Abstract:
A sliding friction-free gear has cams of specific profiles, rollers separating cooperating cams that roll over the cams of the gears, thus eliminating the sliding friction between the gears. The rollers are mounted rotatably e.g. in sliders or eccentrics, wherein the sliders, resp. eccentrics, are mounted slidingly, resp. rotatably, in a yoke attached rigidly to a shaft. The rollers are free to execute oscillating motion relative the yoke while rolling over the cam profiles of the cams of the gears. The cooperating members of the gear contact along a line like in conventional evolvent gears, and unlike in the Wildhaber-Novikov&#39;s gear (where the momentary contact between teeth of cooperating toothed wheels is pointwise); therefore the gear according to the present invention offers much larger transmission capacity and is much less prone to seizure than known gears.

Description:
TECHNICAL FIELD OF THE INVENTION 
       [0001]    The invention relates to gears, and is particularly concerned with a method for elimination of sliding friction between cooperating gear members. 
       STATE OF THE ART AND BACKGROUND OF THE INVENTION 
       [0002]    Power transmitting devices known in the prior art are naturally divided into the following four categories: friction drives, hydraulic (hydrostatic or hydrodynamic) transmissions, electric transmissions, and cogwheels or gears. All these drives have important disadvantages. Thus friction drives that depend upon friction force acting between moving parts in contact to transmit motion have very limited transmission capacity and therefore are usually bulky and heavy, and suffer from slippage between cooperating members which diminishes their durability and efficiency, and renders their reliability questionable. Hydraulic transmissions are capable of coping with large loads while retaining compact size, but are complicated and expensive, and feature poor efficiency and not always satisfactory reliability. Electric transmissions are complicated, heavy and expensive, and definitely not suitable for application in high power drive lines of aerial vehicles. Gears are far superior to all aforementioned power transmitting devices in terms of transmission capacity/weight, efficiency and reliability, and are used as the only power transmitting devices in aviation drives (with exception of those of very small power). However also gears have some disadvantages: Cooperating gear teeth slide against each other (rarely used Wildhaber-Novikov&#39;s meshing being an exception), which diminishes gear&#39;s transmission capacity and reliability, and causes gears require good lubrication, as lack of lubrication leads to seizure and catastrophic failure. Recent trends in large bypass ratio turbofan engines technology (large diameter fans driven by turbine via a gear) impose particularly high demands on the gears as far as reliability, safety and transmission capacity are concerned. 
         [0003]    Gears, in which teeth on the pinion are replaced by rollers, as described e.g. in U.S. Pat. No. 5,247,847 haven&#39;t earned wider acceptance, and rollers that cyclically engage and disengage spur gear teeth seems to be the main reason of this state of affairs. 
         [0004]    Thus there is a need for a compact, lightweight, extremely reliable gear, insusceptible to seizure, capable of meeting stringent requirements of aviation industry. 
       SUMMARY OF THE INVENTION 
       [0005]    Thus the principal objective of the present invention is to provide a gear that outperforms known power transmitting devices particularly in aspects of compactness, lightness, reliability, insusceptibility to seizure, and transmission capacity, thus being capable of meeting most stringent requirements of aviation industry. 
         [0006]    Another objective of the present invention is to provide a one stage gear of compact structure having large transmission ratio (16:1 to 50:1 and more). 
         [0007]    These and other objectives are achieved according to the present invention by providing sliding friction-free gears, the cooperating members of which contact along a line like in conventional evolvent gears, and unlike Wildhaber-Novikov&#39;s gear (where the momentary contact between teeth of cooperating toothed wheels is pointwise), which therefore offer much larger transmission capacity and are much less prone to seizure than known gears. 
         [0008]    The method for achieving sliding friction-free gears according to the present invention consists in providing gears with a specific toothing (or rather cams of specific profiles, as described below and claimed in separate patent claims), and separating cooperating gears by rollers that roll over the cams of the gears, thus eliminating the sliding friction between the gears. The rollers are mounted rotatably e.g. in sliders or eccentrics, wherein the sliders, resp. eccentrics, are mounted slidingly, resp. rotatably, in a yoke. The rollers are free to execute oscillating motion relative the yoke while rolling over the cam profiles of the cams of the gears. 
         [0009]    As mentioned above, thanks to the virtual absence of the sliding friction between the transmission elements meeting along the lines, the gear of the present invention features substantially greater transmission capacity than ordinary gears of the same dimensions. Thus the gear according to the present invention is expected to offer substantial gains in transmission&#39;s capacity/weight and transmission&#39;s capacity/size ratios and transmission&#39;s reliability (e.g. ability to work properly in the absence of lubrication) in comparison with conventional gears. 
         [0010]    An interesting and important advantage of the gear according to the present invention is that the rollers separating cams remain in constant contact with cams, which contributes to the gear reliability and durability; moreover the rollers do the job of bearing, which renders separate bearings supporting output shaft unnecessary. 
         [0011]    As indicated above, the gear of the instant invention is destined mainly for aviation drives (e.g. geared turbofan engines like Pratt&amp;Whitney&#39;s “Pure Power” engine, final drives of rotorcrafts) and other most demanding applications like drivelines of high speed military vessels, ferries, hydrofoils, hovercrafts and pleasure crafts, and highly loaded winches. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0012]      FIG. 1  is a general view of a gear according to the present invention. 
           [0013]      FIG. 2  is an exploded view of the gear according to the present invention showing its all principal elements. 
