# Patent

## Title
Variable ratio epicyclic gear drives

## Abstract

A transmission for transferring torque from a prime mover includes a first input shaft adapted to be driven by the prime mover, a second input shaft, an output shaft and an epicyclic gearset. The gearset includes a ring gear being driven by the first input shaft, pinion gears being driven by the ring gear, a sun gear driven by the second input shaft and driving the pinion gears, and a carrier driving the output shaft. A reaction motor drives the second input shaft. A controller controls the reaction motor to vary the speed of the second input shaft and define a gear ratio between the first input shaft and the output shaft based on the second input shaft speed.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
This application is a continuation-in-part of U.S. patent application Ser. No. 12/617,031 filed on Nov. 12, 2009. The entire disclosure of the above application is incorporated herein by reference.


FIELD
The present disclosure relates to power transmission devices. More particularly, a transmission for transferring torque at a variable speed reduction ratio includes a planetary gear drive driven by two sources of power.
BACKGROUND
Geared transmissions typically function to change the rotational speed of a prime mover output shaft and an input shaft of a desired work output. In a vehicle, the prime mover may include a diesel or gasoline internal combustion engine. It should be noted that there are many more applications than automobiles and trucks. Locomotives are equipped with transmissions between their engines and their wheels. Bicycles and motorcycles also include a transmission. Speed-increasing transmissions allow the large, slow-moving blades of a windmill to generate power much closer to a desired AC frequency. Other industrial applications exist. In each case, the motor and transmission act together to provide power at a desired speed and torque to do useful work. Geared transmissions have also been used in combination with electric motors acting as the prime mover.
Multiple speed transmissions have been coupled to high torque prime movers that typically operate within a narrow speed range, most notably structured as large displacement diesel engines of tractor trailers. Electric motors have a much wider speed range in which they operate effectively. However, the motor operates most efficiently at a single speed. Known multiple speed transmissions attempt to maintain an optimum operating speed and torque of the prime mover output shaft, but only approach this condition due to the discrete gear ratios provided. Accordingly, a need for a simplified variable speed ratio power transmission device exists.
Many existing transmissions incorporate planetary gearsets within the torque path. A traditional planetary gear drive has three major components: a sun gear, an annulus ring gear and a planet carrier. When one of those components is connected to the prime mover, another is used as the output and the third component is not allowed to rotate, the input and output rotate at different speeds, and may also rotate in opposite directions, with the ratio of input to output speeds being a fixed value. If the previously fixed third component is connected to a second input and forced to rotate, the transmission will have a continuously varying speed ratio dependent on the speeds of both the prime mover and this new second input. One example of such a planetary gear drive is made by Toyota. While planetary gearsets have been successfully used in vehicle power transmissions in the past, a need exists for a planetary drive and control system for optimizing the gear drive's efficiency and power density.
SUMMARY
This section provides a general summary of the disclosure, and is not a comprehensive disclosure of its full scope or all of its features.
A transmission for transferring torque from a prime mover includes a first input shaft adapted to be driven by the prime mover, a second input shaft and an output shaft. A compound planetary gearset includes a sun gear driven by the first input shaft, first pinion gears being driven by the sun gear, a ring gear fixed for rotation with the second input shaft and being meshed with second pinion gears, and a carrier driving the output shaft. A reaction motor drives the second input shaft. A controller controls the reaction motor to vary the speed of the second input shaft and define a gear ratio between the first input shaft and the output shaft based on the second input shaft speed.
A transmission for transferring torque from a prime mover includes a first input shaft adapted to be driven by the prime mover, a second input shaft, an output shaft and an epicyclic gearset. The gearset includes a ring gear being driven by the first input shaft, pinion gears being driven by the ring gear, a sun gear driven by the second input shaft and driving the pinion gears, and a carrier driving the output shaft. A reaction motor drives the second input shaft. A controller controls the reaction motor to vary the speed of the second input shaft and define a gear ratio between the first input shaft and the output shaft based on the second input shaft speed.
A transmission includes a first input shaft adapted to be driven by the prime mover, a second input shaft, an output shaft, and an epicyclic gearset. The gearset includes a sun gear being driven by the first input shaft, pinion gears in meshed engagement with the sun gear, a carrier driven by the second input shaft and rotatably supporting the pinion gears, and a ring gear fixed for rotation with the output shaft. A reaction motor drives the second input shaft. A controller controls the reaction motor to vary the speed of the second input shaft and define a gear ratio between the first input and the output shaft based on the a second input shaft speed.
Further areas of applicability will become apparent from the description provided herein. The description and specific examples in this summary are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure.



DRAWINGS
The drawings described herein are for illustrative purposes only of selected embodiments and not all possible implementations, and are not intended to limit the scope of the present disclosure.
FIG. 1 is a schematic depicting an exemplary vehicle equipped with a transmission constructed in accordance with the teachings of the present disclosure;
FIG. 2 is a schematic representation of a transmission having a simple planetary gearset;
FIG. 3 is a graph depicting sun speed to carrier speed ratio versus ring speed to sun speed ratio for a number of fixed ring ratios of a simple planetary gearset;
FIG. 4 depicts relative diameters of sun, planet and annulus ring gears for a planetary gearset having a fixed ring ratio of 9.706.
FIG. 5 is a schematic depicting a transmission equipped with a compound planetary gearset;
FIG. 6 is a graph depicting sun speed to carrier speed ratio versus ring speed to sun speed ratio for a number of compound planetary gearsets;
FIG. 7 is a schematic of an alternate transmission including a simple planetary gearset and an offset motor and speed reduction unit;
FIG. 8 is a schematic depicting another transmission having a compound planetary gearset driven by an offset reaction motor and speed reduction unit;
FIG. 9 is a schematic of another transmission including two simple planetary gearsets;
FIG. 10 is a schematic of another transmission equipped with a compound planetary gearset and a simple planetary reduction gearset;
FIGS. 11 and 12 depict alternate transmissions including worm and worm wheel drives;
FIGS. 13 and 14 depict alternative transmissions including concentrically arranged worm drive mechanisms;
FIG. 15 is a schematic representation of a transmission having a simple solar epicyclic gearset;
FIG. 16 is a graph depicting ring speed to carrier speed ratio versus sun speed to ring speed ratio for a number of fixed sun speed ratios of a simple solar epicyclic gearset;
FIG. 17 is a schematic representation of a transmission having a simple star epicyclic gearset;
FIG. 18 is a graph depicting carrier speed to ring speed ratio versus sun speed to ring speed ratio for a number of fixed sun speed ratios of a simple star epicyclic gearset;
FIG. 19 is a graph depicting speed ratio sensitivity versus ring speed for a compound solar epicyclic gearset; and
FIG. 20 is a graph depicting speed ratio sensitivity versus carrier speed for a compound star epicyclic gearset.



