Program Compound Epicyclic Gear Design (Parallel Axis) Introduction

Program 60-1162 Compound Epicyclic Gear Design (Parallel Axis) Introduction The compound epicyclic gear unit consists of a central external gear (sun gear) meshed with one or more external gears (sun planet gears). The sun planet gears are part of a two-gear cluster on the same shaft or axis. The second gear is the ring planet gear. The ring planet gears are then meshed with an internal gear (ring gear) which encloses the system. The planet gears and planet gear support bearings are held in a carrier which rotates about the geometrical center of the unit.(the term epicyclic comes from the path of a point on a planet gear which traces out an epicycloid in space.) There are three basic types of compound epicyclic gear units. Planetary: When the ring gear is fixed or grounded and the sun and carrier are input/output members the unit is called a planetary gear. Star: When the carrier is fixed and the sun and ring are input/output members the unit is a star gear. The star gear is not an epicyclic gear as the planet gear centers do not rotate about the unit central axis but since the construction is basically the same it is included in the family. For high speed units the star gear is often used in cases where planetary gears are not practical because of the high centripetal acceleration loads on the planet gears. Solar: When the sun is fixed and the carrier and ring are input/output members the unit is a solar gear. The range of speed reduction ratios for which these units can be designed with reasonable proportions is as follows: Planetary Gear: 6:1 to 25:1 Star Gear: 5:1 to 24:1 Solar Gear: 1.05:1 to 2.2:1 Below these ranges the planet gears become quite small and it becomes difficult to design the gears and planet bearings for reasonable life. Above these ranges the sun gear becomes small and the number of planets that can be used without interference is limited. This, again, makes the design of the bearings difficult. If more than one planet gear is used, the number of planets that will assemble between the sun and ring is limited by the numbers of teeth in the gears and by the possibility of interference between the tips of the planet gear teeth. For a number of

UTS Integrated Gear Software identical planets to assemble equally spaced around the center, the following relationship between the tooth numbers in the gears must be met: [(N ring N pl-sun )+(N sun N pl-ring )]/n p =integer where: N ring = Number of teeth in ring gear N pl-sun = Number of teeth in sun gear planet N sun = Number of teeth in sun gear N pl-ring = Number of teeth in ring gear planet n p = Number of planet gears The distance between the planet gear centers in the carrier must, of course, be greater than the outside diameter of the planet gears or tooth tip interference will result (assuming the planet gears are in the same plane). It is not necessary that the planets be equally spaced. However, to make assembly possible, they must be spaced at multiples of the least mesh angle. e p /β = integer β = 360 / [(N ring N pl-sun )+(N sun N pl-ring )] where: e p = Angle between adjacent planet gears, deg β = Least mesh angle, deg It is not necessary (or even desirable) that PD ring = PD sun +PD pl-sun +PD pl-ring (where PD = Teeth/(TransPitch)). If this relationship is met, the center distance is standard and the operating pressure angles at the sun/planet external mesh, φ ext, and the planet/ring internal mesh, φ int,will be equal to the nominal pressure angle of the systems. A higher operating pressure angle at the external mesh, φ ext, will often increase the strength of the gear set while not affecting the operating characteristics of the mesh adversely. (If the ring gear rim thickness is 2 tooth depths or more, a highφ int will tend to reduce the bending stress. If the ring gear rim thickness is 1.5 tooth depths or less, a low φ int will tend to reduce the bending stress.) Epicyclic units are often used as differentials. UTS model 60-1162 is restricted to systems where one element is fixed, and it does not consider use as a differential. UTS models 60-1163 and 60-1164 treat simple and compound epicyclic units used as differentials. 2

60-1162 Compound Epicyclic Gear Design (Parallel Axis) Examples If you are using model 60-1162 for the first time you may wish to run the following example. Assume we wish to design a spur gear compound planetary set with about 15.5 to 1 reduction ratio with a ring gear diameter of about 12 inches. Assume also that the smallest number of teeth we wish to use is 20 for the sun gear and 28 for the internal mesh planetary gears. (The number of teeth would be selected based upon material and duty cycle. See UTS model 60-180.) Open a new analysis in 60-1162. Enter 20 in the input column for Sun Gear Teeth, 28 for Ring Planet Teeth and 15.5 for Planetary Gear ratio. Figure 1 shows the data input form with this data. Report 1 shows the solution. Fig. 1 3

UTS Integrated Gear Software Report 1 MESSAGE FIELD ERROR MESSAGE, internal mesh ERROR MESSAGE, external mesh ERROR MESSAGE, mesh - general Prime factors greater than 100 NUMBER OF TEETH Ring Gear Teeth Ring Planet Teeth (Internal mesh) 28 Sun Planet Teeth (External mesh) Sun Gear Teeth 20 RATIOS Planetary Gear (Forward rotation) 15.5000 Star Gear (Reverse rotation) Solar Gear (Forward rotation) Planet/Sun Ratio Ring/Planet Ratio EXTERNAL MESH (SUN/PLANET) Normal Pitch Normal Pressure Angle Nominal helix Angle 1.000000 1/in 20.000000 deg 0.000000 deg 4

