gear force components at the mesh positions. Also, find the thrust force applied to the counter shaft bearing.

Solution:

1. Shaft distance:
   
   
2. , so long‐short addendum system can be designed. Standard tooth thickness:
   

   Since ,

   
   
   
   

   Pitch radii:

   

   Pitch radii:

   

   Addendum radii:

   
   

   Contact ratio:

   
   
   
3. Gears N_2c and N_2o must be designed as general non‐standard gears to accommodate the shaft distance determined in (a).
   
   
   
   
   
   
4. Gear forces: For gears N_i and

   Contact Force at the mesh of gears N_i and N_c:

   

   Gear force components on gear N_c:

   
   
   

   For gears N_i and

   Contact Force at the mesh of gears N_i and N_c:

   

   Gear force components on gear N_1c:

   
   
   

   Resultant thrust on the counter shaft:

   Directions of gear forces on the counter shaft:

   

3.8 Planetary Gear Trains

Gears can be designed in series where all gear axes are fixed with respect to each other or in systems where one or more gear axes rotate with respect to one another. Such systems are called planetary gear trains (PGT). There are many PGT types categorized in terms of their structure or configuration. Two PGT types are widely applied in automotive automatic transmissions: simple PGT and Ravigneaux PGT. Simple PGTs can be designed with one planet gear that meshes with the sun gear and the ring gear, as shown in Figure 3.20, or two planet gears in series as shown in Figure 3.21.

Figure 3.20 Simple planetary gear train.

Figure 3.21 Dual‐planet simple planetary gear train.

In the simple PGT shown in Figure 3.20 there are four elements: sun gear, planet gear, ring gear, and carrier. The number of planet gears does not change the kinematics, but multiple planet gears, usually four, are used in automatic transmissions for load sharing. The sun gear and the ring gear rotate about the same axis. A planet gear participates in two rotations, one about its own axis and the other about the axis of the sun gear. In other words, the planet gear rotates about its axis which rotates about the sun. The angular velocities of the sun gear ω_S, the carrier ω_C, and the ring gear ω_R of the simple PGT in Figure 3.20 are governed by the so‐called characteristic equation as follows:

(3.96)

where β is the planetary parameter that is equal to the ratio between the numbers of teeth in the ring gear and in the sun gear (). As can be observed from Eq. (3.96), the simple planetary train has two degrees of freedom, since only the characteristic equation constrains the three angular velocities. If one of the three angular velocities is given as input, the other two are not defined unless an additional constraint is imposed. In automatic transmissions designed with planetary gear trains, the additional constraints are from hydraulically actuated clutches.

In the dual‐planet PGT shown in Figure 3.21, the carrier supports two planet gears in series, and the mesh path is sun gear – planet gear 1 – planet gear 2 – ring gear. Each planet gear group, planet 1 or planet 2, contains multiple planet gears for load sharing in automatic transmission applications. The characteristic equation governing the three angular velocities differs from Eq. (3.96) only by the sign before the planetary parameter β:

(3.97)

The structure of a Ravigneaux planetary gear train is shown in Figure 3.22. A Ravigneaux PGT consists of sun gear 1, sun gear 2, long planet gear, short planet gear, carrier, and ring gear. In terms of kinematics, a Ravigneaux PGT can be considered as the combination of a simple planetary PGT and a dual‐planet PGT with a shared carrier and a shared ring gear. There are two mesh paths in the Ravigneaux PGT: sun gear 1 – long planet gear – ring gear; and sun gear 2 – short planet gear – long planet gear – ring gear. Here, Eq. (3.96) applies to the first mesh path and Eq. (3.97) applies to the second mesh path. The characteristic equations that govern the angular velocities of a Ravigneaux PGT are:

(3.98)

(3.99)

where ω_S1 and ω_S2 are the angular velocities of sun gear 1 and sun gear 2 respectively. ω_C and ω_R are the angular velocities of the carrier and the ring gear, β₁ is the ratio between the number of teeth in the ring gear and the number of teeth the sun gear 1 , and β₂ is the ratio between the number of teeth in the ring gear and the number of teeth in the sun gear 2 . It can be observed from Eqs (3.98) and (3.99) that the Ravigneaux PGT also has two degrees of freedom. The four angular velocities, ω_S1, ω_S2, ω_C, and ω_R, are constrained by two equations. When one angular velocity is given as the input, there must be an additional constraint for the motions to be defined. The Ravigneaux PGT offers a compact design in automatic transmissions since it provides the functionality of two simple PGTs.

Figure 3.22 Ravigneaux planetary gear train.

