Patent Publication Number: US-RE41313-E

Title: Planet pinion carrier assembly for ravigneaux gearset

Description:
BACKGROUND OF THE INVENTION 
     The invention relates to gearsets for motor vehicle transmissions, particular to Ravigneaux gearsets for automatic transmissions. 
     A conventional Ravigneaux gearset includes a single carrier containing two sets of planet pinions, long pinions, and short pinions. Each pinion set includes three pinions, each pinion being mutually spaced angularly about a central axis, each short pinion meshing with a corresponding long pinion. There are two sun gears, a short sun gear meshing with the three short pinions, and a long sun gear meshing with the three long pinions. There is one ring gear, which meshes with the three long pinions. 
     Both planet pinion sets are supported for rotation on the carrier. Each long pinion is in mesh with a short pinion, the ring gear, and the long sun gear. Each short pinion is in mesh with a long pinion and the short sun gear. 
     Ravigneaux gearsets are used in many automatic transmissions due to their compact size. A Ravigneaux gearset can be used instead of two simple planetary gearsets to produce four forward speeds and a reverse gear. They are smaller, and require fewer components than two simple planetary gearsets because they have a common pinion, one ring gear, and one carrier. Their manufacturing and assembly costs compare favorably to those of simple planetary gearsets. 
     Ravigneaux gearsets, however, are limited in torque capacity due to the pairing requirement for the pinions. Unlike conventional planetary gear sets, which can be made more robust by adding an extra pinion, the pairing requirement of a Ravigneaux gearset requires that two pinions be added to increase torque capacity. Because there is not enough space in the annular zone surrounding the carrier to add a long pinion and a short pinion, the toque and power capacity of a Ravigneaux gearset is usually increased by making the gearset larger, widening the gears and pinions and increasing the diameter of the shafts, or by incorporating premium processes to the manufacture of the gears, pinions and carrier. But this increases the cost weight and package size of the transmission. Heretofore, Ravigneaux gearsets have been limited to three pairs of planet pinions, six pinions. 
     There is a need to increases the power and torque capacity of a Ravigneaux gearset, preferably by adding pinions, but without increasing the size of the gearset. 
     SUMMARY OF THE INVENTION 
     A gearset arranged and sized by the method of this invention increases the power and torque capacity by adding extra sets of pinions to a conventional six-pinion Ravigneaux gearset, each additional pinion set including a long planet pinion and a short planet pinion. Each pinion of the gearset is in meshing engagement with two adjacent pinions. This arrangement improves torque capacity by distributing the torsional load across an extra set of pinions, and by sharing the load across multiple mesh points within the pinions. 
     In a gearset having one additional short pinion in mesh with the short sun gear, the load on each pinion is reduced by 25 percent. Likewise, there is a 25 percent reduction in load at the long pinion-to-ring gear mesh points. With each short pinion in mesh with two long pinions, the load at each mesh point is reduced by 62.5 percent. The load cycle of all of the gears increases since the number of mesh points increases, but the reduced loading significantly increases gear life and allows the gearset to be used in higher torque applications. This allows existing gear designs to be modified for higher torque applications without affecting packaging. Future designs can likewise be packaged in a smaller space, thereby saving cost and easing packaging constraints on the vehicle. 
     The eight-pinion Ravigneaux gearset design of this invention has been shown to increase significantly its torque capacity and enables a wider gear ratio to be produced. The wider gear ratio range improves vehicle performance and fuel economy. 
     A gearset according to the present invention includes a planet pinion carrier, long pinions having external gear teeth, mutually spaced angularly about an axis, and supported for rotation on the carrier, and short pinions having external gear teeth and supported for rotation on the carrier, each short pinion being spaced about the axis, located between two long pinions and meshing with two long pinions. 
     The carrier includes a disc disposed substantially perpendicular to, and aligned with the axis, having a first set of axially directed, angularly spaced holes, and a second set of axially directed, angularly spaced holes; posts radially spaced from the axis, extending axially from the disc toward the pinions, each post mutually angularly spaced about the axis; and a pad secured to each post, axially spaced from the disc, extending radially toward the axis, and having a axially directed hole aligned with a hole of the second set of holes. 
    
