Epicyclic Load Sharing Map-Application As a Design Tool

In this paper, the concept of an Epicyclic Load Sharing Map (ELSM) will be explained. The ELSM is a physics-based tool that is derived from a physical explanation of the load-sharing phenomenon.

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AGMA Paper #11FTM05

One of the main advantages of planetary transmissions is that the input torque is split into a number of parallel paths. Therefore, in an n planet planetary system, each sun-pinion-ring path is designed to carry 1/n of the input torque. However, equal load sharing between the planets is possible only in the ideal case. In the presence of positional type manufacturing errors, equal load sharing is not realized, and the degree of inequality in load sharing has major implications for gear system sizing, tolerancing schemes, and torque ratings.

In this paper, the concept of an Epicyclic Load Sharing Map (ELSM) will be explained. The ELSM is a physics-based tool that is derived from a physical explanation of the load-sharing phenomenon. It is a plot of the Load ratio (or % of input torque) versus a non-dimensional parameter Xe. The non-dimensional parameter is a function of combined system stiffness, tolerance level, and operating torque. The ELSM maps out the operating space of any epicyclic gear set, and a given gear set at a given operating condition maps to a point on the ELSM. Once a gear set is located on the ELSM, its behavior under any load and error condition can be quickly predicted. Also, the advantages of adding extra planets can be accurately estimated.

In this paper, the application of the ELSM as a design tool will be discussed. The general case when there are errors on the position of every carrier pinhole will be considered. Statistical simulations will be performed for a given manufacturing error distribution for three to seven planet systems.
 
Introduction
Epicyclic transmissions are compact as the input torque is split into a number of parallel sun-pinion-ring paths, and each path is designed to transmit a fraction of the input torque. In the absence of manufacturing variations, perfect load sharing between the different parallel paths is possible. The power density of such epicyclic gearsets can be improved by simply adding additional planets (up to the maximum number that can fit).

However, in reality due to the presence of various manufacturing variations that cause positional differences in the location of the individual planets, such equal load sharing cannot be achieved [1-14]. Some of the planets will transmit higher than nominal loads, while others transmit lower than nominal loads. Previous experimental [1-6] and computational [6-13] studies of varying complexities have demonstrated the significant load sharing inequalities that result from these errors. This load sharing inequality needs to be accurately estimated in order to properly size epicyclic gearsets and reliably estimate their torque capacity [14].

Recent research work has shown that the load sharing behavior is associated with positional deviations from ideal location that causes one or more planets to lead or lag the other planets. These deviations from ideal location are due to manufacturing variations, and will be referred to as “positional error” or simply “error” in this paper. Bodas and Kahraman [11] classify the manufacturing errors into time invariant, assembly independent errors (pinhole position error, pinhole diameter error), time invariant, assembly dependent errors (planet tooth thickness, planet pin and bore eccentricities), and time variant, assembly dependent errors (run-outs of the gears). They also offer a way to combine all these errors into a cumulative positional error. In this work, “error” will refer to this cumulative positional error that includes the contributions from all sources.

There are several key factors that influence the load sharing behavior. Some of these factors are the transmitted torque, error level, directionality of error, system flexibility, number of planets, and amount of float in the system. The sensitivity of load sharing inequality to many of these variables has been studied [1-5, 10-14]. These factors are also recognized in ANSI/AGMA 6123-B06 [15].

While the research activities have revealed much about the factors influencing load sharing, and provided computational means of quantifying the load inequalities, a basic physical understanding of the true mechanism that leads to the load sharing behavior was lacking. In recent papers [16-18], the author has proposed a physical mechanism that explains all known load sharing behavior. Both floating and non-floating (fixed centers) systems were treated. The physical explanation leads to simple expressions that seem to completely describe the complex load sharing behavior. These expressions are in non-dimensional terms and can be applied to any epicyclic gear set under any operating condition. Comparisons to computational models and experimental results have shown excellent correlation.

The proposed physical explanation also leads to the concept of an epicyclic load-sharing map (ELSM). The ELSM is a plot of the Load ratio (or % of input torque) versus a non-dimensional parameter Xe. The non-dimensional parameter is a function of combined system stiffness, tolerance level, and operating torque. The ELSM maps out the operating space of any epicyclic gear set, and a given gear set at a given operating condition maps to a point on the ELSM. The ELSM contains curves for 3, 4, 5, 6, and 7 (and more) planet systems. Once a gear set is located on the ELSM, its behavior under any load and error condition can be quickly predicted. Also, the advantages of adding extra planets can be accurately estimated.

The Load Ratio term used in the ELSM is defined similar to the mesh load factor, Kγ, defined in the AGMA standards [15]. AGMA recommends estimating Kγ by measurement, or using a table provided in [15]. The ELSM provides an alternate method of defining the load sharing inequality which is based on an understanding of the physical behavior, and implicitly includes the influence of the key variables like error, stiffness, number of planets, transmitted torque, etc.