When a planetary gear train is loaded in the transmission of power, members in the PGT are subjected to externally or internally applied torque. In transmission applications, external torque can be applied to the sun gear, carrier, or ring gear through the input, reaction, or output, as will be detailed in Chapter 5. The internal torque is applied on the other PGT members. Similar to the angular velocities, the torque magnitudes on PGT members are related to the planetary parameter, and the direction of the internal torque on each member depends on the PGT type.

3.8.1 Simple Planetary Gear Train

The directions of the internal torques on the sun gear, the carrier, and the ring gear of a simple PGT are shown in Figure 3.23a. Note that only the relative directions are shown in the figure for the internal torques. For the simple PGT, the internal torques on the sun gear and on the ring gear have the same direction, but the internal torque on the carrier is in the opposite direction, as shown in Figure 3.23a. The internal torque on the sun gear with a magnitude of T_S is applied to the sun gear via the mesh between the planet gear and the sun gear. The internal torque on the ring gear with a magnitude of T_R is applied to the ring gear via the mesh between the planet gear and the ring gear. The magnitude of the torque on the carrier is the algebraic sum of the torques on the sun gear and on the ring gear. For convenience and clarity the carrier and the planet can be considered as one body in drawing the torque diagrams for automatic transmissions that involve multiple planetary gear trains. The torque on the sun gear and the torque on the carrier–planet assembly are action and reaction, and so have the same magnitude but opposite directions. Similarly, the torque on the ring gear and the torque on the carrier‐planet assembly are also action and reaction, as shown in Figure 3.23a. The magnitudes of the torque on the sun gear, the torque on the ring gear, and the torque on the carrier are related by the following equations:

(3.100)

(3.101)

Figure 3.23 Internal torque directions for simple and dual‐planet PGTs.

3.8.2 Dual‐Planet Planetary Gear Train

The directions of the internal torques on the sun gear, the carrier, and the ring gear of a dual‐planet PGT are shown in Figure 3.23b. For the dual‐planet PGT, the internal torques on the sun gear and on the ring gear have the opposite direction. The internal torque on the sun gear with a magnitude of T_S is applied to the sun gear via the mesh between the planet gear 1 and the sun gear. The internal torque on the ring gear, with a magnitude of T_R, is applied to the ring gear via the mesh between the planet gear 2 and the ring gear. The magnitude of the torque on the carrier is the algebraic sum of the torques on the sun gear and the ring gear. The magnitudes of the torques on the sun gear, the ring gear, and the carrier are related by the following:

(3.102)

(3.103)

Since the planetary parameter β is always larger than one, the direction of the internal torque on the carrier is always the same as the direction of the torque on the sun gear in a dual‐planet PGT, and the magnitude of the ring gear torque is the sum of the sun gear torque and the carrier torque.

3.8.3 Ravigneaux Planetary Gear Train

The directions of the internal torques on the members of a Ravigneaux PGT are shown in Figure 3.24. Note that the Ravigneaux PGT is treated as the combination of a simple PGT and a dual‐planet PGT. The torque magnitudes for a Ravigneaux PGT are related by the following:

(3.104)

(3.105)

(3.106)

(3.107)

Figure 3.24 Internal torque directions for Ravigneaux PGTs.

In automatic transmissions, there are multiple PGTs with members structurally interconnected. Equations (3.96–3.99) are used to constrain the angular velocities and accelerations of PGT members and other drive train components in the powertrain system. Equations (3.100–107) are used to relate the transmission input, output torque, and clutch torques, as detailed in Chapter 5.

References

1. 1 Gear – Wikipedia https://en.wikipedia.org/wiki/Gear.
2. 2 Involute – Wikipedia https://en.wikipedia.org/wiki/Involute.
3. 3 Buckingham, E.: Analytical Mechanics of Gears, Dover Publications, 1988, ISBN 13‐9780486657127.
4. 4 Litvin, F.L. and Fuentes, A.: Gear Geometry and Applied Theory, Second Edition, Cambridge University Press, 2004, ISBN 0‐521‐81517 7.
5. 5 Radzevich, S.P.: Dudley’s Handbook of Practical Gear Design and Manufacture, 3rd Edition, CRC Press, ISBN 13‐978‐1498753104.
6. 6 AGMA 933‐B03: Basic Gear Geometry.
7. 7 AGMA 913‐A98: Method for Specifying the Geometry of Spur and Helical Gears.
8. 8 American Gear Manufacturers Association: Design Manual for Bevel Gears, ANSI/AGMA 2005‐B88, Ma9 1988, ISBN 1‐55589‐496‐8.
9. 9 AGMA 2000‐A88: Gear Classification and Inspection Handbook.
10. 10 AGMA 2002‐B88: Tooth Thickness Specification and Measurement.
11. 11 AGMA 2001‐D04: Fundamental Rating factors and Calculation Methods for Involute Spur and Helical Gear Teeth.
12. 12 AGMA 918‐A93: A summary of Numerical Examples Demonstrating the Procedures for Calculating Geometry Factors for Spur and Helical Gears.
13. 13 AGMA 908‐B89: Information Sheet – Geometry Factors for Determining the Pitting Resistance and Bending Strength for Spur, Helical and Herringbone Gear Teeth, Parts 1 & 2.
14. 14 American Gear Manufacturers Association: Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur, Helical and Herringbone Gear Teeth, AGMA 908‐B89, April 1989, ISBN 1‐55589‐525‐5.
15. 15 AGMA 6002‐B93: Design Guide for Vehicle Spur and Helical Gear Drives.