Corresponding reference numerals indicate corresponding parts throughout the several views of the drawings.
DETAILED DESCRIPTION
The present disclosure is directed to a transmission that can be adaptively controlled to transfer torque between a first rotary member and a second rotary member. The transmission finds particular application in motor vehicle drivelines such as, for example, a continuously variable torque transfer mechanism. Thus, while the transmission of the present disclosure is hereinafter described in association with particular arrangements for use in specific driveline applications, it will be understood that the arrangements shown and described are merely intended to illustrate embodiments of the present disclosure.
With particular reference to FIG. 1 of the drawings, a drivetrain 10 for an all-wheel drive vehicle is shown. Drivetrain 10 includes a driveline 12 and a powertrain 16 for delivering rotary tractive power (i.e., drive torque) to the driveline. In the particular arrangement shown, driveline 12 is the rear driveline. Powertrain 16 is shown to include an engine 18 and a transmission 20. A pair of front wheels 24L and 24R are not driven. Driveline 12 includes a propshaft 30 driven by transmission 20 and a rear axle assembly 32 for transferring drive torque from engine 18 to a rear differential 34. A pair of rear axleshafts 38L and 38R interconnect rear differential 34 to corresponding rear wheels 36L and 36R.
As shown in FIGS. 1 and 2, transmission 20 includes a planetary gearset 40 and a reaction motor 42. Drivetrain 10 is shown to further include vehicle sensors 44 for detecting certain dynamic and operational characteristics of the motor vehicle and a controller 45 for controlling actuation of reaction motor 42 in response to input signals from vehicle sensors 44.
Planetary gearset 40 includes a sun gear 46 fixed for rotation with an output shaft 48 of engine 18. An annulus ring gear 50 is fixed for rotation with an output shaft 52 of reaction motor 42. Planetary gearset 40 also includes a carrier 54 rotatably supporting a plurality of pinion gears 56 that are each in constant meshed engagement with annulus ring gear 50 and sun gear 46. An output shaft 58 is fixed for rotation with carrier 54. The remainder of this disclosure discusses how the sun gear speed ωS to carrier speed ωC ratio is a function of an annulus ring gear speed ωR to sun speed ωS ratio in simple and compound planetary gearsets and how the asymptotic nature of this speed ratio may be exploited to improve the gear drive's efficiency and power density.
If a positive direction of annulus ring rotation is defined to be in the same direction as that of the sun gear and carrier assembly, it can be shown that in the general case, the ratio of sun to carrier speeds is given by:




  
    
      
        
          
            ω
            S
          
          
            ω
            C
          
        
        =
        
          
            
              
                z
                S
              
              +
              
                z
                R
              
            
            
              z
              S
            
          
          ⁡
          
            [
            
              1
              -
              
                (
                
                  
                    
                      ω
                      R
                    
                    ⁢
                    
                      z
                      R
                    
                  
                  
                    
                      
                        ω
                        S
                      
                      ⁢
                      
                        z
                        S
                      
                    
                    +
                    
                      
                        ω
                        R
                      
                      ⁢
                      
                        z
                        R
                      
                    
                  
                
                )
              
            
            ]
          
        
      
    
    
      
        (
        1
        )
      
    
  




where ωS, ωC, and ωR are the sun, carrier and annulus ring angular velocities and zR and zS are the number of teeth in the annulus ring and sun gears, respectively. Note that if ωR=0, equation (1) simplifies to the familiar relationship between sun and carrier speeds for a fixed annulus ring.

We define the ratio of ring speed to sun speed as





  
    
      
        
          ω
          
            R
            /
            S
          
        
        =
        
          
            ω
            R
          
          
            ω
            S
          
        
      
    
    
      
        (
        2
        )
      
    
  




Equation (1) may then be rewritten as





  
    
      
        
          
            ω
            S
          
          
            ω
            C
          
        
        =
        
          
            
              z
              R
            
            +
            
              z
              S
            
          
          
            
              z
              S
            
            +
            
              
                ω
                
                  R
                  /
                  S
                
              
              ·
              
                z
                R
              
            
          
        
      
    
    
      
        (
        3
        )
      
    
  




It can be noted that there will be a value of ωR/S for which ωS/ωC will become asymptotic. FIG. 3 shows equation (3) plotted against the ring to sun speed ratio for different numbers of ring and sun gear teeth. Each curve is labeled with its carrier to sun fixed ring ratio (FRR). Table 1 lists the number of teeth in the sun, planets and ring of each planetary combination, as well as each tooth combination's FRR. The combinations of sun, planet and ring gear tooth numbers were selected to span the practical limits of geometry constraints.
FIG. 3 plots the speed ratio ωS/ωC against ωR/S for various FRR values listed in Table 1 and shows that the vertical asymptote increases from ωR/S=−0.650 to −0.11.5 as the FRR increases. More importantly, the magnitude of the ωS/ωC slope as the curve crosses the ordinate axis ωR/S=0 increases as well. As the magnitude of the ωS/ωC slope increases, the annulus ring speed required to effect a change in the speed ratio ωS/ωC decreases dramatically. The annulus ring gear 50, however, is the largest of all component gears in the planetary gearset 40. As such, a relatively large torque may need to be reacted by reaction motor 42. The torque necessary to supply a sufficient reaction to pinion gears 56 may be quite large as well.
The sensitivity of the speed ratio to its fixed ring ratio is quantified by defining the ratio of the highest to lowest speed ratios as Δ for an arbitrary value of ωR/S selected as +/−10% of the sun's speed, as well as the value of ωR/S for the vertical asymptote. Table 1 presents this data.







TABLE 1







Simple Planetary Tooth Combinations










Numbers of Teeth
Fixed Ring
Ratio Spread
Vertical












Sun
Planets
Ring
Speed Ratio
Δ = ωS/ωC)MAX/(ωS/ωC)MIN
Asypmptote


zS
zP
zR
(zS + zR)/zS
−0.1 < ωR/S < 0.1
ωR/S = ωR/ωS















17
65
148
9.706
14.455
−0.115


20
62
145
8.250
6.273
−0.138


23
59
142
7.174
4.227
−0.162


28
54
137
5.893
2.916
−0.204


34
48
131
4.853
2.254
−0.260


41
41
124
4.024
1.867
−0.331


52
30
113
3.173
1.555
−0.460


65
17
100
2.538
1.364
−0.650









Table 1 and FIG. 3 illustrate that a large fixed ring ratio (FRR) is necessary to have the desirable feature of ωS/ωC speed ratio sensitivity. Design constraints may exist where such a large FRR is not practical. FIG. 4 shows the relative diameters of the sun, planets and annulus ring of the 9.706 FRR planetary. The relatively small sun size will limit the strength of the shaft on which the sun gear is fixed. Furthermore, just as there is a minimum practical FRR, below which the planet pinions are too small to be supported with rolling element bearings, there is a maximum FRR, above which the tips of the planet pinions will interfere.
FIG. 5 depicts an alternative transmission 20a having a compound planetary gearset 40a in lieu of simple planetary gearset 40. Alternative transmission 20a is substantially similar to transmission 20. Accordingly, like elements will be identified with reference numerals including an “a” suffix. Planetary gearset 40a differs from simple planetary gearset 40 in that compound pinion gears 60 replace pinion gears 56. Each compound pinion gear 60 includes a first pinion gear 62 in meshed engagement with sun gear 46a as well as a reduced diameter second pinion gear 64 in constant meshed engagement with annulus ring gear 50a. First pinion gears 62 have a predetermined number of teeth, module, pressure angle and helix angle based on the mesh with sun gear 46a. Second pinion gears 64 have a reduced number of teeth, a different module, pressure angle and helix angle for the gear meshes with annulus ring gear 50a. The compound planetary gearset provides a minimized inner and outer radial packaging. Furthermore, the compound planetary gearset provides a greater reduction gear ratio. It should be appreciated that first pinion gears 62 and second pinion gears 64 are aligned in pairs to rotate on common pinion centers.
To operate on the same pinion centers, the module, helix angle and number of teeth must satisfy this constraint:




