Zero dynamic increment gearing

Spectrum analyses of gear noise show that the predominant frequencies are the tooth contact frequency (1.times.TCF) and twice the tooth contact frequency (2.times.TCF). The reason for this is that at two positions in the tooth engagement cycle, the effective mesh stiffness of conventional gearing differs substantially from the ideal mesh stiffness that eliminates the dynamic increment of load. One of these two positions occurs once per tooth engagement cycle and the other one twice, giving rise to the 1.times.TCF and 2.times.TCF excitations respectively. These periodic fluctuations in the effective mesh stiffness, which are in proportion to what is called "static transmission error," produce inertia forces between engaged teeth that increase with both speed and proximity to the critical (resonance) speed of the gear pair. In conventional gearing the dynamic increment of load generated by these inertia forces can be as large as or even larger than the useful transmitted load. Not only is this dynamic increment non-useful, it is actually detrimental, since it increases operating noise and diminishes the torque capacity available for useful power transmission. Accordingly, the optimization of any gear pair requires that the dynamic increment be eliminated. This is done by introducing a special pattern of topological modifications to the tooth working surfaces. There is also an optimum form of these modifications which maximizes the torque capacity of the gear pair without sacrificing any of the quietness attainable through minimization of the transmission error.

BACKGROUND OF THE INVENTION 
FIELD OF THE INVENTION 
This invention relates to the shape of gear teeth. Specifically it relates 
to the utilization of particular gear tooth characteristics that greatly 
smooth gear meshing action so that the critical speeds of gear pairs with 
respect to noise and dynamic load are substantially eliminated while the 
torque capacity is substantially increased. The invention compromises 
additions to and improvements on the concepts disclosed in U.S. Pat. No. 
5,083,474 (hereinafter Reference 1). 
SUMMARY OF THE INVENTION 
Spectrum analyses of gear noise show that the predominant frequencies are 
the tooth contact frequency (1.times.TCF) and twice the tooth contact 
frequency (2.times.TCF). The reason for this is that at two positions in 
the tooth engagement cycle (i.e. at two positions along each base pitch 
length of the path of contact), the effective mesh stiffness of 
conventional gearing differs substantially from the average mesh 
stiffness. One of these two positions occurs once per tooth engagement 
cycle and the other one twice, giving rise to the 1.times.TCF and 
2.times.TCF excitations respectively. These periodic fluctuations in the 
effective mesh stiffness, which are in proportion to what is called 
"static transmission error," produce inertia forces between engaged teeth 
that increase with both speed and proximity to the critical (resonance) 
speed of the gear pair. In conventional gearing the dynamic increment of 
load generated by these inertia forces can be as large or even larger than 
the useful transmitted load. Not only is this dynamic increment 
non-useful, it is actually very detrimental, since it increases operating 
noise and diminishes the torque capacity available for useful power 
transmission. 
Accordingly, two major objects of the invention are to optimize gear 
performance with respect to both quietness of operation and maximization 
of power density (allowable torque per unit of weight). Both of these 
objectives are attained by reducing the static transmission error to such 
a degree and in such a manner that the gearing is substantially free of 
dynamic increment at all loads and all speeds, and to carry out this 
minimization of transmission error by means of a particular set of 
topological modifications that not only maximizes the torque capacity of 
the gear pair, but also maintains quietness at part load, as well as full 
load, allows for a minimum amount of material to be removed by the 
finishing operation (grinding or shaving), and lends itself to embodiments 
that have complete interchangeability. These and other objects and 
advantages of the invention will be evident from the drawings as explained 
in the specification that follows:

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
FIG. 11 is a partial section of a pair of mating gears embodying the 
invention, showing mating teeth in transverse (direction of motion) 
section. 
In detail and referring to the drawings, FIGS. 1,2 and 3 are introduced in 
order to explain the nature and origin of gear dynamic increment and 
noise. The explanation presented with the aid of these three figures 
appears to account fully for the main characteristics of gear excitation 
and to provide a rational basis for the method disclosed below for 
eliminating from any gear pair the dynamic increment of load and the noise 
that it generates. 
FIG. 1 shows a typical spectrum of the dynamic transmission error for a 
conventional prior art spur gear pair. The predominant frequencies are the 
tooth contact frequency (1.times.TCF), appearing as spike 11 at 224 Hertz, 
and twice the tooth contact frequency (2.times.TCF), appearing as spike 12 
at 448 Hertz. DTE at higher frequencies, at 3,4,5, etc. times TCF (spikes 
13,14,15 etc.), is too small to have a significant effect on gear set 
performance. 
Gear engineers have long appreciated that the 1.times.TCF is associated 
with tooth loading and unloading, but the 2.times.TCF excitation has long 
been a mystery. Of these two main excitation frequencies, the 2.times.TCF 
is nearly always as prominent as the 1.times.TCF and often more so, 
because it reaches the resonance (critical) speed at half at high a 
driving speed as the 1.times.TCF excitation and is therefore much more 
often within the operating speed range of the gear set. 
FIGS. 2 and 3 show where this 2.times.TCF excitation comes from. FIG. 2 
shows two tooth pair stiffness curves. The first is an approximately 
parabolic curve 21 (e,f,g,d,g',f',e') that plots what in this 
specification will be called a "conventional tooth pair stiffness curve." 
It has lead crowning, which most power train gearing has, but the usual 
tip relief, which would typically be straight lines connecting points f 
and a, and f' and a', respectively, has been omitted in the interest of 
simplicity. The parabola-like shape of curve 21 is the result of the 
increased compliance of the teeth at the outer portions of the path of 
contact, as a result of tip loading of the teeth of one of the mating 
pair. (See also FIGS. 5 and 6, and text relating thereto, in Reference 1.) 
The second tooth pair stiffness curve 22 in FIG. 2 is what was called in 
Reference 1 a "zero transmission error" tooth pair stiffness curve. The 
term "zero transmission error" (ZTE) means that when the ordinates of the 
curve are added to the ordinates of all possible overlapping identical ZTE 
curves offset by one pitch angle, the ordinate sum is constant for all 
roll angles. A ZTE curve is therefore a useful tooth pair stiffness curve 
for all forms of toothed gearing, since it generates no variations in mesh 
stiffness and no static transmission error, and therefore no noise or 
dynamic increment of load. 