In this paper, we will first briefly review the physical explanation of the load sharing phenomenon for fixed and floating systems. We will also summarize the previously published findings on the detailed mechanism of load sharing in 3 – 7 planet systems. A detailed derivation of a five-planet system will be provided for the sake of completeness. Next, the concept of the Epicyclic Load Sharing Map will be discussed. An equivalent error metric that captures the cumulative effect of errors on the position of each planet in the system will also be discussed. A comparison between the values predicted by the ELSM and those found in [15] will also be discussed.

Finally a statistical simulation will be performed to demonstrate the application of the ELSM to actual gearsets with varying levels of manufacturing accuracy.

Key Elements of the Proposed Framework
The following are the key elements of the framework that will be used to describe the planetary load sharing behavior:

• Tangential position error is the root cause
• System float partially neutralizes the errors
• Elastic deformation under load neutralizes the remaining portion of the errors
• Non-dimensional Neutralizing Ratio
• Equal load sharing in the absence of errors

Tangential Position Error As Root Cause
Position error is defined as the deviation in the location of the center of the planets from their ideal locations. It has been widely reported that the presence of positional error results in the phenomenon of unequal load sharing between the planets. Several recent publications [5,12] have also shown that the epicyclic system is sensitive to errors in the tangential direction and insensitive to errors in the radial direction.

Consider an epicyclic system with an error ẽ on the location of one of the planets, while all the other planets are at their ideal location. Figure 1 shows a schematic of the planet with the error. Under unloaded conditions, the error will cause the planet contacting surfaces to come closer to, or move farther away from, their mating surfaces. If the error causes the planet to come in contact earlier than the other planets, then the error is considered to be positive (planet leads all the other planets) and the planet with the error will carry more load than all the other planets. On the other hand, if the error is negative, the planet will lag all the other planets and carry a lighter load than the other planets. The magnitude of inequality in the load sharing will depend upon the magnitude by which the planet error causes the mating sun-planet and ring-planet surfaces to come closer to (or move away from) each other.

Let OS’ and OR’ be lines parallel to the sun-pin and pin-ring planes of action; Φ be the operating pressure angle; ẽ be the pinhole position error; ẽr, ẽs, ẽT be the error components along the sun LOA, ring LOA and tangential direction; and θ be the orientation of the error with respect to the tangential direction OX.

Then, the component of error along the planet-ring plane of action is:

r = ẽ cos (θ – Φ)            (1)

The component of error along the planet-sun plane of action is:

s = ẽ cos (θ – Φ)            (2)

These are the amounts by which the planet surfaces come closer to (or move farther away from) their mating surfaces. When ẽr ≠ ẽs, the first surface pair (say sun mesh) that comes in contact cannot carry load until the planet rotates about its axis and the other surface pair (say ring mesh) also comes in contact. In general, the planet comes closer to its mating surfaces by an amount:

p = (ẽr + ẽs )/2 = ẽ cos θ cos Φ
p = ẽT cos Φ            (3)

Eq. 3 shows that when the error is in the radial direction, θ = 90º and ẽp = 0. This explains why an error in the radial direction has no influence on epicyclic load sharing. Also, the magnitude of the error is maximum when θ = 0º or 180º. When θ = 0º, ẽp is positive and the planet will lead all the other planets, and when θ = 180º,  ẽp is negative and the planet will lag all the other planets. For any arbitrary error direction, the magnitude of error in the tangential direction is the only relevant parameter.

System Float
In non-floating systems, no movement is possible between the centers of the coaxial members (sun, ring, and carrier). In these systems, all the error has to be neutralized by the elastic deformation in the system. On the other hand, in floating systems the center of at least one of the coaxial members is free to move radially, and thus relative motion between the coaxial members is possible. The major advantage of floating systems over non-floating systems is that a portion of the positional error is neutralized by system float. The remaining error is neutralized by system deflections. The portion of the error that is neutralized by system deflection is the cause of the load sharing inequality.

Elastic Deformation Under Load
In a rigid system (rigid gear tooth surfaces and rigid bearing supports), the presence of a positive error will cause the entire load to be carried by the planet with the error, and all other planets will remain unloaded. However, in elastic systems, as the planet with the error gets loaded, the tooth flanks in mesh and the planet on the needle bearing supports undergo elastic deformation, and this causes the error to be neutralized. The force required to neutralize a given error will be called the ‘neutralizing force’ and the corresponding torque will be called the ‘neutralizing torque’. Since the only relevant error is the component in the tangential direction, and the net resultant of forces acting on the planet center is in the tangential direction, all computations of stiffnesses and deflections will be performed in the tangential direction and at the center of the planet.

Let, Kb, Ks and Kr be the bearing stiffness (includes needles and planetary pin), the sun-planet mesh stiffness (due to deformation of both the sun and pinion members), and the planet-ring mesh stiffness (due to deformation of the pinion and ring members), respectively. The effective stiffness of the sun-planet-ring–bearing system in the tangential direction is:

       (4)           

Keff is the cumulative stiffness due to Hertzian contact at the sun–planet and planet–ring meshes, the tooth bending deflections, the tooth base rotation, and the planet bearing and pin stiffnesses. Keff can be considered to be lumped at the center of the planet, and the rest of the system can be considered to be rigid. Keff is a property of the sun-planet-ring-needle bearing system and is generally invariant with the number of planets in the system.