60-1162 Compound Epicyclic Gear Design (Parallel Axis) Transverse Pitch Transverse Press Angle Normal Module Transverse Module Opr Pitch Dia, planet Opr Pitch Dia, sun gear INTERNAL MESH (RING/PLANET) 1.0000 1/in 20.0000 deg 25.4 mm' 25.4 mm' in in Normal Pitch Normal Pressure Angle Nominal helix Angle Transverse Pitch Transverse Press Angle Normal Module Transverse Module Opr Pitch Dia, ring gear Opr Pitch Dia, planet 1.000000 1/in 20.000000 deg 0.000000 deg 1.0000 1/in 20.0000 deg 25.4 mm' 25.4 mm' The model has set the diametral pitch to 1, the pressure angle to 20 degrees and the helix angle to 0 degrees since no values were input. A solution for the numbers of teeth in the ring gear and sun planet gears requires an iterative solution, since the equations solved in this direction cannot be solved directly. We also need to temporarily fix an operating pitch diameter to provide enough input for the solution. For now we will set the operating pitch diameter of the sun to the standard pitch diameter. To do this we must work directly in the TK Solver Variable Sheet. Toggle to this, move the mouse pointer to the input column for Opr Pitch Dia, sun gear and type ns/pt_s. When you press the enter key you should have 20 in the input column. We will make a guess of 100 teeth for the ring gear. Type 100 in the input column for this variable. Move the mouse pointer to the status column and type G, or doubleclick the Status cell and pick Guess from the drop-down list that appears. This will in in 5

UTS Integrated Gear Software tell TK to start an iteration with a first guess of 100. Your screen should be like Sheet 1. Sheet 1 60-1162 COMPOUND EPICYCLIC (Ver 6.01) Use Tools, GwzUnits to Change Units m_i ERROR MESSAGE, internal mesh m_x ERROR MESSAGE, external mesh m_g ERROR MESSAGE, mesh - general m_p Prime factors greater than 100 NUMBER OF TEETH: G 100 nr Ring Gear Teeth 28 np_r Ring Planet Teeth (Internal mesh) np_s Sun Planet Teeth (External mesh) 20 ns Sun GearTeeth RATIOS: 15.5 pl_mg Planetary Gear (Forward rotation) st_mg StarGear (Reverse rotation) sl_mg SolarGear (Forward rotation) mx Planet/Sun Ratio mi Ring/Planet Ratio opr_cd in Operating Center Distance std_cds in "Standard" CD - External mesh std_cdr in "Standard" CD - Internal mesh tpa_x deg Opr Press Angle - External mesh tpa_i deg Opr Press Angle - Internal mesh RATIO RANGE: (Speed Reduction Gears) ms_pl Planetary Range (Normal = 6 to 25) In: Sun Fixed: Ring Out: Carrier ms_st Star Range (Normal = 5 to 24) In: Sun Fixed: Carrier Out: Ring ms_sl Solar Range (Normal = 1.05 to 2.2) In: Ring Fixed: Sun Out: Carrier PLANETSPACING: lma deg Least mesh angle (IDENTICAL planets must be spaced at increments of the least mesh angle for assembly) NUMBER OF EQUALLY SPACED PLANETS p1 (These are the 1st 4 up to 50 that p2 will assemble without interference - p3 planets will assemble equally if p4 (ring*sun planet)+(sun*ring planet) divided by planets is an integer EXTERNAL MESH (Sun/Planet): pn_s 1 1/in Normal Pitch npa_s 20 deg Normal Pressure Angle ha_s 0 deg Nominal helix Angle pt_s 1 1/in Transverse Pitch tpa_s 20 deg Transverse Press Angle 6

60-1162 Compound Epicyclic Gear Design (Parallel Axis) mod_n_s 25.4 mm Normal Module mod_t_s 25.4 mm Transverse Module opr_pdp in Opr Pitch Dia, planet 20 opr_pds in Opr Pitch Dia, sun gear Solve and you should have Report 2. Report 2 MESSAGE FIELD ERROR MESSAGE, internal mesh ERROR MESSAGE, external mesh ERROR MESSAGE, mesh - general Prime factors greater than 100 NUMBER OF TEETH non_integer_teeth unknown Ring Gear Teeth 117 Ring Planet Teeth (Internal mesh) 28 Sun Planet Teeth (External mesh) 69 Sun Gear Teeth 20 RATIOS Planetary Gear (Forward rotation) 15.5000 Star Gear (Reverse rotation) 14.5000 7

UTS Integrated Gear Software Solar Gear (Forward rotation) 1.0690 Planet/Sun Ratio 3.4626 Ring/Planet Ratio 4.1876 EXTERNAL MESH (SUN/PLANET) Opr Pitch Dia, sun gear 20.0000 in With 20 teeth in the sun, 28 teeth in the ring planet and a planetary ratio of about 15.5 we need about 117 teeth in the ring and about 69 teeth in the sun planet. We enter the tooth numbers and blank the ratio and the operating pitch diameter of the sun. The completed data input form is shown in Figure 2. 8