    
     
       Various objects and advantages of this invention will become apparent to those skilled in the art from the following detailed description of the preferred embodiment, when read in light of the accompanying drawings. 
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is an end view of a conventional Ravigneaux gearset; 
         FIG. 2  is a front view of the conventional Ravigneaux gearset of  FIG. 1 ; 
         FIG. 3  is an end view of an eight-pinion, load sharing Ravigneaux gearset; 
         FIG. 4  is a view of a ten-pinion, load sharing Ravigneaux gearset; 
         FIG. 5  is a cutaway isometric view of a planetary pinion carrier for a load sharing gearset with the pinions absent from the carrier; 
         FIG. 6  is a cutaway isometric view of the carrier of  FIG. 5  showing short pinions installed; 
         FIG. 7  is a cutaway isometric view of the carrier of  FIG. 5  showing short pinions and long pinions; 
         FIG. 8  is a cutaway isometric view of the carrier of  FIG. 5  showing short pinions and long pinions installed and the carrier components in mutually spaced relation; 
         FIG. 9  is an end view of the carrier with the backing plate removed showing the long pinion pockets and the short pinion pads; 
         FIG. 10  is an isometric view of the assembled carrier of  FIG. 5 ; 
         FIG. 11  is a schematic diagram of a portion of load sharing gearset showing a short sun gear, two short pinions and a long pinion in meshing relation; and 
         FIG. 12  is a schematic diagram similar to that of FIG.  11 . 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     Referring now to the drawings, there is illustrated in  FIGS. 1 and 2a  conventional Ravigneaux gearset, which includes a planet pinion carrier  10  containing three pairs of meshing planet pinions having external gear teeth, the pairs being formed of long pinions  12 ,  14 ,  16  and short pinions  18 ,  20 ,  22 . Each pinion pair is equally spaced angularly about a central axis  24  and includes a short pinion meshing with a corresponding long pinion. Each long pinion  12 ,  14 ,  16  is supported for rotation on the carrier  10  on a long pinion shaft  26 . Each short pinion  18 ,  20 ,  22  is supported for rotation on the carrier  10  on a short pinion shaft  27 . 
     There are two sun gears having external gear teeth, a short sun gear  28  meshing with the three short pinions  18 ,  20 ,  22 , and a long sun gear  30  meshing with the three long pinions  12 ,  14 , and  16 . A ring gear  32 , surrounding the sun gears and planet pinions and having internal gear teeth, meshes with the three long pinions. The sun gears  28 ,  30  and the ring gear  32  are arranged coaxially about axis  24 . 
     Each long pinion  12 ,  14 ,  16  is in mesh with a short pinion  18 ,  20 ,  22 , the ring gear  32 , and long sun gear  30 . Each short pinion  18 ,  20 ,  22  is in mesh with a long pinion  12 ,  14 ,  16  and the short sun gear  28 . Preferably, the gear teeth of each gear and pinion are helical, involute teeth. 
     There are two sun gears having external gear teeth, a short sun gear  28  meshing with the three short pinions  18 ,  20 ,  22 , and a long sun gear  30  meshing with the three long pinions  12 ,  14 ,  16 . A ring gear  32 , surrounding the sun gears and planet pinions and having internal gear teeth, meshes with the three long pinions. The sun gears  28 ,  30  and the ring gear  32  are arranged coaxially about axis  24 . 
     Referring now to  FIG. 3 , a load sharing gearset  34  according to this invention includes a planet pinion carrier containing two sets of planet pinions, long pinions  42 ,  44 ,  46 ,  48  and short pinions  50 ,  52 ,  54 ,  56 , each pinion having external involute gear teeth. Each pinion set includes four pinions, each pinion being mutually spaced angularly about a central axis  24 . Each short pinion  50 - 56  is in meshing engagement with two adjacent long pinions, and is supported on the carrier  40  for rotation on a pinion shaft  58 . Each long pinion  42 - 48  is in meshing engagement with two adjacent short pinions, and is supported on the carrier  40  for rotation on a pinion shaft  60 . The pinions are supported on the pinion shafts by needle bearings. 
     There are two sun gears having external involute gear teeth, a short sun gear  62  meshing with the four short pinions, and a long sun gear  64  meshing with the four long pinions. A ring gear  66 , surrounding the sun gears and planet pinions and having internal involute gear teeth, meshes with the four long pinions. The sun gears  62 ,  64  and the ring gear  66  are arranged coaxially about axis  24 . 
     Each long pinion  42 ,  44 ,  46 ,  48  is in mesh with two short pinions, the ring gear  66 , and the long sun gear  64 . Each short pinion  50 ,  52 ,  54 ,  56  is in mesh with two long pinions and the short sun gear  62 . Preferably the gear teeth of each gear and pinion are helical or bevel gears. 
       FIGS. 5-10  show a planetary carrier assembly  40  that includes a drum  70 , support disc  72 , and backing plate  74 , each formed preferably of powered metal. To facilitate assembly, the long pinions  42 ,  44 ,  46 ,  48  are inserted in the axial direction into the carrier  40 . The drum and support are assembled and secured mutually by brazing to form the subassembly of  FIG. 5 , the drum  70  and support  72 . A backing plate  74  is then installed in the drum-support subassembly. 
     Drum  70  includes a cylindrical shell  78  and four radially directed carrier posts  79 , each post being mutually spaced angularly about axis  24  and extending axially between support disc  70  and a radial extending pad  80 . Each pad  80  is formed with an axially directed pinion shaft hole  82 , extending through the thickness of the pad, sized and located to receive a short pinion shaft  58  on the pad  80 . The lateral surfaces  84 ,  86  of each pad, which face an adjacent pad, have a partial circular cylindrical form, such that the lateral surfaces of adjacent pads form a pocket  87  surrounding a portion of a long pinion. This arrangement is seen best in FIG.  8 . 
     The support  72  is formed with four axially directed, angularly spaced pinion shaft holes  88 , each extending through the thickness of the support, aligned with a hole  82 , and sized and located to receive the opposite end of the short pinion shaft  58  that is supported at the aligned hole  82  on the pad  80 . Support  72  is also formed with four axially directed, angularly-spaced pinion shaft holes  90 , each extending through the thickness of the support, aligned with a cylindrical pocket  87 , and sized and located to receive a long pinion shaft  60 . 