Comparison of the curves 21 and 22 in FIG. 2 shows that the conventional 
tooth pair stiffness curve 21 is too stiff at the roll angles of points f 
and f', by the stiffness increments f-b and f'-b' respectively, and too 
compliant at the roll angles of and adjacent to points c and c', by 
stiffness increments g-c and g'-c' respectively. These deviations from the 
ZTE stiffness curve 22 produce a periodic fluctuation in the mesh 
stiffness that is diagrammed in FIG. 3. This figure, which is to a smaller 
scale than FIG. 2 and covers two pitch angles of rotation instead of one, 
shows the explanation to the origin of the 2.times.TCF excitation 
frequency: Each pitch angle of rotation (i.e., each full meshing cycle) 
includes one excessive stiffness peak (31 or 32), but two excessive 
compliance valleys (33,34 or 35,36). This indicates that it is the 
insufficient tooth pair stiffness at the inner ends of the loading ramps, 
that is to say, at the roll angles in the vicinity of points c and c', 
that produces the 2.times.TCF excitation indicated as spike 12 in FIG. 1. 
The amplitude of spikes 11 and 12 in FIG. 1 will be seen to be fairly 
small, of the order of one ten thousandth of an inch (0.025 mm). However 
the frequency of the excitation is quite low, relative to the frequencies 
at which most gears exhibit serious problems of noise and high dynamic 
increment. The frequency of 24 Hz. corresponds to a 22 tooth pinion 
turning at about 10 revolutions per second or 600 rpm. In many gear 
applications this is at the lower end of the operating speed range and 
would represent a domain where the "magnification factor" is very small 
and is increasing only very little faster than linearly with speed. When 
the resonance or "critical" speed is approached, on the other hand, the 
magnification factor increases rapidly, and the DTE amplitudes may be 
several times those shown in FIG. 1. At low loads these large DTE 
amplitudes can cause tooth separation, which generates a serious noise 
problem called "gear rattle," and at high loads the dynamic increment is 
of such a magnitude that it can diminish the useful torque capacity of the 
gear set by 20 to 50%, or in some cases even more. 
Reference 1 disclosed methods of eliminating this kind of excitation by 
using "differential crowning" to transform curves such as 61 in FIG. 13 of 
the reference into "zero transmission error" curves such as 131. The 
present invention elaborates and extends the disclosures of Reference 1 by 
disclosing how the methods proposed in that patent can be improved on so 
as to increase the torque capacity by 30 to 50%, and also to maintain the 
freedom from significant transmission error at all loads from zero to 
maximum allowable load, and to achieve these advantages with embodiments 
that minimize finishing costs and also have full interchangeability. 
In the present specification the term "differential crowning" means local 
changes in the amount or shape of the total lengthwise crowning that 
increase or decrease the tooth stiffness at particular roll angles. "Net" 
crowning is the sum of the "lead" (or "basic") crowning, which is the same 
at all roll angles, and the "supplementary crowning," which is different 
at different roll angles. For this specification, crowning thus has two 
components, one of which is constant and one variable. If the net crowning 
at a particular roll angle is greater than the lead crowning, the 
difference is called "positive supplementary crowning." If the net 
crowning is smaller than the lead crowning, the difference is called 
"negative supplementary crowning." 
At any given pinion roll angle the "composite net crowning" is the sum of 
the net crowning for the pinion at that roll angle and the net crowning 
for the gear at the corresponding gear roll angle. Composite net crowning 
can thus have as many as four components, since it can include lead 
crowning for both members and possibly also supplementary crowning for 
both members. The term "initial separation" (I.S.) refers to the distance 
between mating points-on the gear and pinion teeth when the working 
surfaces are in contact at the high point but are not loaded (i.e., zero 
applied torque). I.S. will include the effects of as many of the four 
crowning components as are present in the design, plus any other effects 
that are present and increase or decrease the distance between the mating 
points, such as misalignment, composite lead error, tip relief, root 
relief, or manufacturing errors. 
In the foregoing descriptions and those below it will be noted that the 
term "roll angle" and "transverse displacement" (z) are used more or less 
interchangeably. This is because these two paramaters are linearly 
related, since the distance of a point on the transverse profile from the 
point of tangency of the pressure line with the base circle is equal to 
the roll angle times the base radius. As is well known, the base radius is 
one of the two design paramaters for involute gearing that does not vary 
with operating center distance (the other being the base helix or spiral 
angle). Accordingly, any diagram that shows tooth pair stiffness as a 
function of transverse displacement z along the path of contact will have 
exactly the same shape if plotted as a function of pinion or gear roll 
angle. Only the abscissa scale and units will differ, by the factor of the 
pinion or gear base radius, as the case may be. 
The kind of tooth pair stiffness curve that was called a "zero transmission 
error curve" in Reference 1 has in the present specification been renamed 
as a "self-complementary" tooth pair stiffness curve to distinguish it 
from its earlier form, the difference being that the "self-complementary" 
curve is an optimum zero transmission error (ZTE) curve. That is to say, 
the new curve still has a zero transmission error characteristic, but is 
the particular ZTE curve that affords the maximum torque capacity, allows 
for the least expensive finishing, and lends itself to embodiments that 
have complete interchangeability. 
The reasons for making this distinction lie primarily in the potentially 
detrimental effects of crowning on torque capacity. A limited amount of 
crowning has been found to be advantageous in power train gearing because 
it reduces the load applied to the tooth ends as a result of lead error 
and misalignment. But when this crowning is increased beyond the minimum 
amount needed to control tooth end loading, torque capacity is 
substantially diminished. For example, if the crowning is say doubled, 
this will increase the specific load (load per linear inch or millimeter 
of tooth length) by 30 to 35%. This increases tooth stresses by a 
comparable amount and therefore reduces torque capacity by about 25%. When 
this loss of torque capacity is taken into consideration, it becomes 
immediately apparent why it is essential to minimize the magnitude of the 
maximum corrections that must be made when the conventional tooth pair 
stiffness curve is transformed into an optimum zero transmission error 
curve, if that curve is to be what is called in this specification a 
"self-complementary" tooth pair stiffness curve. Ideally, these maximum 
corrections should not exceed twenty percent, and preferably not more than 
ten percent, of the ordinates of the conventional tooth pair stiffness 
curve. 
