This paper relates to the load-equalizing capabilities of planetary gearboxes, and more generally, addresses the load-equalizing capabilities of multi-flow gear trains, that is, gear trains that feature two or more flows of transmitted power. Typically due to manufacturing errors, displacements under operating load, or other factors, planetary pinions share power disproportionately. Enhanced accuracy in the machining of the components is a straightforward, less-than-economical approach to equalizing the load share in planetary gearboxes. Therefore, a need continues to exist for a load-equalizing mechanism for planetary pinions.
Overview of Equal Power Sharing in Planetary Gearboxes
Planetary gearboxes are widely used in the industry; examples are readily found in the automotive, aerospace, and other industries. (An example of a five-pinion planetary gearbox is shown in Figure 1.)
Planetary gearboxes with multiple planetary pinions enable a substantial reduction in the size and weight of the gear drive — concurrent with the number of pinions in the gearbox — and only when the transmitted load is equally shared among all the planet pinions.
Epicyclic gear systems have typically been equipped with straddle-mounted planetary pinions with pins supported on the input and output sides of the carrier. The torsional wind-up of the carrier, position accuracy of the pins, machining tolerances of the planetary gear system components, and bearings clearances all can contribute to poor load sharing among the planetary pinions, as well as misaligned gear contacts in the deflected state. In the traditional epicyclical gearing system — i.e., where the distance between planetary pinion centerlines is specified by the design to be within a fixed range — it is widely recognized that load sharing is not evenly distributed among planetary gear meshes. Furthermore, stress, too, is distributed invariably at mesh points. Load sharing and stress distribution at each mesh point are heavily influenced by global design configuration; backlash tolerance; component design tolerances; manufacturing accuracy; component deflection; and thermal distortion. Figure 2 shows in an exaggerated form that contact is made at the mesh point Ksg.p of the planet pinion before any contact is made at the mesh points of the other planets (assuming the ring gear makes contact with all pinions at points Krg.p). In a rigid system, this condition imposes unbalanced loading among the planetary pinions.
To share torque equally among all the planetary pinions, the utilization of flexible components in the design of a planetary gearbox shows promise. For example, the Stoeckicht system from 1940 solves this problem by making the annulus ring flexible while designing both it and the sun gear to float without bearings so that they are both supported by their respective mesh points.
Other designers have applied a number of novel designs with varying degrees of success for epicyclical gearing systems that help distribute load among planet pinions more evenly thus increasing power density. In general, such improvements use components in the gear train that are elastically compliant and are intended to compensate clearance variations without imparting any negative operating characteristics, including:
- Flexible ring gears have been applied, but the effectiveness of this approach is not universal because radial deflections of the ring gear are not enough to compensate clearance (backlash) variations present at the various mesh points
- Floating ring gear system (used in some off-highway applications)
- Floating sun gear
- Floating planet carrier
- Double-helical gear with floating members
- Floating planetary pinion (flexible pin or abbreviated to "flex-pin")
Note: A brief overview of known load-equalizing means for planetary pinions in a planetary gearbox can be found in References 1-10. This paper discusses gearboxes that feature flex-pins.
Planetary gear drives with flexible pins.
The application of gearboxes with flexible pins is based on the ideas of British inventor Raymond J. Hicks . In 1964, Hicks developed a method of providing load sharing between the planet pinions of an epicyclic gearbox, i.e., the flexible pin, which has since been applied to a large variety of industrial, aerospace, and marine gearboxes.
The original Hicks invention is shown in Figure 3 as a fragmentary and diagrammatic part-sectional elevation of an epicyclic gear with flexure, greatly enlarged for illustration. Referring to Figure 3, which illustrates the invention diagrammatically, the epicyclic gear broadly comprises a sun wheel 10, an annulus gear ring 11, and a plurality of planet pinions 12, which mesh with both sun and annulus. The planets are supported on spindles 13 fast with a carrier 14. The effect of the gear depends upon whether the sun, annulus, or carrier is the input or output and which of these three is fixed either permanently or optionally. In any event, the center 15 of the planet teeth, measured axially, is at an equal distance from the points 16 and 17, which lie in planes contacting the point of emergence of spindle 13 from the carrier and the planet, respectively. Thus, the couples will be equal, as previously explained.
Similarly, the planet does not skew when misaligned; rather, it deflects the shaft (Figure 3) until each planet is equally loaded; the annular gap 18 permits this. (It is understood that, in practice, the deflection involved is relatively slight.)
The planet 12 may seat directly on spindle 13, or, as shown in Figure 3, may seat and be journalled on a sleeve 19 that provides the gap. In this case, the sleeve is fast with the spindle. The spindle is a press or shrink-fit in the carrier 14 and possibly in the sleeve 19, but a system of c-clips 20 is also used as a precaution against damage through fit relaxation.