       FIG. 6  shows two short pinions  52 ,  54  having been installed in the carrier  40 , and supported for rotation on pinion shafts  58  at a hole  82  on a radial pad  80  and at a hole  88  on support  80 . The other short pinions  50 ,  56  are similarly installed and supported.  FIG. 7  shows two long pinions  44 ,  46  having been installed in the carrier  40 , and supported for rotation on pinion shafts  60 . Long pinion  44  is meshing with short pinions  52 ,  54 ; long pinion  46  is meshing with short pinions  54 ,  56 ; long pinion  42  is meshing with short pinions  52 ,  50 . The other long pinion  48  is similarly installed and supported. To facilitate assembly of the long pinions in the carrier  40 , the long pinions, needle bearings, and pinion shafts  60  are first formed as a subassembly. That subassembly is then inserted axially in the direction from the axial end of the carrier  40  that is opposite support  72 , through the cylindrical pockets  27  between adjacent pads  80  of the drum  70 , toward support  72 . Each pinion shaft  60  engages a hole  90  in the support  72 . As the long pinions are inserted in the carrier, they mesh with the short pinions. 
     A backing plate  98  is inserted into the drum such that the long pinion shafts  60  are supported on the backing plate  98  at angularly spaced holes  100 , aligned with the shafts  60 . The short pinion shafts  58  are supported on the backing plate  98  at angularly spaced holes  102 , aligned with the shafts  58 . The backing plate  98  locates onto the carrier assembly  40  using extensions  104  of the short pinion shafts  58 . These extensions  104  allow the short pinion  58  to act as locating dowels to guide the backing plate into correct angular position about the axis  24 . Backing plate  98  prevents axial movement of the long and short pinion shafts  60 ,  58 , and supports the adjacent end of each long pinion shaft  60 . The shafts  60  are secured at each respective end to support disc  72  and backing plate  98 , thereby securing disc  72  to plate  98 . 
     As  FIG. 10  shows, four openings  96  extending axially between support  72  and shell  78  and extending circumferentially about axis  24  between the posts are formed on the support  70 . The gear teeth on the external surface of each long pinion  42 - 48  pass through a respective opening  96  so that each the long pinions can mesh with the gear teeth on the internal surface of the ring gear  66 . 
     The torque capacity of a conventional Ravigneaux gearset can be further increased by adding another yet pair of pinions to the gearset of FIG.  3 .  FIG. 4  shows a ten-pinion, load-sharing gearset. Due to spatial limitations, the short pinions are made smaller than those of  FIG. 3  to provide space for the extra pinion pair and to provide space for the carrier posts  79 . The size of the short pinions and torque capacity of the gearset are maximized by positioning the pinions for load sharing. Each short pinion meshes with two long pinions as well as the short sun gear  62 . Each long pinion mates with two short pinions, the long sun gear  64 , and the ring gear  66 . The carrier posts  79  occupy the space between the short pinions and the ring gear. 
     When designing a conventional Ravigneaux gear set, standard gear design practices can be used to position the gears and design the gear teeth. Often it is sufficient to analyze the gears in mesh-pairs to establish the distance between centers of the gears, i.e., the center distances. When designing an eight-pinion gearset with load sharing, standard design techniques are insufficient; getting all of the gears to mesh properly will be a problem. The most likely outcome will be that seven of the pinions can be positioned in mesh, but the eighth pinion will not assemble. A method for determining the relative positions, center distance and size of gears and pinions, and other gearset parameters of a load sharing gearset that will assemble is described next. 
     In  FIG. 11 , each circle represents the base circle of each gear. Lines  114 ,  115 ,  116 ,  117  are tangential to the base circles and represent the lines-of-action of the various mesh pairs, respectively. Input gear A drives gears B and D, and gears B and D drive output gear C. It is unimportant to the method which gears are designated ‘input’, ‘output’, ‘driven’, or ‘driver’. Although gears B and D are located symmetrically with respect to a line connecting the centers of gears A and C, gears B and D need not be positioned symmetrically about that line or have the same size. All gears in the system must have the same ratio of base circle diameter to number of teeth, and all gears have involute gear tooth geometry. 
     Additional gears can be added to the system incrementally and analyzed in groups of four. Lines  114 ,  115 ,  116  and  117  are lines-of-action at gear mesh points. Lines  128 ,  129 ,  130 ,  131 ,  132 ,  133 ,  134 ,  135  are radial lines on the base circle diameters, each radial line being drawn from a gear center, and each radial line intersecting a corresponding line-of-action  114 ,  115 ,  116 ,  118  at a base circle. Lines  136 ,  138 ,  140 ,  142  connect the centers of the gear base circles. 
     Each interior angle between a line connecting the centers A, B, C, D and a radial line  128 - 135  that intersects a line-of-action  114 - 117  is equal to the transverse pressure angle of the related mesh. For example, angle C 1  is the transverse pressure angle of the mesh between gears B and C. 
     The points of intersection of the lines-of-action  114 - 117  and the base circle radii  128 - 135  are numbered  1 - 8 . These points give a common positional reference to any tooth within its mesh cycle with respect to another mesh point on the same gear. 
     N A , N B , N C , and N D  are the number of teeth on gears A, B, C, and D, respectively.  FIG. 11  is a starting layout developed to define center distances and angles. 
     In order to develop a phase error term, the relative position into the mesh cycle of the meshes on a gear must be determined. If a gear has N equally spaced teeth, then each mesh cycle is 360/N degrees of rotation. A relative position between two mesh cycles on the same gear can be derived using the intersection of the lines of action  114 - 117  and the base circle radii as a reference. Using gear C in  FIG. 11  as an example, suppose the mesh between gear pair B-C is positioned such that the involute of a tooth on gear C is at the point  1 , the intersection of a radius of the base circle C and the line of action  115 . As a reference, point  1  is zero degrees into the mesh cycle. 
     What is the position of a tooth on gear C in its the mesh cycle for the CD mesh? The teeth on gear  112  are spaced 360/N C  degrees apart. A phase into the mesh cycle is the difference between the angular distance between point  1  and point  8 , i.e., angle C 1 +C 2 , and an integral multiple of the tooth spacing on gear C: 
               Pos     1   -   8       =       Pos   1     +     mod   ⁡     (         C   ⁢           ⁢   1     +     C   ⁢           ⁢   2       ,     360     N   C         )                 (   1   )             
 
