Epicyclic Gearing: A Handbook For design engineers who are just beginning their careers—and even old pros who could use a refresher course—the following article takes a basic approach to discussing epicyclic gearing.

By Amy Flanagan and Jim Marsch Recent articles in Solutions have people designing epicyclic gear trains discussed epicyclic gearing, but often for the first time—and perhaps, if you in the context of experienced engi- will, ease their degree of suffering. neers. As more and more of these We will begin by defining types and engineers reach retirement age young- arrangements and then discuss why er engineers must pick up where they epicyclic gear sets are used. Next left off, and for many epicyclic gearing we’ll look at what’s unique to epicy- is an area where they lack experience. clic , including relative speeds, Epicyclic gearing requires a step-by- torque splits, and multiple mesh con- step process to make it work, and siderations. Finally we’ll discuss “dos some of the steps are not necessarily and don’ts” and share some design intuitive. As such, this article aims to tips and pitfalls associated with epi- provide assistance and guidelines for cyclic gears. three types: simple planetary epicyclic; com- Types and Arrangements pound epicyclic; and coupled epicyclic sets. Why Epicyclic Gearing? Let’s begin by examining some basic terminol- The reasons why epicyclic gearing is used There are several possibilities for epicyclic ogy. Epicyclic gears consist of several compo- have been covered in this magazine, so we’ll arrangements: nents: sun, carrier, planets, and rings. The sun is expand on the topic in just a few places. Let’s the center gear, meshing with the planets, while • Planetary, with ratios between 3:1 and 12:1 begin by examining an important aspect of any the carrier houses the planet gear shaft. As the (see fig. 1) project: cost. Epicyclic gearing is generally less carrier rotates, planets rotate on planet gear • Star, with ratios between -2:1 and -11:1 (see shafts while orbiting the sun. Finally, the ring is fig. 2) the internal gear that meshes with the planets. • Solar, with ratios between 1.2:1 and 1.7:1 Epicyclic gear systems can be divided into (see fig. 3)

Fig. 1: Planetary, with ratios between 3:1 and 12:1

#B-7108 #D-4609

Fig. 2: Star, with ratios between -2:1 and -11:1

26 gearsolutions.com expensive, when tooled properly. Just as one ratios, derived from taking the square root would not consider making a 100-piece lot of of the final ratio (7.70). (See fig. 4.) In the gears on an N/C milling with a form process of reviewing this solution we notice cutter or ball end mill, one should not consider its size and weight is very large. To reduce the making a 100-piece lot of epicyclic carriers on weight we then explore the possibility of mak- an N/C mill. To keep carriers within reasonable ing two branches of a similar arrangement, as manufacturing costs they should be made seen in the second solutions. This cuts tooth from castings and tooled on single-purpose loading and reduces both size and weight with multiple cutters simultaneously considerably (see fig. 5). We finally arrive removing material. at our third solution, which is the two-stage Size is another factor. Epicyclic gear sets star epicyclic. With three planets this gear are used because they are smaller than offset train reduces tooth loading significantly from gear sets since the load is shared among the the first approach, and a somewhat smaller planed gears. This makes them lighter and amount from solution two (see “methodology” more compact, versus countershaft gearboxes. at end, and fig. 6). Also, when configured properly, epicyclic gear The unique design characteristics of epi- sets are more efficient. The following example cyclic gears are a large part of what makes illustrates these benefits. Let’s assume that them so useful, yet these very characteristics we’re designing a high-speed gearbox to satisfy can make designing them a challenge. In the the following requirements: next sections we’ll explore relative speeds, torque splits, and meshing considerations. • A turbine delivers 6,000 horsepower at Our objective is to make it easy for you to 16,000 RPM to the input shaft. understand and work with epicyclic gearing’s • The output from the gearbox must drive a unique design characteristics. generator at 900 RPM. • The design life is to be 10,000 hours. Relative Speeds Let’s begin by looking at how relative speeds With these requirements in mind, let’s look work in conjunction with different arrange- at three possible solutions, one involving a ments. In the star arrangement the carrier single branch, two-stage helical gear set. A is fixed, and the relative speeds of the sun, second solution takes the original gear set planet, and ring are simply determined by and splits the two-stage reduction into two the speed of one member and the number of branches, and the third calls for using a teeth in each gear. two-stage planetary or star epicyclic. In this In a planetary arrangement the ring gear is instance, we chose the star. Let’s examine fixed, and planets orbit the sun while rotating each of these in greater detail, looking at their on the planet shaft. In this arrangement the ratios and resulting weights. relative speeds of the sun and planets are The first solution—a single branch, two- determined by the number of teeth in each stage helical gear set—has two identical gear and the speed of the carrier. Things get a bit trickier when working with coupled epicyclic gears, since relative speeds may not be intuitive. It is therefore imperative to always calculate the speed of the sun, plan- et, and ring relative to the carrier. Remember that even in a solar arrangement where the sun is fixed it has a speed relationship with the planet—it is not zero RPM at the mesh. Torque Splits When considering torque splits one assumes the torque to be divided among the planets equally, but this may not be a valid assump- tion. Member support and the number of plan- ets determine the torque split represented by an “effective” number of planets. This number in epicyclic sets constructed with two or three planets is in most cases equal to the actual number of planets. When more than three Fig. 3: Solar, with ratios between 1.2:1 planets are used, however, the effective num- #B-7005 and 1.7:1