Examination of FIG. 10 in Reference 1 shows that when the crowning is 
doubled (curve 104 as compared to curve 101 in that figure), the mesh 
deflection for a given torque load increases by about 15%. This percentage 
increase varies inversely with the magnitude of the initial crowning but 
tends to be fairly independent of load. To put this 15% increase into 
perspective, the required correction in the case of the diagram of FIG. 2 
is increment f-b, which is about 25% of ordinate f-i. The analogous 
correction required in the case of the diagram of FIG. 9 of Reference 1 is 
increment i-b, which is about 30% of ordinate i-r. Corrections of this 
magnitude require major increases in the net crowning at b and b', to the 
degree that a critical Hertz stress may be produced at that roll angle, 
reducing the torque capacity in a manner that is totally unnecessary. The 
present invention discloses an improved method of correcting the 
deviations of the self-complementary curve from the conventional tooth 
pair stiffness curve at b and c (and of course b' and c'). This improved 
method is explained with the aid of FIG. 4. 
The improved self-complementary curve 42 shown in FIG. 4 will be seen to 
have a synchronization ratio of two. (The synchronization ratio is defined 
in this specification as the synchronization length in accordance with 
Reference 1 (distance b--b') divided by the base pitch. In all cases the 
synchronization ratio is an exact integer.) Because the synchronization 
ratio is greater than unity, it allows the corrections to be spread over 
at least one additional tooth pair, so that each correction may be 
smaller. In effect, it is a way to take maximum advantage of the fact that 
when the synchronization ratio is two, there is always one more tooth pair 
in contact than when the ratio is one. If that extra tooth pair is given 
the same type of differential crowning as the outer teeth (at b,b',c and 
c'), the extra tooth pair no longer tends to mitigate the corrections made 
at the ends of the ramps, but instead augments them, or if desired can 
even supplant them. 
For example, if the net crowning at b and b' is increased in order to 
reduce the stiffness of ordinates m and m' to that at b and b', 
respectively, the reduced stiffness transfers a substantial amount of load 
to the tooth pair at t, which is already substantially stiffer than the 
teeth at b and b' and therefore carrying much more load. This shift of 
load to the tooth pair at the center of the path means that the 
corrections at b and b' will be that much less effective, so even more 
crowning must be employed to obtain the needed reduction in mesh 
stiffness. Unless the face is extremely wide, the crowning at b and b' can 
approach a gable shape, such as curve 103 or 105 in FIG. 10 of Reference 
1, and this produces extremely high stresses at the peak of the crown when 
the transmitted torque is large. This serious stress concentration can be 
entirely avoided if the crowning at t is increased by a sufficient amount. 
The stiffness ordinate v-t will then be reduced to that represented by 
ordinate v-j, and the required reduction in mesh stiffness achieved 
without excessive crowning at b, b'. 
This increased crowning at the roll angle of point t will of course 
concentrate more of the transmitted load at the center of the tooth, as 
compared the ends. This would be undesirable in most cases, particularly 
since the pitting hazard is at its worst near the pitch point. In this 
case, however, the increase in crowning at this roll angle reduces the 
tooth pair stiffness which in turn shifts load back to points b and b'. 
The net effect on Hertz and bending stresses will be favorable, because 
the reduction in the maximum amount of correction needed (increments 
m-b,t-j, and m'-b') will be greatly reduced, and this allows the basic or 
lead crowning for the set to be significantly reduced. 
It will be noted that the self-complementary curve 42 in FIG. 4 shows 
additional deviations from the conventional tooth pair stiffness curve 41 
at points s and s'. These points, which are one pitch angle inside the 
roll angles at c' and c, respectively, have been given increased 
stiffness, by the ordinate increments s-i and s'-i', by reducing the 
crowning at their roll angles. (There is, however, a limitation on the 
crowning at c and c' that is different from the limitation at b and b'. 
Instead of a problem of high local load, the problem is one of the 
undesirability, for reasons of fabrication cost, of using negative 
supplementary crowning that has an absolute value greater than the lead 
crowning. This would result in a net crowning that was negative, that is 
to say concave, and this would result in a need to grind or shave off a 
considerable amount of material over the entire remainder of the tooth 
working surface. So, while negative crowning is physically possible if the 
mating surface has lead crowning of greater absolute value, the deterrent 
is an economic one.) 
The use of zero net crowning at c or c' or both, on the other hand, can be 
highly desirable, because it gives the maximum stiffness at such points 
for a given amount of lead crowning. Since it is an increase in mesh 
stiffness at c and c' that is needed to eliminate the 2.times.TCF 
excitation (see FIG. 3), zero net crowning at c or c' is obviously a 
desirable feature. But again there are limitations. Perfect symmetry of 
the conventional tooth pair stiffness curve is not often realized, so it 
becomes necessary to use different corrections on the pinion and gear in 
order to impart the perfect symmetry that the self-complementary curve 
must have. This of course rules out zero net crowning at all of points c, 
c', i, and i'. At best one or two of these points can have it. In 
addition, the composite total crowning at c and c' can rarely be smaller 
than the net lead crowning, which is usually half of the composite lead 
crowning, because differential crowning near the start of active profile 
is limited by the need to maintain profile convexity at roll angles 
smaller than four or five degrees. Zero net crowning is possible, on the 
other hand, at points i and i' for both the pinion and the gear. However 
this would leave these roll angles with zero composite crowning, 
eliminating the main safeguard against excessive tooth end loads that 
result from misalignment and lead error. Accordingly, it is recommended 
that if zero net crowning is used at either i or i' on either member, the 
crowning of the other member at the mating roll angle should be the full 
lead crowning. That is to say, there should be zero supplementary crowning 
for that member at that roll angle. The minimum composite crowning at i or 
i' would thus be half the composite lead crowning, which is the same 
minimum crowning recommended for points c and c'. 