           [0014]      FIG. 3  is a perspective view of the gear according to the present invention with a section of the gear body removed. 
           [0015]      FIG. 4 . is a cut-away section of the gear according to the present invention showing a roller&#39;s yoke. 
           [0016]      FIG. 5 . is a frontal view of the gear according to the present invention with a section of the body removed. 
           [0017]      FIG. 6  is a perspective/frontal view of the gear according to the present invention with a section of the body and a section of a roller&#39;s yoke removed. 
           [0018]      FIG. 7  is another cut-away view of the gear according to the present invention showing the arrangement of its internal parts. 
           [0019]      FIG. 8  is another cut away view of the gear according to the present invention with a section of the body and a section of the roller&#39;s yoke removed. 
           [0020]      FIG. 9  is a transverse section of the gear according to the present invention showing an assembly of the rollers and a cam. 
           [0021]      FIG. 10  is another transverse section of the gear according to the present invention showing an assembly of the rollers and an input shaft eccentric. 
           [0022]      FIG. 11  is a general view of an assembly of the roller&#39;s yoke and an output shaft. 
           [0023]      FIG. 12  is a general perspective view of the roller&#39;s yoke with the rollers removed. 
           [0024]      FIG. 13  is an exploded view of the assembly of the yoke and the output shaft. 
           [0025]      FIG. 14  is a general perspective view of an assembly of the rollers and associated sliders. 
           [0026]      FIG. 15  is an exploded view of the assembly of the rollers and associated sliders. 
           [0027]      FIG. 16  is a frontal perspective view of another version of the gear according to the present invention with a section of the body removed and differently arranged rollers. 
           [0028]      FIG. 17  shows the assembly of the rollers and the cam of the other version of the gear according to the present invention. 
           [0029]      FIG. 18  is a perspective cut-away view of yet another version of the gear according to the instant invention showing its internal elements. 
           [0030]      FIG. 19  is a perspective partially exploded view of said version of the gear according to the present invention with the assembly of rollers and the output shaft ejected from the body. 
           [0031]      FIG. 20  is another perspective partially exploded view of said version of the gear according to the present invention with the assembly of rollers and the output shaft ejected from the body. 
           [0032]      FIG. 21  is a partially exploded view of the assembly of rollers and the output shaft of said gear according to the instant invention with one roller assembly offset from the output shaft. 
           [0033]      FIG. 22  is a longitudinal cross-section of said version of the gear according to the present invention. 
           [0034]      FIG. 23  is an exploded view of the roller assembly of said version of the gear according to the present invention. 
           [0035]      FIG. 24  is a perspective view of yet another version of the gear according to the present invention partially disassembled. 
           [0036]      FIG. 2S  is another perspective view of the version of the gear shown in  FIG. 24  with one section of the body removed, and with the output shaft offset from the body. 
           [0037]      FIG. 26  is a perspective cut-away view of the version of the gear according to the present invention illustrated in  FIGS. 24, 25 . 
           [0038]      FIG. 27  is another perspective cut-away view of said version of the gear according to the present invention showing an assembly of input and output shafts, rollers and levers. 
           [0039]      FIG. 28  is yet another perspective cut-away view of said version of the gear according to the present invention exhibiting the assembly of input and output shafts and rollers, with levers removed, and showing a lever&#39;s socked. 
           [0040]      FIG. 29  is another perspective view of the last version of the gear according to the present invention with input and output shafts offset from the gear body. 
           [0041]      FIG. 30  is a general view of the assembly of input and output shafts, rollers and levers of the last version of the gear according to the instant invention. 
           [0042]      FIG. 31  is a side view of the assembly of two main rollers and the associated lever. 
           [0043]      FIG. 32  is another view of the assembly of two main rollers and the associated lever shown in  FIG. 31 . 
           [0044]      FIG. 33  is an exploded view of the assembly of two main rollers and the associated lever; 
           [0045]      FIG. 34  is a general view of a gear being the fourth preferred embodiment of the present invention; 
           [0046]      FIG. 35  is another general view of the gear being the fourth preferred embodiment of the present invention; 
           [0047]      FIG. 36  is an exploded view of the gear being the fourth preferred embodiment of the present invention; 
           [0048]      FIG. 37  is another exploded view of the gear being the fourth preferred embodiment of the present invention; 
           [0049]      FIG. 38  is a view of the gear being the fourth preferred embodiment of the present invention partially assembled; 
           [0050]      FIG. 39  is another view of the gear being the fourth preferred embodiment of the present invention partially assembled; 
           [0051]      FIG. 40  shows the assembly of input shaft, rollers and output shaft of the gear being the fourth preferred embodiment of the present invention; 
           [0052]      FIG. 41  is a cut-away section of the gear being the fourth preferred embodiment of the present invention; 
           [0053]      FIG. 42  is a transverse cross section of the gear being the fourth preferred embodiment of the present invention showing intermediate rollers meeting a cam placed on the output shaft, and secondary rollers meeting an eccentric placed on an input shaft; 
           [0054]      FIG. 43  is another transverse cross section of the gear being the fourth preferred embodiment of the present invention showing primary rollers meeting a cam placed in the gear body, and intermediate rollers meeting a cam placed on the output shaft; 
           [0055]      FIG. 44  is a general view of a rollers assembly; 
           [0056]      FIG. 45  is an exploded view of the rollers assembly; 
           [0057]      FIG. 46  is an auxiliary drawing explaining main geometric dependences between various geometric parameters of profiles of cams. 
       
