Gear Tooth Design
3.1 Basic Requirements for Gear Tooth Design
3.1.3 Long- and Short-Addendum Gear Design
The previous section shows the geometric limitations on the amount that gears can be made long or short addendum. This section indicates cases in which long and short adden-dum should be considered.
Modification of the addendum of the pinion, and in most cases the gear number, is rec-ommended for gears serving the following applications:
• Meshes in which the pinion has a few teeth
• Meshes operating on nonstandard center distances because of limitations on ratio or center distances
• Meshes of speed-increasing drives
• Meshes designed to carry maximum power for the given weight allowance (This type of gearing is usually designed to achieve the best balance in strength, wear, specific sliding, pitting, or scoring.)
Standard pitch
line of cutter Standard pitch
line of cutter
Cutting short-addendum teeth. (a) Standard tooth; (b) short-addendum tooth; (c) short-addendum (abnormal) tooth.
• Meshes in which an absolute minimum of energy loss through friction is to be achieved
3.1.3.1 Addendum Modification for Gears Having a Few Teeth
Undercutting is one of the most serious problems occurring in gearing having small num-bers of teeth. The amount that gears with small numnum-bers of teeth should be enlarged (made long addendum) to avoid undercut, has been standardized by the AGMA. The values of modification are based on the use of a hob or rack-type cutter and as a result are more than adequate for gears cut with circular, shaper-type cutters. The values of addendum modifi-cation recommended for each number of teeth are shown in Table 3.1.
When two gears, each containing a few teeth, must be operated together, it may be nec-essary to make both members long addendum to avoid undercut. Such teeth will have a tooth thickness that is larger than standard and that will necessitate the use of greater-than-standard center distance for the pair.
Modifications to avoid undercut fall into three categories.
• If both members have fewer teeth than the number critical to avoid undercut, increase the center distance so that the operating pressure angle is increased.
Then get appropriate values of the addendum and whole depth for the pinion and for the gear.
10 1.468 0.532 25 1.184 0.816 15 1.4151 0.5849 33
11 1.409 0.591 24 1.095 0.905 14 1.3566 0.6434 30
12 1.351 0.649 23 1.000 1.000 12 1.2982 0.7019 27
13 1.292 0.708 22 1.000 1.000 12 1.2397 0.7604 25
14 1.234 0.766 21 1.000 1.000 12 1.1812 0.8189 23
15 1.175 0.825 20 1.000 1.000 12 1.1227 0.8774 21
16 1.117 0.883 19 1.000 1.000 12 1.0642 0.9358 19
17 1.058 0.942 18 1.000 1.000 12 1.0057 0.9943 18
18 1.000 1.000 – 1.000 1.000 12 1.0000 1.0000 –
Note: The values in this table are for gears of 1 diametral pitch. For other sizes, divide by the required diametral pitch. The values of addendum shown are the minimum increase necessary to avoid undercut. Additional addendum can be provided for special applications to balance strength. See Table 3.2.
a These values are less than the proportional amount that the tooth thickness is increased (see Table 3.15) to pro-vide a reasonable top land.
• If the pinion member has fewer teeth than the critical number, and the mating gear has considerably more, the usual practice is to decrease the addendum of the gear by the amount proportional to the amount that the pinion is increased. This results in a pair of gears free from undercut that will operate on a standard center distance.
• If the pinion member has fewer teeth than the critical number, and the gear just slightly more, a combination of the first two practices above may be employed.
An alternate is to increase the pinion addendum by the required amount and increase the center distance by an equivalent amount to make it possible to use a standard gear.
3.1.3.2 Speed-Increasing Drives
Most gear trains are speed reducing (torque increasing), and most data on gear tooth pro-portions are based on the requirements of this type of gear application. The kinematics of speed-increasing drives is somewhat different, and as a result, special tooth propor-tions should be considered for this type of gear application. As in the case of conventional drives, the problems to be discussed here are most serious in meshes involving a few teeth.
The first of these problems involves the tendency of the tip edge of the pinion tooth to gouge into the flank of the driving gear tooth. This gouging can come about as a result of spacing errors in the teeth of either member, which allow the flank of the gear tooth to arrive at the theoretical contact point on the line of action before the pinion does. The pin-ion tooth has to deflect to get into the right positpin-ion or it will gouge off a sliver of gear tooth side. If the gears are highly loaded, the unloaded pinion tooth entering the mesh will be out of position (lagging) since it is not deflected. The gear tooth is in effect slightly ahead of where it should be. The result is the same as if the gear tooth had an angular position error.