In the rest of this paper, we will focus on the tangential error ẽT (or simply error e) and the tangential neutralizing force or torque required to neutralize this error. Also, all the stiffnesses will be lumped in the Keff  term, and the rest of the system will be assumed to be rigid.

Non-dimensional Neutralizing Ratio
If an n planet epicyclic gear set has an error e on the position of one of its planets, and all other planets are at their ideal location, then the Neutralizing Force is given by:
        (5)

The corresponding non-dimensional Neutralizing Ratio is the ratio of the Neutralizing Force to the Total Input Force, or the Neutralizing Torque to the Total Input Torque:

    (6)

The Neutralizing Ratio captures the influence of the system flexibility, the amount of error, and the loading on the gear set. In this paper, we will express the load sharing behavior in terms of the Neutralizing Ratio. The developed expressions will be applicable for any epicyclic gear set, regardless of error or loading level, system stiffness or number of planets.

Equal Load Sharing in the Absence of Errors
Consider an n planet epicyclic gear set and assume the meshes to be in-phase (phasing has a transient secondary effect on mesh load sharing). Under ideal conditions, all n mesh paths are simultaneously in contact. Under this condition, all meshes will share the load equally, and 1/n of the load will pass through each sun-planet-ring path. Equally spaced planets, in the absence of location errors, always share the load equally amongst all the planets. Even slightly unequally spaced planets (small deviations from equally spaced due to assembly considerations) share the load approximately equally.

Schematic Representation
We will use a schematic representation to describe the load sharing behavior. Figure 2(a) shows an example of a 5 planet epicyclic gear set with no position errors and all planets in contact, and  Figure 2(b) shows its equivalent schematic. In the schematic, the centers of the planets are connected to the center of the carrier by rigid arms and the sun-ring bodies are represented by another rigid body. The combined stiffness,Keff of all the elastic bodies (sun, planet, ring, bearings, planet pin, and carrier) is lumped at the interface between the planet and the sun-ring bodies. Though this stiffness is not explicitly shown in the schematics, it is assumed to always be present at the planet centers. The regions of contact are represented by  as shown in the figure, and all loaded elastic deformations (δ) take place in these regions. Interference will be represented by as shown in Figure 2(c), and such material penetration will not be allowed.

Load Sharing Mechanism: Non-Floating System
In non-floating systems, the centers of the coaxial members (sun, ring, carrier) are fixed and no relative motion is possible between their reference frames (other than rotation about their axis). Here, the total positional errors can only be neutralized by elastic deformation under load. The load needed to neutralize the error is the cause of the load sharing inequality.

Here, we will consider a five-planet epicyclic gear set, though the methodology works with systems with any arbitrary number of planets [16]. When there are positional errors, all planets are in contact and experience equal load sharing (Figure 3(a)). Now, consider the case when there is a positive error ẽ1 on planet P1 as shown in Figure 3(b), and let all other planets be at their ideal location. The gearset cannot be assembled in this state as there would be interference at planets P2-P5. During assembly, the carrier assembly (or sun and ring gears) will rotate about their axis and the initial unloaded position will be as shown in Figure 3(c). Now let us start applying load through the epicyclic gearset. At first, only planet P1 will transmit torque and all the other planets are unloaded. This is what will happen under lightly loaded conditions.

Now, let us start increasing the torque flowing through the gearset. As the P1 mesh gets loaded, there is elastic deformation at the mesh, and the carrier assembly rotates about its center by the magnitude allowed by the deformation. When the amount of deformation equals the originally introduced error ẽ1, the error is completely neutralized and all other planets come in contact. The torque required to neutralize the error will be the Neutralizing Torque Tẽ1. Any further increase in torque will be equally shared by all the planets.

If the error is in the opposite direction (- ẽ1), then in the unloaded case only the P1 path would be unloaded and all the other paths would be loaded equally. The neutralizing torque in each path will still be Tẽ1. After the error has been neutralized, the remaining torque will be shared equally among all the planet paths.

The detailed calculations for load sharing can be found in [16]. A brief summary is provided below:

Input load (7)

where Yn is the ideal load sharing.

Stage 1 – Only path P1 is loaded until error ẽ1 is neutralized. Remaining planets unloaded.

Neutralizing Force (8)

Remaining Force is (9)

Stage 2 – All pinions are in contact and share the remaining load equally.
Remaining load per pinion (10)

Table 1 shows the steps in the calculation for an epicyclic gearset with an arbitrary number of planets ‘n’. Summing up the torque transmitted by each planet, the load sharing can be determined. As seen in Table 1, the final expressions can be formulated in terms of the non-dimensional Neutralizing Ratio Xe, and the same equations are valid for systems with any number of planets. It can also be seen that a positive error (planet with error leads) is more detrimental than a negative error (planet with error lags). The load ratio term has similar definition to the AGMA mesh load factor, Kγ.