    
    
       [0058]    Like numerals denote like gear&#39;s elements throughout all the drawings, where:
       Numeral  10  refers generally to the gear of the instant invention;   numeral  11  refers generally to the gear body;   numerals  111 ,  112 ,  113 ,  114 , and  115  indicate the body parts;   numeral  1121 , resp.  1141 , indicates a cam placed in the part  112 , resp.  114 , of the gear body  11 .   numeral  12  indicates the input shaft;   numeral  121  indicates an eccentric on the input shaft  12 ;   numeral  13  refers to the output shaft;   numeral  131  indicates a disc attached to the output shaft;   numeral  16  refers to primary rollers;   numeral  161  indicates secondary rollers;   numerals  151  and  152  refer to sliders supporting rollers  16 ;   numeral  14  refers generally to the roller&#39;s yoke;   numerals  141 ,  142 ,  143 ,  144 , and  145  indicate parts of the yoke  14 ;   numeral  1411 , resp. numeral  1451 , refers to sockets placed in the part  141 , resp.  145 , of the yoke  14 , and accommodating sliders  151 , resp.  152 ;   numeral  1421 , resp.  1441 , indicates recesses placed in the part  142 , resp.  144 , of the yoke  14  and accommodating the primary rollers  16 ;   numeral  1431  refers to recesses placed in the part  143  of the yoke  14  and accommodating the secondary rollers  161 ;   numeral  162  refers generally to some other rollers of a second preferred embodiment of the invention;   numeral  17  refers to axles supporting rollers  161  and  162  of the second preferred embodiment of the invention;   numeral  18  indicates eccentrics supporting axles  17 ;   numeral  19  refers to webs joining axles  17  with eccentrics  18 ;   numeral  132  refers to a cam attached to the output shaft  13  of a third preferred embodiment of the invention;   numeral  23  indicates levers of the third preferred embodiment of the invention;   numeral  21 , resp.  22 , indicates generally to holders supporting rollers  161 , resp.  162  of the third embodiment of the invention;   numeral  116  refers to sockets placed in body  11  of the third preferred embodiment of the invention that accommodating levers  23 ;   numeral  163  refers to auxiliary rollers used to mount pivotally holders  21  and  22  in the levers  23 ;   numeral  231 , resp.  232 , refers to sockets placed in the levers  23  and accommodating auxiliary rollers  163 ;   numeral  211 , resp.  221 , indicates sockets placed in the holders  21 , resp.  22 , and accommodating auxiliary rollers  163 .       
 
       DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     First Embodiment 
     Variant A (FIGS.  1 - 15 ) 
       [0086]    A gear  10  according to the present invention, being the first preferred embodiment of the invention (see  FIGS. 1-17 , and particularly  FIGS. 1-15 ), has a body  11 , an input shaft  12 , an output shaft  13 , and a roller&#39;s yoke  14  ( FIGS. 1, 2 ). Body  11  is composed of five parts  111 ,  112 ,  113 ,  114 , and  115  joined by screws (not shown) or in any other suitable manner ( FIGS. 1, 2, 3, 4, 7, 8 ). Input shaft  12  is supported pivotally in body part  111 ; output shaft  13  is supported pivotally in body part  115 , coaxially with the input shaft  12  ( FIGS. 2, 7, 8 ). Output shaft  13  has a disc  131  fastened thereon ( FIGS. 2, 7, 11 ). Roller&#39;s yoke  14  is fastened with the help of screws (not shown), or in any other suitable manner, to the disc  113 . Roller&#39;s yoke  14  is composed of two principal elements  141 , and  145 , and three intermediate elements  142 ,  143 ,  144 , connected together by screws (not shown), or in any other suitable manner (FIGS.  7 ,  8 , and particularly  FIGS. 11, 12, 13 ). Arranged at equal angular distances there is a plurality (in this case  15 ) of roller assemblies ( FIGS. 7, 8, 13 ) mounted movably in yoke  14 , each roller assembly being composed of a primary roller  16 , a secondary roller  161 , and two sliders  151  and  152  ( FIGS. 14, 15 ). Primary rollers  16  are supported at their both ends in sliders  151 ,  152  in any suitable manner, e.g. using needle bearings (not shown); similarly, secondary rollers  161  are mounted rotatably on their respective primary rollers  16  in any suitable manner, e.g. with the help of needle bearings (not shown). Sliders  151  are mounted movably in their respective sockets  1411  placed at equal angular distances in principal yoke part  141  of yoke  14 ; analogously, sliders  152  are mounted movably in their respective sockets  1451  placed at equal angular distances in principal yoke part  145  of yoke  14  ( FIGS. 3, 4, 5, 7, 8, 13 ). The purpose of recesses  1421  placed in intermediate part  142  of yoke  14  is to accommodate primary rollers  16 ; analogously, the purpose of recesses  1441  placed in intermediate part  144  of yoke  14  is to accommodate primary rollers  16 ; the purpose of recesses  1431  placed in intermediate part  143  of yoke  14  is to accommodate secondary rollers  161 . The purpose of three intermediate yoke parts  142 ,  143 , and  144  is to connect two principal yoke parts  141  and  145 . Placed on input shaft  12  there is a cam  121  (in the case of the first preferred embodiment of the invention, by way of an example but not in a restricted sense, cam  121  is a circular eccentric, i.e. its profile is a circle placed eccentrically relative the axis of rotation of input shaft  12 , but it may be any suitable cam, the profile of which is given in a cylindrical coordinate system (r,φ,z), rε&lt;0,∞&gt;, φε&lt;0,2π), zε&lt;−∞,∞&gt;, by a differentiable function r=f(φ,z)) ( FIGS. 2, 7, 8, 10 ). Placed in part  112  of body  11  there is a cam  1121 , the profile of which is given in a cylindrical coordinate system (r,Φ,z), rε&lt;0,∞&gt;, φε&lt;0,2π), zε&lt;−∞,∞&gt;, by a differentiable function r=F(φ,z)) ( FIGS. 2, 6, 9 ); analogously, placed in part  114  of body  11  there is a cam  1141 , the profile of which is given in a cylindrical coordinate system (r,φ,z), rε&lt;0,∞&gt;, φε&lt;0,2π), zε&lt;−∞,∞&gt;, by a differentiable function r=G(φ,z)) ( FIGS. 2, 10 ). The cam profile F(φ,z) of cam  1121  and the cam profile G(φ,z) of cam  1141  are determined by the cam profile f(φ,z) of cam  121 , required transmission ratio k, and method of mounting of rollers assemblies in yoke  14  (in this example with the help of sliders), so as during operation all primary rollers  16  are kept in constant contact with cams  1121  and  1141  placed in parts  112  and  114  of body  11  ( FIGS. 6, 7, 8, 9 ), and secondary rollers  161  remain in constant contact with cam  121  on input shaft ( FIGS. 6, 7, 8, 10 ); equations binding together the profiles F(φ,z), G(φ,z) and f(φ,z) are presented below. In the case of the first preferred embodiment of the invention, cam  121  is a circular eccentric, and each of cams  1121  and  1141  has a number (in this example 16) of suitably shaped lobes and the same number of suitably shaped valleys; thus in this example the number of roller assemblies  16 - 161  is by one smaller than the number of lobes of cams  1121  and  1141 . 
         [0087]    Here is a description of functioning of the gear according to the present invention. 
         [0088]    Input shaft  12  is driven by any suitable source of rotary power (internal combustion engine, electric motor, hydraulic motor, or any other suitable means), and cam  121  (in this instance eccentric) presses on secondary rollers  161 . Secondary rollers  161  press on primary rollers  16 , which in turn press on cams  1121  and  1141  placed in parts  112  and  114  of the gear body  11 , and generate a force F acting upon cams  1121  and  1141 . By the third Newton law of dynamics, cams  1121  and  1141  exert the force −F acting upon primary rollers  16 . The tangential component of the force −F is transferred via sliders  151  and  152  to yoke  14  and, via disc  131 , to output shaft  13  forcing it to rotate, in this case with the rotational velocity equal to −1/n, the rotational speed of the input shaft  12 , where n is the number of roller assemblies (−1/n=−1/15 in this example; in particular, output shaft  13  rotates in the opposite direction than input shaft  12 ). 
         [0089]    By stopping output shaft  13  and letting body  11  to rotate, a gear in accordance with the present invention with transmission ratio 1/16 is obtained (some changes, that would be obvious to those skilled in the art, of the structure of the gear are necessary to get a fully operational device). 
       First Embodiment 
     Variant B (FIGS.  16 ,  17 ) 
       [0090]    Another preferred embodiment of the gear according to the present invention, illustrated in  FIGS. 16, 17 , deffers from the variant A of the first embodiment only by the number of roller assemblies  16 - 161 , which in the variant B is by one greater than the number of lobes of cams  1121  and  1141  (and in this example equals 17). In particular, the profile of cams  1121  and  1141  used in the variant B of the gear is the same as the profile of cams  1121  and  1141  used in variant A (it is a non-trivial fact that both the profiles are identical; a discussion of this issue is given below). The discussion of the functioning of the variant A of the gear given above applies almost literally to the variant B, the only difference is that the rotational speed of output shaft  13  of the variant B of the first embodiment of the gear equals 1/17 of the rotational speed of input shaft  12 ; in particular, both input shaft  12  and output shaft  13  rotate in the same direction. 
         [0091]    By stopping output shaft  13  and letting body  11  to rotate, a gear in accordance with the present invention with transmission ratio −1/16 is obtained (again some changes, obvious to those skilled in the art, of the structure of the gear are necessary to get a fully operational device). 
       Second Embodiment 
     FIGS.  18 - 23   
       [0092]    The second embodiment of the gear according to the present invention differs from the first embodiment by the method of connecting the rollers with output shaft  13 . This circumstance forces other profiles of cams, but renders the roller&#39;s yoke unnecessary. 