The use of flexible pins eliminates the need for straddle mounting and so allows the maximum possible number of planet pinions to be used, subject to tip-to-tip clearance for any particular gear ratio. The number of planet pinions varies with the ratio between the annulus and sun gear tooth numbers.
Load sharing is achieved by ensuring that deflection of the planet pinion spindle under its normal load is considerably greater than the manufacturing errors that cause misallocation. Put another way, if one planet pinion tends to take more load than the others, it will deflect until the others assume their share.
Later, the Hicks' flex-pin concept was enhanced .
Figure 4a depicts a typical planet gear supported by a planet spindle mounted on a flexible pin cantilevered from a simple carrier plate. The two ends of the pin are fitted to the carrier plate and the spindle; the latter is counter-bored to allow the pin to deflect freely.
Figure 4b shows that a uniform tooth load — the centroid is symmetrical with the teeth length of the flexible pin — exerts equal and opposite moments on the built-in ends so that they remain parallel during deflection.
Figure 4c shows that a point load concentrated on either end of the tooth face produces a relative angular deflection of that end, with respect to the other, which is six times the finite deflection that occurs when it is loaded at the center.
The deflections shown are theoretical values that assume that the built-in portions of the pin at either end are supported so rigidly that they have zero slopes. However, static tests have shown that elastic deflections in the joints between the spindle and the carrier plane produce complementary finite slopes such that the effective flexibility of the pin is more than doubled — without affecting the parallel movement of the spindle.
With the proportions given, the relative rigidity of the spindle is such that its own cantilever deflection in terms of the total is so small that it has virtually no effect on tooth load distribution. If a thinner planet spindle is used with a significant flexibility, it is possible to compensate for this by reducing the length of the counterbore.
An important feature of this design is that because the planet spindle and flexible pin are co-axial, it is capable of deflection about two axes, making it virtually self-aligning. This means that the pin is influenced by radial as well as tangential tooth loads, and it is able to compensate helix errors of different magnitudes or sense at the sun and annulus mesh points. Therefore, it is also capable of compensating torsional deflection of the sun gear, which takes place in gearboxes of large tooth ratio. However, if the resultant load of the sun and annulus mesh points is not in the same plane as the midpoint of the unsupported portion of the flexible pin, there are two restoring effects:
- The offset tangential load tips the spindle in the tangential plane in a manner that tends to offset the respective load points an equal amount to either side of the midpoint of the pin.
- The radial couple resulting from the offset radial loads tilts the spindle in the radial plane until the residual couple is reduced to an amount compatible with the angular flexibility of the spindle assembly.
In short, there is a complex movement in two planes as the spindle takes up a position of minimum strain energy. This complex movement is in fact beneficial, as it promotes a slight crowing effect as a result of the skewed or nonparallel axes.
On the other hand, if the planet pinion is cross-cornered so that the resultant tangential load is in the same plane as the midpoint of the pin, there is still a radial tilting couple to provide a restoring action.
When a gearbox has a rotating planet carrier, additional radial loads and deflections are imposed on the flexible pin assembly due to the combined centrifugal weight of planet pinion, spindle, and pin.
The flexible pin eliminates the need for straddle mounting and, therefore, enables the maximum possible number of planet pinions to be used that are subject to tip-to-tip clearance for any particular epicyclic ratio. Load sharing is achieved by ensuring that deflection of the planet spindle under its normal load is considerably greater than the manufacturing errors that cause misallocation. If one planet tends to take more load than the others, it will deflect until the others take their share.
Simply put, the flexible pin is designed to use high deflections to provide uniform tooth loads between planet pinions and across sun-to-planet and planet-to-annulus tooth face widths. An added benefit of producing equal loads across the tooth contact face widths is the occurrence of equal loading along the planet pinion bearings — the most critical element of a high-capacity, low-speed epicyclic gear.
Conversely, the industrial design of epicyclic gears requires high carrier rigidity relative to the gear tooth stiffness, but this is impractical as it induces misallocation of load across the teeth and bearing, leading to premature failure.
Because a supporting shaft of the planet gear is of a flexible, double-cantilever construction (flexible pin system), a planet pinion that receives more load moves in parallel due to sagging of the pinion and in order that all planet pinions receive equal load. Consequently, an excellent equal sharing effect is shown in such cases, and the whole system is of a smaller size.
Due to the flexible pin system, the shock-absorbing effect for torque variation of a prime mover or a load is expected.
