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Gear Types and Axial Arrangements; Details of Involute Gearing, Elements of Metric Gear Technology (Cont.)

Table 2-2 Table 2-2
Table 2-2 cont. Table 2-2
Table 2-3 Table 2-3
Metric Module and Inch Gear Preferences: Because there is no direct equivalence between the pitches in metric and inch systems, it is not possible to make direct substitutions. Further, there are preferred modules in the metric system. As an aid in using metric gears, Table 2-2, Table 2-2 continued, presents nearest equivalents for both systems, with the preferred sizes in bold type.

2.7 Gear Types And Axial Arrangements

In accordance with the orientation of axes, there are three categories of gears:
  1. Parallel Axes Gears
  2. Intersecting Axes Gears
  3. Nonparallel and Nonintersecting Axes Gears
Spur and helical gears are the parallel axes gears. Bevel gears are the intersecting axes gears. Screw or crossed helical, worm and hypoid gears handle the third category. Table 2-3 lists the gear types per axes orientation.

Also, included in Table 2-3 is the theoretical efficiency range of the various gear types. These figures do not include bearing and lubricant losses. Also, they assume ideal mounting in regard to axis orientation and center distance. Inclusion of these realistic considerations will downgrade the efficiency numbers.

2.7.1 Parallel Axes Gears

Figure 2-8: Spur Gear1. Spur Gear, Figure 2-8
This is a cylindrical shaped gear in which the teeth are parallel to the axis. It has the largest applications and, also, it is the easiest to manufacture.

Figure 2-9: Spur Rack2. Spur Rack, Figure 2-9
This is a linear shaped gear which can mesh with a spur gear with any number of teeth. The spur rack is a portion of a spur gear with an infinite radius.

Figure 2-10: Internal Gear and Spur Gear3. Internal Gear, Figure 2-10
This is a cylindrical shaped gear but with the teeth inside the circular ring. It can mesh with a spur gear. Internal gears are often used in planetary gear systems.

Figure 2-11: Helical Gear4. Helical Gear, Figure 2-11
This is a cylindrical shaped gear with helicoid teeth. Helical gears can bear more load than spur gears, and work more quietly. They are widely used in industry. A disadvantage is the axial thrust force the helix form causes.

Figure 2-12: Helical Rack5. Helical Rack, Figure 2-12
This is a linear shaped gear which meshes with a helical gear. Again, it can be regarded as a portion of a helical gear with infinite radius.

Figure 2-13: Double Helical Gear6. Double Helical Gear, Figure 2-13
This is a gear with both left-hand and right-hand helical teeth. The double helical form balances the inherent thrust forces.

2.7.2 Intersecting Axes Gears

Figure 2-14: Straight Bevel Gear1. Straight Bevel Gear, Figure 2-14
This is a gear in which the teeth have tapered conical elements that have the same direction as the pitch cone base line (generatrix). The straight bevel gear is both the simplest to produce and the most widely applied in the bevel gear family.

Figure 2-15: Spiral Bevel Gear2. Spiral Bevel Gear, Figure 2-15
This is a bevel gear with a helical angle of spiral teeth. It is much more complex to manufacture, but offers a higher strength and lower noise.

Figure 2-16: Zerol Gear3. Zerol Gear, Figure 2-16
Zerol gear is a special case of spiral bevel gear. It is a spiral bevel with zero degree of spiral angle tooth advance. It has the characteristics of both the straight and spiral bevel gears. The forces acting upon the tooth are the same as for a straight bevel gear.

2.7.3 Nonparallel And Nonintersecting Axes Gears

Figure 2-17: Worm Gear1. Worm And Worm Gear, Figure 2-17
Worm set is the name for a meshed worm and worm gear. The worm resembles a screw thread; and the mating worm gear a helical gear, except that it is made to envelope the worm as seen along the worm's axis. The outstanding feature is that the worm offers a very large gear ratio in a single mesh. However, transmission efficiency is very poor due to a great amount of sliding as the worm tooth engages with its mating worm gear tooth and forces rotation by pushing and sliding. With proper choices of materials and lubrication, wear can be contained and noise is reduced.

