Noncircular drives, including chain drives with noncircular sprockets, achieve a desired functional relationship and minimize slack variation in the elongate flexible member thereof. Noncircular gears having the desired functional relationship are first designed. Noncircular members having the same pitch curve or operative surface configuration as the pitch curves of the noncircular gears are theoretically located at the desired center-to-center distance of the noncircular rotational members of the drive being designed. The noncircular members are manipulated such that points initially in contact on the gears are not joined by a common tangent. Using a computer, the drive is incremented with the common tangent acting as the driving span of the flexible member. At each increment functional relationship error and overall length of a taut elongate flexible member are calculated. The variation in taut flexible member length is the slack variation. In the event of excess functional relationship error, the design procedure is iterated modifying the functional relationship on which the original gear pair was based by adding the functional relationship error. This iterative procedure is continued until functional relationship is acceptable. If the slack variation is excessive, the same procedure continues with the other span, i.e., common tangent, driving until again the functional relationship is acceptable. If slack variation is still unacceptable, the design procedure continues using first the one run of the flexible member and then the other until both functional relationship and slack variation is acceptable.

BACKGROUND OF THE INVENTION 
This invention relates to a drive such as a chain, belt, band, or tape 
drive, with spaced noncircular, rotational members interconnected in 
driving relation by a flexible endless chain, belt, band, tape or the 
like. More particularly, the invention relates to a drive of this nature 
that accomplishes the desired functional relationship, and in which the 
noncircular members and elongate flexible drive member entrained about 
them in driving relation operate with a minimum of slack variation in the 
flexible drive member. 
Over the years, chain drives with noncircular sprockets, particularly for 
bicycles, have been proposed. Many or all proposals of this nature lack 
any teaching of how to accomplish a preferred functional relationship 
between driving and driven noncircular rotational members. Many such 
proposals entirely fail to address the likely variations in slack in the 
chain and how to eliminate that or compensate for that. Some proposed 
drives with one slightly elliptical sprocket simply rely upon the 
conventional chain drive's limited tolerance for variation in slack. Other 
proposed noncircular drives of this nature utilize a movable compensatory 
slack take-up idler sprocket or roller to maintain the chain taut as the 
sprockets turn, by taking up the excessive slack that occurs. Another 
proposed approach has been to vary the distance between the sprockets 
during each rotation to take up the excessive slack that would otherwise 
periodically occur. On occasion, it has been suggested to make both of the 
pair of sprockets noncircular, but without any clear indication how this 
might be accomplished. 
In this last category of chain and sprocket drives with two noncircular 
sprockets, one suggestion in the patent literature was to use on a bicycle 
two elliptical sprockets of differing sizes about which is entrained the 
typical chain. No consideration was given to the differing rates of 
rotation of the two different sized sprockets, or the resulting changing 
angular relationship between sprockets, whereby one sprocket would not 
necessarily be in the correct position to take up slack when the other 
tended to contribute to the slack. A similar proposal was to mount 
eccentrically the rear sprocket of a bicycle so that, as that sprocket 
turned, it was to move eccentrically about its axis of rotation to take up 
the slack that resulted from an elliptical drive sprocket. Again, there 
was no recognition that the changing angular relationship between the 
sprockets throughout the rotation thereof would prevent operation of the 
drive as desired. 
Only simple functional relationships between driving and driven rotational 
members have been sought for the prior art noncircular drives mentioned 
above. Writings on the subject have expressed no understanding that 
accomplishment of a variety of functional relationships between driving 
and driven members could be sought while minimizing slack variations. 
BRIEF SUMMARY OF THE INVENTION 
In accordance with this invention, there is provided a drive, and a method 
of manufacturing a drive, with noncircular rotational members, such as 
sprockets, pulleys, rollers or the like, about which is entrained an 
endless drive chain, belt, or other flexible endless member, the drive 
having a desired functional relationship of the rotational members and 
limited variation in the slack of the endless member. More particularly, 
the methods and apparatus according to the invention use, initially, the 
design of noncircular gears to arrive at noncircular member shapes 
correctly functionally related as to size and shape and having, in many 
embodiments, a sufficiently taut endless member for all angular 
relationships of the noncircular members. 
As used herein, noncircular, when applied to gears and sprockets, refers to 
the pitch curve of that element. The term noncircular member as used 
herein means a member that has a noncircular surface or pitch curve, or a 
member that has a circular surface or circular pitch curve, but is 
eccentric respecting its point or axis of rotation. The term operative 
surface is occasionally used herein. It means the pitch curve in the case 
of gears, sprockets or other toothed or notched rotational members or the 
actual smooth surface of rollers pulleys and the like. "Center" as used 
herein means the point or axis of rotation of a noncircular member. 
The drive and its method of manufacture can be applied to the typical 
two-sprocket chain and sprocket drive of the kind used on bicycles, but it 
is applicable, as well, to many drives in which one rotary member drives 
another, via an entrained endless band, belt or other flexible, endless 
member. The drive may have more than two rotational members about which 
the endless flexible member is entrained, and all may be noncircular. In 
drives with just one driving and one driven noncircular member, the 
rotational member shapes can very often achieve the desired functionality 
and sufficient tautness throughout cycling of the drive. Drives according 
to the invention can be used for machinery in which there is desired a 
functional relationship that varies in speed, angular displacement, or 
mechanical advantage as the rotational members turn through a single cycle 
of operation. In other words, while the drive may accomplish three 
revolutions of the driven rotational member for each individual revolution 
of the drive member, the difference in speed between the two members and 
the mechanical advantage may vary with the angle of the rotational 
members, rather than maintaining a constant 3 to 1 ratio, as in drives 
with circular members. 