SEPTEMBER 2008 27 ber of planets is always less than the actual and carrier will not be coincident due to manu- freedom or float, which allows the sun, ring, number of planets. facturing tolerances. Because of this fewer and carrier to seek a position where their cen- Let’s look at torque splits in terms of fixed planets are simultaneously in mesh, resulting ters are coincident. This float could be as little support and floating support of the members. in a lower effective number of planets sharing as .001-.002 inches. With floating support With fixed support, all members are support- the load. With floating support, one or two three planets will always be in mesh, resulting ed in bearings. The centers of the sun, ring, members are allowed a small amount of radial in a higher effective number of planets sharing the load. Multiple Mesh Considerations At this time let’s explore the multiple mesh considerations that should be made when designing epicyclic gears. First we must trans- late RPM into mesh velocities and determine the number of load application cycles per unit of time for each member. The first step in this determination is to calculate the speeds of each of the members relative to the - rier. For example, if the sun gear is rotating at +1700 RPM and the carrier is rotating at +400 RPM the speed of the sun gear relative to the carrier is +1300 RPM, and the speeds of planet and ring gears can be calculated by that speed and the numbers of teeth in each of the gears. The use of signs to represent clockwise and counter-clockwise rotation is Fig. 4: Ratio 1 = 4.216, Ratio 2 = 4.216, Weight = 5,293# important here. If the sun is rotating at +1700 RPM (clockwise) and the carrier is rotating -400 RPM (counter-clockwise), the relative speed between the two members is +1700-(- 400), or +2100 RPM. The second step is to determine the num- ber of load application cycles. Since the sun and ring gears mesh with multiple planets, the number of load cycles per revolution relative to the carrier will be equal to the number of planets. The planets, however, will experience only one bi-directional load application per relative revolution. It meshes with the sun and ring, but the load is on opposite sides of the teeth, resulting in one fully reversed stress cycle. Thus the planet is considered an idler, and the allowable stress must be reduced 30 percent from the value for a unidirectional load application. As noted above, the torque on the epicyclic members is divided among the planets. In analyzing the stress and life of the members we must look at the resultant loading at each mesh. We find the concept of torque per mesh to be somewhat confusing in epicyclic gear analysis and prefer to look at the tangential load at each mesh. For example, in looking at the tangential load at the sun-planet mesh, we take the torque on the sun gear and divide it by the effective number of planets and the operating pitch radius. This tangential load, combined with the peripheral speed, is used Fig. 5: Ratio 1 = 3.925, Ratio 2 = 4.536, Weight = 3,228# to compute the power transmitted at each

28 gearsolutions.com mesh and, adjusted by the load cycles per must be used to compute power loss. For sim- revolution, the life expectancy of each com- ple epicyclic sets, the total power transmitted ponent. through the sun-planet mesh and ring-planet In addition to these issues there may mesh may be less than input power. This also be assembly complications that need is one of the reasons that simple planetary addressing. For example, placing one planet epicyclic sets are more efficient than other in a position between sun and ring fixes the reducer arrangements. In contrast, for many angular position of the sun to the ring. The coupled epicyclic sets total power transmitted next planet(s) can now be assembled only internally through each mesh may be greater in discreet locations where the sun and ring than input power. can be simultaneously engaged. The “least What of power at the mesh? For simple and mesh angle” from the first planet that will compound epicyclic sets, calculate pitch line accommodate simultaneous mesh of the next velocities and tangential loads to compute planet is equal to 360° divided by the sum of power at each mesh. Values can be obtained the numbers of teeth in the sun and the ring. from the planet torque relative speed, and the Thus, in order to assemble additional planets, operating pitch diameters with sun and ring. they must be spaced at multiples of this least Coupled epicyclic sets present more complex mesh angle. If one wishes to have equal issues. Elements of two epicyclic sets can be spacing of the planets in a simple epicyclic coupled 36 different ways using one input, set, planets may be spaced equally when the one output, and one reaction. Some arrange- sum of the number of teeth in the sun and ments split the power, while some recirculate ring is divisible by the number of planets to an power internally. For these types of epicyclic integer. The same rules apply in a compound sets, tangential loads at each mesh can only epicyclic, but the fixed of the planets be determined through the use of free-body adds another level of complexity, and proper diagrams. Additionally, the elements of two planet spacing may require match marking epicyclic sets can be coupled nine different of teeth. ways in a series, using one input, one out- With multiple components in mesh, losses put, and two reactions. Let’s look at some need to be considered at each mesh in order examples. to evaluate the efficiency of the unit. Power In the “split-power” coupled set shown in transmitted at each mesh, not input power, figure 7, 85 percent of the transmitted power