     Pos 1  is the relative starting position (assumed, but not required to be zero). The relative position of the mating tooth on gear D into its mesh cycle is: 
               Pos     1   -   9       =         Pos     1   -   8       ⁢       N   C       N   D         =       Pos   1     +         N   C       N   D       ⁢     mod   ⁡     (         C   ⁢           ⁢   1     +     C   ⁢           ⁢   2       ,     360     N   C         )                     (   2   )             
 
     There now exists the ability to calculate the position of any tooth into its mesh cycle through two distinct paths among the gears. The position of the teeth on gear C into the mesh pair C-D can be calculated using Equation (1), and that position can be calculated with reference to the path in the opposite direction from gear B to gear A to gear D, as expressed in Equation (3): 
               Pos     1   -   8       =       (         (         (         Pos   1     ⁢       N   C       N   B         +     (       B   ⁢           ⁢   1     +     B   ⁢           ⁢   2       )       )     ⁢       N   B       N   A         -     (       A   ⁢           ⁢   2     +     A   ⁢           ⁢   3       )       )     ⁢       N   A       N   D         +     (       D   ⁢           ⁢   2     +     D   ⁢           ⁢   3       )       )     ⁢       N   D       N   C                 (   3   )             
 
     Taking the difference between Equation (1) and Equation (3), and using the Mod function to express it in mesh cycles generates an error term: 
             Error   =           mod   (     (     (           (                   Pos   1     ⁢       N   C       N   B         +     (       B   ⁢           ⁢   1     +     B   ⁢           ⁢   2       )       )     ⁢       N   B       N   A         -             (       A   ⁢           ⁢   2     +     A   ⁢           ⁢   3       )     )     ⁢       N   A       N   D         +     (       D   ⁢           ⁢   2     +     D   ⁢           ⁢   3       )       )     ⁢       N   D       N   C         -     (       C   ⁢           ⁢   1     +     C   ⁢           ⁢   2     -     Pos   1       )       ,     360     N   C         )                   (   4   )             
 