Load Sharing Mechanism: Floating System
In floating systems, the center of at least one of the coaxial members is free to move radially and thus relative motion between the coaxial members is possible. It is well established that floating one or more of these bodies leads to significantly better load sharing characteristics. In this section, that benefit will be explained and quantified.

The major advantage of floating systems over non-floating systems is that a portion of the positional error is neutralized by system float. The remaining error is neutralized by system deflections. Only the error that is neutralized by system deflection causes load-sharing inequalities. In order to fully understand the way floating systems work, we need to individually analyze systems with different number of planets.
 
Detailed derivation of systems with 3 – 6 planets can be found in [16] and systems with 7 planets can be found in [17]. Here we will schematically describe the load sharing mechanism in 3 – 6 planet systems, and illustrate the calculation steps for a 5 planet system.

The extra step involved in floating systems is to mathematically compute the portion of the error that will be neutralized by system float (without resulting in any load sharing penalty). It will be seen that in floating systems with equally spaced planets, at the end of the floating stage there are two pinions in contact for systems with even number of planets, and three pinions in contact for systems with odd number of planets. The amount of load required to move from one loading stage to the next loading stage also needs to be mathematically computed. This was done in [16-17]. Here, only a description of the different stages is provided, to help understand the resulting Epicyclic Load Sharing Map.

Figure 4 shows a three planet epicyclic gearset. In the absence of positional errors, all planets are in contact and share the load equally. Let planet P1 now have the prescribed amount of positive error, as shown in Figure 4(b). The gearset cannot be assembled in this configuration, and the carrier assembly will rotate during the assembly process, and initial orientation will be as shown in Figure 4(c). In a floating system, one or more of the coaxial members will now float until other pinions come in contact. In this case, both the other planets, P2 and P3, come in contact at the end of the floating process. The entire error is therefore neutralized by the system float, and three planet floating gearsets experience equal loading on all the planets.

Figure 5 shows the different calculation stages for a four planet epicyclic gearset. Again, in the absence of error all planets equally share the load. In the presence of a positive error on P1, at the end of the assembly process, only planet P1 is in contact. The system then floats until some other planet(s) come in contact. In this case, opposing planet P3 comes in contact, and from force balance it is obvious that they will experience equal loading. The meshes will then deform under load, and the carrier assembly will rotate about its center until planets P2 and P4 also come in contact. Any further loading will then be equally shared by all the planets.

In a five planet epicyclic gearset (Figure 6), the system floats until planets P1, P3, and P4 are in contact. Since these three planets are not equally spaced, the loading will not be equal (unlike the three planet case). The loading ratios on these planets can be determined from force balance. As the loading is increased, the planets will deform under load, and the planet carrier will move until the other planets P2 and P5 also come in contact. Any further increase in load will be equally shared by all the five planets.

Figure 7 shows a six planet system with an error on planet P1. Due to system float, initially opposing planets P1 and P4 come in contact and equally share the load. The system then deforms under load, until additional planets P3 and P5 also come in contact. The loading on P1 and P4 before additional planets come in contact can be computed. In this next stage there are 4 loaded planets and the relative loading on each planet can be determined. This stage will continue until remaining planets P2 and P6 also come in contact. Any further increase in load will be equally shared by all the six planets.

In order to model the initial floating condition, a few simple geometrical calculations are needed [16]. In Figure 8, the bounding circle of the planet centers is shown as a circle of radius R, and the center point is O. Any point I on the bounding circle at an angle βi is at a radius Ri given by:
(11)

If the bounding circle rotates by an amount θ about A, then point I rotates to I’, and its projection in the tangential direction is I’’. This changes the separation at point I by I-I’’ which is given by:
(12)

Let us now examine the detailed calculation steps for the five planet case. Let there be an error ẽ1 on the location of planet P1, all other planets being at their ideal locations. After initial assembly (Figure 6(c)), P1 will be in contact and the separations at planets P2, P3, P4 and P5 will be:

Stage 1 – Float
After the floating stage, P1, P3, and P4 will be in contact. This will happen when the rotation about P1 equals the separation at planets P3 and P4. The rotation θ about P1 at the end of the floating stage can be calculated using Eq. 12 as:

The gap at P2 and P5 at the end of the floating stage will be:

Similarly,

The amount of carrier float needed in this stage in order to realize the fully floating condition is:

   
Stage 2 – 3 planet contact
Relative forces on P1, P3 and P4 can be computed from the force balance as:

This stage will continue until additional planets come in contact. The movement at P1, P3, and P4 is given by:

As the forces at each planet mesh are proportional to the displacements at that mesh:

Additional planets come in contact when the separation at P2 and P5 are neutralized, ie.

The neutralizing torque is given by:

The carrier motion needed in this stage is back towards its original center and the magnitude is given by:

Stage 3 – 5 planet contact
Once all five planets come in contact, the remaining load is distributed equally among all the planets. In this stage the carrier rotates about its own center.