         [0093]    Gear  10  being the second embodiment of the present invention ( FIGS. 18-23 ) has a body  11 , input shaft  12 , output shaft  13 , and a plurality (in this example 9) of roller assemblies  161 - 162 - 17 - 18 - 19 . Body  11  ( FIGS. 18-20 and 22 ) is composed of four parts  111 ,  112 ,  113 , and  114 . Placed in part  112  of body  11  there is cam  1121  with a number ( 10  in this example) of lobes; thus the number of lobes is greater by one than the number of roller assemblies, so output shaft  13  and input shaft  12  rotate in mutually opposite directions, and the transmission ratio of the gear equals −n, where n is the number of roller assemblies (thus in this example the transmission ratio equals −9). Input shaft  12  is mounted pivotally in part  111  of body  11 . Output shaft  13  is mounted pivotally in part  114  of body  11  ( FIGS. 18-22 ). Input shaft  12  has a external cam (in this example an eccentric)  121  that remains in constant contact with rollers  162  ( FIGS. 18-22 ). Output shaft is equipped with disc  131 . Each roller assembly  161 - 162 - 17 - 18 - 19  ( FIGS. 18-22 , and particularly  FIGS. 21, 23 ) is composed of eccentric  18 , axle  17 , web  19  joining axle  17  and eccentric  18 , and two rollers  161 ,  162  mounted pivotally on axle  17 . Each roller  161  remains in constant contact with cam  1121  placed in part  112  of body  11 . Each roller assembly  161 - 162 - 17 - 18 - 19  is mounted pivotally in disc  131  of output shaft  13  through eccentric  18  ( FIGS. 18-22 ). 
         [0094]    The discussion of the functioning of the variant A of the first embodiment of the gear given above applies almost literally to the second embodiment, the only difference is that the trajectory in the output shaft reference system of center of rollers  161 ,  162  is a circular arc rather than a segment of a straight line, which, as mentioned above, forces different profile of cam  1121 . 
         [0095]    Like in the case of the first embodiment of the gear, there is also a variant of the second embodiment of the gear according to the present invention, in which the number n of roller assemblies is by one greater than the number of lobes on cam  1121 . In this case the transmission ratio equals n (in particular both input shaft and output shaft rotate in the same direction), however the profile of cam  1121  differs from that of the first variant of the second embodiment. 
       Third Embodiment 
     FIGS.  24 - 33   
       [0096]    The third preferred embodiment of the gear according to the instant invention differs from the first and the second embodiment in that cam  1121  placed in body  11  of the first and second embodiment is replaced by cam  132  placed on output shaft; this difference also forces a change in the method of mounting of roller assemblies, which in the present example are “stationary”, i.e. mounted directly in the gear body  11 . 
         [0097]    Thus gear  10  being the third embodiment of the invention has body  11 , input shaft  12 , output shaft  13 , and a number n (in this example 9) of roller assemblies  21 - 22 - 23 - 161 - 162 - 163  ( FIGS. 24-30 ). Input  12  and output  13  shafts are mounted pivotally and coaxially in body  11 . Arranged in body  11  there are n sockets  116 , in which semi-circular levers  23  of roller assemblies  21 - 22 - 23 - 161 - 162 - 163  are mounted pivotally ( FIGS. 24-29 , and particularly  FIG. 28 ). Input shaft has cam (in this example an eccentric)  121  placed thereon. Output shaft  13  has cam  132  placed thereon. Cam  132  has a number m of lobes and valleys (in this example m=n−1=8). Each roller assembly  21 - 22 - 23 - 161 - 162 - 163  ( FIGS. 24-33 , and particularly  29 - 33 ) consists of semi-circular lever  23 , two holders  21  and  22 , two auxiliary rollers  163 , through which holders  21  and  22  are mounted pivotally on lever  23 , and two rollers  161 ,  162  cooperating with cams  121 ,  132  respectively. Roller  161  is mounted pivotally in holder  21 , and roller  162  is mounted pivotally in holder  22 . Holder  21 , resp.  22 , has a socket  211 , resp.  221 , accommodating their respective auxiliary roller  163 . Roller assemblies  21 - 22 - 23 - 161 - 162 - 163 , input shaft  12  and output shaft  13  are mounted in body  11  so that rollers  161  remain in constant contact with cam  121  on input shaft  12 , and rollers  162  remain in constant contact with cam  132  on output shaft  13 . Transmission ratio of the gear equals ±k/m, where k is the number of lobes of cam  121  (in this example k=1, and the transmission ratio equals −1/8; in particular input shaft  12  and output shaft  13  revolve in opposite directions). 
         [0098]    Again there are versions of the third embodiment of the gear with different numbers of lobes on cams  121  and  132 ; in particular, there are versions with both shafts rotating in the same direction. 
         [0099]    Work of the gear is as follows: As input shaft  12  rotates, cam  121  presses rollers  161 , which trough levers  23  press rollers  162 ; rollers  162  in turn exert force on cam  132  on output shaft  13 . This force, thanks to the specific profile of cam  132 , has a non-zero tangential component that generates positive torque on output shaft  13 . 
       Fourth Embodiment 
     FIGS.  34 - 45   
       [0100]    This embodiment of the present invention is of particular interest, because it allows for obtaining very large transmission ratios in very compact one-stage gears. 
         [0101]    The gear (which, in this example, assumes the form of winch) has a body  1 , input shaft  12  mounted rotatably in the body  11 , output shaft  13  mounted rotatably in the body  11  coaxially with input shaft  12  and assuming, in this example, the form of spool, and a plurality (15 in this example) of roller assemblies  16 - 161 - 162 . Placed in body  11  there is an external cam  1121  having a number n of lobes and valleys (n=14 in this example). Placed on the output shaft  13  there is a cam  132  having a number m of lobes and valleys (m=16 in this example). Input shaft  12  has a cam  121  (with yet another number k of lobes and valleys; k=1, and cam  121  assumes the shape of eccentric in this example) placed thereon. Primary rollers  16  roll over cam  1121  placed in gear body  11 , secondary rollers  161  roll over eccentric  121  placed on input shaft  12 , and rollers  162  roll over cam  132  placed on output shaft (spool)  13 . Profiles of cams  121 ,  132 , and  1121  are linked together by certain mathematical formulas (not formulated explicitly in this patent specification, but similar to those given in the mathematical appendix below), which assure constant contact of rollers  16 ,  161 ,  162  with cams  1121 ,  121 ,  132  respectively, during operation of the gear. 
         [0102]    Work of this gear is similar to that of previously described preferred embodiments: As input shaft  12  and its cam  121  revolves, rollers  16  follow the contour of cam  121 , which forces rollers  162  to follow the contour of cam  1121  placed in body  11 , and rollers  161 - to follow the contour of cam  132  placed on output shaft  13 ; this forces output shaft to rotate. The assemblies of rollers rotate in unison in one direction, while output shaft  13  rotates in the opposite direction with the rotational speed equal to 1/n×m of the rotational speed of input shaft  12 ; thus the transmission ratio of the gear equals 1/14×16=1/224 in this example. 
       Establishment of the Relationship Between Profiles of the Cams Placed on the Input Shaft and in the Gear Body 
       [0103]    Now, having described preferred embodiments of the present invention, I embark on a discussion of relationship between profiles of cam  121  on input shaft and cams  1121  and  1141  placed in the gear body, or on input and output shafts (which will provide a proof that the construction of my gears is feasible), ( FIG. 46 ). 
         [0104]    I assume that I am given the following set of data: 
         [0105]    1. A polar coordinate system (r, φ), rε&lt;0,∞), φε&lt;0,2π), a cylindrical coordinate system (r, φ, z), zε(−∞,∞), a first (“quiescent”) Cartesian coordinate system (x,y,z), x=r cos φ, y=r sin φ; and a second (“rotating”) Cartesian coordinate system (x 1 ,y 1 ,z 1 ), where the z-axis of the first Cartesian coordinate system coincides with the z 1 -axis of the second Cartesian coordinate system, and the origin of the first Cartesian coordinate system coincides with the origin of the second Cartesian coordinate system; 
         [0106]    2. The axes of rotation of input shaft  12  and output shaft  13  coincide with the z-axis of the first Cartesian coordinate system and the cylindrical coordinate system (and hence with the z 1 -axis of the second Cartesian coordinate system; 
         [0107]    3. Yoke  14  and output shaft  13  remain at rest in the first Cartesian coordinate system; the gear body  11  (and hence cams  1121  and  1141 ) remains at rest in the second Cartesian coordinate system; 
         [0108]    4. Base transmission ratio of the gear, understood as the ratio 
         [0000]    
       