If the load is distributed evenly among the teeth faces, it is the same as when a concentrated load is applied to the center; the pins flex as double-cantilever beams, and parallelism relative to other planetary pinions is retained. If there is any error in relative positioning between flexible pins — due, for example, to errors in machining or assembly — the planetary pinion positioned here receives more load than the others, and the flexible pin supporting that gear flexes further to absorb the error. Thus, the uniform load distribution mechanism keeps load distribution even.
If an eccentric load is applied to the left end of a tooth face, the flexible pin flexes (Figure 4c) and the load on the right side of the tooth face increases, mitigating the eccentric distribution of the load across the width of the tooth. The effect of gear tooth trace errors, gear casing deformation, misalignment, and other problems can be absorbed and mitigated.
However, for just about all equipment types, economics dictate the need for increased power density and improved reliability. A common approach is an attempt to build in more planets, thereby reducing forces and stresses at each mesh point. But, as planets are added, so is uncertainty about just how much power each planet is transmitting. Instead of fixing the angular positions of the planet pinions, the flexible pins were designed so that they deflect independently in a circumferential direction. This ultimately helps equalize the force distribution among the planets while transmitting torque at various levels; here on, this feature will be referred to as torsional compliancy.
Torsional compliancy is achieved by applying the double-cantilevered beam design (Figure 4b), i.e., when two tangential forces are applied to the flex-pin pinion, the angular deflection caused by the bending of the pin cantilevered from a carrier wall can be offset in the opposite direction by the angular deflection caused by bending of the sleeve cantilevered from the other end of the pin.
Flexible pins have been incorporated into various types of equipment with designs typically including assembly of separable components, including gears, pins, mounting sleeves, backing plates, cap screws, and various types of rolling element bearing races and bushings.
Such designs achieve the objective of creating a torsionally compliant system. Additionally, because gears are less prone to be tipped off axis because the single-sided planetary carrier can no longer wind, it can be argued that gear contacts have a much higher probability of remaining centered at all meshes. It follows then that the flexible pin permits the designer to specify narrower gears and still avoid stress concentration at the face ends. Therefore, power density is improved in the axial direction.
An elastic deformation of the flex-pin allows for overcoming manufacturing errors, displacements under operating load, and so forth. The elastic deformation must be large enough to accommodate such manufacturing errors and displacements, yet not exceed a particular given value. When zero torque is applied to the driving member of the planetary gearbox, no force is exerted from the flex-pin and no deformation of the flex-pin is observed. When the maximum torque is applied, maximum force is exerted from the flex-pin and maximum deformation of the flex-pin occurs. Because only elastic deformations of the flex-pin are considered valid (see Young's Law), the loaded diagram is represented by a linear function (Figure 5). Huge displacements y of the flex-pin are necessary to attain the operating load that acts against the flex-pin. It is best to keep the displacements y as small as possible; however, they need to be sufficient to address aforementioned manufacturing errors and displacements.
The flex-pin concept is presented here in order to make a correct comparison of this concept with the concept of elastic absorbers of manufacturing errors (EAME).
Elastic Absorbers of Manufacturing Errors (EAME)
The purpose of the elastic absorbers of manufacturing errors
- 1. Reduce required minimum displacement of planetary pinions.
- 2. Make a gear train with split torque insensitive to manufacturing errors as well as displacements of other kinds.
Therefore, the EAME is a component of a gear train with split torque that is:
- Preloaded by a precalculated load
- Mounted between any two components of the gear train where an additional degree of freedom is required to override the manufacturing errors
The capability of an elastic absorber to override manufacturing errors relies strongly upon its stiffness. For most of the materials used in the production of gears and gear units, the relationship between the applied load and the displacement caused by the load is linear (Figure 5). In the worst-case scenario, the accuracy, A, of load sharing among the planet pinions can be calculated from the following formula:
is the allowed variation of the load per planet pinion
is the total number of planet pinions
Shown in Figure 6 is a worst-case scenario of load sharing among the planet pinions in a multi-flow gear train, that is, when only one of npp planet pinions is loaded the most heavily, while the rest of the planet pinions withhold the lowest permissible torque. Remember that all the loads per planet pinion are within the allowed band of variation, k. With that said, the accuracy, A, in Equation 1 is derived based on an arithmetic average of different loads.
The calculations reveal that in a worst-case scenario for a planetary gearbox — eight planet pinions (npp = 8) and the allowed variation of the load per planet pinion k = 0.1 — the deviation of the transmitted load from the desired value does not exceed 8.75 percent. Thus, for a planetary gear drive with three planet pinions (npp = 3) and the allowed variation of the load per planet pinion k = 0.05, deviation of the transmitted load from the desired value does not exceed 3.33 percent. The actual deviations are less than those calculated for the worst-case scenario.