The steps in the calculation for the five planet epicyclic gearset are shown in  Table 2. The load ratios on each individual planet is expressed in terms of the non-dimensional neutralizing ratio Xe, which is a function of system stiffness, error level, and operating torque (as given by eq. 6). These load ratios are valid for every 5 planet epicyclic gearset under any operating condition, though the value of Xe varies from gearset to gearset (based on stiffness) and with operating conditions.

Similar computations for 4, 6, and 7 planet epicyclic gearsets are documented in [16-17]. Both positive and negative errors were treated, and it was shown that the final load ratio expressions are the same, whether the errors are in the positive or negative direction. The value of Xe would be negative when the error is negative. For the five pinion case, for negative errors P1, P3, and P4 would see a reduction if load and P2 and P5 would see an increase in their load. It is however clear that the positive error case is more detrimental as the magnitude of the increase is 2Xe for positive errors and 1.618Xe for negative errors.

Table 3 documents the load ratio at the individual meshes of 3 – 7 planet systems. These equations are valid when there is enough loading to ensure that all planets are in contact. In each case, the error is only on planet P1, and all other planets are at their ideal location. There are a number of observations that can be made from  Table 3 and other results presented so far, and many of these observations match previously reported epicyclic load sharing behavior in floating systems.
 
• Tangential positional errors influence load sharing behavior, and radial positional errors have no effect on load sharing. [12]
• Three planet epicyclic gearsets experience equal load sharing [1-4, 7]
• In four planet systems, opposing planets experience equal load sharing [5, 6, 12, 13]. It can be seen that the increase in load on planets P1 and P3 matches the decrease in load on planets P2 and P4.
• In all cases with positive errors, the planet with the error (P1) experience maximum loading. For negative error on P1, adjacent planets P2 and Pn-1 experience maximum loading.
• Sensitivity to positional error increases as the number of planet increases [5, 10-13]

From Table 1 and  Table 3, it can be seen that in an n planet epicyclic gearset, the load ratio on the most heavily loaded planet is be given by:

Load Ratio non-floating = 1 + (n-1) X   (13)

Load Ratio floating = 1 + (n-3) X   (14)

These load ratios can be directly compared to the mesh load factor, Kγ in AGMA 6123-B06 [15]. The above equations quantify the beneficial effects of having adequate float in the system. The amount of float that is needed can also be derived from the detailed solution.  Table 4 shows the amount of carrier float needed to achieve a truly floating condition. In reality, it is only the relative motion that matters, and the float can come from movement of any of the bodies. Based on the physical explanation provided, it can also be concluded that once freely floating condition is reached, further increase in system float will not have any beneficial effects on load sharing.

Epicyclic Load Sharing Map
The load sharing behavior shown in  Table 3 is valid when the operating conditions are such that all planets are in contact. The loading on planet P1 represents the worst case load when the error is positive and the loading on planet P2 represents the worst case loading when the error is negative (Xe is negative in these cases).

The condition that all pinions are in contact can be reduced to requirements on Xe. As the positive error ẽ1 increases, Xe  increases and after some critical value of Xe  some of the planets may lose contact. Similarly, for negative errors, as the magnitude of the error increases, the magnitude of |Xe| increases. Beyond a critical value of |Xe|, one or more of the planets will lose contact.

The load ratios when some pinions are not in contact can also be derived from calculations such as those shown in  Table 2. The number of planets in contact, and the corresponding load ratio expressions, will depend upon the stage reached. For example, in the five planet case, if the value of Xe  (error, load and stiffness) is such that we reach stage 3, then all five planets will be in contact. However, if only stage 2 is reached, then only planets P1, P3, and P4 will be loaded, and planets P2 and P5 will be unloaded.

The general form of the load ratio expression is:

3 Planet systems
   LR = 1+ 0 Xe  any  Xe       3P contact

4 Planet systems

The above equation defines the entire operating range of any epicyclic gearset. It also clearly defines the transition points where some planets lose contact. As shown in the equation, the worst case loading when the error is negative is different from the worst case loading for positive errors.

The ELSM equations can also be written as a percentage of the total input torque transmitted through the most heavily loaded planet.

3 Planet systems
    LR = 33.33                    any Xe          3P contact

4 Planet systems
    =50.0                       Xe ≤ -1.0           2P contact
    =25.0 + 25.0 |Xe|   -1.0 ≤ Xe ≤ 1.0     4P contact
    =50.0                       Xe ≥ 1.0            2P contact

Figure 9 plots the above equations and will be referred to as the Epicyclic Load Sharing Maps (ELSM). Figure 9(a) shows the entire operating space in terms of the load ratio, whereas Figure 9(b) shows the corresponding plot in terms of the percentage of input torque for the positive Xe region (as that is the more detrimental condition. The ELSM is valid for every epicyclic gearset operating under floating conditions. An epicyclic gearset operating under a given torque and error level maps to a point on the corresponding nP line in the ELSM. Under any other torque or error condition, the corresponding Xe can simple be recalculated using:
(15)   

The critical values of Xe at which some planets start to lose contact can also be seen in the ELSM. For a 4 planet epicyclic gearset, two planets become unloaded at torque and error conditions such that Xe>1.0. Similarly, the 5P gearset has only 3 loaded planets above Xe of 0.618. A 6 planet gearset has four loaded planets above Xe of 0.5, and only two loaded planets above Xe of 2.0.