Figure 2-18: Screw Gear2. Screw Gear (Crossed Helical Gear), Figure 2-18
Two helical gears of opposite helix angle will mesh if their axes are crossed. As separate gear components, they are merely conventional helical gears. Installation on crossed axes converts them to screw gears. They offer a simple means of gearing skew axes at any angle. Because they have point contact, their load carrying capacity is very limited.

2.7.4 Other Special Gears

Figure 2-19: Face Gear1. Face Gear, Figure 2-19
This is a pseudobevel gear that is limited to 90° intersecting axes. The face gear is a circular disc with a ring of teeth cut in its side face; hence the name face gear. Tooth elements are tapered towards its center. The mate is an ordinary spur gear. It offers no advantages over the standard bevel gear, except that it can be fabricated on an ordinary shaper gear generating machine.

Figure 2-20: Double Enveloping Worm Gear2. Double Enveloping Worm Gear, Figure 2-20
This worm set uses a special worm shape in that it partially envelops the worm gear as viewed in the direction of the worm gear axis. Its big advantage over the standard worm is much higher load capacity. However, the worm gear is very complicated to design and produce, and sources for manufacture are few.

Figure 2-21: Hypoid Gear3. Hypoid Gear, Figure 2-21
This is a deviation from a bevel gear that originated as a special development for the automobile industry. This permitted the drive to the rear axle to be nonintersecting, and thus allowed the auto body to be lowered. It looks very much like the spiral bevel gear. However, it is complicated to design and is the most difficult to produce on a bevel gear generator.


3.1 Pressure Angle

Figure 2-3 Figure 2-3
Figure 2-4 Figure 2-4
Figure 3-1 Figure 3-1
The pressure angle is defined as the angle between the line-of-action (common tangent to the base circles in Figures 2-3 and 2-4) and a perpendicular to the line-of-centers. See Figure 3-1. From the geometry of these figures, it is obvious that the pressure angle varies (slightly) as the center distance of a gear pair is altered. The base circle is related to the pressure angle and pitch diameter by the equation:
Equation 3-1

where d and α are the standard values, or alternately: Equation 3-2

where d' and α' are the exact operating values.

The basic formula shows that the larger the pressure angle the smaller the base circle. Thus, for standard gears, 14.5° pressure angle gears have base circles much nearer to the roots of teeth than 20° gears. It is for this reason that 14.5° gears encounter greater undercutting problems than 20° gears. This is further elaborated on in SECTION 4.3.

Figure 3-2 Figure 3-2

3.2 Proper Meshing And Contact Ratio

Figure 3-2 shows a pair of standard gears meshing together. The contact point of the two involutes, as Figure 3-2 shows, slides along the common tangent of the two base circles as rotation occurs. The common tangent is called the line-of-contact, or line-of-action.

A pair of gears can only mesh correctly if the pitches and the pressure angles are the same.
Pitch comparison can be module (m), circular (p), or base Pb. That the pressure angles must be identical becomes obvious from the following equation for base pitch:
Equation 3-3

Thus, if the pressure angles are different, the base pitches cannot be identical.
The length of the line-of-action is shown as ab in Figure 3-2.

Figure 3-3 Figure 3-3

3.2.1 Contact Ratio

To assure smooth continuous tooth action, as one pair of teeth ceases contact a succeeding pair of teeth must already have come into engagement. It is desirable to have as much overlap as possible. The measure of this overlapping is the contact ratio. This is a ratio of the length of the line-of-action to the base pitch. Figure 3-3 shows the geometry. The length-of-action is determined from the intersection of the line-of-action and the outside radii. For the simple case of a pair of spur gears, the ratio of the length-of-action to the base pitch is determined from:

It is good practice to maintain a contact ratio of 1.2 or greater. Under no circumstances should the ratio drop below 1.1, calculated for all tolerances at their worst-case values.