         
           
             
               
                 ω 
                 2 
               
               
                 ω 
                 1 
               
             
             , 
           
         
       
     
         [0000]    where ω 1  is the rotational speed of input shaft  12  relative yoke  14  and output shaft, (i.e. relative the first Cartesian coordinate system), and ω 2  is the rotational speed of the gear body  11  relative yoke  14  and output shaft  13  (i.e. in the first Cartesian coordinate system), is given by a rational (strictly positive or strictly negative) number k; 
         [0109]    5. For certain fixed position of input shaft  12  relative the cylindrical coordinate system (r, φ, z), rε&lt;0,∞), φε&lt;0,2π), zε(−∞,∞), understood as a position corresponding to a zero angle of rotation of input shaft  12  relative the cylindrical coordinate system, the profile of cam  121  on input shaft  12  is given in the cylindrical coordinate system (r, φ, z), rε&lt;0,∞), φε&lt;0,2π), zε(−∞,∞) by a (continuously differentiable) function r(φ,z)=f(φ) for some continuously differentiable function r=f(φ), and zε&lt;a,b&gt; for some real numbers a&lt;0&lt;b, where 
         [0000]    
       
         
           
             
               
                 
                   lim 
                   
                     ϕ 
                     → 
                     0 
                   
                 
                  
                 
                   f 
                    
                   
                     ( 
                     ϕ 
                     ) 
                   
                 
               
               = 
               
                 
                   lim 
                   
                     ϕ 
                     → 
                     
                       2 
                        
                       π 
                     
                   
                 
                  
                 
                   f 
                    
                   
                     ( 
                     ϕ 
                     ) 
                   
                 
               
             
             , 
             
               
 
             
              
             
               
                 
                   
                     lim 
                     
                       ϕ 
                       → 
                       0 
                     
                   
                    
                   
                     
                       
                          
                         
                             
                         
                       
                       
                          
                         ϕ 
                       
                     
                      
                     
                       f 
                        
                       
                         ( 
                         ϕ 
                         ) 
                       
                     
                   
                 
                 = 
                 
                   
                     lim 
                     
                       ϕ 
                       → 
                       
                         2 
                          
                         π 
                       
                     
                   
                    
                   
                     
                       
                          
                         
                             
                         
                       
                       
                          
                         ϕ 
                       
                     
                      
                     
                       f 
                        
                       
                         ( 
                         ϕ 
                         ) 
                       
                     
                   
                 
               
               ; 
             
           
         
       
     
         [0000]    thus the profile of cam  121  is given in the first Cartesian coordinate system by x(φ,z)=f(φ)cos φ, y(φ,z)=f(φ)sin φ, z=z; 
         [0110]    6. For the position of input shaft  12  relative the cylindrical coordinate system as described in paragraph 5 above (i.e. corresponding to the zero angle of rotation of input shaft  12  relative the cylindrical coordinate system) the x-axis of the first Cartesian coordinate system coincides with the x 1 -axis of the second Cartesian coordinate system, and the y-axis of the first Cartesian coordinate system coincides with the y 1 -axis of the second Cartesian coordinate system 
         [0111]    7. Radius of roller  161  rolling over cam  121  on input shaft  12  equals r 0  for some strictly positive real number r 0 , and radius of roller  16  rolling over cam  1121  placed in the gear body  11  equals r 1  for some strictly positive real number r 1 ; 
         [0112]    8. The center of the circle being the intersection of a fixed roller  161  with the plane (x,y,0) of the first Cartesian coordinate system moves during the gear operation along a curve γ given in the first Cartesian coordinate system by continuously differentiable functions x=γ 1 (t), y=γ 2 (t), z=0. 
         [0113]    9. r(φ,ψ)=f(φ−ψ) is the profile in the cylindrical coordinate system of cam  121  rotated by the angle ψ relative the first Cartesian coordinate system; I write f(φ−ψ)=f ψ (φ); thus in the first Cartesian coordinate system the profile of cam  121  rotated by the angle ψ relative the first Cartesian coordinate system is given by (x ψ (φ)=f ψ (φ)cos φ, y ψ (φ)=f ψ (φ)sin φ); 
         [0114]    Principal unknowns are: 
         [0115]    0. Functions {tilde over (x)}(φ) and {tilde over (y)}(φ) describing the profile of cam  1121  (or, to be more precise, the profile of the intersection of cam  1121  with the plane (x,y,0)) in the first Cartesian coordinate system. 
         [0116]    I must also introduce some auxiliary unknowns: 
         [0000]    Auxiliary unknowns:
       1. ψ is an angle of rotation of input shaft  12  relative yoke  14 , i.e. relative the first Cartesian coordinate system.   2. (x(ψ),y(ψ),0) are coordinates of the center of roller  161  (remaining in constant contact with cam  121 ) in the first Cartesian coordinate system corresponding to the angle ψ of rotation of input shaft  12  relative the first Cartesian coordinate system;   3. (x t (ψ),y= t (ψ),0) are coordinates of the point of contact of the profile (a curve) of roller  161  and the profile (a curve) of cam  121  in the first Cartesian coordinate system corresponding to the angle ψ of rotation of input shaft  12  relative the first Cartesian coordinate system;       
 
         [0120]    The following two assertions, mathematical of nature, will be crucial for the determination of profile  1121 : 
         [0121]    Let δ(ψ) will be the profile in the second Cartesian coordinate system rotated relative the first Cartesian coordinate system by the angle kψ of roller  161  assuming the position corresponding to the angle ψ of rotation of input shaft  12  relative the first Cartesian coordinate system; thus δ(ψ) is a circle of radius r 0  with the center placed at the point having in the second Cartesian coordinate system coordinates ({tilde over (x)} 1 (ψ),{tilde over (y)} 1 (ψ))=(cos(kψ)x(ψ)+sin(kψ)y(ψ);−sin(kψ)x(ψ)+cos(kψ)y(ψ)). 
         [0122]    ASSERTION 1. The profile of cam  1121  (and  1141 ) in the second Cartesian coordinate system is an envelope of the family of circles δ(ψ). 
         [0123]    ASSERTION 2. There exist precisely two envelopes of a family of circles of a given fixed radius r&gt;0, the centers of which are placed along a planar curve ξ given in a Cartesian coordinate system ({circumflex over (x)}, ŷ) by a continuously differentiable function ({circumflex over (x)}=ξ 1 (s), ŷ=ξ 2 (s)) (where s is a real parameter)); these two envelopes are two components of the (zero dimensional) sphere bundle of the radius r of normal bundle to the curve ξ. In other words, the two envelopes are the two curves given in the Cartesian coordinate system ({circumflex over (x)},ŷ) by the following functions: 
         [0000]    
       
         
           
             
               
                 x 
                 ^ 
               
                
               
                 ( 
                 s 
                 ) 
               
             
             = 
             
               
                 
                   ξ 
                   1 
                 
                  
                 
                   ( 
                   s 
                   ) 
                 
               
               ± 
               
                 r 
                 ( 
                 
                   
                     
                        
                       
                         
                           ξ 
                           2 
                         
                          
                         
                           ( 
                           s 
                           ) 
                         
                       
                     
                     
                        
                       s 
                     
                   
                   / 
                   
                     
                       
                         
                           ( 
                           
                             
                                
                               
                                 