These considerations lead to a valuable prescription for optimizing the 
shape of the self-complementary curve. If we refer to the roll angles at 
c, c' i, and i' as the "stiffening-needed" roll angles, and note that one 
of them in all probability has a greater deviation between its ordinates 
on curves 41 and 42 than the others, that roll angle should on one or both 
members be given composite supplementary crowning within plus or minus 
twenty percent, and preferably within plus or minus ten percent, of half 
the composite lead crowning. If the composite lead crowning is evenly 
divided between the pinion and the gear, this would give the effect of a 
net crowning on one member reasonably close to zero and insure that at 
this critical roll angle the self-complementary curve will have the 
optimum ordinate. The other three stiffening-needed roll angles, because 
they will probably require smaller corrections, will all have either equal 
or smaller magnitudes of composite negative supplementary crowning. 
Two critical geometric properties of curve 42 in FIG. 4 should be noted. 
The first is that when a pair of identical self-complementary curves are 
displaced laterally by two base pitch lengths, the right hand ramp 
c'-b'-a' will occupy the dotted line position at left shown as p-b-n. 
Adding the overlapped ordinates of the triangle a-b-n to those of curve 42 
exactly doubles the combined stiffness to the dotted line p-l-c, since the 
ordinate o-b is exactly equal to ordinate b-l. This produces what is 
herein called the "overlapped self-complementary curve," l-c-i-j-i'-c'-l. 
The critical characteristic of this curve is that it insures that the 
ramps are shaped and proportioned so the rates of mesh loading and 
unloading are (a) identical in absolute value and (b) synchronized, so 
that the process of tooth entry into and exit from the field of contact 
produces no significant transmission errors. Accentuation of the word 
"mesh" in this statement is intended to draw attention to the fact that 
when the corrections are spread to an additional tooth pair at the center 
of the stiffness curve, increases or decreases in the stiffness of that 
tooth pair affect the rate of loading or unloading of the mesh at least as 
much as stiffness changes at the ramp roll angles. 
The second critical geometric property of curve 42 is the special kind of 
symmetry that is common to all ZTE curves including self-complementary 
curves. This is called in this specification "inverse offset symmetry," 
and means that for every point on the overlapped ZTE or self-complementary 
curve there is a point that is (a) a mirror image about the horizontal 
axis of the curve, and (b), offset by one base pitch. This type of 
symmetry, which was discussed in connection with FIG. 7 of Reference 1, is 
more general than right-left symmetry about the vertical centerline in 
that it allows for asymmetrical ZTE or self-complementary curves of all 
kinds. Such ZTE or self-complementary curves occur whenever the 
conventional tooth pair stiffness curve is slightly asymmetrical, as it 
tends to be when the gear ratio is substantially greater than unity. 
A mathematical description of an overlapped self-complementary curve would 
be as follows: it is any curve that is the algebraic sum of one or more 
periodic curves each of which has a period of one base pitch times the 
synchronous ratio divided by an odd integer, includes the overlapping 
ordinate contributions of ramp portions at its ends, and is made up 
entirely of continuous, piecemeal continuous or discontinuous segments all 
of which have offset inverse symmetry of one base pitch offset. Further, a 
basic characteristic of this overlapped self-complementary curve is that 
in order to minimize the required corrections, the area under the curve 
from one end of the synchronization length to the other should be within 
ten percent, and preferably within five percent, of the area under the 
conventional tooth pair stiffness curve over that length. 
The diagram of FIG. 4 is complex, and it may be helpful to review and to 
justify its special features. Basically it is related to the corresponding 
tooth pair stiffness diagram of FIG. 9 of Reference 1. The principal 
difference is that the spreading of the corrections to all the teeth in 
simultaneous contact adds a kind of flat M-shaped correction h-i-j-i'-h' 
at the apex of the undulating curve 71 in FIG. 7 of Reference 1, and this 
requires identical inverted corrections at the left and right nadirs, 
p-l-c-d and p'-l'-c'-d', respectively. (The outermost leg of the inverted 
M's at the nadirs are beyond the diagram left and right boundaries.) The 
reason why corrections must be made at the nadirs if one has been made at 
the apex is that if corrections to a self-complementary curve are not to 
destroy the self-complementary characteristic, they must embody the same 
two geometric features that distinguish all self-complementary curves. As 
noted above, these are (a) synchronized mesh loading and unloading, and 
(b) inverse offset symmetry. 
Confirmation that curve 42 in FIG. 4 does in fact maintain a constant mesh 
stiffness may be obtained in two ways. One way is to consider that curves 
that are identical to 42 but are offset by one normal base pitch to the 
right or the left, so their centerlines are at o-A and o'-A' instead of 
v-B, may be supposed to be translating, along with curve 42, laterally 
past a fixed vertical line. (The abscissa variable will in this case be a 
time variable instead of the displacement variable z shown.) If this 
translating construction is carried out, it will be found that at any 
lateral position the sum of the several ordinates that are colinear will 
be constant. 
The second method of confirming that curve 42 produces a constant mesh 
stiffness is to consider that the curve is stationary and the stiffness 
ordinates of the tooth pairs that occupy the field of contact 
simultaneously are translating laterally. In this kind of analysis one has 
only to add together these ordinates for any of the eight positions (or 
other desired number) into which the base pitch has been divided. One thus 
adds the ordinates at a, i and c'; b, j and b'; c, i' and a'; d and h'; e 
and g'; f and f'; g and e'; and h and d'. For all these arrays of two and 
three tooth pairs the sum of the stiffness ordinates will be found to be 
exactly twice the ordinate of the average ordinate line A--A'. 