The bearing and lubricating problems at the beginning point of contact are particularly bad. The edge of the pinion tooth tends to act as a scraper and remove any lubricating film that may be present, for some distance along the flank of the tooth.
One possible solution to this problem is to give the tip of the pinion tooth a moderate amount of tip relief. This provides a sort of sled-runner condition that is easier to lubricate and that helps the pinion tooth find the proper position relative to the gear tooth with less impact.
A better solution, which may also be combined with the tip modification, is to modify the tooth thickness and addendums so as to get as much of the gear tooth contact zone in the arc of recess as possible.
The action of gear teeth in the arc of approach and recess may be likened to a boy pushing or dragging a stick down the street. A gear tooth driving a pinion in the arc of approach is like the case where the boy pushes the stick along ahead of him. It tends to gouge into the ground. The gear tooth action in the arc of recess is like the case where the boy drags the stick along behind him. It has no tendency to gouge in; it rides up over bumps and is easier to pull. The relative gear efficiencies in each case are discussed below under Section 3.1.3.4.
3.1.3.3 Power Drives (Optimal Design)
As shown in other chapters, gears fail in one or more of the following ways: actual break-ing of the teeth, pittbreak-ing, scorbreak-ing, or by wear. In the case of drives with gears of standard tooth proportions and similar metallurgy, the weakest member is the pinion, and if tooth breakage does occur, it is generally in the pinion. This is a result of the weaker shape of
the pinion tooth, as well as the larger number of fatigue cycles that it accumulates. This problem can be relieved to a considerable degree by making the pinion somewhat longer and, in so doing, increasing the thickness of the teeth and also improving their shapes. If a standard center distance is to be maintained, the gear addendum is reduced a proportional amount. If the proper values are chosen, the pinion tooth strength will be increased and the gear tooth strength somewhat reduced, which will result in almost equal gear and pin-ion tooth strength. This will result in an overall increase in the strength of the gear pair.
Several authorities have suggested addendum modifications that will balance scor-ing, specific slidscor-ing, and tooth strength. Unfortunately, each balance results in different tooth proportions so that the designer has to use proportions that will balance only one feature or else proportions that are a compromise. Table 3.2 gives values that are such a compromise.
Experimental data seem to indicate that a pair that is corrected to the degree that seems to be indicated by tooth layouts or by calculation for balanced tooth strength will usually result in an overcorrection to the pinion member. The gear is not as strong as form factors seem to indicate. Notch sensitivity in higher hardness ranges seems to be a problem, espe-cially if the gear is to experience a great number of cycles of loading.
TABLE 3.2
Values of Addendum for Balance Strength mG, (Gear Ratio)
NG/NP a (Addendum) mG, (Gear Ratio) NG/NP a (Addendum)
From To Pinion, aP Gear, aG From To Pinion, aP Gear, aG
1.000 1.000 1.000 1.000 1.421 1.450 1.240 0.760
1.001 1.020 1.010 0.990 1.451 1.480 1.250 0.750
1.021 1.030 1.020 0.980 1.481 1.520 1.260 0.0740
1.031 1.040 1.030 0.970 1.521 1.560 1.270 0.730
1.041 1.050 1.040 0.960 1.561 1.600 1.280 0.720
1.051 1.060 1.050 0.950 1.601 1.650 1.290 0.710
1.061 1.080 1.060 0.940 1.651 1.700 1.300 0.700
1.081 1.090 1.070 0.930 1.701 1.760 1.310 0.690
1.091 1.110 1.080 0.920 1.761 1.820 1.320 0.680
1.111 1.120 1.090 0.910 1.821 1.890 1.330 0.670
1.121 1.140 1.100 0.900 1.891 1.970 1.340 0.660
1.141 1.150 1.110 0.890 1.971 2.060 1.350 0.650
1.150 1.170 1.120 0.880 2.061 2.160 1.360 0.640
1.170 1.190 1.130 0.870 2.161 2.270 1.370 0.630
1.190 1.210 1.140 0.860 2.271 2.410 1.380 0.620
1.210 1.230 1.150 0.850 2.411 2.580 1.390 0.610
1.231 1.250 1.160 0.840 2.581 2.780 1.400 0.600
1.251 1.270 1.170 0.830 2.781 3.050 1.410 0.590
1.271 1.290 1.180 0.820 3.051 3.410 1.420 0.580
1.291 1.310 1.190 0.810 3.411 3.940 1.430 0.570
1.311 1.330 1.200 0.800 3.941 4.820 1.440 0.560
1.331 1.360 1.210 0.790 4.821 6.810 1.450 0.550
1.361 1.390 1.220 0.780 6.811 ∞ 1.460 0.540
1.391 1.420 1.230 0.770 – – – –
Note: Do not select values from this table for the pinion members that are smaller than those given in Table 3.1.