     The angular distances A 2 +A 3  and C 2 +C 3  are subtracted because gears A and C rotate opposite the direction of rotation of gears B and D. The starting position Pos 1  subtracts out, so Equation (4) can be simplified into Equation (5): 
             Error   =           mod   (     (           (         (       B   ⁢           ⁢   1     +     B   ⁢           ⁢   2       )     ⁢       N   B       N   A         -             (       A   ⁢           ⁢   2     +     A   ⁢           ⁢   3       )     )     ⁢       N   A       N   D         +     (       D   ⁢           ⁢   2     +     D   ⁢           ⁢   3       )       )     ⁢       N   D       N   C         -     (       C   ⁢           ⁢   1     +     C   ⁢           ⁢   2       )       ,     360     N   C         )                 (   5   )             
 
     The next step, modifying one or more dimensions until the error term is acceptably low, is best achieved with the use of a computer program or a spreadsheet. A minimal set of design constraints can be developed, and then the remainder of the dimensions can be calculated. For example, the following set of design inputs is sufficient to calculate all distances and angles for the system in FIG.  11 :
         1) Tooth counts and base-circle diameters for all four gears.   2) Angle BAD   3) Distances AB, AD, and AC       

     The remaining distances and angles can be calculated from the above inputs. The phase error function can then be calculated. The error function can then be driven to zero by incrementally adjusting one of the inputs. In the above example, distance AC could be the adjustment. It is important to realize that there will be multiple solutions for each set of conditions. In the above example, there will be several distances AC that drive the error to zero. In addition, modifying the inputs can generate new sets of solutions. For example, simply changing distances AB and/or AD will produce a new set of solution (with distance AC as the adjustment). Tooth counts can also be changed to yield a new layout. 
     An numerical example of an application of the method is described next with reference to FIG.  11 . 
     Develop Gear Ratios and Tooth Counts 
     To design a load sharing Ravigneaux gearset, first the desired gear ratios to be produced by the gearset must be determined. The gear ratios are determined by conventional means and are used to determine the number of teeth for both sun gears  62 ,  64 , the long pinions  42 ,  44 ,  46 ,  48 , and the ring gear  66 . For equal spacing between like pinions, the tooth counts must also satisfy the following conditions: 
                   N   R     +     N   shortSun         #   ⁢           ⁢   ShortPinions       =   Integer           (   6   )             
                   N   R     +     N   LongSun         #   ⁢           ⁢   LongPinions       =   Intege           (   7   )             
 
In Equations (6) and (7), N is the number of teeth on the respective gear (ring gear, long sun, etc.).
 
     The gear ratios and the requirements imposed by equal spacing among the short and long pinions are used to determine the number of teeth for all gears except the short pinion. For example, suppose it is desired to build an eight-pinion gearset to produce the following gear ratios: 2.92, 1.57, 1.0, 0.71, and −2.4 (four forward gear ratios and one reverse). The following tooth counts would produce such ratios:
         N Ring =79   N LongSun =33   N ShortSun =27       

     The gear ratios are calculated from the following equations using the selected number of teeth for the sun gears and ring gear to demonstrate that the required gear ratios will be produced. Unity is always available as a gear ratio regardless of the size of the sun gears and ring gear by locking the gearset. 
           N   R       N   ShortSun       =   2.92       
               N   R     ⁢       N   LongSun       N   ShortSun         +     N   R           N   R     +     N   LongSun         =   1.57       
           N   R         N   R     +     N   LongSun         =       0.71   ⁢     
     -       N   R       N   ShortSun         =     -   2.39           
 
     The number of teeth on the long pinion can also be determined from Equation (8). 
               N   LongPinion     =           N   Ring     -     N   LongSun       2     =   23             (   8   )             
 
     It is not necessary to use Equation 8 to determine the long pinion tooth counts. Alternate methods exist and they are equally acceptable and immaterial to the positioning algorithm. The number of teeth for all gears and pinions is determined except for the short pinion. 
     Define Gear Parameters 
     Conventional methods are used to develop basic parameters and center distances. For example, suppose it is desired to design a gearset whose pinions and gears have the number of teeth mentioned above, a module of 1.66, a helix angle of 22 degrees, and a minimum gear pair backlash of 25 microns. Standard gear pair center distances can be calculated with the following equation: 
             D   =         (       N   1     +     N   2       )     *   Module       2   *     cos   ⁡     (   HelixAngle   )                   (   9   )             
 
     Using Equation (9), the standard center distance between the long sun gear  64  and each of the long pinions  42 ,  44 ,  46 ,  48  is 50.130 mm. A set of base circles is also developed. The positioning method does not require the actual base circles, but any set of circles that follows the relationship: 
                 N   1         B   ⁢             1       =       N   2       B   2               (   10   )             
 
for the entire set, wherein N is the number of teeth, and B is the base circle diameter, mm. Irrational values (extremely large or negative) may create computational problems, so it is best to use realistic values. For this example, a ring gear base circle diameter of 127 mm. is selected as an appropriate first assumption. The base circle diameter is selected large enough to provide sufficient circumferential length to accommodate the 79 teeth of the ring gear with a realistic pressure angle. Pressure angles are typically 16-22 degrees for planetary gearsets. Once the center distances are established, the base circle size can be changed to target a desired pressure angle without affecting assembly.
 