A contact ratio between 1 and 2 means that part of the time two pairs of teeth are in contact and during the remaining time one pair is in contact. A ratio between 2 and 3 means 2 or 3 pairs of teeth are always in contact. Such a high contact ratio generally is not obtained with external spur gears, but can be developed in the meshing of an internal and external spur gear pair or specially designed nonstandard external spur gears.

More detail is presented about contact ratio, including calculation equations for specific gear types, in SECTION 11.

Equation 3-4

Figure 3-4 Figure 3-4

3.3 The Involute Function

Figure 3-4 shows an element of involute curve. The definition of involute curve is the curve traced by a point on a straight line which rolls without slipping on the circle.

The circle is called the base circle of the involutes. Two opposite hand involute curves meeting at a cusp form a gear tooth curve. We can see, from Figure 3-4, the length of base circle arc ac equals the length of straight line bc.

Equation 3-5

The θ in Figure 3-4 can be expressed as inv α + α, then Formula (3-5) will become:

Equation 3-6

Function of α, or inv α, is known as involute function. Involute function is very important in gear design. Involute function values can be obtained from appropriate tables. With the center of the base circle O at the origin of a coordinate system, the involute curve can be expressed by values of x and y as follows:
Equation 3-7

Figure 4-1 Figure 4-1
Table 4-1 Table 4-1
Table 4-2 Table 4-2


4.1 Standard Spur Gear

Figure 4-1 shows the meshing of standard spur gears. The meshing of standard spur gears means pitch circles of two gears contact and roll with each other. The calculation formulas are in Table 4-1.

All calculated values in Table 4-1 are based upon given module (m) and number of teeth (z1 and z2 ). If instead module (m), center distance (a) and speed ratio (i ) are given, then the number of teeth, z1 and z2, would be calculated with the formulas as shown in Table 4-2.

Note that the numbers of teeth probably will not be integer values by calculation with the formulas in Table 4-2. Then it is incumbent upon the designer to choose a set of integer numbers of teeth that are as close as possible to the theoretical values. This will likely result in both slightly changed gear ratio and center distance. Should the center distance be inviolable, it will then be necessary to resort to profile shifting. This will be discussed later in this section.

Figure 4-2 Figure 4-2

4.2 The Generating Of A Spur Gear

Involute gears can be readily generated by rack type cutters. The hob is in effect a rack cutter. Gear generation is also accomplished with gear type cutters using a shaper or planer machine.

Figure 4-2 illustrates how an involute gear tooth profile is generated. It shows how the pitch line of a rack cutter rolling on a pitch circle generates a spur gear.

4.3 Undercutting

Figure 4-3 Figure 4-3
Figure 4-4 Figure 4-4
From Figure 4-3, it can be seen that the maximum length of the line-of-contact is limited to the length of the common tangent. Any tooth addendum that extends beyond the tangent points (T and T') is not only useless, but interferes with the root fillet area of the mating tooth. This results in the typical undercut tooth, shown in Figure 4-4. The undercut not only weakens the tooth with a wasp-like waist, but also removes some of the useful involute adjacent to the base circle.

From the geometry of the limiting length-of-contact (T-T', Figure 4-3), it is evident that interference is first encountered by the addenda of the gear teeth digging into the mating-pinion tooth flanks. Since addenda are standardized by a fixed value (ha = m), the interference condition becomes more severe as the number of teeth on the mating gear increases. The limit is reached when the gear becomes a rack. This is a realistic case since the hob is a rack-type cutter. The result is that standard gears with teeth numbers below a critical value are automatically undercut in the generating process. The condition for no undercutting in a standard spur gear is given by the expression:
Equation 4-1

This indicates that the minimum number of teeth free of undercutting decreases with increasing pressure angle. For 14.5° the value of zc is 32, and for 20° it is 18. Thus, 20° pressure angle gears with low numbers of teeth have the advantage of much less undercutting and, therefore, are both stronger and smoother acting.

» Enlarged Pinions - Continued on page 3