In small numbers of teeth, the correction required to avoid undercut on gears operated on standard center distances is excessive, in many cases, in respect to equal tooth strength.
An overcorrection in pinion tooth thickness can lead to an excessive tendency to score.
As a result, the values of addendum recommended in Table 3.2 represent a compromise among balanced strength, sliding, and scoring.
Gears with teeth finer than about 20 diametral pitch (generally) cannot score since the tooth is not strong enough to support a scoring load; therefore, the values for addendum increase in fine-pitch gears are somewhat larger than the values for coarse-pitch power gearing.
3.1.3.4 Low-Friction Gearing
In cases where a speed-increasing gear train is to transmit power or motion with the least possible loss of energy, the selection of the tooth proportions is of considerable importance.
The sliding should be kept as low as possible, and as much of the tooth action should be put into the arc of recess as possible.
Figure 3.9 shows two involute curves (tooth profiles) in contact at two different points along the line of action. The direction in which the driven pinion tooth slides along the driving gear tooth is shown by the arrows. This example is a speed-increasing drive that is the most sensitive to friction between the teeth. At the pitch point (where the line of centers crosses the line of action) there is no sliding, and it is at this point that the direction of rela-tive sliding of one tooth on the other changes.
The forces shown in Figure 3.9 are those acting on the driven pinion. The subscripts “a”
are the values considered in the arc of approach and “r” are those considered in the arc of recess. The normal driving force WN is the force that occurs at the pitch point; if there were no friction at the point of gear tooth contact, it would be the force at all other points of con-tact along the line of action. Since there is friction, the friction vectors fa and fr oppose the sliding of the gear teeth in the arcs of approach and recess. Note the change in direction
RfPa
RfGr RNG Wfa RfPr
RfGa
TaR fr
Wfr WN Φ
Φ WN fa RNP
FIGURE 3.9
Effect of friction on tooth reactions.
due to the change in direction of sliding. The angle of friction is Φ and is assumed to be the same in both cases. The torque exerted by the shaft driving the driving gear TDR manifests itself in arc of approach as follows:
TDRa=WN×RNG if no friction (3.20) TDR Wf RfG
a= a× a if friction is assumed (3.21)
and in the arc of approach as:
TDRa=WN×RNG if no friction (3.22) TDRr =Wfr×RfGr if friction is assumed (3.23)
The resisting moments are, in the arc of approach:
TDNa=WN×RNP if no friction (3.24) TDNa=Wfa×RfPa if friction is assumed (3.25)
and the corresponding moments are, in the arc of recess:
TDNr=WN×RNP if no friction (3.26)
TDNr =Wfr ×RfPr if friction is assumed (3.27) Note that, in all cases above, single tooth contact is assumed.
Efficiency is output divided by input and in this case is the torque that would appear on the driven shaft when friction losses are considered, compared with the torque that would result if no losses occurred.
Equation 3.28 below shows the efficiency of the mesh (single tooth contact) for the con-tacts occurring in the arc of approach, and Equation 3.29 shows the efficiency in the arc of recess:
E R
R R
approach fP R
fG NG NP a a
= ⋅ (3.28)*
E R
R R
recess fP R
fG NG NP r
r
= ⋅ (3.29)
*Equations 3.28 and 3.29 can be used in the metric system by using newtons for force and millimeters for dis-tance. This makes the units of torque N · mm.
In the case of speed-increasing drives, the increased efficiency can have considerable significance; cases have occurred in which, for any high ratios and poor lubrication, the speed increase actually became self-locking.