Errors On All Planet Locations
In the discussions so far, we have analyzed the case when the error exists on one planet location, and all the other planets are at their ideal location. However, in general, all planet locations are subject to positional errors, and the final load distribution is the net effect of all the individual planet errors. In [17], it was shown that the cumulative errors on individual planets  ẽi could be expressed in terms of the equivalent error Ẽi. Table 5 shows the equivalent error expressions for 3-7 planet gearsets. It should be noted that each planet location will have an equivalent error term, and the corresponding load ratio or fraction of input torque transmitted through that pinion can be calculated from the corresponding expressions in  Table 5. The maximum value of the equivalent error Ẽmax can then be used to calculate the equivalent non-dimensional neutralizing ratio XẼ-max. Figure 9 and Figure 10, when used with XẼ-max give the ELSM in terms of equivalent errors which capture the influence of errors on all planet locations.

Figure 10 shows the ELSM in terms of the equivalent error. Only the regions where all planets are in contact are shown in Figure 10. Also, as the positive error condition is more severe in terms of load sharing, only the positive error portion is shown.

Design Implications
The ELSMs shown in Figure 9 and Figure 10 carry a lot of information that can have significant design implications. These plots completely characterize the load sharing behavior of every epicyclic gearset.

Worst-Case Loading
In the worst case, in a 4-planet system there are only two loaded planets, each transmitting 50% of the input torque (Load Ratio =2.0). In a 5-planet system, there are three loaded planets, with the worst planet transmitting about 45% of the input torque (Load Ratio=2.24). For the 6 planet case, there are again 2 loaded planets, each transmitting 50% of the load (Load Ratio=3.0). For the 7 planet case, there are three loaded planets, with the worst planet transmitting 47.4% of the load (Load Ratio=3.32). Of course these happen under high error conditions, and/or low torque conditions, when the value of Xe is above the individual thresholds. Under these conditions, a three planet floating system performs better, with each planet transmitting 33.3% of the input torque.

Effect of Adding Additional Planets
The torque capacity of epicyclic gearsets depends upon the number of planets in the system, and the torque capacity can be increased simply by increasing the number of planets (if physically possible). However the gains that are achieved by adding planets are reduced due to the load sharing behavior. The ELSM can help quantify the torque capacity improvements.

A family of epicyclic gearsets with varying number of planets will be represented by a vertical line in Figure 10. The intersection of the vertical line and the nP planet lines gives the load sharing for a given value of XE. The non-dimensional variable XE depends upon the error level, torque value, and system stiffness. The straight-line behavior represents the case where the addition of planets does not influence the system stiffness. The gear mesh and bearing stiffnesses may vary slightly with changes in nominal load, and the carrier stiffness may vary as the construction changes to accommodate more planets. In these cases, the straight line will slightly curve to the right if the system gets stiffer with the addition of planets, and to the left if the system gets more flexible. The beneficial effects of adding additional planets can be easily estimated from the ELSM.

Figure 10(b) shows very interesting behavior. If the effective stiffness is invariant with number of planets, all the nP lines intersect at a critical value of XE=1/3. At this operating condition, the most heavily loaded planets carries 1/3rd of the input torque, regardless of the number of planets in the system.

Below this critical value of XE, there are benefits to adding additional planets. In the range of 0 ≤ XE ≤ 0.33, the maximum planet loading drops as the number of planet increases. In this region, the torque capacity of epicyclic gears can be improved by the addition of planets. The improvement is of course the greatest when there are no manufacturing positional errors (XE =0). The magnitude of the drop rapidly decreases as XE approaches a value of 0.33.

When XE ≥ 0.33 and all planets are in contact, there is no advantage to adding extra planets. In fact, the loading becomes worse as the number of planets increases. The 3 planet case gives the best loading condition in these cases. When XE is large enough for planets to lose contact, the relative ranking of 4, 5, and 6 planet systems varies in different regions. The 3 planet system still ranks the best in terms of load sharing.

Comparison with AGMA Mesh Load Factor
AGMA recommends experimentally quantifying the mesh load factor. That is of course expensive, and can be only done after hardware has been built. In the design stage, an accurate estimate of the load sharing inequalities is therefore extremely valuable.
Table 8 in [15] provides some estimates (based on experience) for the mesh load factor, and is partially reproduced below as Table 6. The Mesh Load Factor Kγ is shown to be dependent upon the number of planets in the system, manufacturing quality of gearsets, float in the system, and system flexibility. These are the same influences that we considered in this paper. The significant difference is that the method in this paper is based on a physical explanation of the load sharing mechanism, and provides a method to scientifically quantify the influence of these parameters. The applied torque was also identified in the ELSM as a key influence.