In the bicycle art it is known that it would be of advantage to have an 
elliptical driving sprocket that (1) imparts greater speed to the rear 
sprocket, but lower mechanical advantage, during each downward stroke of a 
pedal, where the rider more easily applies greater force, and that (2) has 
a higher mechanical advantage, but imparts less speed to the rear 
sprocket, at the bottom (and top) of each pedal's path, where the rider is 
less capable of applying force. Another application of the noncircular 
drive of the invention is a stationary exercise bicycle in which the force 
that must be applied to the pedal varies through the drive sprocket's 
rotation to achieve increased exercise benefit. The drives can find 
application in other exercise machines, wherein, as in known machines of 
other construction, the force required varies with the position of the 
member to which force is applied. Drives according to this invention can 
find application in shaker conveyors. Other applications are nonuniform 
motion transmissions using a chain, tape, belt, or band drives, or timing 
belts. And harmonic motion generators driven by drives according to this 
invention can have their operation appropriately refined by attention to 
the size and shape of the noncircular members. 
In the method of making drives pursuant to the invention, a central step is 
the establishment, by known techniques available from standard texts, of a 
noncircular gear pair. This gear pair, which is a design tool, not an 
actual physical gear pair, can be externally meshing with exterior pitch 
curves in contact, or it can be internally meshing with the exterior 
geared surface of the one gear engaging an interior geared surface of the 
other. One first determines the functional relationship desired from the 
ultimate noncircular drive. Then a gear pair that satisfies that 
relationship is designed using known techniques. Such techniques are 
reported, for example, in Bloomfield, "Noncircular Gears," Product 
Engineering, Mar. 14, 1960, incorporated herein by reference. Thereafter 
occurs a series of manipulative steps. Several are described in detail 
below. The gears are first separated a distance equal to the desired 
center-to-center distance between the noncircular rotational members of 
the drive being designed. In the case in which externally meshing gears 
are chosen for use in the design process, the pitch curves of the 
separated gears represent the shapes of the two rotational members. In the 
case of the internally meshing gears, the internal geared surface's pitch 
curve, and the meshed externally geared surface's pitch curve represent 
the shapes of the two rotational members. Spaced apart the desired 
distance, the rotational members' sizes are reduced uniformly, keeping 
their relative sizes, and thus their functional relationship. This is to 
reduce the variation in the inclination of the upper and lower runs of an 
endless member entrained about their exterior surfaces. In the case of a 
chain drive, the sprocket sizes, measured along the pitch curves of the 
sprockets, must also correspond to an integral number of chain pitches. 
Next in the manipulative steps, in the case in which one started with 
externally meshing noncircular gears of the desired functional 
relationship, each noncircular member is rotated through an angle such 
that the points on the gears that had been in contact now lie above the 
"center" , or axis of rotation of their respective noncircular members. 
One gear is flipped, or turned over, about an axis from its center to that 
previously contacting point that is now directly above its center. The 
gears are rotated slightly until a common tangent line can be drawn 
between the previously contacting points. This tangent represents the 
upper run of the endless member at this point in the design. The procedure 
can be effected, if desired, so as to first produce a lower common 
tangent, representing the lower run of the endless member in driving 
relationship between the noncircular members. This can be done by 
following the above procedure, but bringing the points that were in 
contact on the gears directly below the centers, flipping the one gear, 
and rotating slightly as needed to draw the tangent. 
In the case of the internally meshed gears that have been separated to 
define the sprockets' shapes at the desired distance, the gears are 
similarly manipulated, but without the step of flipping or turning over 
the one gear. In other words, the gears are again reduced in size 
proportionally, so as not to disturb the relationship therebetween, but 
reducing the variation in inclination of the upper and lower runs of the 
endless member that occur during operation, and being certain in the case 
of a chain drive to arrive at gear sizes equalling an integral number of 
chain pitches for a given chain. Again, rotating the noncircular members 
slightly as necessary, the points above the centers of the rotational 
members that had been the contacting points of the gears' pitch curves are 
connected by a tangent line representing the upper (or lower) run of the 
endless member. 
Using either of the above procedures, preferably using a suitably 
programmed computer, the functional relationship of the drive is examined 
throughout rotation of the noncircular members. This is first done with 
one run of the endless member driving. The error in the generated 
functional relationship is measured at each incremental rotation of the 
driving rotational member. The length of a taut endless member is measured 
at each incremental rotation, and the excess slack is calculated by 
subtracting the shortest length of a perfectly taut endless member in a 
complete cycle from the longest perfectly taut endless member in that 
cycle. If the error in the functional relationship, e.g., the error in 
angular displacement, is within acceptable bounds, and the length of the 
taut endless member is acceptable, the design procedure is complete. If 
the functional relationship error is excessive, then a second iteration of 
the design procedure is effected using a design functional relationship 
modified to lessen the functional error. The new design functional 
relationship can be the original, desired functional relationship plus the 
functional relationship error at each of the incremental angular positions 
where the functional relationship was observed. Again the functional 
relationship of the drive formulated in this iteration is compared to the 
original, desired functional relationship, and the error is measured at 
incremental angular positions of the driving rotational member. The slack 
variation is again measured in the same way. If both errors are acceptable 
the design is complete. If the functional relationship error is still 
excessive, a further iteration is undertaken, and so on until the 
functional relationship error is acceptable. 
If after the original incremental measuring routine, or any iteration 
thereof, the functional relationship error is acceptable, but the endless 
member slack variation is unacceptably large, then the drive is 
incremented throughout a complete cycle with the other run of the endless 
member driving. The same procedure is followed, i.e., measuring endless 
member slack variation and functional relationship error and iterative 
revision to bring functional relationship error within bounds. This can 
result in an acceptable drive, or repeated applications of these 
techniques, first with one run driving and then the other run driving, may 
be necessary until both functional relationship and slack variation are 
within bounds. 