  
    
      
        
          
            
              m
              R
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              β
              S
            
          
          
            
              m
              S
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              β
              R
            
          
        
        =
        
          
            
              z
              S
            
            +
            
              z
              PS
            
          
          
            
              z
              R
            
            -
            
              z
              PR
            
          
        
      
    
    
      
        (
        4
        )
      
    
  




where mR and mS are the normal modules of the ring and sun meshes, respectively. The planet pinions zPS and zPR mesh with the sun and annulus, respectively. In addition to the geometry constraint of equation (4), each of the compound planet pinions independent meshes must have the same torque, but because each torque will act at different pitch geometries, the tooth loads may differ significantly and require largely different modules as a result.

If the design of a compound planetary gear set is modified to allow for an annulus gear that may move at a controlled angular speed while still providing the necessary reaction torque for the planet pinions, a similar asymptotic behavior to that seen in FIG. 3 exists. With the same reasoning used to develop equations (1) and (2), it can be shown that the speed ratio ωS/ωC of a compound planetary gear set, in which the annulus gear is allowed to rotate is given by:




  
    
      
        
          
            ω
            S
          
          
            ω
            C
          
        
        =
        
          
            
              
                z
                R
              
              ·
              
                z
                PS
              
            
            +
            
              
                z
                S
              
              ·
              
                z
                PR
              
            
          
          
            
              
                z
                S
              
              ·
              
                z
                PR
              
            
            +
            
              
                ω
                
                  R
                  /
                  S
                
              
              ·
              
                z
                R
              
              ·
              
                z
                PS
              
            
          
        
      
    
    
      
        (
        5
        )
      
    
  




As with equations (1) and (2), equation (5) reduces to the familiar speed relationship for a fixed annulus ring when ωR/S=0, Since a compound planetary gear set is capable for a larger speed ratio ωS/ωC, the benefits of the asymptotic nature of equation (5) can be more fully exploited. FIG. 6 shows the sun-carrier speed ratios again plotted against ring-sun speed ratios for different fixed ring ratios.







TABLE 2







Compound Planetary Tooth Combinations








Numbers of Teeth














Planets
Planets


Ratio Spread


Sun
(Sun)
(Ring)
Ring
Fixed Ring Speed Ratio
Δ = (ωS/ωC)MAX/(ωS/ωC)MIN


zS
zPS
zPR
zR
(zSzPR + zRzPS)/(zSzPR)
−0.03 < ωR/S < 0.03















19
79
17
115
29.127
11.805


22
79
17
115
25.291
6.373


23
76
17
98
20.049
3.667


29
68
17
100
14.793
2.412


35
64
19
94
10.047
1.745


37
50
23
107
7.287
1.465


47
46
23
95
5.043
1.276









As was done for simple planetary gear drives, tooth combinations shown in Table 2 were selected to attempt to span the practical FRR limits. A FRR less than 5.043 would most likely not justify the additional complexity and expense of a compound planetary over a simple planetary and a FRR larger than 30 may not be practical, as can be seen from FIG. 6. It is also noted that in comparing FIGS. 3 and 6, it can be seen for a given FRR a compound planetary will have a larger asymptote value than a simple planetary.
FIG. 7 depicts an alternate transmission 20b and is constructed substantially similarly to transmission 20. Similar elements will be identified with like reference numerals including a “b” suffix. A reaction motor 42b includes an output shaft 52b that extends offset and parallel to an axis of rotation of output shaft 58b. Reaction motor 42b drives a reduction gearset 72 to rotate ring gear 50b. Reduction gearset 72 includes a first gear 76 fixed for rotation with output shaft 52b. A second gear 78 is in constant meshed engagement with first gear 76 and is fixed for rotation with a concentric shaft 80. Annulus ring gear 50b is also fixed for rotation with concentric shaft 80.
FIG. 8 depicts an alternate transmission identified at reference numeral 20c. Transmission 20c includes the offset motor and speed reduction unit arrangement shown in FIG. 7 being used in conjunction with the compound planetary gearset first described at FIG. 5. Accordingly, similar elements will be identified with like reference numerals including a “c” suffix. In operation, reaction motor 42c drives first gear 76c and second gear 78c to rotate annulus ring gear 50c and vary the output ratio provided to output shaft 58c. 
FIG. 9 depicts another alternate transmission identified at reference numeral 20d. Transmission 20d is substantially similar to transmission 20 with the addition of a planetary reduction gearset 90. Similar elements will be identified with like reference numerals having a “d” suffix. Planetary gearset 90 includes a sun gear 92 fixed for rotation with reaction motor output shaft 52d. Reaction motor output shaft 52d is concentrically aligned with and circumscribes output shaft 58d. A ring gear 94 is restricted from rotation. A plurality of pinion gears 96 are supported for rotation on a carrier 98. Pinion gears 96 are each meshed with sun gear 92 and ring gear 94. Carrier 98 is fixed for rotation with a concentric shaft 100. Annulus ring gear 50d is also fixed for rotation with concentric shaft 100.
FIG. 10 depicts another alternate transmission 20e that incorporates the planetary reduction gearset of FIG. 9 and mates it with the compound planetary gear arrangement shown in FIG. 5. Similar elements will be identified with like reference numerals having a “e” suffix. Reaction motor 42e includes an output shaft 52e transferring torque to planetary reduction unit 90e. Carrier 98e is fixed for rotation with concentric shaft 100e and annulus ring gear 50e. 
FIGS. 11 and 12 depict alternate transmissions 20f and 20g, respectively. Each of transmissions 20f, 20g include a worm drive 110 including a reaction motor 42f, 42g, driving a worm gear 112 along an axis of rotation that extends substantially perpendicular to an axis of rotation of output shaft 58f, 58g. Worm gear 112 is in constant meshed engagement with a worm wheel 114. Worm wheel 114 is fixed for rotation with a concentric shaft 116. In FIG. 11, concentric shaft 116 is fixed for rotation with annulus ring gear 50f. In similar fashion, concentric shaft 116 of FIG. 12 is fixed for rotation with annulus ring gear 50g. 
FIGS. 13 and 14 also depict alternative transmissions identified at reference numerals 20h and 20i, respectively. FIGS. 13 and 14 are substantially similar to FIGS. 11 and 12 except that worm wheels 114h, 114i concentrically surround annular ring gears 50h and 50i. 
FIG. 15 depicts a solar epicyclic gear arrangement 120 defined by sun gear 46, annulus ring gear 50, carrier 54, pinion gears 56 and reaction motor 42 when output shaft 52 of reaction motor 42 is fixed for rotation with sun gear 46. Engine 18 acts as the prime mover supplying torque to driveline 12. Output shaft 58 is fixed to carrier 54. Since the reaction member of a solar arrangement is sun gear 46, we define the sun gear's speed in terms of annulus ring gear 50:

ωS=ωS/R·ωR  (6)

The input to output speed ratio u may be expressed as:





  
    
      
        u
        =
        
          
            
              ω
              R
            
            
              ω
              C
            
          
          =
          
            
              
                z
                S
              
              +
              
                z
                R
              
            
            
              
                z
                R
              
              +
              
                
                  ω
                  
                    S
                    /
                    R
                  
                
                ·
                
                  z
                  S
                
              
            
          
        
      
    
    
      
        (
        7
        )
      
    
  




At FIG. 16, we plot the above equation for the same sun, planet and annulus ring tooth combinations in an attempt to span the limits of practicality. For simplicity, the same sun, planet and ring tooth combinations used for planetary arrangements are used again for solar arrangements. Note that both the scale and the fixed ring ratios have been updated to agree with the above equation, but the general asymptotic nature is the same as FIG. 3. However, the sun-to-ring speed ratios at which the vertical asymptotes exist are observed to occur at different values.
As was with planetary gearset 40, the power and steady-state torque into the gear drive must balance that exiting the gear drive.

τR+τS=τC  (8)

τR·ωR+τS·ωS=τC·ωC  (9)

As before, we make the kinematics and steady-state torque substitutions into the power balance equation, and obtain the carrier and sun torques in terms of the ring torque.





  
    
      
        
          
            τ
            C
          
          
            τ
            R
          
        
        =
        
          
            
              z
              S
            
            +
            
              z
              R
            
          
          
            z
            R
          
        
      
    
    
      
        (
        10
        )
      
    
  
  
    
      
        
          
            τ
            S
          
          
            τ
            R
          
        
        =
        
          
            z
            S
          
          
            z
            R
          
        
      
    
    
      
        (
        11
        )
      
    
  




As was done in the previous section with planetary gearset 40, we again define the ratio of the reaction power to the input power as κ, and write that for the solar arrangement variable speed ratio,




  
    
      
        u
        =
        
          
            
              
                z
                S
              
              +
              
                z
                R
              
            
            
              
                z
                R
              
              +
              
                
                  ω
                  
                    S
                    /
                    R
                  
                
                ·
                
                  z
                  S
                
              
            
          
          =
          
            
              
                z
                S
              
              +
              
                z
                R
              
            
            
              
                z
                R
              
              ⁡
              
                (
                
                  1
                  +
                  κ
                
                )
              
            
          
        
      
    
    
      
        (
        12
        )
      
    
  




The same relationship for transmission ratio spread as was discussed earlier exists as well. Therefore, for both a planetary and solar epicyclic arrangement, the kinematics exhibits an asymptotic behavior. However, the speeds of the reaction member at which the asymptote is observed will depend on the type of epicyclic arrangement, as well as the fixed reaction member speed ratios. In both cases, the transmission ratio spread required for the application will define the power requirements of the reaction motor/generator.

Another alternate transmission is shown in FIG. 17 and identified as a star epicyclic gear arrangement having each of sun gear 46, annulus ring gear 50, carrier 54, pinion gears 56 and reaction motor 42. Prime mover or engine 18 drives sun gear 46. Reaction motor 42 drives carrier 54. Output shaft 58 is fixed to annulus ring gear 50. As with the planetary and solar epicyclic gear drives, we define the reaction member's speed in terms of the input's. For a star gear drive, the reaction member is the carrier.

ωC=ωC/S·ωS  (13)

If we follow the same steps as we did for the planetary and solar epicyclic gear drives, we can rearrange the equation to give us the sun to ring speed ratio.





  
    
      
        u
        =
        
          
            
              ω
              S
            
            
              ω
              R
            
          
          =
          
            
              -
              
                z
                R
              
            
            
              
                z
                S
              
              -
              
                
                  ω
                  
                    C
                    /
                    S
                  
                
                ·
                
                  (
                  
                    
                      z
                      S
                    
                    +
                    
                      z
                      R
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        14
        )
      
    
  




Note that the negative sign in front of the right hand side means the input and output shafts will rotate in opposite directions when the carrier speed is zero. FIG. 18 plots the speed ratio against ωC/S.

τS−τC=−τR  (15)

τS·ωS−τC·ωC=τR·ωR  (16)

After substitute and rearranging terms of the above equations, the ring and carrier torques may be expressed in terms of the sun torque. The ring and carrier torques must be in a direction opposite to that of the sun. This has been addressed with the negative signs in the equation above.





  
    
      
        
          
            τ
            R
          
          
            τ
            S
          
        
        =
        
          
            -
            
              z
              R
            
          
          
            z
            S
          
        
      
    
    
      
        (
        17
        )
      
    
  
  
    
      
        
          
            τ
            C
          
          
            τ
            S
          
        
        =
        
          
            -
            
              (
              
                
                  z
                  S
                
                +
                
                  z
                  R
                
              
              )
            
          
          
            z
            S
          
        
      
    
    
      
        (
        18
        )
      
    
  




The ratio of the reactive power to the input power as defined as κ. However, since the carrier and sun torques are in opposite directions, the same will be true of their powers. Therefore, when the carrier and sun speeds are in the same direction κ.<0, and when the carrier and sun are in opposite directions, κ.>0 and we have as was true for the planetary and solar arrangements:





  
    
      
        u
        =
        
          
            
              -
              
                z
                R
              
            
            
              
                z
                S
              
              ⁡
              
                (
                
                  1
                  +
                  κ
                
                )
              
            
          
          =
          
            
              u
              F
            
            
              (
              
                1
                +
                κ
              
              )
            
          
        
      
    
    
      
        (
        19
        )
      
    
  




Thus, in all three epicyclic arrangements, when it is desirable to have a greater speed reduction than the fixed reaction-member ratio, the device that powers the input member must also power the reaction device to provide this additional speed reduction ratio. Conversely, if it is desired to have a speed reduction ratio that is less than that of the fixed reaction-member ratio, the reaction device must power the reaction member to achieve this kinematic relationship. The reaction power will exit the gear drive through the output member, adding to the power output of the gear drive.

It should be appreciated that solar gear arrangement 120 and star gear arrangement 150 may be modified to function with compound planet gears as previously described in relation to FIG. 5. The offset drive and speed reduction units of FIGS. 7-14 may also be used with the solar and star simple or compound epicyclic gearset arrangements.
The kinematics relationships for the compound solar and star epicyclic arrangements are derived in a manner similar to that previously described. FIGS. 19 and 20 show the speed reduction ratios plotted against the ratio of the reaction member to input member speed for compound solar and star arrangements, respectively. It should be noted that the fixed ring speed ratios of the compound solar epicyclic gearsets range between 1.03 and 1.25. Furthermore, the vertical asymptote of a ratio of carrier speed to sun gear speed versus a ratio of sun gear speed to ring gear speed for the compound solar gearset ranges between 5 and 28.
Regarding FIG. 20 and the star epicyclic gearsets, the fixed carrier speed ratios of the compound star epicyclic gearsets range between −4.0 and 28.0. The vertical asymptote of a ratio of ring gear speed to sun gear speed versus a ratio of carrier speed to sun gear speed ranges for the compound solar gearsets between 0.02 and 0.20. The planetary, solar and star arrangement speed reduction ratios can be simplified using the reactive power quotients and, as with their simple epicyclic counterparts, can all be written as the fixed reaction member speed ratio divided by the quantity 1+κ.
Tables 3, 4 and 5 summarize the speed reduction ratios and planet speeds, torque and power for each epicyclic gear drive considered. Table 6 summarizes the speed reduction ratios using the reactive power quotients for each arrangement and Table 7 summarizes the asymptotic characteristics of where the asymptote occurs and what is the slope as the curve crosses the vertical axis.