Define the Short Pinion
 
       FIG. 5  is a subset of the gearset of  FIG. 3  or  FIG. 4 , although it is discussed here with reference to FIG.  3 . Gear A is the forward sun gear  62 . Gears B and D are short pinions  50 ,  52 . Gear C is a long pinion  42 . The circles shown are the base circles for these pinions and the gear. The lines  114 ,  115 ,  116 ,  117  are tangential to the base circles and represent the lines-of-action of the various mesh pairs. Distances AB (short sun to short pinion) and BC (short pinion to long pinion) are calculated using trial values of the number of teeth of the short pinions B and D. This can easily be accomplished with a spreadsheet: 
     
       
         
           
               
               
               
             
               
                   
                   
               
               
                   
                  Short Pinion 
                   
               
               
                   
                  Tooth Count 
               
               
                   
                  16 
                 17 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Distance - Short Pinion to Forward Sun 
                 38.49 
                 39.39 
               
               
                   
                 Distance - Short Pinion to Long Pinion 
                 34.91 
                 35.81 
               
               
                   
                 Angle BAC 
                 44.01 
                 45.21 
               
               
                   
                   
               
            
           
         
       
     
     The pinions are equally spaced about the sun gears, and Equations 1 and 2 assume this equal spacing. This assumption infers that angle BAD is 90 degrees (or A 1 +A 2 =90°). Further, for equal torsional loading of each short pinion-long pinion mesh, and equal wear among the pinions, the layout must be symmetric. This infers that BC=CD and AB=AD. From symmetry, Angle BAD=2*Angle BAC. But this is not the case with either a 16 tooth short pinion (2*BAC=88.02°) or a 17 tooth short pinion (2*BAC=90.42°). 
     It is often beneficial (but not necessary) to select a short pinion tooth count that produces an angle BAC as close to the desired result as possible. In this case, a 17 tooth short pinion is more desirable than a 16 tooth short pinion. 
     Calculate Phase Error 
     The gear teeth create a repeating pattern about the circumference of the gear. This pattern repeats every mesh cycle, i.e., 360/N degrees, where N is the number of teeth. Mesh cycle can be used to track the relative positions of any mating gears. Equation 11 governs the positions of mating gears:
 
Θ1*N1=Θ2*N2  (11)
 
wherein Θ is the angular rotation of gears  1  and  2 , respectively. If one gear rotates one mesh cycle, a mating gear will also rotate one of its own mesh cycles. Two mesh engagements on the same gear are related mutually based on angular spacing relative to incremental counts of mesh cycles: two mesh engagements on a gear, spaced an integral number of mesh cycles apart, will be in the same relative position within their cycles, i.e., they will be in phase. Other phasings can be calculated using the Mod function.
 
     The phasing must be calculated using lines tangential to the base circles of gear pairs in mesh. Such tangential lines must be of the same hand for any particular gear. Lines of Action fit this condition. As an exception, lines connecting the centerlines of mating gears can be used if two identical gears are each in mesh with a third gear. For consistency, lines of action are used for calculating mesh cycles. 
     Referring to  FIG. 11 , if line AD  142  is arbitrarily picked as position  0  for the mesh cycles between the sun gear A and pinion D, then the phase of the mesh cycle at the mesh between sun gear A and pinion B (AB) relative to the mesh cycle of the mesh between sun gear A and pinion D (AD) can be calculated along two paths. First, calculate the phase of the mesh cycle based on the drive path through angle A 2 +A 3 . Second, calculate the phase of the mesh cycle based on the drive path through angles D 2 +D 3 , C 1 +C 2 , and B 1 +B 2 .  FIG. 12  shows the directions of rotation of the gears, and the two drive paths, which are referenced in the following calculations of the phase of the mesh cycle. For the gearset to assemble and all of the gears to mesh properly, the phasing calculated by both methods must be equal or approximately equal. Calculating the difference between the two paths generates an error term that is used to change the current gearset parameters and determine the optimal gearset parameters that will permit the gearset to assemble and all the gears to mesh properly. 
     Before calculating the error term, it is useful to summarize the known values and relationships: 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
             
            
               
                   
                 Module 
                 1.66 
                   
               
               
                   
                 Helix Angle 
                 22 
               
            
           
           
               
               
               
               
               
               
            