There will be those functional relationships for which excessive slack 
cannot be eliminated. There will also be those design applications for 
which no amount of slack will be acceptable. In these cases, a tensioner, 
such as the mentioned moveable slack take-up roller or idler, may be used 
to eliminate slack virtually entirely. In the event a tensioner is 
employed, the described design process has minimized the amount of 
translational movement necessary for the roller to eliminate slack.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
In FIG. 1 a generalized noncircular drive 10 is shown. A first noncircular 
rotational member 1 and a second noncircular rotational member 2 are 
interconnected by an endless, elongate flexible member 30. In the 
preferred, exemplary embodiments described, the design steps and the 
resulting mechanical characteristics of the preferred embodiments of 
drives are applied to chain and sprocket drives, with the understanding 
that these same principles and mechanical characteristics enable the 
design and manufacture of various other drives such as belt, band, or tape 
drives. In chain and sprocket drives the noncircular members 1 and 2 are, 
of course, sprockets, and their surfaces 11 and 21, shown in the 
generalized drive of FIG. 1, are the pitch curves of the sprockets. In the 
case of belt, band or tape drives, in which the noncircular rotational 
members 10 and 20 are smooth surfaced rollers or pulleys, then the outer 
surfaces 11 and 21 of the noncircular members can be the smooth outer 
surfaces of the rotational members. If the drive is of another kind, say a 
toothed belt, then these surfaces may be the pitch curves for that 
particular type of rotational member and flexible endless member. The 
chain 30 has an upper run 31 that is assumed to be the driving run of the 
drive illustrated in FIG. 1. A lower run 32 is the "slack" or nondriving 
run of the chain. 
In FIG. 1: 
O.sub.1, O.sub.2 denote the "centers" of sprockets 1, 2, respectively, 
which is to say the locations of the axes about which the sprockets 
rotate. 
T.sub.1, T.sub.2 denote the points of tangency between the upper, tangent 
run T.sub.2 T.sub.1 of the taut endless member and the pitch curves of the 
sprockets 1, 2, respectively. 
V.sub.1, V.sub.2 denote the velocities of the points T.sub.1 , T.sub.2 on 
sprockets 1, 2 respectively. 
.omega..sub.1 .omega..sub.2 denote the angular velocities of sprockets 1, 
2, respectively, positive being counterclockwise (C.C.W.). 
.theta..sub.1, .theta..sub.2, denote the angular displacements of sprockets 
1, 2, repectively, measured C.C.W. from O.sub.1 O.sub.2, the line 
connecting centers O.sub.1 and O.sub.2 in the direction from O.sub.1 to 
O.sub.2. 
.alpha..sub.1, .alpha..sub.2 denote the angles between velocity vectors, 
V.sub.1, V.sub.2, and the directed span, T.sub.2 T.sub.1 measured C.C.W. 
from V.sub.1 and V.sub.2 to T.sub.2 T.sub.1. 
V denotes the component of belt velocity in the direction of the upper 
tangent vector T.sub.2 T.sub.1. 
P.sub.1, P.sub.2 denote the normals from points O.sub.1 and O.sub.2 to 
tangent T.sub.2 T.sub.1, or its extension. 
I denotes the points of intersection of O.sub.1 T.sub.1 and O.sub.2 
T.sub.2. 
##EQU1## 
Hence, we have theorem 1. 
Theorem 1 - The angular velocity ration W.sub.1 /W.sub.2 of the sprockets 
is equal to the inverse ration P.sub.2 /P.sub.1 of the normals from the 
sprocket centers to the span. 
The Instant Center in Noncircular Chain Drives 
To facilitate analysis of the motion of the sprocket 10 relative to the 
sprocket 2 in the chain drive shown in FIG. 1, FIG. 2 shows the drive with 
driven sprocket 2 fixed, the arm 40 (O.sub.1 O.sub.2) now rotating C.C.W. 
about O.sub.2, and the span 31 driving. The angular-velocity relationships 
between sprocket 1, arm 40, and sprocket 2, can be determined as in 
planetary gearing from the following table, considering the stationary 
sprocket 1 as the sun, and the moving sprocket 2 as the planet: 
TABLE 1 
______________________________________ 
Angular Velocities in Planetary Operation of Chain Drive 
Planet Arm 
Angular Displacement 
Sun (.omega..sub.PL) (.omega..sub.A) 
______________________________________ 
Motion with arm 
.omega..sub.A 
.omega..sub.A .omega..sub.A 
Motion relative to arm 
-.omega..sub.A 
##STR1## 0 
Sum (= actual motion) 
0 
##STR2## .omega..sub.A 
______________________________________ 
Hence, the angular velocity, .omega..sub.PL, of the planet is given by 
EQU .omega..sub.PL =.sup..omega..sub.A (1-p.sub.2 /p.sub.1). [5] 
P denotes the point of contact between the centrodes of the relative motion 
of the sprockets, as shown in FIG. 2. Since the velocity of arm 40 is 
perpendicular to line O.sub.1 O.sub.2, point P must be collinear with line 
O.sub.1 O.sub.2. Referring to FIG. 2, 
EQU let V.sub.p = velocity of point of contact P between centrodes; 
EQU V.sub.01 = velocity of center, O.sub.1, of sprocket 1; 
EQU and V.sub.02 = velocity of center, 02, of sprocket 2. 
EQU Then V.sub.p =0, V.sub.01 +V.sub.p/01 =0, [6] 
EQU where, V.sub.p/01 denotes the velocity of P relative to O.sub.1. 
EQU It follows from [6] that 
EQU V.sub.p =.omega..sub.A C +R.sub.1 .omega..sub.PL =O, [7] 
EQU where .omega..sub.A =angular velocity of "arm" O.sub.1 O.sub.2, 
EQU .omega..sub.A =C =angular velocity of "arm" O.sub.1 O.sub.2, 
EQU C =center distance, O.sub.1 O.sub.2, 
EQU and R.sub.1 =Po.sub.1. 
Solving [7]for R.sub.1 : 
##EQU2## 
Substituting [5] 
##EQU3## 
If, however, P is the point of intersection of lines O.sub.1 O.sub.2 and 
T.sub.1 T.sub.2, then from similar triangles in FIG. 2: 
##EQU4## 
which coincides with [9]. 