Problems

1. A six‐speed RWD MT is shown in the figure. The following data are given in the gear design:  Input gears: Ni = 26, Nc = 29, normal diametral pitch: 12, normal pressure angle: 20°; helical angle: 25°, tooth length: 0.95 inch.

    First speed: N10/N1c = 41/13, normal diametral pitch: 12, normal pressure angle: 20°, helical angle: 25°.

    Sixth speed: N60/N6c = 21/34, normal diametral pitch: 12, normal pressure angle: 20°, helical angle: 25°.

   

   All the gears on the counter shaft are right handed. The transmission efficiency is 0.98 and the final drive efficiency is 0.97.

   1. The input gears Ni and Nc are designed as standard gears. Determine the contact ratio.
   2. The tooth thickness of gear N1c is designed to be 20% larger than the standard tooth thickness. Determine the tooth thickness of gears N1c and N10.
   3. The vehicle is running in 6th gear at a speed of 65 miles on a 1% slope and with an acceleration of 2 ft/s². Determine the gear forces on the counter shafts, and the resultant thrust on the bearing on the counter shaft. You must show the direction of each gear force and the thrust.
   4. An optional 6th gear ratio of 0.7215 is to be made available for the transmission by redesigning the six gears N60 and N6c only. The redesigned 6th gears must be standard helical gears with normal diametral pitch equal to 12 and normal pressure angle equal to 20°. The number of teeth N6c must be kept the same at 34. Determine the helical angle of the resigned 6th gears.

   Vehicle data:

   Front axle weight: 1900 lb	Rear axle weight: 1500 lb
   Center of gravity height: 14 in.	Wheel base: 108 in.
   Air drag coefficient: 0.30	Frontal projected area: 21 sq. ft
   Tire radius: 11.0 in	Roll resistance coefficient: 0.02
   Mass factor: 1.0	Final drive ratio 3.21

1. Prob. 2: A six‐speed FWD MT is shown in the figure. The gears for the 1st speed, the 6th speed, and the final drive are standard helical gears. The following data are given in gear design: First speed: gear ratio: 3.3846, number of teeth of gear N_1i: 13, normal diametral pitch: 12, normal pressure angle: 20°

   Sixth speed: numbers of teeth: N6i/N60 = 36/23, normal diametral pitch: 12, normal pressure angle: 20°, helical angle: 25°, tooth length: 0.8 in.

   Final drive: numbers of teeth: 59/18, normal diametral pitch: 10, normal pressure angle: 20°; helical angle: 21°

   All gears on the output shaft are left handed. The overall drive line efficiency from transmission input to final drive output is .96 and the final drive efficiency is .97.

   1. Determine the contact ratio of the 6th gears.
   2. Determine the pitch radii for the gears of the 1st speed.
   3. The vehicle is running in 6th gear at a speed of 65 mph on a 1% slope and with an acceleration of 2 ft/s². Determine the gear forces on the output shafts and the resultant thrust on the bearing on the output shaft. You must show the direction of each gear force and the thrust.
   4. An optional final drive is fitted into the same transmission housing. The numbers of teeth of the optional final drive gears are: 59/16, the normal diametral pitch is 10, the normal pressure angle is 20°, and the helical angle is 21°. The tooth thickness of the pinion (the gear with 16 teeth) is to be designed 40% thicker than the standard tooth; determine the cutter displacements for the two gears of the final drive.

   Vehicle data:

   Front axle weight: 1900 lb	Rear axle weight: 1500 lb
   Center of gravity height: 14 in.	Wheel base: 108 in.
   Air drag coefficient: 0.30	Frontal projected area: 21 sq. ft
   Tire radius: 11.0 in.	Roll resistance coefficient: 0.02
   Mass factor: 1.0