                                   ξ 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                             
                                
                               s 
                             
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             
                                
                               
                                 
                                   ξ 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                             
                                
                               s 
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 y 
                 ^ 
               
                
               
                 ( 
                 s 
                 ) 
               
             
             = 
             
               
                 
                   ξ 
                   2 
                 
                  
                 
                   ( 
                   s 
                   ) 
                 
               
                
               • 
                
               
                   
               
                
               
                 r 
                 ( 
                 
                   
                     
                        
                       
                         
                           ξ 
                           1 
                         
                          
                         
                           ( 
                           s 
                           ) 
                         
                       
                     
                     
                        
                       s 
                     
                   
                   / 
                   
                     
                       
                         
                           ( 
                           
                             
                                
                               
                                 
                                   ξ 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                             
                                
                               s 
                             
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             
                                
                               
                                 
                                   ξ 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                             
                                
                               s 
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
         [0124]    Now I am in a position to formulate a relationship between profile r(φ,z)=f(φ) of cam  121  and the profile of cam  1121 . 
         [0125]    The unknowns and given data as described above satisfy the following system of equations: 
         [0000]        x (ψ)=γ 1 ( t (ψ)),  y (ψ)=γ 2 ( t (ψ)) for some auxiliary unknown  t (ψ);  (1)
 
         [0126]    these two equations express the fact, that the center (x(ψ),y(ψ)) of roller  161  lies on the curve γ; 
         [0000]        x   t (ψ)= f   ψ (φ(ψ))cos(φ(ψ)),  y   t (ψ)= f   ψ (φ(ψ))cos(φ(ψ)), for some auxiliary unknown function φ(ψ);  (2)
 
         [0127]    these two equations express the fact, that the point of contact of the profile of roller  161  and profile  121  rotated by the angle ψ lies on this last profile (curve), i.e. on the curve (x ψ (φ), y ψ (φ)); 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             - 
                             
                               y 
                                
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
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                               y 
                               i 
                             
                              
                             
                               ( 
                               ψ 
                               ) 
                             
                           
                         
                         ) 
                       
                       / 
                       
                         r 
                         0 
                       
                     
                     = 
                     
                       
                         
                            
                           
                             x 
                             ψ 
                           
                         
                         
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                           ϕ 
                         
                       
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                           ( 
                           
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                                 ( 
                                 
                                   
                                     
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                                     ) 
                                   
                                 
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                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     
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                                         y 
                                         ψ 
                                       
                                     
                                     
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                                     ( 
                                     
                                       ϕ 
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                                         ( 
                                         ψ 
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                                     ) 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       
                         
                           ( 
                           
                             
                               - 
                               
                                 x 
                                  
                                 
                                   ( 
                                   ψ 
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 x 
                                 i 
                               
                                
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           r 
                           0 
                         
                       
                       = 
                       
                         
                           
                              
                             
                               x 
                               ψ 
                             
                           
                           
                              
                             ϕ 
                           
                         
                          
                         
                           
                             ( 
                             
                               ϕ 
                                
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                             ) 
                           
                           / 
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                          
                                         
                                           x 
                                           ψ 
                                         
                                       
                                       
                                          
                                         ϕ 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         ϕ 
                                          
                                         
                                           ( 
                                           ψ 
                                           ) 
                                         
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                          
                                         
                                           y 
                                           ψ 
                                         
                                       
                                       
                                          
                                         ϕ 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         ϕ 
                                          
                                         
                                           ( 
                                           ψ 
                                           ) 
                                         
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                     ; 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0128]    these two equations express the fact, that the profile of roller  161  and profile  121  rotated by the angle ψ are tangent to one another at the point (x t (ψ),y t (ψ)). 
         [0129]    Thus we have a system of six equations for six unknowns x(ψ), y(ψ), x t (ψ), y(ψ), t(ψ), φ(ψ), which, under quite general conditions, can be solved (e.g. in the case of all four preferred embodiments presented above). 
         [0130]    Let x(ψ)= x (ψ),y(ψ)= y (φ) etc. be a solution of the system of equations (1)-(3); thus ( x (ψ), y (φ)) is the trajectory of the center of roller  161  in the first Cartesian coordinate system (expressed as a function of the angle ψ of rotation of input shaft  12  relative the first Cartesian coordinate system), and ({tilde over (x)} 1 (ψ),{tilde over (y)} 1 (ψ))=(cos(kψ) x (ψ)+sin(kψ) y (ψ);−sin(kψ) x (ψ)+cos(kψ) y (ψ)) is the trajectory of the center of roller  161  in the second Cartesian coordinate system. Now, applying Assertions 1 and particularly 2, we get the following formula for the profile of cam  1121  in the second Cartesian coordinate system: 
         [0000]    
       
         
           
             
               
                 
                   x 
                   ~ 
                 
                 2 
                 1 
               
                
               
                 ( 
                 ψ 
                 ) 
               
             
             = 
             
               
                 
                   
                     x 
                     ~ 
                   
                   2 
                 
                  
                 
                   ( 
                   ψ 
                   ) 
                 
               
               ± 
               
                 
                   ρ 
                   1 
                 
                 ( 
                 
                   
                     
                        
                       
                         
                           
                             y 
                             ~ 
                           
                           2 
                         
                          
                         
                           ( 
                           ψ 
                           ) 
                         
                       
                     
                     
                        
                       ψ 
                     
                   
                   / 
                   
                     
                       
                         
                           ( 
                           
                             
                                
                               
                                 
                                   
                                     x 
                                     ~ 
                                   
                                   2 
                                 
                                  
                                 
                                   ( 
                                   ψ 
                                   ) 
                                 
                               
                             
                             
                                
                               ψ 
                             
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             
                                
                               
                                 
                                   
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                                     ~ 
                                   
                                   2 
                                 
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                                   ( 
                                   ψ 
                                   ) 
                                 
                               
                             
                             
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                           ) 
                         
                         2 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     y 
                     ~ 
                   
                   2 
                   1 
                 
                  
                 
                   ( 
                   ψ 
                   ) 
                 
               
               = 
               
                 
                   
                     
                       y 
                       ~ 
                     
                     2 
                   
                    
                   