##EQU5## 
and: R=p.sub.1 /p.sub.2 = where O.sub.1, O.sub.2, denote the sprocket 
rotations, and R.sub.1 (O.sub.1), 
R.sub.2 (O.sub.2) the polar coordinates of the sprocket pitch curves. 
The same type of analysis is applicable when the left span T'.sub.1 
T'.sub.2 of FIG. 2 is driving. However, the point of intersection, P', of 
T'.sub.1 T'.sub.2 with O.sub.1 O.sub.2 does not in general coincide with 
P. It follows from equation [11c ] that the angular-velocity ratio of the 
sprockets when the left span is driving is not in general the same as that 
when the right span is driving. Hence, for a taut chain drive, it is 
necessary that points P and P' be coincident at all times. Summarizing, 
the following theorem has been established: 
Theorem 2 - In a taut chain drive the instant center of the relative motion 
of the sprockets is the point of intersection of the two chain spans and 
this point lies on the line of centers, O.sub.2 O.sub.2. 
The Analogy Between Noncircular Gears and Noncircular Chain Drives 
Suppose, however, that axes O.sub.1 and O.sub.2 of FIG. 2 were the axes of 
rotation of two noncircular gears having the same absolute angular 
displacements as the sprockets. The equations defining these gears are 
well known. See, for instance the above-referenced Product Engineering 
article by Bloomfield. They are precisely the equations [11b,c] just 
derived for the noncircular sprockets, except that R.sub.1, R.sub.2 now 
refer to the pitch radii of the noncircular gears. It follows that the two 
relationships are in fact identical, recognizing the fact that in both 
cases radii R.sub.1 and R.sub.2 are measured from the axes of rotation to 
the instant center of the relative motion between the rotating members. 
Summarizing, this provides the following theorem: 
Theorem 3 - To any chain drive with noncircular sprockets there corresponds 
a unique internal and external noncircular gear pair having the same 
center distance and the same absolute angular velocity relationship 
between the paired rotational members and the radii from the centers of 
rotation to the instant center of the relative motion of the noncircular 
members. 
Basic Principles of the Technique for Synthesizing the Sprocket Pitch 
Curves 
The kinematic correspondence established above between noncircular 
sprockets and noncircular gears can be used to fabricate noncircular chain 
and sprocket drives. Given a desired relationship between the sprocket 
rotations of a variable ratio chain drive, one synthesizes, using for 
example the techniques of standard texts or Bloomfield, supra., the pitch 
curves of a noncircular gear pair (either externally or internally 
meshing) satisfying the desired angular relationship. Several examples 
appear below. FIG. 3a illustrates the pitch curves of a generalized, 
externally meshing, noncircular gear pair 50. The sectors 53 and 54, shown 
in heavy solid lines in the figure, identify arcuate mating portions of 
the pitch curves of the two gears 51 and 52, respectively. These sectors 
are to facilitate illustration of the manipulative steps that follow. 
Points S.sub.1 and S.sub.2 are the corresponding points of contact on the 
gears. In the algorithm, which is embodied in Fortran in the program 
attached as Appendix A, the desired functional relationship is input 1. 
The gears are generated and the gears are separated to a desired sprocket 
center-to-center distance as shown in FIG. 3b. They are then rotated until 
points S.sub.l and S.sub.2 are vertically above gear "centers " O.sub.1, 
O.sub.2, the points (or axis locations) about which the gears rotate. The 
gears are now in the positions illustrated in FIG. 3c. Gear 51 is then 
rotated 180.degree. , i.e., flipped or turned over, about axis O.sub.1 
S.sub.1. The gears are then in the positions shown in FIG. 3d. The gears 
are rotated slightly, until points S.sub.1 and S.sub.2 are the points of 
tangency on a common tangent S.sub.1 S.sub.2 to the pitch curves of the 
two gears, as illustrated in FIG. 3e. The tangent is the center line of a 
taut driving run of a sprocket drive thus defined. 
The procedure using internally meshing, noncircular gears as the starting 
point is similar. These are shown in FIGS. 4a -4d. The gears are designed 
to provide the sought after functional relationship using known 
techniques, typified again by Bloomfield, supra., for example. The 
technique differs from that just described for externally meshing gears in 
that neither gear 62 or 61 in FIG. 4 is turned over. In the gear pair 60, 
the gears 61 and 62 are shown contacting at points S'.sub.1 and S'.sub.2 
in FIG. 4a. The pitch curve of gear 62, shown in FIGS. 4a -4d, it should 
be understood, is that of the internally geared surface of that gear. The 
desired center-to-center distance C of FIG. 4b is established by moving 
the gears apart. The interior shape of the gear 62 becomes the external 
shape of the pitch curve of the corresponding sprocket. The gears are 
rotated to bring S'.sub.1 over O.sub.1 and S".sub.2 over O.sub.2, as 
before. The gears are then as shown in FIG. 4 c. The gears are rotated as 
needed to enable a common tangent S'.sub.2 S'.sub.1, being established. 
This tangent, FIG. 4a, is representative of the chain center line. 
The sequence of contacting point pairs on the respective sprockets are now 
as nearly as possible identical with those on the noncircular gears from 
which they were derived. One can now begin to simulate operation of the 
drive of either FIG. 3e or FIG. 4d with, say, the upper tangent driving, 
again preferably by computer, as is facilitated by the appendix/program. 
If the distance C.sub.1 between centers O.sub.1 and O.sub.2 in either 
FIGS. 3e or 4d, were infinite, the conditions of Theorem 2 would be 
satisfied, inasmuch as any three parallel lines meet at a point at 
infinity. Thus we can summarize the preceding consideration as follows: 
Theorem 4 - A noncircular sprocket chain drive derived from noncircular 
gear pitch curves, as described with respect to FIG. 3 or FIG. 4, and with 
infinite center distance, synthesizes the desired relationship between 
sprocket rotations with a chain drive which is always taut. 