               
                   
                   
                   
                 Base 
                   
                   
               
               
                   
                   
                 Tooth 
                 Circle 
               
               
                   
                 Gear 
                 Count 
                 Radius 
               
               
                   
                 Ring 
                 79 
                 63.50 
                 Angle BAD 
                 90.41433 
               
               
                   
                 Forward Sun 
                 27 
                 21.70 
                 Angle ABC 
                 83.47502 
               
               
                   
                 Reverse Sun 
                 33 
                 26.53 
                 Angle ADC 
                 83.47502 
               
               
                   
                 Long Pinion 
                 23 
                 18.49 
                 Angle BCD 
                 102.6356 
               
               
                   
                 Short Pinion 
                 17 
                 13.66 
               
               
                   
                   
                   
                   
                 Angle B1 
                 26.11488 
               
               
                   
                   
                   
                   
                 Angle B3 
                 26.11488 
               
               
                   
                   
                   
                   
                 Angle D1 
                 26.11488 
               
            
           
           
               
               
               
            
               
                 Center Distances 
                 Angle D3 
                 26.11488 
               
            
           
           
               
               
               
               
            
               
                 Forward Sun to Short Pinion 
                 39.38809 
                   
                   
               
               
                 Short Pinion to Long Pinion 
                 35.80735 
                 Angle A2 + A3 
                 90.41433 
               
               
                 Forward Sun to Long Pinion 
                 50.13029 
                 Angle B1 + B2 
                 83.47502 
               
               
                   
                   
                 Angle C1 + C2 
                 102.6356 
               
               
                   
                   
                 Angle D2 + D3 
                 83.47502 
               
               
                   
               
            
           
         
       
     
     Although Angles B 1 , B 3 , D 1 , and D 3  are equal, they need not be equal, and likely will not be equal when the method is completed. 
     Phase across path 1: 
         Phase   1     =       Mod   ⁡     (     ∠A2A3   ,     360     N   ForwardSun         )       =   10.4143         
         360   /     N   ShortSun       =       13.333   ⁢     °   /   tooth     ⁢           ⁢   ∠A2A3     =   90.41433         
         (     90.41433   /   13.333     )     =       6.7810   ⁢           ⁢     teeth   ⁢     
     (     0.7810   *   13.333     )       =     10.4143   ⁢   °           
 
     Phase Across path 2: 
         Phase   2     =     MOD   (         (         (       ∠B1B2   ⁢       N   SP       N   LP         +   360   -   ∠C1C2     )     ⁢       N   LP       N   SP         +   ∠D2D3     )     ⁢       N   SP       N   FwdSun         ,     360     N   FwdSun         )         
         Phase   2     =       MOD   (         (         (         8   ⁢   3.475     ⁢     17   23       +   360   -   102.6356     )     ⁢     23   17       +   83.475     )     ⁢     17   24       ,     360   27       )     =   4.353         
 
     The angles used are those that are in the same direction as the direction of rotation of the corresponding gear. For example, if sun gear A is arbitrarily assigned a counterclockwise rotation, the angles used in the error calculation are the interior angles for all gears except the long pinion C (hence the 360-Angle BCD term). If sun gear A were arbitrarily assigned a clockwise rotation, the opposite would hold true, and the result would be equally valid.
 
Error=Phase 1 −Phase 2 =10.4143−4.353=6.06
 
     The error value for this layout does not equal zero and is not acceptable close to zero. This indicates that the gear arrangement will not assemble. To modify the layout such that it will assemble, one or more of the values in Table 1 require adjustment. To create a symmetric layout of pinions about the sun gear A, angle BAD must be 90 degrees, as discussed above. To keep the gear ratios constant, the gear tooth counts should not be changed. Also, because of the symmetry requirement, the relationships AB=AD and CB=CD will be maintained. With these constraints established, different values of AB and CB can be investigated until the error term is zero or acceptable close to zero. For this example, only distance A-B, the center distance between short sun gear A and short pinion B is adjusted. With the constraint that angle BAD=90 degrees, CB, the center distance between long pinion C and short pinion B will be a calculated value (BAC=45 degrees, AC is known &amp; fixed). 
     