If the preloaded elastic absorber is loaded by a precalculated value, actual displacement of the pinions does not exceed the allowed displacement tolerance.
Consider a planetary gearbox for which the permissible range of variation of load share among the planetary pinions is equal to ±ORl (Figure 7). The operating range of the deformation ORd of the elastic absorber needs to be the smallest possible; however, it must also be large enough to allow for the manufacturing errors and the planetary pinions displacement under operating load. The two ranges ±ORl and ORd specify a rectangle. In Figure 7, a diagonal of this rectangle forms an angle y with the horizontal axis. The desirable stiffness, c, of the elastic absorber must be equal to (or less than):
Once the stiffness, c, is determined, a straight line through the origin can then be constructed. This line is at the angle y with respect to the horizontal axis. Point of interception of the constructed straight line and the straight line of the nominal (operating) load, NLop, specifies the desirable pre-deformation of the elastic absorber of manufacturing errors (Figure 7). The pre-deformation, PDea, of the elastic absorber is calculated from the equation:
Calculation of the design parameters of the elastic absorber of manufacturing errors is based on two parameters, i.e., of the ±ORl and ORd.
An Illustrative Example of Application of the Preloaded Elastic Absorber of Manufacturing Errors
In a two-stage planetary reducer, the preloaded elastic absorber of manufacturing errors can be placed in between the first-stage planet pinion and the second-stage planet pinion1 (Figure 8). It is common practice to hob both the planet pinions of the cluster planet pinion. For this purpose, it is convenient to assemble the cluster planet pinion comprising two planet pinions. However, note that proper phasing of the pieces in relation to one another while assembling the cluster planet pinion is critical; misphasing errors with planet pinions can be catastrophic. The preloaded elastic absorber of the manufacturing errors is installed between the two planet pinions of the cluster planet pinion (Figure 8).
For equal torque sharing among the planetary pinions, the misphasing Dj must be zero. As the misphasing Dj cannot be eliminated, it must be absorbed. For this purpose, it is necessary to introduce an additional degree of freedom for one of the planetary pinions in relation to another and, in this way, to make the planet pinions self-aligning. Self-alignment of the planet pinions can be ensured by implementation of the preloaded elastic absorber of manufacturing errors. An angular displacement, Dj, to be absorbed by the elastic absorber can be eliminated when the linear displacement, Dl, is equal to:
In Equation 4, the radial location of the preloaded elastic absorber is specified by the distance r. Deformation, Dl, of an elastic body under load usually (but not necessarily) relates to the applied load, T, linearly or (at least) almost linearly, D = c .T (c is a proportionality factor equal to the rigidity of the preloaded elastic absorber). In such a case, the angle f can be calculated from the formula:
In the general case, when c ≠ const, the current value of c is equal to:
The variation interval for the applied load should be known for the calculation of the design parameter of the preloaded elastic absorber of manufacturing errors.
Note: It should be stressed here that the elastic absorbers of manufacturing errors can be applied in the design of multi-flow gear trains (including, but not limited to, planetary gearboxes) of any and all types and sizes: small, medium, and large size.
The advantage of the elastic absorber of manufacturing errors over the flex-pin concept is clearly illustrated in Figure 9. When the flex-pin concept is used, only a small fraction, D(ORd), is used to overcome the unfavorable displacement of the planetary pinions under operating load (Figure 9a). This is because the flex-pins are not preloaded and, therefore, the stiffness angle jfp is relatively small. When the concept of the elastic absorber of manufacturing errors is used, the entire ORd is used to accommodate for the unfavorable displacement of planetary pinions under operating load (Figure 9b). As such, the stiffness angle jea is significant. Typically, the inequality jea >> jfp is always observed. This advantage of the elastic absorber of manufacturing errors is significant in that it enables a higher power density for transmission through the planetary gearbox. Evidently, the presented concept of the elastic absorber of manufacturing errors can be used for the improvement of power density in any and all gear trains that feature split torque.
In this brief overview of known approaches for equal power sharing in planetary gearboxes, planetary gear drives with flexible pins (the Hicks' approach) were discussed, and a novel method for equal power sharing in planetary gearboxes was proposed. The method is based on the use of elastic absorbers of manufacturing errors (EAME), and an illustrative example of the application of EAME was provided.
Use of the concept of elastic absorbers of manufacturing errors allows a significant improvement in power density being transmitted through the planetary gearbox. The elastic absorbers of manufacturing errors can be applied in the design of multi-flow gear trains of any type.
Numerous patents on inventions are granted in the U.S. and other countries all around the world on the inventions of gear trains with split torque that are based on the implementation of EAME.
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