A straight comparison between the data in Table 6 and the ELSM proposed in this paper is not possible due to differences listed.

In order to illustrate how the ELSM could be used to more accurately estimate the load sharing influence, some comparisons are provided by matching the mesh load factor for the 6 planet level 2 case with the corresponding value from ELSM, and then predicting the load sharing for the 3 – 9 planet cases. Table 7 shows the values predicted by ELSM. In order to match the value of 1.44 for a floating 6P case, the value of XE is 0.1467. At this value of XE, the load ratio prediction for all the other planet cases is shown. It will be seen that other than the 3P case, there are significant differences between Table 6 and Table 7.

The level 3 row in Table 6 has better quality and a more flexible ring gear. This will cause the errors to be lower and the system stiffness to be lower. The corresponding value of XE will then be lower. A value of XE =0.0767 is needed to match the 6P predictions. This is directionally correct, but the drop in XE may be larger than what can be achieved by the improvements in accuracy and flexibility. This is of course assuming that we are using the same base hardware/gear system design, and the numbers in Table 6 may have been based on drastically different designs.

Level 4 in Table 6 has flexible mounts in addition to better quality gears and flexible ring gear. This will further reduce the system stiffness. In order to match the 6P load sharing behavior, the value of XE shows a further drop. This is again directionally correct. The 5 and 7 planet cases also show good match. The 3P planet case has perfect load sharing. However, the 4, 8, and 9 planet cases have significant differences.

Finally we will consider the non-floating case. Level 1 in Table 6 has low quality gears, in a low speed application, and non-floating members. But for 5, 6, 7, and 8 planet cases, it shows the same or better load ratios than level 2 cases. This is not possible based on the explanation provided in this paper. The ELSM predicts an extremely harsh load sharing effect in non-floating cases. However, it should be pointed out that it is very hard to achieve the theoretically perfectly non-floating case. Even in simulations reported in earlier works, it was found to be extremely hard to achieve a truly non-floating system (no clearances, stiff supports). There is always some amount of relative movement possible.

Strategies to Improve Load Sharing
As seen in Figure 10, the load sharing in a given epicyclic gearset depends upon the computed value of the non-dimensional Neutralizing Ratio (based on equivalent error) XE. This variable captures the effect of operating torque, cumulative positional errors from all manufacturing tolerances, and the combined system stiffness. As stated earlier, Figure 10 is valid for a floating system, and the load sharing will be significantly worse for systems without adequate float.

The load sharing improves when the magnitude of XE is reduced. XE is lower if the relevant tolerances are reduced or the system stiffness is reduced. On the other hand, if the tolerances are higher, or the system is made stiffer the magnitude of XE is increased, and load sharing gets worse.

The input torque also has an influence, and load sharing appears to be better at higher torques, and worse at lower torque. The proposed physical explanation offered in this paper shows that the amount of load sharing inequality remains the same and is derived from the amount of load it takes to neutralize the error on the planets due to elastic deformations. Any further increase in load is shared equally by all the planets. As the load increases beyond that needed to neutralize the errors, the load sharing does not increase and becomes a smaller percentage of the total torque. Also, if the torque is sufficiently reduced, XE will increase to the point that one or more planets start losing contact. Under these conditions, the errors on the planets are not completely neutralized by elastic deformations.

There are many techniques that have been developed to improve load sharing in planetary gears. An extensive list of methods is listed in [15]. Based on the methodology presented in this paper, these methods can be classified into three categories:

1). Ensure adequate system float
The differences in load sharing between floating and non-floating systems are so significant, that systems should be designed to have a floating member, wherever possible. This is also generally easy to achieve, and has minimal cost impact.

2) Reduce system manufacturing errors that impact tangential positional errors.
a. Increased precision of tangential carrier pin-hole position errors
b. Minimize planet tooth thickness variations
c. Classifying planets and assembling epicyclic gearsets with matched pinion sizes
d. Reduce runouts of all members
e. Reduce out-of-roundness of the internal gear
3) Improve system flexibility
a. Lower internal gear and sun gear support stiffness
b. Lower needle bearing radial stiffness
c. Lower gear mesh stiffness
d. Flexible pinion pins [19-21]
e. Lower tangential stiffness of the carrier

Statistical Simulation
The equations in Table 5 can be used to evaluate the impact of any given error combination on epicyclic load sharing behavior. In actual gearsets, the errors on individual pinions will be random, but for a given manufacturing process, the values will have a certain distribution. We will assume that the cumulative tangential position errors on individual planets have a normal distribution centered about the no error condition. The methodology can of course handle any other error distribution. In order to evaluate the impact of the distributed error conditions, a Monte Carlo simulation will be performed.

In the example case that follows:
    Mean error = 0.000 mm
    Standard deviation = 0.0233 mm
    Tolerance band = 6 * std. dev = 0.140 mm
                              (-70 μm to + 70 μm)

Let us assume that at an input torque on the sun gear of 1000 Nm, and error of +70 μm, the non-dimensional neutralizing ratio is:

XE = X70= 0.2625

This can be estimated using analytical tools or experimentally. XE at any other error or torque condition can be computed using Eq 14.