TABLE 3







Speed Reduction Ratio and Planet Speeds










SPEED REDUCTION RATIO
PLANET SPEEDS





Simple Planetary




  
    
      ω
      S
    
    
      ω
      C
    
  
  =
  
    
      
        z
        R
      
      +
      
        z
        S
      
    
    
      
        z
        S
      
      +
      
        
          ω
          
            R
            /
            S
          
        
        ·
        
          z
          R
        
      
    
  








  
    
      ω
      P
    
    
      ω
      S
    
  
  =
  
    
      
        (
        
          
            ω
            
              R
              /
              S
            
          
          ·
          
            z
            R
          
        
        )
      
      -
      
        z
        S
      
    
    
      2
      ·
      
        z
        P
      
    
  






 


Compound Planetary




  
    
      ω
      S
    
    
      ω
      C
    
  
  =
  
    
      
        
          z
          R
        
        ·
        
          z
          PS
        
      
      +
      
        
          z
          S
        
        ·
        
          z
          PR
        
      
    
    
      
        
          z
          S
        
        ·
        
          z
          PR
        
      
      +
      
        
          ω
          
            R
            /
            S
          
        
        ·
        
          z
          R
        
        ·
        
          z
          PS
        
      
    
  








  
    
      ω
      P
    
    
      ω
      S
    
  
  =
  
    
      
        (
        
          
            ω
            
              R
              /
              S
            
          
          ·
          
            z
            R
          
          ·
          
            z
            PS
          
        
        )
      
      -
      
        
          z
          S
        
        ·
        
          z
          PR
        
      
    
    
      2
      ·
      
        z
        PR
      
      ·
      
        z
        PS
      
    
  






 


Simple Solar




  
    
      ω
      R
    
    
      ω
      C
    
  
  =
  
    
      
        z
        R
      
      +
      
        z
        S
      
    
    
      
        z
        R
      
      +
      
        
          ω
          
            S
            /
            R
          
        
        ·
        
          z
          S
        
      
    
  








  
    
      ω
      P
    
    
      ω
      R
    
  
  =
  
    
      
        z
        R
      
      -
      
        (
        
          
            z
            S
          
          ·
          
            ω
            
              S
              /
              R
            
          
        
        )
      
    
    
      2
      ·
      
        z
        P
      
    
  






 


Compound Solar




  
    
      ω
      R
    
    
      ω
      C
    
  
  =
  
    
      
        
          z
          R
        
        ·
        
          z
          PS
        
      
      +
      
        
          z
          S
        
        ·
        
          z
          PR
        
      
    
    
      
        
          z
          R
        
        ·
        
          z
          PS
        
      
      +
      
        
          ω
          
            S
            /
            R
          
        
        ·
        
          z
          S
        
        ·
        
          z
          PR
        
      
    
  








  
    
      ω
      P
    
    
      ω
      R
    
  
  =
  
    
      
        
          z
          R
        
        ·
        
          z
          PS
        
      
      -
      
        (
        
          
            ω
            
              R
              /
              S
            
          
          ·
          
            z
            S
          
          ·
          
            z
            PR
          
        
        )
      
    
    
      2
      ·
      
        z
        PR
      
      ·
      
        z
        PS
      
    
  






 


Simple Star




  
    
      ω
      R
    
    
      ω
      S
    
  
  =
  
    
      -
      
        z
        R
      
    
    
      
        z
        S
      
      -
      
        
          ω
          
            C
            /
            S
          
        
        ·
        
          (
          
            
              z
              R
            
            +
            
              z
              S
            
          
          )
        
      
    
  








  
    
      ω
      P
    
    
      ω
      S
    
  
  =
  
    
      
        [
        
          
            ω
            
              C
              /
              S
            
          
          ·
          
            (
            
              
                z
                R
              
              +
              
                z
                S
              
            
            )
          
        
        ]
      
      -
      
        z
        S
      
    
    
      2
      ·
      
        z
        P
      
    
  






 


Compound Star




  
    
      ω
      R
    
    
      ω
      S
    
  
  =
  
    
      
        -
        
          z
          R
        
      
      ·
      
        z
        PS
      
    
    
      
        
          z
          S
        
        ·
        
          z
          PR
        
      
      -
      
        
          ω
          
            C
            /
            S
          
        
        ·
        
          (
          
            
              
                z
                R
              
              ·
              
                z
                PS
              
            
            +
            
              
                z
                S
              
              ·
              
                z
                PR
              
            
          
          )
        
      
    
  








  
    
      ω
      P
    
    
      ω
      S
    
  
  =
  
    
      
        [
        
          
            ω
            
              C
              /
              S
            
          
          ·
          
            (
            
              
                
                  z
                  R
                
                ·
                
                  z
                  PS
                
              
              +
              
                
                  z
                  S
                
                ·
                
                  z
                  PR
                
              
            
            )
          
        
        ]
      
      -
      
        
          z
          S
        
        ·
        
          z
          PR
        
      
    
    
      2
      ·
      
        z
        PR
      
      ·
      
        z
        PS
      
    
  




















TABLE 4







Torque Summary











SUN
RING
CARRIER





Simple Planet- ary
τS




  
    
      τ
      R
    
    
      τ
      S
    
  
  =
  
    
      z
      R
    
    
      z
      S
    
  








  
    
      τ
      C
    
    
      τ
      S
    
  
  =
  
    
      
        z
        R
      
      +
      
        z
        S
      
    
    
      z
      S
    
  






 


Com- pound Planet- ary
τS




  
    
      τ
      R
    
    
      τ
      S
    
  
  =
  
    
      
        z
        R
      
      ·
      
        z
        PS
      
    
    
      
        z
        S
      
      ·
      
        z
        PR
      
    
  








  
    
      τ
      C
    
    
      τ
      S
    
  
  =
  
    
      
        
          z
          R
        
        ·
        
          z
          PS
        
      
      +
      
        
          z
          S
        
        ·
        
          z
          PR
        
      
    
    
      
        z
        S
      
      ·
      
        z
        PR
      
    
  






 


Simple Solar




  
    
      τ
      S
    
    
      τ
      R
    
  
  =
  
    
      z
      S
    
    
      z
      R
    
  




τR




  
    
      τ
      C
    
    
      τ
      S
    
  
  =
  
    
      
        z
        R
      
      +
      
        z
        S
      
    
    
      z
      R
    
  






 


