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Calculations of Internal Gears and The Fundamentals of Helical Gears, Elements of Metric Gear Technology (Cont.)

Figure 4-5 Figure 4-5
Figure 4-6 Figure 4-6
Figure 4-2 Figure 4-2

4.4 Enlarged Pinions

Undercutting of pinion teeth is undesirable because of losses of strength, contact ratio and smoothness of action. The severity of these faults depends upon how far below zc the teeth number is. Undercutting for the first few numbers is small and in many applications its adverse effects can be neglected.

For very small numbers of teeth, such as ten and smaller, and for high-precision applications, undercutting should be avoided. This is achieved by pinion enlargement (or correction as often termed), wherein the pinion teeth, still generated with a standard cutter, are shifted radially outward to form a full involute tooth free of undercut. The tooth is enlarged both radially and circumferentially. Comparison of a tooth form before and after enlargement is shown in Figure 4-5.

Figure 4-7 Figure 4-7

4.5 Profile Shifting

As Figure 4-2 shows, a gear with 20 degrees of pressure angle and 10 teeth will have a huge undercut volume. To prevent undercut, a positive correction must be introduced. A positive correction, as in Figure 4-6, can prevent undercut.

Undercutting will get worse if a negative correction is applied. See Figure 4-7. The extra feed of gear cutter (xm) in Figures 4-6 and 4-7 is the amount of shift or correction. And x is the shift coefficient.

The condition to prevent undercut in a spur gear is:
Equation 4-2
The number of teeth without undercut will be:
Equation 4-3
The coefficient without undercut is:
Equation 4-4

Table 4-3 Table 4-3
Figure 4-8 Figure 4-8
Profile shift is not merely used to prevent undercut. It can be used to adjust center distance between two gears.

If a positive correction is applied, such as to prevent undercut in a pinion, the tooth thickness at top is thinner.

Table 4-3 presents the calculation of top land thickness.

4.6 Profile Shifted Spur Gear

Figure 4-8 shows the meshing of a pair of profile shifted gears. The key items in profile shifted gears are the operating (working) pitch diameters (dw) and the working (operating) pressure angle (αw). These values are obtainable from the operating (or i.e., actual) center distance and the following formulas:
Equation 4-5

Table 4-4 Table 4-4
Table 4-5 Table 4-5
In the meshing of profile shifted gears, it is the operating pitch circles that are in contact and roll on each other that portrays gear action. The standard pitch circles no longer are of significance; and the operating pressure angle is what matters.

A standard spur gear is, according to Table 4-4, a profile shifted gear with 0 coefficient of shift; that is, x1 = x2 = 0. Table 4-5 is the inverse formula of items from 4 to 8 of Table 4-4.

There are several theories concerning how to distribute the sum of coefficient of profile shift, (x1 + x2) into pinion, (x1) and gear, (x2) separately. BSS (British) and DIN (German) standards are the most often used. In the example above, the 12 tooth pinion was given sufficient correction to prevent undercut, and the residual profile shift was given to the mating gear.

Table 4-6 Table 4-6

4.7 Rack And Spur Gear

Table 4-6 presents the method for calculating the mesh of a rack and spur gear. Figure 4-9a shows the pitch circle of a standard gear and the pitch line of the rack.

One rotation of the spur gear will displace the rack (l) one circumferential length of the gear's pitch circle, per the formula:
Equation 4-6

Figure 4-9a Figure 4-9a
Figure 4-9b Figure 4-9b
Figure 5-1 Figure 5-1
Figure 4-9b shows a profile shifted spur gear, with positive correction xm, meshed with a rack. The spur gear has a larger pitch radius than standard, by the amount xm. Also, the pitch line of the rack has shifted outward by the amount xm. Table 4-6 presents the calculation of a meshed profile shifted spur gear and rack. If the correction factor x1 is 0, then it is the case of a standard gear meshed with the rack. The rack displacement, l, is not changed in any way by the profile shifting. Equation (4-6) remains applicable for any amount of profile shift.