Fig. 6: Ratio 1 = 4.865, Ratio 2 = 3.655, Weight = 2,422# #B-7005

SEPTEMBER 2008 29 manner similar to what happens in a “four- square” test procedure for drive . With the torque locked in the system, the horsepower at each mesh within the loop increases as speed increases. Consequently, this set will experience much higher power losses at each mesh, resulting in significantly lower unit efficiency (see fig. 8). Figure 9 depicts a free-body diagram of an epicyclic arrangement that experiences power recirculation. A cursory analysis of this free- body diagram explains the 60 percent efficien- cy of the recirculating set shown in figure 8. Since the planets are rigidly coupled together, the summation of forces on the two gears must equal zero. The force at the sun gear mesh results from the torque input to the sun gear. The force at the second ring gear mesh results from the output torque on the ring gear. The ratio being 41.1:1, output torque is 41.1 times input torque. Adjusting for a pitch radius difference of, say, 3:1, the force on the second planet will be approximately 14 times the force on the first planet at the sun gear mesh. Therefore, for the summation of Fig. 7: Split Powered Coupled Set, Ratio = -40.9, Efficiency = 97.4%. Calculations forces to equate to zero, the tangential load derived using Integrated Gear Software. at the first ring gear must be approximately 13 times the tangential load at the sun gear. If we assume the pitch line velocities to be the same at the sun mesh and ring mesh, the power loss at the ring mesh will be approxi- mately 13 times higher than the power loss at the sun mesh (see fig. 9). Additional Considerations As carrier speeds increase, centrifugal forces on the planet gears become more and more significant; especially if they have a relatively large mass. These forces must be resolved by the planet bearing and oftentimes they are higher than the forces that transmit torque to the carrier. They must be considered in the planet bearing calculations. Lubrication of the planet bearings can be challenging, especially at higher carrier speeds. These challenges have led to many highly creative solutions. Researching patents on this subject will prove beneficial. Retention of planet pins in heavily loaded sets can also prove quite challenging. Deflections will loos- en press fits and crack welds. Loose fits may “wallow” out the bores in the carrier, causing Fig. 8: Set with Power Recirculation, Ratio = 41.1, Efficiency = 61.6%. Calculations more than desired float. Again, researching derived using Integrated Gear Software. patents will be fruitful. flows to ring gear #1 and 15 percent to ring series, 0 percent of the power will be transmit- A final check that must be made—especial- gear #2. The result is that this coupled gear ted through each set (see fig. 7). ly in high ratio planetaries—is tip clearance set can be smaller than series coupled sets Our next example depicts a set with “power between adjacent planets. The time to find because the power is split between the two recirculation.” This gear set comes about this answer is at the design stage… not when elements. When coupling epicyclic sets in a when torque gets locked in the system in a it adds a complication at assembly.

30 gearsolutions.com Fig. 9: Free Body Diagram of a System Be sure and allow “float” or specify very Dos and Don’ts tight location and run-out tolerances or load Now that we’ve looked at epicyclic gear types sharing will be less than anticipated. Finally, and arrangements and their unique design use tangential loads and pitch-line velocities characteristics, as well as several examples, to determine mesh power and let’s discuss the dos and don’ts of epicyclic losses. gear design. Like any skill, designing epicyclic gears is something that becomes easier with practice. Do: As retiring engineers take their know-how • Calculate planet locations with them, younger engineers remain to pick • Define assembly match marks on up where they left off. Although this short drawing primer cannot possibly cover every nuance • Address relative speeds of epicyclic gearing, hopefully it will serve as • Divide torques correctly a jumping-off point for engineers tasked with • Analyze planets as idlers in simple epi- designing their first epicyclic gear set—and cyclic sets perhaps even act as an occasional refresher • Check planets for OD interference for the more experienced designer. • Use free-body diagrams

Don’t: METHODOLOGY: • Rigidly fix all members unless the applica- The UTS Integrated Gear Software (IGS) was tion requires it used to perform the calculations shown in • Assume power splits Figures 7 and 8. IGS is a comprehensive • Use coupled sets that have internal power gear knowledge system that helps design- ers optimize their designs, eliminate noise recirculation and premature failure, lower design and • Forget centrifugal loads on planet production costs, and shorten time to mar- bearings ket. See ANSI/AGMA 6023-A-88 or ASME Paper 68-MECH-45 by P.W. Jensen for more Design Tips and Pitfalls information about epicyclic gears. In closing, here are some design tips to embrace and pitfalls to avoid as you design About the authors: epicyclic gears. Remember that designing on Amy Flanagan is spe- standard centers will result in higher specific cialist and Jim Marsch is gear product sliding and lower efficiency. If struggling with manager at Universal Technical Systems, meshes, removing one tooth from the planet Inc. To learn more call (815) 963-2220 gear will enhance both sun and ring meshes. or go to [www.uts.us.com]. #B-7005

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