When analyses such as these are made, the purpose of the dotted lines in 
FIG. 4 will become evident: They are intended to show the effect on 
portions of curve 42 at the loading ramps a-c and a'-c' produced by the 
identical tooth stiffness curves of tooth pairs offset by two pitch 
angles. The dotted line p-b-n is simply the same line as the right hand 
loading ramp c'-b'-a'. Both will be seen to have the same two segments 
that are unequal in length and non-colinear. When the ordinates of line 
p-b-n are added to those of the left hand loading ramp a-b-c, the sum 
gives the saw-tooth dotted line p-l-c. The segment l-c will have polar 
symmetry with solid line i-j about point f, and the same relationship will 
exist for segments l'-c' and i'-j, giving a saw-tooth line over the 
synchronization length b--b' made up of these four short segments and two 
long segments, c-i and c'-i'. 
It might be considered that this undulating saw-tooth form is an odd and 
unexpected shape for what appears to be the optimum tooth pair stiffness 
curve for all mating gears. But this irregular shape becomes immediately 
understandable when it is considered that gear excitation is generated by 
deviations of the tooth pair stiffness curve from an ideal curve primarily 
at the loading ramps. This was explained above in conjunction with FIGS. 
1,2 and 3. And secondly, the irregularity at the center of curve 42, at 
h-i-j-i'-h', is strictly a result of taking advantage, in order to 
maximize torque capacity, of the possibility of spreading the corrections 
of these loading ramp deviations (at b, b', c and c') over all of the 
tooth pairs that are simultaneously in contact. The reason this produces 
irregularities at the center of the stiffness curve is that when the 
synchronization ratio is two, the center of the stiffness curve 42 is 
automatically one pitch angle inside the ends of the synchronization 
length. These ends, which are at b and b', must always be in the central 
region of the loading ramp, because this is where the purely elastic means 
for reducing the mesh stiffness will in most designs give way to 
conventional tip relief (segments a-b and a'-b'). In some designs it may 
be found that extending this relief slightly into the synchronization 
length reduces the peak stresses at b and b' without seriously increasing 
the transmission error. 
FIG. 4 may be somewhat misleading in one respect, and that is in the sharp 
cusps and V-shaped valleys that it shows. The stiffness curve 42 is given 
this shape so as to make its features easier to explain. In an actual gear 
pair stiffness curve all these sharp changes in slope would be rounded 
because of the fact that the contact surface of the grinding wheel used to 
make the gears or the shaving cutters that make them has fairly limited 
curvature. As a result, the sharp angles shown would become precision 
points maintained by the grinder control program, but between these 
precision points the slope variations would be much more gradual. The 
effect of this will be to make the ramps have more of an S-shaped such as 
shown in FIG. 2, with a mirror image of an S-shape at the opposite end of 
the synchronization length. Similarly, the cusps and valleys on the 
stiffness curve 42 will in an actual gear pair take on the form of 
well-rounded undulations. 
Moving on to FIG. 5, this diagram is introduced in order to provide a 
graphical explanation of why initial separation (I.S.) reduces tooth pair 
stiffness if it is increased, and conversely, increases the stiffness if 
it is decreased. (Tooth pair and mesh stiffness of gearing, as well as 
deflection, are always in the transverse direction, even when the gears 
are helical or spiral bevel.) This figure shows what happens to a purely 
elastic load-deflection curve 51, which has an initial separation 
(I.S.).sub.1, and an effective stiffness (or slope) K.sub.1, when its 
initial separation is doubled so as to shift it to position 52. The 
original stiffness K.sub.1, which is the ratio of load to deflection, 
F.sub.1 /.DELTA..sub.1, is reduced to K.sub.2, which is F.sub.2 
/.DELTA..sub.2. In effect, the stiffness (slope) K.sub.2 has been reduced 
by the ratio of .DELTA..sub.1 /.DELTA..sub.2, or .DELTA..sub.1 
/(.DELTA..sub.1 +the increase in initial separation). It is this effect 
that explains why crowning, which contributes the major portion of initial 
separation, can be utilized to introduce useful adjustments to tooth pair 
stiffness. 
FIG. 6 shows how the composite net crowning for a gear pair embodying the 
invention may vary at different roll angles. It is thus a graphical 
representation of what is meant by the term "differential crowning". The 
horizontal and vertical scales in this diagram are made very different so 
as to accentuate the crowning differences. While the horizontal 
magnification is perhaps two or three, the vertical magnification may be 
two or three thousand. In the curves labeled 61 (zone B), the composite 
lead crowning is plotted. This is essentially conventional crowning in the 
sense that it is constant for the full working height of the tooth. 
Ideally it is just sufficient to insure that anticipated misalignment and 
composite lead error does not produce a peak specific tooth load at either 
end of the tooth. The curve band of zone B is intended to be construed as 
indicating that this type of crowning does not exceed five, and preferably 
not exceed two, times the module times one ten thousandth of an inch (two 
point five thousandths of a millimeter). 
In zone C of FIG. 6 (curve 62), the much greater crowning at b,j or b' is 
indicated. As FIG. 4 shows, at these roll angles, the conventional tooth 
pair stiffness curve 41 lies above the self-complementary curve 42, by the 
increments m-b, t-j, and m'-b', respectively. This means that at the roll 
angles of these increments curve 41 needs to have its local stiffness 
reduced, and this calls for increased crowning. This increased crowning is 
obtained by adding positive supplementary crowning at roll angles b and b' 
or j, or, preferably, both. Typically this positive supplementary crowning 
may at b or b' be as great as ten times the composite lead crowning, 
unless positive supplementary crowning at j is also provided, in which 
case the amounts will seldom be more than three times the composite lead 
crowning. 