5.1 Internal Gear Calculations

Calculation of a Profile Shifted Internal Gear
Figure 5-1 presents the mesh of an internal gear and external gear. Of vital importance is the operating (working) pitch diameters, dw, and operating (working) pressure angle, αw. They can be derived from center distance, ax, and Equations (5-1). Equation 5-1

Table 5-1 Table 5-1
Table 5-2 Table 5-2
Table 5-1 shows the calculation steps.
It will become a standard gear calculation if x1 = x2 = 0.

If the center distance, ax, is given, x1 and x2 would be obtained from the inverse calculation from item 4 to item 8 of Table 5-1. These inverse formulas are in Table 5-2.

Pinion cutters are often used in cutting internal gears and external gears. The actual value of tooth depth and root diameter, after cutting, will be slightly different from the calculation. That is because the cutter has a coefficient of shifted profile. In order to get a correct tooth profile, the coefficient of cutter should be taken into consideration.

5.2 Interference In Internal Gears

Three different types of interference can occur with internal gears:

(a) Involute Interference
(b) Trochoid Interference
(c) Trimming Interference

(a) Involute Interference
This occurs between the dedendum of the external gear and the addendum of the internal gear. It is prevalent when the number of teeth of the external gear is small. Involute interference can be avoided by the conditions cited below:
Equation 5-2

where αa2 is the pressure angle seen at a tip of the internal gear tooth.
Equation 5-3

and αw is working pressure angle:
Equation 5-4

Equation (5-3) is true only if the outside diameter of the internal gear is bigger than the base circle:
Equation 5-5

For a standard internal gear, where α = 20°, Equation (5-5) is valid only if the number of teeth is z2 > 34.

(b) Trochoid Interference

This refers to an interference occurring at the addendum of the external gear and the dedendum of the internal gear during recess tooth action. It tends to happen when the difference between the numbers of teeth of the two gears is small. Equation (5-6) presents the condition for avoiding trochoidal interference.
Equation 5-6

Equation 5-7

where αa1 is the pressure angle of the spur gear tooth tip:
Equation 5-8

In the meshing of an external gear and a standard internal gear α = 20°, trochoid interference is avoided if the difference of the number of teeth, z1z2, is larger than 9.

(c) Trimming Interference

This occurs in the radial direction in that it prevents pulling the gears apart. Thus, the mesh must be assembled by sliding the gears together with an axial motion. It tends to happen when the numbers of teeth of the two gears are very close. Equation (5-9) indicates how to prevent this type of interference.
Equation 5-9
Equation 5-10

Table 5-3a Table 5-3a
Table 5-3b Table 5-3b
Table 5-4 Table 5-4
This type of interference can occur in the process of cutting an internal gear with a pinion cutter. Should that happen, there is danger of breaking the tooling. Table 5-3a shows the limit for the pinion cutter to prevent trimming interference when cutting a standard internal gear, with pressure angle 20°, and no profile shift, i.e., xc = 0.

There will be an involute interference between the internal gear and the pinion cutter if the number of teeth of the pinion cutter ranges from 15 to 22 (zc = 15 to 22). Table 5-3b shows the limit for a profile shifted pinion cutter to prevent trimming interference while cutting a standard internal gear. The correction, xc, is the magnitude of shift which was assumed to be: xc = 0.0075 zc + 0.05.

There will be an involute interference between the internal gear and the pinion cutter if the number of teeth of the pinion cutter ranges from 15 to 19 (zc = 15 to 19).

5.3 Internal Gear With Small Differences In Numbers Of Teeth
In the meshing of an internal gear and an external gear, if the difference in numbers of teeth of two gears is quite small, a profile shifted gear could prevent the interference. Table 5-4 is an example of how to prevent interference under the conditions of z2 = 50 and the difference of numbers of teeth of two gears ranges from 1 to 8.

All combinations above will not cause involute interference or trochoid interference, but trimming interference is still there. In order to assemble successfully, the external gear should be assembled by inserting in the axial direction.

A profile shifted internal gear and external gear, in which the difference of numbers of teeth is small, belong to the field of hypocyclic mechanism, which can produce a large reduction ratio in one step, such as 1/100.