Every pinion on the 4, 5, 6, and 7 pinion variants of the example gear set was assumed to have a random normally distributed error. 10000 combinations of errors were analyzed in each case. The torque increase/decrease on any pinion due to the introduced error has a normal distribution, as seen in Figure 11. The width of the distribution is different in each case. The 4 pinion gear set has the narrowest distribution of loading change. As the number of pinion increases, so does the width of the distribution of the change in loading.

While individual planets can have an increase or decrease in load, depending upon whether the combined error is positive or negative, the planetary gear set as a whole always sees an increase in load (on one or more pinion). In order to evaluate the impact of pinhole position errors, the maximum equivalent error and the corresponding load on the most heavily loaded pinion should be considered.

Figure 12(a) shows the cumulative probability distribution of the equivalent error for the 4, 5, 6, and 7 planet cases. As seen in the figure, a significant portion of the population has an equivalent error that is greater than 70 μm. This means that statistically there is a significant portion of the production population that has load sharing that is worse than the case when there was one pinion at extreme tolerance and all the other pinions are at their ideal location. Figure 12(b) shows a “zoomed in” view of the cumulative probability distribution. In this case, it can be seen that roughly 8 to 14% of the population has an equivalent error condition that is greater than 70μm. Figure 12(c) shows the resulting loading on the most heavily loaded planet. The nominal load under ideal condition decreases as the number of pinion increases. This can be seen in the figure. But the advantages of additional pinions are partially (and often significantly) eroded by positional errors. From a design standpoint, the loading on the worst parts is of critical importance. The 90th percentile gear set experience a total load (on the most heavily loaded planet) of 320 Nm, 310 Nm, 297 Nm, and 285 Nm in the 4, 5, 6, and 7 planet systems, respectively. For the 99th percentile gearset, the corresponding loads are 364Nm, 357Nm, 348 Nm, and 342 Nm. These calculations show that at these positional tolerances and for this input torque, the parts with the highest percentile errors will experience very little benefit by adding additional planets. Also, the actual load transmitted by the most heavily loaded planet is significantly greater than the nominal loads. Using this procedure, design loads can be calculated for any reliability level.

The above discussion was based on a tolerance level of ± 70 μm on each pinion. Next, the effect of different tolerance bands is explored. Simulations were performed for tolerance bands of ± 35 μm, ± 50 μm, ± 70 μm, and ± 100 μm. In each case, the error was assumed to have a normal distribution with a standard deviation equal to 1/6 of the tolerance band. The cumulative probability curves for the total load for the ± 35 μm and ± 100 μm cases are shown in Figure 13. As seen in the figure, at an error band of ± 35 μm, having more than 4 pinions offers significant benefits in terms of reduction in maximum load. This benefit gradually decreases as the error band is increased. At an equivalent error of ~ 90 μm, the value of XE approached 1/3. It was earlier shown that at this critical XE value, the most heavily loaded planet will carry 1/3 of the torque, regardless of the number of pinions in the system. It can therefore be expected that when the error band is ± 100 μm, there will be a significant number of parts that will have an error higher than the critical value. The cumulative probability corresponding to a part with the critical amount of error will be different in 4, 5, 6, and 7 planet systems. Figure 13(c) shows a close up view of a portion of the plot. It can be seen that above the 75th percentile part, the advantages of having more than 4 pinions is fully eroded and gear sets with higher number of planets even experience higher loads than gear sets with fewer number of planets.

Conclusions
In this paper, a physical explanation of the epicyclic load sharing behavior is provided. Load sharing behavior in gearsets with 3 to 7 planets are shown. Both floating and non-floating systems are treated. Closed form load sharing expressions were developed for epicyclic gearsets with an arbitrary number of planets.  The physical explanation and developed expressions lead to the concept of Epicyclic Load Sharing Maps. These non-dimensional ELSM captures the load sharing behavior of every epicyclic gearset, and maps the entire operating space of any epicyclic gearset.

Cumulative error expressions that capture the combined effect of random errors on every pinion position were also developed. Finally, statistical simulations to evaluate the impact of random distributed errors on individual pinholes have been performed. The derived expressions and the ELSM can be used to quickly evaluate the impact of manufacturing tolerances on any epicyclic gearset. 

References
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[2] Hidaka, T., Terauchi, Y., and Dohi, K., “On the Relation between the Run Out Errors and the Motion of the Center of Sun Gear in a Stoeckicht Planetary Gear,” Bull. JSME, 22, pp. 748–754, 1979
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[15] ANSI/AGMA 6123-B06 Design Manual for Enclosed Epicyclic Gear Drives
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Acknowledgments:
Printed with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, 5th Floor, Alexandria, Virginia 22314. Statements presented in this paper are those of the Authors and may not represent the position or opinion of the American Gear Manufacturers Association. AGMA Paper #11FTM05