       
         
           
               
             
               
                 TABLE 2 
               
               
                   
               
             
            
               
                 Module 1.66 
               
               
                 Helix Angle 22 
               
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                   
                   
                 Base 
               
               
                   
                   
                 Tooth 
                 Circle 
               
               
                   
                 Gear 
                 Count 
                 Radius 
               
               
                   
                   
               
               
                   
                 Ring 
                 79 
                 63.50 
               
               
                   
                 Forward Sun 
                 27 
                 21.70 
               
               
                   
                 Reverse Sun 
                 33 
                 26.53 
               
               
                   
                 Long Pinion 
                 23 
                 18.49 
               
               
                   
                 Short Pinion 
                 17 
                 13.66 
               
               
                   
                 Angle BAD 
                 90 
               
               
                   
                 Angle ABC 
                 85.5 
               
               
                   
                 Angle ADC 
                 85.5 
               
               
                   
                 Angle BCD 
                 99 
               
               
                   
                 Angle B1 
                 22.34097 
               
               
                   
                 Angle B3 
                 25.27978 
               
               
                   
                 Angle D1 
                 22.34097 
               
               
                   
                 Angle D3 
                 25.27978 
               
               
                   
                 Angle A2 + A3 
                 90 
               
               
                   
                 Angle B1 + B2 
                 82.5612 
               
               
                   
                 Angle C1 + C2 
                 99 
               
               
                   
                 Angle D2 + D3 
                 88.4388 
               
               
                   
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Center Distances 
                   
                 Notes 
               
               
                   
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Short Sun to Long Pinion 
                 50.130295 
                 Fixed 
               
               
                   
                 Angle BAD 
                 90 
                 Fixed 
               
               
                   
                 Short Pinion to Long Pinion 
                 35.557082 
                 Calculated 
               
               
                   
                 Short Sun to Short Pinion 
                 38.237248 
                 Adjusted 
               
               
                   
                 Path 1 
                 10.0000 
                 Calculated 
               
               
                   
                 Path 2 
                 10.0000 
                 Calculated 
               
               
                   
                 Error 
                 2.332E-08 
                 Calculated 
               
               
                   
                   
               
            
           
         
       
     
     After changing the current gearset parameters in this way, the calculated error becomes 2.332 E-08, indicating that paths 1 and 2 are substantially equal. The values in Table 2 can be used to produce an eight-pinion Ravigneaux gearset with load sharing as diagrammed in  FIGS. 3 and 8 . The layout is symmetric for load sharing, and the gears will assemble and mesh properly. Even though only four gears were analyzed out of eleven total (four short pinions, four long pinions, one short sun gear, one long sun gear, and one ring gear), the entire gearset will assembly because it is symmetric, evenly spaced, and the tooth counts all satisfy the constraints for an evenly spaced set of pinions (Equations 1 &amp; 2). 
     In the numerical example, a single pair backlash of 25 microns was selected, although the selected magnitude of single pair backlash may vary depending on its compatibility with manufacturing capabilities and other factors. If the magnitude of the error is equal to or greater than this backlash, the gears and pinions of the gearset may be too tightly spaced. In service, a drive flank of a first gear at one mesh may be in contact with its mating second gear, and a coast flank of the first gear at another mesh may be in contact with a mating third gear. This condition is avoided when the magnitude of the error is less than the magnitude of the selected single pair backlash. 
     If the physical dimensions of the gearset are symmetric about line AC, i.e., gears B and D are dimensionally identical, AB=AD, BC=BD, and angle BAD is an integral factor of 360, then the design can be repeated about gear A. For example, if angle BAD is 90 degrees, then four gears identical to B and D can be placed 90 degrees apart about the center of gear A at a radial distance AB from the center of gear A, or four gears identical to C can also be place 90 degrees about the center of gear A at radial distance AC. This layout can be used to generate a multi-pinion Ravigneaux gearset with load sharing among the pinions by placing all gears B and C on a carrier that rotates about the center of gear A. 
     Driving the solution to a value of phase error other than zero can bias the load sharing. In the example above with AC as the adjustment, a slightly positive phase error would imply no contact between gears C and D under nominal loads. Since gear teeth deflect under load, a nominal gap could be targeted such that load sharing does not occur until a certain load threshold is achieved. Backlash in the system would have to be increased to account for this additional ‘forward lash’. With this type of biasing, a torque reversal will cause the primary torque path to switch. 
     The gearset could also be biased with non-symmetry. If gears B and D are identical but distances CD and BD are not the same, one mesh pair will have additional backlash. That increased backlash could then be used for a ‘forward lash’ to bias the torque path. In this case, the primary torque path would not necessarily change if the load through the system reversed. 
     To address the inherent limitations for adding pinions to a Ravigneaux gearset, an additional pair of long and short pinions was added to a conventional gearset. To maximize pinion size, each pinion is in mesh with both adjacent pinions. Each short pinion meshes with two long pinions, and each long pinion meshes with two short pinions. This load sharing increases the service life of the gearset by reducing the bending stress of the pinion teeth. In addition, by maximizing the pinion size and number of pinion teeth, service life is further increased. 
     In accordance with the provisions of the patent statutes, the principle and mode of operation of this invention have been explained and illustrated in its preferred embodiment. However, it must be understood that this invention may be practiced otherwise than as specifically explained and illustrated without departing from its spirit or scope.