Equation 5-11

Figure 5-2
In Figure 5-2 the gear train has a difference of numbers of teeth of only 1; z1 = 30 and z2 = 31. This results in a reduction ratio of 1/30.


Figure 6-1The helical gear differs from the spur gear in that its teeth are twisted along a helical path in the axial direction. It resembles the spur gear in the plane of rotation, but in the axial direction it is as if there were a series of staggered spur gears. See Figure 6-1. This design brings forth a number of different features relative to the spur gear, two of the most important being as follows:

  1. Tooth strength is improved because of the elongated helical wraparound tooth base support
  2. Contact ratio is increased due to the axial tooth overlap. Helical gears thus tend to have greater load carrying capacity than spur gears of the same size. Spur gears, on the other hand, have a somewhat higher efficiency.
Helical gears are used in two forms:
  1. Parallel shaft applications, which is the largest usage.
  2. Crossed-helicals (also called spiral or screw gears) for connecting skew shafts, usually at right angles.
6.1 Generation Of The Helical Tooth
The helical tooth form is involute in the plane of rotation and can be developed in a manner similar to that of the spur gear. However, unlike the spur gear which can be viewed essentially as two dimensional, the helical gear must be portrayed in three dimensions to show changing axial features.

Figure 6-2 Figure 6-2
Figure 2-2 Figure 2-2
Referring to Figure 6-2, there is a base cylinder from which a taut plane is unwrapped, analogous to the unwinding taut string of the spur gear in Figure 2-2. On the plane there is a straight line AB, which when wrapped on the base cylinder has a helical trace AoBo. As the taut plane is unwrapped, any point on the line AB can be visualized as tracing an involute from the base cylinder. Thus, there is an infinite series of involutes generated by AB, all alike, but displaced in phase along a helix on the base cylinder.

Again, a concept analogous to the spur gear tooth development is to imagine the taut plane being wound from one base cylinder on to another as the base cylinders rotate in opposite directions. The result is the generation of a pair of conjugate helical involutes. If a reverse direction of rotation is assumed and a second tangent plane is arranged so that it crosses the first, a complete involute helicoid tooth is formed.

Figure 6-3 Figure 6-3
Figure 6-4 Figure 6-4
6.2 Fundamentals Of Helical Teeth
In the plane of rotation, the helical gear tooth is involute and all of the relationships governing spur gears apply to the helical. However, the axial twist of the teeth introduces a helix angle. Since the helix angle varies from the base of the tooth to the outside radius, the helix angle β is defined as the angle between the tangent to the helicoidal tooth at the intersection of the pitch cylinder and the tooth profile, and an element of the pitch cylinder. See Figure 6-3.

The direction of the helical twist is designated as either left or right. The direction is defined by the right-hand rule.

For helical gears, there are two related pitches – one in the plane of rotation and the other in a plane normal to the tooth. In addition, there is an axial pitch. Referring to Figure 6-4, the two circular pitches are defined and related as follows:
Equation 6-1
The normal circular pitch is less than the transverse radial pitch, pt, in the plane of rotation; the ratio between the two being equal to the cosine of the helix angle. Consistent with this, the normal module is less than the transverse (radial) module.

Figure 6-5 Figure 6-5
The axial pitch of a helical gear, px, is the distance between corresponding points of adjacent teeth measured parallel to the gear's axis – see Figure 6-5. Axial pitch is related to circular pitch by the expressions:
Equation 6-2

Figure 6-6 Figure 6-6
A helical gear such as shown in Figure 6-6 is a cylindrical gear in which the teeth flank are helicoid. The helix angle in standard pitch circle cylinder is β, and the displacement of one rotation is the lead, L.

The tooth profile of a helical gear is an involute curve from an axial view, or in the plane perpendicular to the axis. The helical gear has two kinds of tooth profiles – one is based on a normal system, the other is based on an axial system.

Circular pitch measured perpendicular to teeth is called normal circular pitch, pn. And pn divided by π is then a normal module, mn.
Equation 6-3

The tooth profile of a helical gear with applied normal module, mn, and normal pressure angle αn belongs to a normal system.