Application to Chain Drive Formulation 
If the center distance is finite, however, the slack in the chain will not 
in general vanish. Provided, however, that the ratio of center distance to 
average sprocket diameter is sufficiently large so that the angularity of 
the chain span does not vary too much, the variation in slack can be 
remarkably small, especially in the case of variable ratio drives with 
multiply periodic angular velocity ratios. 
For a given functional relationship between sprocket rotations, after 
having synthesized and reoriented a noncircular gear pair at the desired 
center distance and sprocket shapes as above, sprocket sizes are uniformly 
reduced until the ratio of average sprocket radius to center distance is 
approximately as desired. Then chain operation is simulated, with, for 
example, the upper tangent or chain run driving. The sprocket displacement 
function is determined, either via the instantaneous angular velocity 
ratio, or via other known kinematic characteristics. As described more 
fully below, the error between desired and generated angular displacement 
functions is then computed for an entire cycle of operation of the drive. 
By "cycle of operation" is meant one entire period of the operation of a 
drive that has a periodic function. Depending on the function, this could 
mean less than one, or more than one complete rotations of either or both 
sprockets, and it could mean less than or more than one complete excursion 
of the chain. 
Unless the displacement function error throughout a cycle of operation is 
below a specified limit, a new noncircular gear pair is generated to 
endeavor to bring the drive closer to the design functional relationship. 
An effective procedure for determining the new noncircular gear pair is to 
assume that the difference between the displacement functions of the 
previous (i.sup.th) gear pair and the new ((i+1).sup.st) gear pair should 
be equal to the observed error, which is to say the difference between 
design displacement function and the displacement function generated, as 
observed during the analysis of the previous (i.sup.th) design. With 
displacement function revised by addition of the error previously noted, 
this teen becomes the starting part for a new iteration of the design 
procedure ((i+1).sup.st). Based on this, a new gear pair is developed, 
again as in the texts or the Bloomfield article, and again the gears are 
manipulated, and the resulting sprockets are analytically driven through a 
cycle of operation, with the upper chain run driving. This process is 
repeated until the maximum displacement error between ideal generated 
sprocket displacement functions is less than a predetermined limit. The 
chain slack is analyzed. For a cycle of operation, the length of a 
perfectly taut chain, i.e. one in which there is no slack, is determined 
for each position of the sprockets. The minimum perfectly taut chain 
length is subtracted from the maximum perfectly taut chain length. The 
result is the variation in slack throughout one cycle of operation. If 
this exceeds a predetermined maximum, the same procedure of iterative 
revision of the sprockets can be repeated with the lower span driving. The 
iterative procedure is repeated with alternate spans driving until the 
functional error and slack variation are within limits. 
Pseudocode 
The following pseudocode utilizes a relatively simple approximation for the 
pitch curves. 
Let .theta..sub.2i =f.sub.ui (.theta..sub.1), i =0,1,2, .... [Al]=angular 
displacement of sprocket #2 (driven sprocket) when upper tangent is 
driving, i.sup.th iteration. 
f.sub.ui (.theta..sub.1) =functional relation between sprocket rotations 
(.theta..sub.1, .theta..sub.2) when upper tangent is driving, i.sup.th 
iteration. 
.DELTA..sup.u 2i =angular error, i.sup.th iteration, in position of driven 
sprocket, when upper tangent is driving. 
EQU .DELTA..sup.u.sub.2i =.theta..sub.20 (.theta..sub.1)-.theta..sup.u.sub.2i 
(.theta..sub.1) i=0,1,2, . . . , [A2 ] 
where 
.theta..sub.20 (.theta.) = desired functional relation between sprocket 
rotations, i.e., angular displacements, 
.theta..sup.g.sub.2i (.theta..sub.1) =function generated by noncircular 
gear pair, i.sup.th iteration, 
.theta..sub.1 = rotation or angular displacement of the drive sprocket 
(sprocket #1), 
.theta..sub.2 = rotation or angular displacement of the driven sprocket 
(sprocket #2), 
C = center distance. 
R.sup.g.sub.1 = active pitch radius of a first gear of a noncircular gear 
pair. 
R.sup.g.sub.2 = active pitch radius of a second gear of a noncircular gear 
pair. 
.theta..sup.g.sub.1 = the polar angle to R.sup.g.sub.1 measured 
counterclockwise on the first gear of the noncircular gear pair from the 
line O.sub.1 O.sub.2 between the points about which the noncircular gears 
rotate as shown in FIG. 3a. 
.theta..sup.g.sub.2 = the polar angle to R.sup.g.sub.2 measured clockwise 
on the second gear of the noncircular gear pair from the line O.sub.1 
O.sub.2 between the points about which the noncircular gears rotate as 
shown in FIG. 3a. 
C.sup.g = the center distance O.sub.1 to O.sub.2 of the gear pair as shown 
in FIG. 3a. 
Design procedure 
1. First using conventional techniques, a noncircular gear pair is designed 
having the functional relationship desired for the ultimate sprocket 
drive. The desired functional relationship can and often will be a desired 
angular relationship between sprockets. In which case, it is given that 
the desired functional relationship: 
EQU .theta..sup.g.sub.2i (.theta..sub.1)=f(.theta..sub.1) , say, for 
.theta..sub.1 = j (degrees), J = 1.degree. , 2.degree. ,..., 
360.degree. and center distance, C. Determine the pitch curves, 
[R.sup.g.sub.li (.theta..sub.1) and R.sup.g.sub.2i (.theta..sub.2)] of the 
noncircular gear pair as follows: 
##EQU6## 
EQU and R.sup.g.sub.2 (.theta..sup.g.sub.1) = C.sup.g -R.sup.g.sub.1 
(.theta..sup.g.sub.1), [4] 
where the first derivative, 
##EQU7## 
Alternatively, various techniques can be used to refine the gear design, to 
the extent desired. For example, the pitch curves can be approximated by 
polygons or by continuous curves with continuous derivatives, such as 
polynomial approximations, spline functions or otherwise. 