Comp- ound Solar




  
    
      τ
      S
    
    
      τ
      R
    
  
  =
  
    
      
        z
        S
      
      ·
      
        z
        PR
      
    
    
      
        z
        R
      
      ·
      
        z
        PS
      
    
  




τR




  
    
      τ
      C
    
    
      τ
      R
    
  
  =
  
    
      
        
          z
          R
        
        ·
        
          z
          PS
        
      
      +
      
        
          z
          S
        
        ·
        
          z
          PR
        
      
    
    
      
        z
        R
      
      ·
      
        z
        PS
      
    
  






 


Simple Star
τS




  
    
      τ
      R
    
    
      τ
      S
    
  
  =
  
    
      -
      
        z
        R
      
    
    
      z
      S
    
  








  
    
      τ
      C
    
    
      τ
      S
    
  
  =
  
    
      -
      
        (
        
          
            z
            R
          
          +
          
            z
            S
          
        
        )
      
    
    
      z
      S
    
  






 


Com- pound Star
τS




  
    
      τ
      R
    
    
      τ
      S
    
  
  =
  
    
      
        -
        
          z
          R
        
      
      ·
      
        z
        PS
      
    
    
      
        z
        S
      
      ·
      
        z
        PR
      
    
  








  
    
      τ
      C
    
    
      τ
      S
    
  
  =
  
    
      -
      
        (
        
          
            
              z
              R
            
            ·
            
              z
              PS
            
          
          +
          
            
              z
              S
            
            ·
            
              z
              PR
            
          
        
        )
      
    
    
      
        z
        S
      
      ·
      
        z
        PR
      
    
  




















TABLE 5







Power Summary











SUN
RING
CARRIER





Simple Planetary
τS · ωS




  
    
      
        τ
        R
      
      ·
      
        ω
        R
      
    
    
      
        τ
        S
      
      ·
      
        ω
        S
      
    
  
  =
  
    
      
        z
        R
      
      
        z
        S
      
    
    ·
    
      ω
      
        R
        /
        S
      
    
  








  
    
      
        τ
        C
      
      ·
      
        ω
        C
      
    
    
      
        τ
        S
      
      ·
      
        ω
        S
      
    
  
  =
  
    
      
        z
        S
      
      +
      
        
          ω
          
            R
            /
            S
          
        
        ·
        
          z
          R
        
      
    
    
      z
      S
    
  






 


Compound Planetary
τS · ωS




  
    
      
        τ
        R
      
      ·
      
        ω
        R
      
    
    
      
        τ
        S
      
      ·
      
        ω
        S
      
    
  
  =
  
    
      
        
          z
          R
        
        ·
        
          z
          PS
        
      
      
        
          z
          S
        
        ·
        
          z
          PR
        
      
    
    ·
    
      ω
      
        R
        /
        S
      
    
  








  
    
      
        τ
        C
      
      ·
      
        ω
        C
      
    
    
      
        τ
        S
      
      ·
      
        ω
        S
      
    
  
  =
  
    
      
        
          z
          S
        
        ·
        
          z
          PR
        
      
      +
      
        
          ω
          
            R
            /
            S
          
        
        ·
        
          z
          R
        
        ·
        
          z
          PS
        
      
    
    
      
        z
        S
      
      ·
      
        z
        PR
      
    
  






 


Simple Solar




  
    
      
        τ
        S
      
      ·
      
        ω
        S
      
    
    
      
        τ
        R
      
      ·
      
        ω
        R
      
    
  
  =
  
    
      
        z
        S
      
      
        z
        R
      
    
    ·
    
      ω
      
        S
        /
        R
      
    
  




τR · ωR




  
    
      
        τ
        C
      
      ·
      
        ω
        C
      
    
    
      
        τ
        R
      
      ·
      
        ω
        R
      
    
  
  =
  
    
      
        z
        R
      
      +
      
        
          ω
          
            S
            /
            R
          
        
        ·
        
          z
          S
        
      
    
    
      z
      R
    
  






 


Compound Solar




  
    
      
        τ
        S
      
      ·
      
        ω
        S
      
    
    
      
        τ
        R
      
      ·
      
        ω
        R
      
    
  
  =
  
    
      
        
          z
          S
        
        ·
        
          z
          PR
        
      
      
        
          z
          R
        
        ·
        
          z
          PS
        
      
    
    ·
    
      ω
      
        S
        /
        R
      
    
  




τR · ωR




  
    
      
        τ
        C
      
      ·
      
        ω
        C
      
    
    
      
        τ
        R
      
      ·
      
        ω
        R
      
    
  
  =
  
    
      
        
          z
          R
        
        ·
        
          z
          PS
        
      
      +
      
        
          ω
          
            S
            /
            R
          
        
        ·
        
          z
          S
        
        ·
        
          z
          PR
        
      
    
    
      
        z
        R
      
      ·
      
        z
        PS
      
    
  






 


Simple Star
τS · ωS




  
    
      
        τ
        R
      
      ·
      
        ω
        R
      
    
    
      
        τ
        S
      
      ·
      
        ω
        S
      
    
  
  =
  
    
      
        z
        S
      
      -
      
        
          ω
          
            C
            /
            S
          
        
        ·
        
          (
          
            
              z
              R
            
            +
            
              z
              S
            
          
          )
        
      
    
    
      z
      S
    
  








  
    
      
        τ
        C
      
      ·
      
        ω
        C
      
    
    
      
        τ
        S
      
      ·
      
        ω
        S
      
    
  
  =
  
    
      
        -
        
          (
          
            
              z
              R
            
            +
            
              z
              S
            
          
          )
        
      
      
        z
        S
      
    
    ·
    
      ω
      
        C
        /
        S
      
    
  






 


Compound Star
τS · ωS




  
    
      
        τ
        R
      
      ·
      
        ω
        R
      
    
    
      
        τ
        S
      
      ·
      
        ω
        S
      
    
  
  =
  
    
      
        
          z
          S
        
        ·
        
          z
          PR
        
      
      -
      
        
          ω
          
            C
            /
            S
          
        
        ·
        
          (
          
            
              
                z
                R
              
              ·
              
                z
                PS
              
            
            +
            
              
                z
                S
              
              ·
              
                z
                PR
              
            
          
          )
        
      
    
    
      
        z
        S
      
      ·
      
        z
        PR
      
    
  








  
    
      
        τ
        C
      
      ·
      
        ω
        C
      
    
    
      
        τ
        S
      
      ·
      
        ω
        S
      
    
  
  =
  
    
      
        -
        
          (
          
            
              
                z
                R
              
              ·
              
                z
                PS
              
            
            +
            
              
                z
                S
              
              ·
              
                z
                PR
              
            
          
          )
        
      
      
        
          z
          S
        
        ·
        
          z
          PR
        
      
    
    ·
    
      ω
      
        C
        /
        S
      
    
  




















TABLE 6







Speed Reduction Ratios Using Reactive Power Quotients










SIMPLE
COMPOUND





Planetary




  
    
      ω
      S
    
    
      ω
      C
    
  
  =
  
    
      
        z
        R
      
      +
      
        z
        S
      
    
    
      
        z
        S
      
      ·
      
        (
        
          1
          +
          
            
              