In the axial view, the circular pitch on the standard pitch circle is called the radial circular pitch, pt. And pt divided by π is the radial module, mt.
Equation 6-4

6.3 Equivalent Spur Gear
The true involute pitch and involute geometry of a helical gear is in the plane of rotation. However, in the normal plane, looking at one tooth, there is a resemblance to an involute tooth of a pitch corresponding to the normal pitch. However, the shape of the tooth corresponds to a spur gear of a larger number of teeth, the exact value depending on the magnitude of the helix angle.

Figure 6-7 Figure 6-7
The geometric basis of deriving the number of teeth in this equivalent tooth form spur gear is given in Figure 6-7. The result of the transposed geometry is an equivalent number of teeth, given as:

Equation 6-5

This equivalent number is also called a virtual number because this spur gear is imaginary. The value of this number is used in determining helical tooth strength.

6.4 Helical Gear Pressure Angle
Figure 6-8 Figure 6-8
Although, strictly speaking, pressure angle exists only for a gear pair, a nominal pressure angle can be considered for an individual gear. For the helical gear there is a normal pressure, αn , angle as well as the usual pressure angle in the plane of rotation, α. Figure 6-8 shows their relationship, which is expressed as:
Equation 6-6

6.5 Importance Of Normal Plane Geometry
Because of the nature of tooth generation with a rack-type hob, a single tool can generate helical gears at all helix angles as well as spur gears. However, this means the normal pitch is the common denominator, and usually is taken as a standard value. Since the true involute features are in the transverse plane, they will differ from the standard normal values. Hence, there is a real need for relating parameters in the two reference planes.

6.6 Helical Tooth Proportions
These follow the same standards as those for spur gears. Addendum, dedendum, whole depth and clearance are the same regardless of whether measured in the plane of rotation or the normal plane. Pressure angle and pitch are usually specified as standard values in the normal plane, but there are times when they are specified as standard in the transverse plane.

6.7 Parallel Shaft Helical Gear Meshes
Fundamental information for the design of gear meshes is as follows:

Helix angle – Both gears of a meshed pair must have the same helix angle. However, the helix direction must be opposite; i.e., a left-hand mates with a right-hand helix.

Pitch diameter – This is given by the same expression as for spur gears, but if the normal module is involved it is a function of the helix angle. The expressions are:
Equation 6-7

Center distance – Utilizing Equation (6-7), the center distance of a helical gear mesh is:
Equation 6-8

Note that for standard parameters in the normal plane, the center distance will not be a standard value compared to standard spur gears. Further, by manipulating the helix angle, β, the center distance can be adjusted over a wide range of values. Conversely, it is possible:

  1. to compensate for significant center distance changes (or errors) without changing the speed ratio between parallel geared shafts; and
  2. to alter the speed ratio between parallel geared shafts, without changing the center distance, by manipulating the helix angle along with the numbers of teeth
6.8 Helical Gear Contact Ratio
The contact ratio of helical gears is enhanced by the axial overlap of the teeth. Thus, the contact ratio is the sum of the transverse contact ratio, calculated in the same manner as for spur gears, and a term involving the axial pitch.
Equation 6-9

Details of contact ratio of helical gearing are given later in a general coverage of the subject; see SECTION 11.1.

6.9 Design Considerations

6.9.1 Involute Interference

Helical gears cut with standard normal pressure angles can have considerably higher pressure angles in the plane of rotation – see Equation (6-6) – depending on the helix angle.

Therefore, the minimum number of teeth without undercutting can be significantly reduced, and helical gears having very low numbers of teeth without undercutting are feasible.

6.9.2 Normal Vs. Radial Module (Pitch)

In the normal system, helical gears can be cut by the same gear hob if module mn and pressure angle αn are constant, no matter what the value of helix angle β.

It is not that simple in the radial system. The gear hob design must be altered in accordance with the changing of helix angle β, even when the module mt and the pressure angle αt are the same.

Obviously, the manufacturing of helical gears is easier with the normal system than with the radial system in the plane perpendicular to the axis.

» Helical Gear Calculations - Continued on page 4