Moreover, although the functional relationship is expressed above in terms 
of angular displacement .theta., the gear pair design can be determined 
based upon, for example, the functional relationship expressed in terms 
of, say, angular velocity .omega., again, as in Bloomfield, supra. 2. To 
reorient the gears to the positions thereof illustrated in FIG. 3(d): 
##EQU8## 
Reduce the pitch curves to the desired ratio of center distance to average 
sprocket pitch diameter. The initial configuration and distance between 
the sprockets have now been established. They are, as in FIG. 3d, nearly 
in their relative rotational position to be connected by the chain. In 
this case the sprockets have been separated by the reduction in size of 
the gear originally designed to have the same center distance C as that 
desired for the chain drive. 3. To determine initial tangent and exact 
starting position of sprockets, as in FIG. 3e for example, 
(i) Let .theta..sub.li = .theta..sub.2i = 90 .degree. . 
(ii) Increment .theta..sub.li .fwdarw..theta..sub.li +.DELTA..sub.1, such 
that tangent, T.sub.li , of sprocket 1 (gear 51) at R.sup.g.sub.2i just 
intersects FIG. 5(a). T.sub.li is not yet tangent to sprocket 2, FIG. 
5a.sub.1. 
(iii) Increment .theta..sub.2i 0.sub.2i +.DELTA..sub.2, such that tangent, 
T.sub.2i, of sprocket 2 (gear 52) at R.sup.g.sub.2i just intersects 
R.sup.g.sub.li, as in FIG. 5b. T.sub.2i is not yet tangent to sprocket 1, 
FIG. 5b.sub.1. 
Iterate until the initial tangent is determined, i.e. until all points on 
the pitch curve (other than the points of tangency) lie below the upper 
tangent, measuring normal to the tangent. The initial sprocket drive, 
upper run driving, FIG. 3e, has now been accomplished. This will be used 
for the first, i=0, analytical operation of the drive through at least one 
complete cycle. 
4. Sprocket 1 is rotated in increments j of 1.degree. from the starting 
position and locations of tangents are determined as in step 3. 
5. Coordinates are determined for point of intersection, P.sub.j, FIG. 2, 
of common tangent with 0.sub.1 0.sub.2 for j=1.degree. ,2.degree. , ... at 
least through a cycle of operation. 
6. Determine 
##EQU9## 
through at least a cycle of operation, and (via numerical integration) 
determine the angular displacements .theta..sub.2j (.theta..sub.1) of 
sprocket 2 for each of j=1.degree. ,2.degree. , ... through a cycle of 
operation. 
7. Knowing the tangent points for taut upper and lower runs, determine the 
length of upper and lower spans and the lengths of the portions of chain 
that engage the sprockets, and hence determine the length of a taut chain 
belt for each j=1.degree., 2.degree. , ... through a cycle of operation. 
Determine by subtraction of minimum from maximum the excess chain length. 
Determine functional relationship error, .theta..sub.20 
-.theta..sup.u.sub.2i (.theta..sub.1). 
8. Increment i.fwdarw.(i+1) with 
EQU .theta..sup.g (i) =.sup.g 2i +.theta.20 -.theta..sup.u 2i (.theta..sub.1) 
[A7] 
and iterate until 
EQU .theta..sup.u.sub.2i (.theta..sub.1) -.theta..sub.20 (.theta..sub.1) 
.gtoreq..epsilon., [A8 ] 
where .epsilon. is a predetermined error bound. 
9. If the excess chain length exceeds a predetermined limit, return to step 
1 with lower span driving and proceed through steps 1-9. 
10. Alternate between upper and lower span driving, redesigning the gears 
and then the sprockets as above, until excess chain length is minimized. 
11. Ratio size of system uniformly so that the two pitch curves correspond 
to an integral number of teeth. Of course this step is unnecessary when 
the above procedure is being applied to a belt or band drive with smooth 
rollers or pulleys. 
Note that steps 10-11 are optional depending on desired degree of reduction 
of chain slack. The steps of the above pseudocode are accomplished in 
Fortran in the attached program of Appendix A. The flow chart of FIG. 13 
is illustrative of the overall procedure. Although the above evaluations 
for functional relationships, i.e. displacement, error and slack is said 
to cover a whole cycle of operation, symmetry may make evaluation through 
a lesser portion of the cycle sufficient. 
Applications 
The process of making chain drives as described enables design of bicycle 
drives providing a variable-ratio drive for optimizing short term power or 
long-term power for aerobic exercise purposes. Such a bicycle drive 70 is 
shown schematically in FIG. 6. A pedal assembly has typical pedals 71, 
attached in driving relation to a drive sprocket 72. The drive sprocket 72 
is connected to a driven sprocket 73, via a chain 75, entrained about the 
two sprockets. A bicycle frame 76 establishes the center-to-center 
distance, called C above. Of course, the driven sprocket is connected to 
apply power to the rear wheel. The pitch curve of the drive sprocket 72 is 
twice as long as that of the driven sprocket 73. 
In comparison with a representative constant bicycle reduction ratio of 
40:18, for example, the speed-ratio variation obtained in the 
variable-ratio drive of noncircular sprockets provides an opportunity for 
maximizing power during that part of the pedaling cycle for which the 
mechanical advantage is a maximum, or for other desired performance 
characteristics. FIG. 7a illustrates schematically and more accurately 
pitch curves 77 and 78 and chain span center line 79 of the noncircular 
chain drive 70 of FIG. 6, with the sprockets 72 and 73 based on fourth and 
second order ellipses satisfying the displacement relationship for a 
fourth and second order noncircular gear pair. The center-to-center 
distance is 17.5", in all of the following examples, and the ratio of 
center distance to average sprocket diameter is 4.3. FIG. 7b shows the 
corresponding angular velocity variation, which is quadruply periodic in 
the rotation of the fourth-order gear, the variation being nearly 
sinusoidal. FIG. 7c shows the corresponding variation in taut chain 
length, which is less than one thousandth of the length of the chain. The 
angular displacement error is shown in FIG. 7d, both initially based on 
the original gear pair design (0.sup.th iteration) and after one design 
revision (lst iteration). The lst iteration maximum angular displacement 
error .gtoreq.0.07.degree.. 