                τ
                R
              
              ·
              
                ω
                R
              
            
            
              
                τ
                S
              
              ·
              
                ω
                S
              
            
          
        
        )
      
    
  








  
    
      ω
      S
    
    
      ω
      C
    
  
  =
  
    
      
        
          z
          R
        
        ·
        
          z
          PS
        
      
      +
      
        
          z
          S
        
        ·
        
          z
          PR
        
      
    
    
      
        z
        S
      
      ·
      
        z
        PR
      
      ·
      
        (
        
          1
          +
          
            
              
                τ
                R
              
              ·
              
                ω
                R
              
            
            
              
                τ
                S
              
              ·
              
                ω
                S
              
            
          
        
        )
      
    
  






 


Solar




  
    
      ω
      R
    
    
      ω
      C
    
  
  =
  
    
      
        z
        R
      
      +
      
        z
        S
      
    
    
      
        z
        R
      
      ·
      
        (
        
          1
          +
          
            
              
                τ
                S
              
              ·
              
                ω
                S
              
            
            
              
                τ
                R
              
              ·
              
                ω
                R
              
            
          
        
        )
      
    
  








  
    
      ω
      R
    
    
      ω
      C
    
  
  =
  
    
      
        
          z
          R
        
        ·
        
          z
          PS
        
      
      +
      
        
          z
          S
        
        ·
        
          z
          PR
        
      
    
    
      
        z
        R
      
      ·
      
        z
        PS
      
      ·
      
        (
        
          1
          +
          
            
              
                τ
                S
              
              ·
              
                ω
                S
              
            
            
              
                τ
                R
              
              ·
              
                ω
                R
              
            
          
        
        )
      
    
  






 


Star




  
    
      ω
      S
    
    
      ω
      R
    
  
  =
  
    
      -
      
        z
        R
      
    
    
      
        z
        S
      
      ·
      
        (
        
          1
          +
          
            
              
                τ
                C
              
              ·
              
                ω
                C
              
            
            
              
                τ
                S
              
              ·
              
                ω
                S
              
            
          
        
        )
      
    
  








  
    
      ω
      S
    
    
      ω
      R
    
  
  =
  
    
      
        -
        
          z
          R
        
      
      ·
      
        z
        PS
      
    
    
      
        z
        S
      
      ·
      
        z
        PR
      
      ·
      
        (
        
          1
          +
          
            
              
                τ
                C
              
              ·
              
                ω
                C
              
            
            
              
                τ
                S
              
              ·
              
                ω
                S
              
            
          
        
        )
      
    
  




















TABLE 7







Vertical Asymptotes and Speed Reduction Ratio Slopes at Ordinate Axes










VERTICAL ASYMPTOTE
ORDINATE AXIS SLOPE





Simple Planetary




  
    ω
    
      R
      /
      S
    
  
  =
  
    
      -
      
        z
        S
      
    
    
      z
      R
    
  








  
    
      
      
        
          d
          u
        
        
          d
          
            ω
            
              R
              /
              S
            
          
        
      
      
    
    
      
        ω
        
          R
          /
          S
        
      
      =
      0
    
  
  =
  
    
      
        -
        
          z
          R
        
      
      ·
      
        (
        
          
            z
            R
          
          +
          
            z
            S
          
        
        )
      
    
    
      z
      S
      2
    
  






 


Compound Planetary




  
    ω
    
      R
      /
      S
    
  
  =
  
    
      
        -
        
          z
          S
        
      
      ·
      
        z
        PR
      
    
    
      
        z
        R
      
      ·
      
        z
        PS
      
    
  








  
    
      
      
        
          d
          u
        
        
          d
          
            ω
            
              R
              /
              S
            
          
        
      
      
    
    
      
        ω
        
          R
          /
          S
        
      
      =
      0
    
  
  =
  
    
      
        -
        
          z
          R
        
      
      ·
      
        z
        PS
      
      ·
      
        (
        
          
            
              z
              R
            
            ·
            
              z
              PS
            
          
          +
          
            
              z
              S
            
            ·
            
              z
              PR
            
          
        
        )
      
    
    
      
        (
        
          
            z
            S
          
          ·
          
            z
            PR
          
        
        )
      
      2
    
  






 


Simple Solar




  
    ω
    
      S
      /
      R
    
  
  =
  
    
      -
      
        z
        R
      
    
    
      z
      S
    
  








  
    
      
      
        
          d
          u
        
        
          d
          
            ω
            
              S
              /
              R
            
          
        
      
      
    
    
      
        ω
        
          S
          /
          R
        
      
      =
      0
    
  
  =
  
    
      
        -
        
          z
          S
        
      
      ·
      
        (
        
          
            z
            R
          
          +
          
            z
            S
          
        
        )
      
    
    
      z
      R
      2
    
  






 


Compound Solar




  
    ω
    
      S
      /
      R
    
  
  =
  
    
      
        -
        
          z
          R
        
      
      ·
      
        z
        PS
      
    
    
      
        z
        S
      
      ·
      
        z
        PR
      
    
  








  
    
      
      
        
          d
          u
        
        
          d
          
            ω
            
              S
              /
              R
            
          
        
      
      
    
    
      
        ω
        
          S
          /
          R
        
      
      =
      0
    
  
  =
  
    
      
        -
        
          z
          S
        
      
      ·
      
        z
        PR
      
      ·
      
        (
        
          
            
              z
              R
            
            ·
            
              z
              PS
            
          
          +
          
            
              z
              S
            
            ·
            
              z
              PR
            
          
        
        )
      
    
    
      
        (
        
          
            z
            R
          
          ·
          
            z
            PS
          
        
        )
      
      2
    
  






 


Simple Star




  
    ω
    
      C
      /
      S
    
  
  =
  
    
      z
      S
    
    
      
        z
        R
      
      +
      
        z
        S
      
    
  








  
    
      
      
        
          d
          u
        
        
          d
          
            ω
            
              C
              /
              S
            
          
        
      
      
    
    
      
        ω
        
          C
          /
          S
        
      
      =
      0
    
  
  =
  
    
      
        -
        
          z
          R
        
      
      ·
      
        (
        
          
            z
            R
          
          +
          
            z
            S
          
        
        )
      
    
    
      z
      S
      2
    
  






 


Compound Star




  
    ω
    
      C
      /
      S
    
  
  =
  
    
      
        z
        S
      
      ·
      
        z
        PR
      
    
    
      
        
          z
          R
        
        ·
        
          z
          PS
        
      
      +
      
        
          z
          S
        
        ·
        
          z
          PR
        
      
    
  








  
    
      
      
        
          d
          u
        
        
          d
          
            ω
            
              C
              /
              S
            
          
        
      
      
    
    
      
        ω
        
          C
          /
          S
        
      
      =
      0
    
  
  =
  
    
      
        -
        
          z
          R
        
      
      ·
      
        z
        PS
      
      ·
      
        (
        
          
            
              z
              R
            
            ·
            
              z
              PS
            
          
          +
          
            
              z
              S
            
            ·
            
              z
              PR
            
          
        
        )
      
    
    
      
        (
        
          
            z
            S
          
          ·
          
            z
            PR
          
        
        )
      
      2
    
  













It should be appreciated that all speed reduction ratios can be written in the form:




  
    
      
        u
        =
        
          
            u
            F
          
          
            1
            +
            κ
          
        
      
    
    
      
        (
        20
        )
      
    
  




where uF is the fixed reaction member speed ratio specific to the particular epicyclic arrangement and κ is the reactive power quotient. This is true regardless of whether the epicyclic arrangement is planetary, solar, star, simple or compound. The reactive power quotient can be a positive or negative quantity. When κ<0, the speed reduction ratio will be larger than that of a gear drive with a fixed reaction member, and the input drive motor must supply power to both the output and the reaction motor/generator. When κ>0, the speed reduction ratio will be smaller than that of a fixed reaction member gear drive, and the output will be powered by both the input drive motor and the reaction motor/generator.