                     ( 
                     ψ 
                     ) 
                   
                 
                 ∓ 
                 
                   
                     ρ 
                     1 
                   
                   ( 
                   
                     
                       
                          
                         
                           
                             
                               x 
                               ~ 
                             
                             2 
                           
                            
                           
                             ( 
                             ψ 
                             ) 
                           
                         
                       
                       
                          
                         ψ 
                       
                     
                     / 
                     
                       
                         
                           
                             ( 
                             
                               
                                  
                                 
                                   
                                     
                                       x 
                                       ~ 
                                     
                                     2 
                                   
                                    
                                   
                                     ( 
                                     ψ 
                                     ) 
                                   
                                 
                               
                               
                                  
                                 ψ 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                  
                                 
                                   
                                     
                                       y 
                                       ~ 
                                     
                                     2 
                                   
                                    
                                   
                                     ( 
                                     ψ 
                                     ) 
                                   
                                 
                               
                               
                                  
                                 ψ 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   ) 
                 
               
             
             ; 
           
         
       
     
         [0131]    or, using the fact that both the first and second Cartesian coordinate systems coincide when the angle of rotation of input shaft equals 0, we finally get the following formula for the profile of cam  1121  in the first Cartesian coordinate system: 
         [0000]    
       
         
           
             
               
                 
                   x 
                   ~ 
                 
                  
                 
                   ( 
                   ϕ 
                   ) 
                 
               
               = 
               
                 
                   
                     
                       x 
                       ~ 
                     
                     1 
                   
                    
                   
                     ( 
                     ϕ 
                     ) 
                   
                 
                 ± 
                 
                   
                     r 
                     1 
                   
                   ( 
                   
                     
                       
                          
                         
                           
                             
                               y 
                               ~ 
                             
                             1 
                           
                            
                           
                             ( 
                             ϕ 
                             ) 
                           
                         
                       
                       
                          
                         ϕ 
                       
                     
                     / 
                     
                       
                         
                           
                             ( 
                             
                               
                                  
                                 
                                   
                                     
                                       x 
                                       ~ 
                                     
                                     1 
                                   
                                    
                                   
                                     ( 
                                     ϕ 
                                     ) 
                                   
                                 
                               
                               
                                  
                                 ϕ 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                  
                                 
                                   
                                     
                                       y 
                                       ~ 
                                     
                                     1 
                                   
                                    
                                   
                                     ( 
                                     ϕ 
                                     ) 
                                   
                                 
                               
                               
                                  
                                 ϕ 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   ) 
                 
               
             
             , 
             
               
 
             
              
             
               
                 
                   y 
                   ~ 
                 
                  
                 
                   ( 
                   ϕ 
                   ) 
                 
               
               = 
               
                 
                   
                     
                       y 
                       ~ 
                     
                     1 
                   
                    
                   
                     ( 
                     ϕ 
                     ) 
                   
                 
                  
                 • 
                  
                 
                     
                 
                  
                 
                   
                     
                       r 
                       1 
                     
                     ( 
                     
                       
                         
                            
                           
                             
                               
                                 x 
                                 ~ 
                               
                               2 
                             
                              
                             
                               ( 
                               ϕ 
                               ) 
                             
                           
                         
                         
                            
                           ϕ 
                         
                       
                       / 
                       
                         
                           
                             
                               ( 
                               
                                 
                                    
                                   
                                     
                                       
                                         x 
                                         ~ 
                                       
                                       1 
                                     
                                      
                                     
                                       ( 
                                       ϕ 
                                       ) 
                                     
                                   
                                 
                                 
                                    
                                   ϕ 
                                 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                    
                                   
                                     
                                       
                                         y 
                                         ~ 
                                       
                                       1 
                                     
                                      
                                     
                                       ( 
                                       ϕ 
                                       ) 
                                     
                                   
                                 
                                 
                                    
                                   ϕ 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                     ) 
                   
                   . 
                 
               
             
           
         
       
     
         [0132]    For the completeness of this patent specification, below I present the formula for the profile of cam  1121  of the first embodiment (presented above) of the gear according to the instant invention; thus now I confine myself to the case, where cam  121  on input shaft  12  is an eccentric, and the path γ of the center of rollers  16 ,  161  is a segment of a straight line. 
         [0133]    Thus I start with the following set of data ( FIG. 45 ):
   1. Cam  121  on input shaft  12  is an eccentric with eccentricity e and radius r e ;   2. Base transmission ratio k=1/s, where s is a natural number (not equal to zero); in the case of the first variant of the first embodiment of the gear according to the present invention s=16;   3. Radius of primary roller  16  equals r 1 ; radius of secondary roller  161  equals r 0 ; Then the profile of cam  1121  is given in the first Cartesian coordinate system by the following formulas:
       Let us introduce the following abbreviations:   
       
 
         [0000]        B (φ)= e  cos( s φ)cos φ+ A (φ)cos φ;
 
         [0000]        A (φ)=√{square root over (( r   e   +r   0 ) 2   −e   2  sin 2 ( s φ))};
 
         [0000]        C (φ)= A (φ)+ e  cos( s φ);  D (φ)= se   2  sin( s φ)cos( s φ)/ A (φ)+ se  sin( s φ);
 
         [0000]        E (φ)= e  cos( s φ)sin φ+ A (φ)sin φ;
 
         [0000]      Then {tilde over ( x )}(φ)= B (φ)+ r   1 ( B (φ)− D (φ)sin φ)/√{square root over ((( C (φ)) 2 +(( D (φ)) 2 )}, and
 
         [0000]      {tilde over ( y )}(φ)= E (φ)+ r   1 ( E (φ)+ D (φ)cos φ)/√{square root over ((( C (φ)) 2 +(( D (φ)) 2 )}.
 
         [0138]    Contour of cam  132  placed on output shaft  13  (preferred embodiments 3 and 4) is determined completely analogously. 
         [0139]    The foregoing description discloses four preferred embodiments of the invention. One skilled in the art will readily recognize from this description and from the accompanying figures and patent claims, that many changes and modifications can be made to the preferred embodiments without departing from the true spirit, scope and nature of the inventive concepts as defined in the following patent claims.