In zone A of FIG. 6 (curve 63), the reduced crowning at the 
stiffening-needed points c, c', i, and i', is indicated. As noted above, 
this increased stiffness is needed to eliminate the 2.times.TCF 
excitation. Limits to how much reduction in crowning is possible at these 
four points has been discussed above, but the advantages of reducing the 
crowning as much as these limits will allow should be noted. One advantage 
is that it reduces the moment arm for tooth bending. For conventional HCR 
("High Contact Ratio") gearing, the critical moment arm is at what is 
called the "HPDTC", which is an acronym for "Highest Point of Double Tooth 
Contact." The corresponding point in ZDI gearing is c or c', but at these 
points the reduced crowning distributes the load more widely over the 
tooth length and reduces the maximum specific tooth load by about 25% as 
compared to that of a conventional HCR gear pair. Consequently the largest 
bending moment is shifted downward from points c and c', usually to points 
f and f'. The resulting reduction in moment arm gives the ZDI set an 
advantage in torque capacity that is in the vicinity of 20%. 
The second advantage that minimizing the crowning at c and c' bestows on 
the ZDI gear pair comes indirectly, by allowing the composite lead 
crowning to be minimized. The use of some typical numbers is illuminating. 
In FIG. 4 it will be noted that the ordinate n-r is about half as great as 
the ordinate u'-s'. This means that for the conventional tooth pair 
stiffness curve 41, the amount of load carried by the tooth pair at r is 
about half that carried by the tooth pair at s', one base pitch (or one 
pitch angle) inside of r. Analysis shows that halving the crowning 
increases the stiffness of a tooth pair about 15%. It has been shown above 
that the maximum correction that can be made at c or i' is that produced 
by halving the composite lead crowning. So, the maximum stiffness 
correction that the minimization of crowning can afford at c is about 15% 
of the ordinate n-r. If one makes this correction at c, the 33% of the 
load carried by the tooth pair at c would be increased to at most 40 %. 
The combined increase of stiffness, however, since the 60% of the load 
carried by the tooth pair at s' is unaffected, would be 15% of 40%, or 
about 6%. More often than not, this is not a sufficient correction at c. 
FIG. 9 of Reference 1 shows the needed correction at c to be about 20%. It 
might be thought that a correction of this magnitude could be obtained if 
the composite lead crowning were to be increased. This is not the case. An 
increase in composite lead crowning reduces stiffness at r and s' in the 
same proportion, so one is still left with the available correction being 
6% when 20% is needed. 
If corrections are divided among all of the teeth that are simultaneously 
in contact, this problem is solved. As FIG. 4 shows, the stiffness 
increments r-c and s'-i' are reduced to about 10%, and substantially the 
entire 15% increase in stiffness from halving the composite lead crowning 
at both r and s' is now available. This means that one now has the 
essential feature needed to eliminate the 2.times.TCF excitation. The 
composite lead crowning can remain at the minimum amount needed to prevent 
excessive tooth end loads, and the achievement of zero dynamic increment 
will have afforded an effective increase in torque capacity of another 20 
to 50%. In general, the goal should be to restrict the composite lead 
crowning to five, and preferably two, times the module times one ten 
thousandth of an inch (two point five thousandths of a millimeter). 
It will be noted in FIG. 6 that the outermost ends of the zone C curves 62 
are shown in broken line. This is done to indicate that in some designs 
the tooth surfaces that would otherwise contain these large crowning 
magnitudes may have been removed. The reasons for this will be discussed 
in connection with FIGS. 9 and 10. 
FIG. 7 is another diagram showing, as in the case of FIG. 6, exaggerated 
crowning curves over a symmetrical half width (F/2) of the tooth length. 
In this case the crowns 71, 72 diagrammed are plotted to show how changes 
in the exponent of a crowning curve can redistribute the specific tooth 
load over the length of the tooth. Curve 71 is a parabolic curve, so the 
exponent is 2.0, whereas curve 72 is a cubic curve, for which the exponent 
is 3.0. The advantage of the cubic curve is that for a given tooth pair 
stiffness, the cubic curve will spread the load more effectively than the 
parabolic curve. That is to say, the peak specific tooth load at the 
center of the tooth will be smaller than if the crowning curve is 
parabolic. 
There is a second reason for varying the exponent of the crowning curve, 
which will be discussed in conjunction with FIG. 8. In all prior art gear 
forms the tooth modifications have been designed to afford a minimum 
transmission error at one particular "design load", and loads higher or 
lower have invariably exhibited increased transmission error. In the 
present invention, however, a way has been found to manipulate the shape 
of the crowning curves so that the transmission error remains negligibly 
small for all loads from zero to peak torque. The principles discussed 
above can be utilized to determine the amounts of crowning at all active 
roll angles that for a particular operating torque (e.g. the "design 
load") will limit the deviation from the average mesh deflection for one 
full pitch angle to less than ten microinches (0.254 micrometers), and 
with a modest amount of additional interpolation to less than five 
microinches (0.127 micrometers) times the module. 
In order to eliminate mesh stiffness variation at all loads instead of a 
single "design load", it is necessary to take into consideration the fact 
that when curved surfaces are pressed together, the contact area enlarges 
as the load increases. This characteristic, which enters into the 
calculation of Hertzian deflection of ball bearing balls or rollers, also 
applies to gear teeth, since relative curvature is present in both the 
transverse direction (profile curvature) and the axial direction (crowning 
curvature). 
The non-Hookian characteristic produced by this bidirectional curvature 
prevents diagrams such as FIG. 2 or FIG. 4 from being valid for more than 
a single torque load. If these diagrams are taken as representing the 
tooth pair stiffness at peak torque, analogous diagrams with reduced 
ordinates are needed to represent the stiffness at half or quarter load. 
However, each of these additional diagrams will be closely similar to the 
diagrams shown, and each will have a ZTE or self-complementary tooth pair 
stiffness curve such as 22 or 42 but with the transverse length of the 
ramps being slightly reduced (e.g. i-a and i'-a' in FIG. 2, and o-a and 
o-n in FIG. 4). Fortunately these part-load ZTE or self-complementary 
curves are not unique in the way the optimum peak load curve is, since 
there is no longer a need to maximize torque capacity. Any stiffness curve 
that fulfills the ZTE or self-complementary requirements will be as 
satisfactory as any other. But in order to determine what modifications 
will produce a self-complementary stiffness curve at part load, it is 
necessary to follow a set procedure. No other procedure is possible. 