Furthermore, the ratio spread, or the quotient between the maximum and minimum speed reduction ratios will determine the power requirements of the reaction motor/generator. If the primary drive motor is one with a narrow operating speed range and demands a large spread between the top and bottom speed ratios, the reactive power requirements will be large, possibly several times larger than that of the primary drive motor. If the primary drive motor has a wide operating speed range the reactive power requirements will be small. A ratio spread of 2.0 results in primary and reaction power requirements that are equal. The motor/generator size can be further reduced by mating it with its own fixed reaction member gear drive, using traditional methods to determine its required speed reduction ratio.
The selection of which type of epicyclic arrangement is optimum (i.e., planetary, solar or star) depends on the details and the required speed reduction ratio and ratio spread of the application. The slope of the speed reduction ratio at the ordinate axis of the star arrangement is equal to that of the planetary arrangement and the vertical asymptote of a star arrangement will always occur at a lower reaction member speed to that of a planetary arrangement.
Furthermore, the foregoing discussion discloses and describes merely exemplary embodiments of the present disclosure. One skilled in the art will readily recognize from such discussion, and from the accompanying drawings and claims, that various changes, modifications and variations may be made therein without departing from the spirit and scope of the disclosure as defined in the following claims.

## Claims

1. A transmission for transferring torque from a prime mover, the transmission comprising:
a first input shaft adapted to be driven by the prime mover;
a second input shaft;
an output shaft;
a compound solar epicyclic gearset including a ring gear being driven by the first input shaft, first pinion gears being driven by the ring gear, a sun gear driven by the second input shaft and driving second pinion gears, and a carrier driving the output shaft;
a reaction motor driving the second input shaft; and
a controller for controlling the reaction motor to vary the speed of the second input shaft and define a gear ratio between the first input shaft and the output shaft based on the second input shaft speed wherein a vertical asymptote of a ratio of carrier speed to sun gear speed versus a ratio of sun gear speed to ring gear speed for the compound solar epicyclic gearset ranges between 5 and 28.

2. The transmission of claim 1 wherein the first pinion gears and the second pinion gears define a compound pinion gear supported by the carrier.

3. The transmission of claim 1 wherein a fixed ring speed ratio of the compound solar epicyclic gearset ranges between 1.03 and 1.25.

4. The transmission of claim 1 operable in a neutral mode where the output shaft is not rotating when the first and second input shafts are rotating.

5. The transmission of claim 1 wherein the output shaft is concentrically arranged with the second input shaft.

6. The transmission of claim 1 wherein the second input shaft coaxially extends with the first input shaft.

7. The transmission of claim 1 further including a reduction gearset driven by the second input shaft and driving the sun gear.

8. The transmission of claim 7 wherein the reduction gearset includes a drive gear meshed with a driven gear rotating about offset axes.

9. The transmission of claim 7 wherein the reduction gearset includes a worm gear fixed for rotation with the second input shaft and a worm wheel driving the sun gear.

10. The transmission of claim 7 wherein the reduction gearset includes a planetary gearset and further wherein the second input shaft, the first input shaft and the output shaft rotate about a common axis.

11. A transmission for transferring torque from a prime mover, the transmission comprising:
a first input shaft adapted to be driven by the prime mover;
a second input shaft;
an output shaft;
a compound star epicyclic gearset including a sun gear being driven by the first input shaft, first pinion gears in meshed engagement with the sun gear, second pinion gears meshed with a ring gear, and a carrier driven by the second input shaft and rotatably supporting the first and second pinion gears, the ring gear being fixed for rotation with the output shaft;
a reaction motor driving the second input shaft; and
a controller for controlling the reaction motor to vary the speed of the second input shaft and define a gear ratio between the first input shaft and the output shaft based on the second input shaft speed, wherein a vertical asymptote of a ratio of ring gear speed to sun gear speed versus a ratio of carrier speed to sun gear speed for the compound star epicyclic gearset ranges between 0.02 and 0.20.

12. The transmission of claim 11 wherein the first and second pinion gears define a set of compound pinion gears.

13. The transmission of claim 11 wherein a fixed carrier speed ratio of the compound star epicyclic gearset ranges between −4.0 and 28.0.

14. The transmission of claim 11 operable in a neutral mode where the output shaft is not rotating when the first and second input shafts are rotating.

15. The transmission of claim 11 wherein the output shaft is concentrically arranged with the second input shaft.

16. The transmission of claim 11 wherein the second input shaft coaxially extends with the first input shaft.

17. The transmission of claim 11 further including a reduction gearset driven by the second input shaft and driving the carrier.

18. The transmission of claim 11 wherein the first input shaft is rotated in an opposite direction to the output shaft.

19. A transmission for transferring torque from a prime mover, comprising:
a first input shaft driven by the prime mover;
a second input shaft driven by a reaction motor;
an output shaft;
a compound epicyclic gearset including a ring gear driven by the first output shaft, a sun gear driven by the second output shaft, a carrier driving the output shaft, and compound pinion gears supported by the carrier and meshed with the ring gear and the sun gear; and
a controller for controlling the reaction motor to vary the speed of the second output shaft to establish a gear ratio between the first input shaft and the output shaft based on the speed of the second output shaft, wherein a vertical asymptote of a ratio of carrier speed to sun gear speed versus a ratio of sun gear speed to ring gear speed ranges between 5 and 28.

20. The transmission of claim 19 wherein a fixed ring speed ratio of the compound epicyclic gearset ranges between 1.03 and 1.25.

21. A transmission for transferring torque from a prime mover, comprising:
a first input shaft driven by the prime mover;
a second input shaft driven by a reaction motor;
an output shaft;
a compound epicyclic gearset including a sun gear driven by the first input shaft, a carrier driven by the second input shaft, a ring gear driving the output shaft, and compound pinion gears supported by the carrier and meshed with the sun gear and the ring gear; and
a controller for controlling the reaction motor to vary the speed of the second input shaft to establish a gear ratio between the first input shaft and the output shaft based on the speed of the second input shaft, wherein a vertical asymptote of a ratio of ring gear speed to sun gear speed versus a ratio of carrier speed to sun gear speed ranges between 0.02 and 0.20.

22. The transmission of claim 21 wherein a fixed carrier speed ratio of the compound epicyclic gearset ranges between −4.0 and 28.0.