FIGS. 8,a,b,c,d illustrate the results for a noncircular chain drive 80 
with sprockets 81 and 82 based on second-order ellipses as shown in FIG. 
8a. The difference between the angular velocity ratio is too small to be 
appreciable in a plot the scale of FIG. 8b. Slack variation of little more 
than 1/4 taut chain length fluctuation out of an overall chain length of 
slightly over 46 inches is apparent from FIG. 8c. And the considerable 
reduction in angular displacement error achieved with just one iteration 
of the design procedure (one revised set of noncircular gears) is 
illustrated in FIG. 8d. 
A drive 90 with sprockets 91 and 92 based on second-order and first-order 
ellipses is illustrated in FIG. 9a. Again, angular velocity ratio error is 
too small to be apparent on the graphical illustration of FIG. 9b. Again, 
the variation in taut chain length over the course of a cycle of 
operation, FIG. 9c is slightly in excess of a quarter inch for a total 
chain length of about 46 inches. The dramatic reduction in displacement 
error after one iteration and the appreciable further reduction by a 
second iteration appears in FIG. 9d. Tables 1 a,b,c summarize the chain 
drive characteristics of the drives of FIGS. 7a, 8a and 9a. 
TABLE 1A 
__________________________________________________________________________ 
CHARACTERISTICS OF NC GEAR PAIRS GENERATING DESIRED 
ANGULAR-DISPLACEMENT FUNCTION 
MAX/MIN MAX/MIN 
NC GEAR PAIR CENTER RADII, RADII, 
TYPE DISTANCE 
LARGER GEAR 
SMALLER GEAR 
__________________________________________________________________________ 
4TH AND 2ND 
ORDER, ELLIPTICAL 
1.392" 0.972"/0.882" 
0.510"/0.420" 
2ND AND 
2ND ORDER ELLIPTICAL 
0.930 0.510"/0.420" 
0.510"/0.420" 
2ND AND 
FIRST ORDER ELLIPTICAL 
1.470" 1.110"/0.830" 
0.640"/0.360" 
HARMONIC-MOTION 
GENERATOR GEARS 5.370"/4.731" 
5.290"/4.650" 
__________________________________________________________________________ 
TABLE 1B 
__________________________________________________________________________ 
NC SPROCKET CHARACTERISTICS 
MAX/MIN MAX/MIN 
NC GEAR PAIR CENTER RADII, RADII, 
TYPE DISTANCE 
DRIVE SPROCKET 
DRIVE SPROCKET 
__________________________________________________________________________ 
4TH AND 2ND 
ORDER, ELLIPTICAL 
17.5" 2.842"/2.579" 
1.491/1.227" 
2ND AND 
2ND ORDER ELLIPTICAL 
17.5" 2.232"/1.836" 
2.233"/1.838" 
2ND AND 
FIRST ORDER ELLIPTICAL 
17.5" 2.465"/1.844" 
1.420"/0.799" 
HARMONIC-MOTION 
GENERATOR 17.5" 2.187"/1.921" 
2.148"/1.889" 
__________________________________________________________________________ 
TABLE 1C 
__________________________________________________________________________ 
NC CHAIN DRIVE CHARACTERISTICS 
E/L 
MAX ANGULAR 
E = MAX. EXCESS 
ANGULAR DISPLACEMENT 
CHAIN LENGTH 
DISPLACEMENT ERROR (E.sub.MIN = 0) 
FUNCTION TYPE (DEGREES) L = CHAIN LENGTH) 
MAX. SAG. 
__________________________________________________________________________ 
4TH AND 2ND 
ORDER, ELLIPTICAL 
.ltoreq.0.07.degree. 
1/1075 0.51" 
(2.93%) 
2ND AND 
2ND ORDER ELLIPTICAL 
.ltoreq.0.04.degree. 
1/184.3 1.28" 
(7.3%) 
2ND AND 
FIRST ORDER ELLIPTICAL 
.ltoreq.0.09.degree. 
1/34.6 2.90" 
(16.6%) 
HARMONIC-MOTION 
GENERATOR GEARS .ltoreq.0.01.degree. 
1/917.9 0.58" 
(3.3%) 
__________________________________________________________________________ 
In these chain drives it can be observed that the displacement error even 
without iteration is often less than one degree. The known displacement 
equations used for the initial design of the noncircular gears that serve 
as the starting point from which the sprockets for each of the foregoing 
exemplary sprocket shapes are as follows. 
The Pitch Curves [R.sub.1 (.theta..sup.G.sub.1)[ of 1st, 2nd and 4th Order 
Noncircular Driven Gears: 
For the 1st order elliptical drive gear: 
##EQU10## 
For the 2nd-order elliptical drive gear: 
##EQU11## 
For the 4th order elliptical drive gear: 
##EQU12## 
The above is expressed generally as: 
EQU Drive gear: R.sub.1 =f(.theta..sup.g.sub.l) [5] 
The pitch curve for the driven gear is: 
EQU R.sub.2 (.theta..sup.g.sub.2) C-R.sub.1 C-f(.theta..sup.g.sub.1), [6] 
where C - center distance. For given values of C and R.sub.1, and drive 
gear rotation (.theta..sub.1), one can obtain the displacement equation of 
the driven gear by solving for .theta..sub.2 and substituting .sup.*.sub.1 
from eq. [B8] below with 4th and 2nd order drive gears, as follows: 
##EQU13## 
where N =1, for second-order gears, or 2, for fourth-order gears. 