The main requirement for determining a satisfactory set of modifications is 
to start at the smallest torque load at which it is desired to have the 
gear set operate silently. The reason for this may be seen in FIG. 8. If 
the load is small enough so that the mesh deflection .DELTA. is smaller 
than the net composite crowning (represented in this figure by the maximum 
y value of curve 81, at x=F/2), there will be no contact on the outer ends 
of the teeth. Consequently for large values of x, the shape of the outer 
portion of the crowning curve is immaterial. The worst that can happen is 
that once the crowning values for a self-complementary curve for a 
part-load condition are selected, it may be found that the amount of 
crowning needed at the given roll angle at high load, including peak load, 
may not follow a smooth exponential curve such as 81, but may require a 
sort of "detour" such as shown in curve 82. The same topological grinding 
equipment that produces curve 81 (or sharpens a plunge shaver to produce 
that curve) can produce curve 82 just as readily, so there is no need to 
experiment with other exponents to see if an exponent other than that of 
curve 81 would have allowed the crowning curve to be less sinuous (as for 
example curve 72 in FIG. 7). 
For all designs there will be a load that is small enough so that the 
conditions illustrated in FIG. 8 are present. However for gears that are 
designed for very heavy loads it will often be found that the mesh 
deflection at say one third of peak load may be greater than the maximum 
composite crowning. This means that the contact lines in the entire 
central part of the field of contact, from c to c', occupy the full width 
of the field. In such cases it cannot be said that the amount of crowning 
immediately adjacent to the tooth ends has no effect on the tooth pair 
stiffness at relatively low loads. Nevertheless, crowning curves that will 
give a self-complementary curve at all loads, from zero to peak load, can, 
with persistence or sufficient program iteration, always be synthesized. 
More often than not, such curves will have at various roll angles one or 
more detours such as shown in curve 82 in FIG. 8. Such curves appear to 
embody the first pattern of gear tooth modifications that eliminates 
transmission error with equal effectiveness at all torque loads without 
requiring superfine teeth and large helix angles. 
In gear pairs for vehicular propulsion or other applications where it may 
be desired that the gear set be substantially silent at all loads, 
including zero load, the primary feature that is needed is a "high line". 
This is a continuous line of zero initial separation of length at least 
one full base pitch, located usually in the central portion of the field 
of contact. The presence of such a line insures that at zero and very 
small loads, the teeth will have a true (unmodified) involute curve mating 
with another true (unmodified) involute curve throughout one complete 
pitch angle, so there will be no significant transmission error or 
self-excitation. In the case of gear sets with synchronization ratio 
greater than unity, it is usually advantageous to make this "high line" of 
zero initial separation continue for the full synchronization length. 
FIGS. 9 and 10 illustrate a minor improvement that may be made to spur or 
straight bevel gears 92 embodying the invention, to increase the life of 
the plunge shaver or grinding wheel that is used to finish the gears. In 
such gear types the large amount of crowning at b and b' in FIG. 4, 
combined with the tip relief that extends from b to a and b' to a' and has 
a magnitude over the full length of the tooth more or less equal to the 
mesh deflection at peak load, produces a total initial separation at the 
four corners of the field that substantially exceeds the maximum mesh 
deflection. As a result, these corners never make contact with the mating 
gear and might as well be removed so they will not needlessly shorten the 
wear life of the finishing tool. The flat bevel that cuts back the topland 
from the center of the tooth 91 outward should not extend further into the 
tooth body than a point on the gear end faces that is about half way 
between the roll angles at b and c, or b' and c' as the case may be. The 
effect of bevels of this kind on the shape of the field of contact of both 
spur and straight bevel gears may be seen in FIGS. 10 and 12, 
respectively, of U.S. Pat. No. 3,982,444. 
FIG. 11 is a partial transverse section through a pair of gears 112, 114 
showing typical teeth 111, 113 embodying the invention, with arrows 
showing the direction of motion (transverse direction). Teeth 111 on the 
smaller of the mating pair (pinion 112) and teeth 113 on file larger of 
the mating pair (gear 114) make contact along a pressure line 115 that 
contains the path of contact extending from a starting point at a' and an 
ending point at a. The active heights at teeth 111, 113 terminate at the 
addendum circles 116, 117 of the pinion 112 and gear 114, respectively, 
and determine the position of points a and a', respectively, along the 
pressure line 115. Other features of pinion 112 and gear 114, such as 
hubs, webs, rims, keyways, etc. are standard and are omitted in the 
interest of clarity. 
The pressure line 115 in FIG. 11 shows the direction of the path of contact 
a'-a that appears, to a larger scale, as the base line in FIGS. 2 and 4. 
It should be noted that both of these figures plot tooth pair stiffness k 
as a function of distance z along the path of contact. As in Reference 1, 
the three coordinates used in this specification are x, y, and z. Of 
these, the x-coordinate defines the position of points in the axial 
direction, as indicated in FIGS. 7 and 8. Normal to this direction and 
defining positions in the direction of movement of the contact points 
along paths of contact such as line a'-a in FIG. 1, is the z-coordinate. 
In most systems of orthogonal coordinates the third variable, y, is normal 
to the x and z directions. In this specification, however, a different 
system of using the y dimension is employed: It represents modifications 
to the theoretical ideal involute profile or lead curves and is in the 
same direction as the z-coordinate, that is to say, in the direction of 
line 115 in FIG. 1. As noted above, the scale used in representing these 
modifications in the y direction is very much greater than that used for x 
and z. 
Several observations about the nature and scope of the invention may be 
made: 
(1) Interchangeability: It is also possible to create a set of any desired 
number of gears all having different tooth numbers, such that any of the 
gears will mate with any other gear of the set and will in addition have 
the "zero dynamic increment" (ZDI) characteristic described in this 
specification. Several basic properties would have to be identical on all 
gears, including base pitch, pressure angle, approximate tooth thickness 
at the pitch point, and topological modifications that produce a 
self-complementary stiffness curve. If helical gears are used, the base 
helix angles must also be identical, and the number of gears in the set 
must be doubled to include both right and left hand helices for each tooth 
number. All of these gears, whether spur or helical, would also have to 
have the same length of contact, evenly divided between approach and 
recess action. To meet these requirements gears with larger tooth numbers 
must have progressively shorter addenda and increased dedenda. As with all 
sets of interchangeable gears, major reductions in cost are attainable 
because of reduced design time and increased production quantities. 