##EQU14## 
It should be noted that, due to symmetry, equation [B7] is an example of an 
equation that can be evaluated only for the range .ltoreq..theta.1 
.ltoreq.90.degree., rather than for a full cycle of operation, to arrive 
at displacement error and slack in the noncircular drive design. A further 
example of an application of a noncircular drive is to obtain a frequently 
desired objective in the general machinery field, the generation of a 
purely sinusoidal reciprocating or oscillating motion. For this purpose 
one modifies the standard slider-crank mechanism by adding a noncircular 
chain drive 100, as shown in FIG. 10. The displacement function that needs 
to be generated by the noncircular gears is derived as follows: 
EQU x =rcos.theta..sub.2 +l [ cos.phi. [C1] 
EQU rsin.theta.=l sin.phi. [C2] 
By differentiation, 
EQU r.theta..sub.2 cos.sub.2 =l.phi. cos.phi. [C3] 
##EQU15## 
EQU Let x -x.sub.ideal =rcos.theta..sub.1 +l.sup.* (say), [C5] 
EQU where l.sup.* is a constant. 
EQU .thrfore. rcos.theta..sub.2 +lcos.phi.=rcos.theta..sub.1 +l.sup.*. [C6] 
EQU Substituting [C2] into [C6], 
##EQU16## 
EQU When .theta..sub.2 =0, .theta..sub.1 =0. 
EQU Hence, l=l.sup.* [C8] 
EQU From [C7] we have 
##EQU17## 
An approximate formula can also be obtained as follows: 
##EQU18## 
Higher harmonics can be determined, if desired, by expanding the square 
root term in eq. [C13] in a power series, if desired. 
As can be seen, the resulting piston displacement is a pure harmonic 
function of crank rotation and the noncircular sprockets have very 
favorable proportions. Tables, 1a,b,c, summarize dimensional data, 
angular-displacement errors and chain slack for this case as well. The 
drive 100 that produces essentially the displacement function of equation 
[C15] is more precisely shown in FIG. 11a. The angular velocity ratio of 
the driven sprocket 102 to the drive sprocket 101 is shown in FIG. 11b. 
The variation in taut chain length appears in FIG. 11c. The angular 
displacement error for the 0th and for the first iteration is shown in 
Fig. llq. And the slider displacement versus drive sprocket rotation 
appears in FIG. 11e. 
Chain Slack 
According to the American Chain Association the permissible chain sag (S) 
should be between 2% and 3% of center distance (C). If E denotes the 
excess chain length, the sag, S, is determined from the equation: 
EQU S=.sqroot.0.375CE. [12] 
For the cases considered in tables 1a, b, and c, in which the center 
distance is 17.5", the excess chain length (E) varies from a minimum of 
0.045"(corresponding to a ratio, R, of excess to belt length of 1/1075) 
for the 4-2 ellipses, to a maximum of 1.287"for the 2:1 ellipses 
(R=1/34.6), while for the harmonic motion generator the excess length of 
0.052" corresponds to a ratio of excess to belt length of R=1/918. 
It is readily seen that the 4-2 ellipses and the harmonic generator have 
the least slack (less than 0.2% of belt length). Considering the ratio 
changes involved, their periodicity and the ratio of average sprocket size 
to center distance, this is a remarkably low figure. 
Referring to Table 1c, the recommended sag (corresponding to an excess 
chain length of 2-3% of center distance) is 0.52". The sag for the 4-2 
ellipsis and the harmonic motion generator are close to this value, while 
for the others it is not. The excess chain length, however, now need to be 
increased so that during operation the chain does not become too tight. 
According to eq. 12 we can increase the sag in each case so that E.sub.min 
=0.041, for example. This then yields the following values for the maximum 
sag: 
EQU 4-2 ellipsis: S=0.751"; E.sub.max =0.086". 
EQU 2-2 ellipses: S=1.382"; E.sub.max -0.191". 
EQU 2-1 ellipsis: S=2.95"; E.sub.max =1.328". 
Harmonic Motion Generator: S=0.781"; E.sub.max =0.093". 
The sag for the 4-2 ellipses and the harmonic motion generator remain 
reasonably close to the previously listed values. The 4-2 ellipses and the 
harmonic generator may or may not need a tensioner, (depending on speed 
and loads). If a tensioner is needed, however, its excursion has been 
minimized. 
Suitable tensioners are known and include a movable mount carrying one or 
more slack take-up idlers and biased in the slack take-up direction. FIG. 
12 illustrates schematically a drive 110 having a simple idler 115 of this 
kind. Other slack take-up arrangements can be employed with drives 
embodying the invention. Generally, the greater the rate of change of the 
drive ratio, the more likely it is that a tensioner will be needed. In any 
case in which it is desired that chain slack be completely eliminated, 
regardless of ratio variation, this can be accomplished by the addition of 
a suitably designed tensioner. 
Conclusion 
The program that has been developed and that is appended demonstrates the 
feasibility of designing variable ratio chain drives with minimum slack. 
Provided the angular velocity variation is not too large and the ratio of 
average sprocket size to center distance is acceptable, such drives can be 
designed for a great variety of performance requirements. The attached 
program has been accomplished in FORTRAN on an IBM P/C AT, and the 
underlying analogy with noncircular gearing has proven to be remarkably 
effective in minimizing chain slack. 
An average run of the program, simulating drive operation with upper 
tangent driving and two iterations, takes approximately 5-6 minutes, and 
in many cases iterations may not be necessary. The algorithm can be used 
in the design of any application in which a periodic variable ratio is 
desired. Sprocket geometry is believed sufficiently robust so that, 
provided the departure from a circular pitch curve is not too great, the 
conventional method of tooth generation remains applicable, just as in 
noncircular gearing. Care must be taken that the pitch curves are 
sufficiently convex so as to prevent disengagement between chain and 
sprocket and to maintain favorable force transmission. While the chain can 
be designed to have hunting-tooth engagement with either or both 
sprockets, the relative position of the sprocket teeth is governed by 
their periodicity. This invention is applicable to any drive rotational 
members the pitch curves of which (including smooth outer surfaces of 
rollers, pulleys, etc.) are noncircular plane curves. 
While specific preferred embodiments of the invention have been illustrated 
and described, it will be appreciated that numerous variations in the 
drives and methods of this invention can be made without departure from 
the spirit and scope of the invention as defined in the appended claims.