(2) Helical gearing: In conventional gearing, a helix angle (or spiral 
angle, in the case of bevel gears) is often needed to reduce the static 
and dynamic transmission error. This is not the case with ZDI PG,20 
gearing, however, which uses differential crowning instead. Nevertheless 
some designers may specify a helix or spiral angle for special purposes. 
When this is done, several features are recommended: (a) The best profile 
contact ratio is close to 2.0. (b) The preferred type of high line is 
non-straight and is the locus of the centers of the contact lines as they 
move through the field of contact. (c) The roll angles used to identify 
contact line positions in FIG. 4 and similar diagrams are, for helical 
gear sets, the roll angles of the contact lines (or their extensions if 
they are truncated by the end boundaries of the field) at the central 
transverse plane of the set, so points a and a' in FIGS. 2 and 4 will 
usually be beyond the field boundaries. (d) As indicated in FIG. 4, the 
self-complementary curve 42 requires substantial stiffening at the inner 
ends of the ramps c and c', and this objective is best served in helical 
gearing through maximizing the length of the contact line at those two 
points by making the transverse lengths of the ramps (a-n, a'-n') 
substantially equal to the face width times the tangent of the base helix 
angle, or spiral angle in the case of bevel gearing. It should be noted 
that in helical and spiral bevel gearing, the contact lines intersect the 
common pitch element (i.e., the line of tangency of the pitch cones) at 
the base helix angle or spiral angle, as the case may be. 
(3) Critical tooth numbers: As in the case of conventional gearing, torque 
capactity as a function of pinion tooth numbers is a falling curve for 
tooth bending fatigue and a rising curve for pitting (Hertzian) failure. 
The tooth number at which these two curves intersect is called the 
"critical number of teeth," and it is at this tooth number that any gear 
set has the maximum torque capacity. This is also true of ZDI gearing, so 
whenever possible pinion tooth numbers should be within four, and 
preferably two, of this critical tooth number. 
(4) Face width: The common practice of making the pinion slightly wider 
than the gear is avoided, as it always should be when crowning is present. 
In no case should the face width of the wider member be greater than that 
of the narrower member by more than one percent or so. The reason lies in 
the serious effects of buttressing on stress distribution. 
(5) Symmetry: By proportioning the operating tooth thickness ratio, 
pressure angle and addendum heights and root clearances so as to make the 
conventional tooth pair stiffness curve substantially symmetrical, ZDI 
gears will be given a substantially symmetrical self-complementary curve 
such as shown in FIG. 4. As in all gearing, minimum peak stresses are 
achieved when the tooth pair stiffness curve is symmetrical. In addition, 
symmetry of this curve allows both members to be finished by the same 
tool, the advantages of which are noted in (7) below. Having the same 
modifications of both members at their mating points will be desirable in 
nearly all designs. 
(6) Manufacturing methods: The kind of tooth working surface modifications 
called for in this specification are a special pattern of what are called 
"topological modifications." (In some literature the preferred usage is 
"topographical modifications.") The most economical way of forming these 
modifications, or making the plunge shavers with which to form them, is 
with the aid of generative gear grinding machines that have the necessary 
topological feature. One of the main advantages of the invention is that 
it discloses a pattern of net crowning curves that because they can be 
identical can be formed on the plurality of teeth that are ground 
simultaneously by the continuous-worm type of grinding wheel when the 
synchronization ratio is greater than unity. 
(7) Manufacturing accuracy: The concepts disclosed in the present 
specification are believed to move the theory of gearing sufficiently far 
in the direction of ideal design so that the practical limitation on 
operation without noise or dynamic increment will in the future lie almost 
entirely with accuracy of manufacture. The minimization of all forms of 
fabrication error therefore becomes a paramount consideration. In order to 
realize the potential benefits of ZDI gearing a number of special 
fabrication techniques are recommended: The most important error in ZDI 
gearing to eliminate is base pitch error. In effect this type of error 
mates gears that have slightly different tooth spacing. What this does is 
to give a diagram such as that of FIG. 4 a pronounced overall slope or 
rotation, so that one end is higher than the other. When this condition is 
present all of the corrections that have been described above may become 
secondary. The slope comes from the fact that unequal base pitch values 
introduce an initial separation, such as that shown in FIG. 5, at either 
the beginning or the end of the contact path, greatly reducing the 
effective tooth pair stiffness at the point where it occurs. The 
self-complementary curve no longer gives a constant ordinate sum, but one 
that has a pronounced upward slope followed by a sudden drop in the 
vicinity of one of the ramps. To avoid this effect, and to maintain an 
average base pitch for one of a gear pair that differs from that of the 
other by not more than twenty, and preferably not more than ten, 
microinches (i.e., by not more than 0.508, and preferably by not more than 
0.254, micrometers) times twice the module, a number of expedients are 
useful. The most convenient method of getting an exactly equal base pitch 
on the pinion and gear is to finish them with the same tool. The 
specification of spur teeth instead of helical allows the use of a common 
plunge shaver or hone. For ground gearing, the use of the same worm-type 
grinding wheel or rocking-type base tangent grinder set-up on both pinion 
and gear will help greatly. The next best alternative is to employ base 
tangent measurements in preference to over pin measurements, combined with 
size sorting of finishing tools or finished gears or both. 
Out-of-tolerance members may be honed or lapped. 
It should be appreciated that the foregoing specification has disclosed a 
gear form that has substantial advantages over prior art gear forms but 
needed to be defined in more precise mathematical terms. Gearing having 
slightly different proportions and characteristics but embodying 
essentially the same concepts can readily be designed and will exhibit 
almost equal advantages. The foregoing specification and the ensuing 
claims are intended to read on such gearing as well as gearing that 
adheres to the precise characteristics enumerated.