60-1162 Compound Epicyclic Gear Design (Parallel Axis) Fig. 2 When we solve with these numbers, however, we get a general mesh error message. Enter 118 for the number of ring gear teeth and solve again. Report 3 shows the solved model. 9

UTS Integrated Gear Software Report 3 MESSAGE FIELD ERROR MESSAGE, internal mesh ERROR MESSAGE, external mesh ERROR MESSAGE, mesh - general Prime factors greater than 100 NUMBER OF TEETH Ring Gear Teeth 118 Ring Planet Teeth (Internal mesh) 28 Sun Planet Teeth (External mesh) 69 Sun Gear Teeth 20 RATIOS Planetary Gear (Forward rotation) 15.5393 Star Gear (Reverse rotation) 14.5393 Solar Gear (Forward rotation) 1.0688 Planet/Sun Ratio 3.4500 Ring/Planet Ratio 4.2143 Operating Center Distance Standard CD - External mesh Standard CD - Internal mesh Opr Press Angle - External mesh 44.7500 in 44.5000 in 45.0000 in 20.8617 deg 10

60-1162 Compound Epicyclic Gear Design (Parallel Axis) Opr Press Angle - Internal mesh RATIO RANGE (SPEED REDUCTION GEARS) Planetary Range (Normal = 6 to 25) Star Range (Normal = 5 to 24) Solar Range (Normal = 1.05 to 2.2) PLANET SPACING Least mesh angle 19.1012 deg 0.0414 deg NUMBER OF EQUALLY SPACED PLANETS Number of equally spaced planets 2.0000 Number of equally spaced planets # Number of equally spaced planets # Number of equally spaced planets # This time there is no error message. The ratio with this arrangement is 15.5393. We can use only 2 equally spaced planets with this tooth combination. The unit would share the load between planets better if we had 3 planets and floated one member. Increase the number of ring gear teeth to 119 and decrease the number of sun planet teeth to 68 (to hold the ratio about the same). Solve again for Report 4. 11

UTS Integrated Gear Software Report 4 MESSAGE FIELD ERROR MESSAGE, internal mesh ERROR MESSAGE, external mesh ERROR MESSAGE, mesh - general Prime factors greater than 100 NUMBER OF TEETH Ring Gear Teeth 119 Ring Planet Teeth (Internal mesh) 28 Sun Planet Teeth (External mesh) 68 Sun Gear Teeth 20 RATIOS Planetary Gear (Forward rotation) 15.4500 Star Gear (Reverse rotation) 14.4500 Solar Gear (Forward rotation) 1.0692 Planet/Sun Ratio 3.4000 Ring/Planet Ratio 4.2500 No Group Operating Center Distance Standard CD - External mesh Standard CD - Internal mesh 44.7500 in 44.0000 in 45.5000 in 12

60-1162 Compound Epicyclic Gear Design (Parallel Axis) Opr Press Angle - External mesh Opr Press Angle - Internal mesh RATIO RANGE (SPEED REDUCTION GEARS) Planetary Range (Normal = 6 to 25) Star Range (Normal = 5 to 24) Solar Range (Normal = 1.05 to 2.2) PLANET SPACING Least mesh angle NUMBER OF EQUALLY SPACED PLANETS 22.4904 deg 17.1683 deg 0.0416 deg Number of equally spaced planets 2.0000 Number of equally spaced planets 3.0000 Number of equally spaced planets # Number of equally spaced planets # EXTERNAL MESH (SUN/PLANET) Normal Pitch Normal Pressure Angle Nominal helix Angle Transverse Pitch Transverse Press Angle Normal Module Transverse Module Opr Pitch Dia, planet Opr Pitch Dia, sun gear 1.000000 1/in 20.000000 deg 0.000000 deg 1.0000 1/in 20.0000 deg 25.4 mm' 25.4 mm' 69.1591 in 20.3409 in 13

UTS Integrated Gear Software INTERNAL MESH (RING/PLANET) Normal Pitch Normal Pressure Angle Nominal helix Angle Transverse Pitch Transverse Press Angle Normal Module Transverse Module Opr Pitch Dia, ring gear Opr Pitch Dia, planet 1.000000 1/in 20.000000 deg 0.000000 deg 1.0000 1/in 20.0000 deg 25.4 mm' 25.4 mm' 117.0385 in 27.5385 in With this tooth combination we can assemble 3 planets equally spaced around the axis of the unit. With a diametral pitch of 1 the operating pitch diameter of the ring gear is 117 inches. Since we want about a 12-inch ring, we need to change the pitch to 10. Input 10 for the pitch of both meshes and solve. The completed data input form is shown in Figure 3, the solved model in Report 5. 14

60-1162 Compound Epicyclic Gear Design (Parallel Axis) Fig. 3 Report 5 MESSAGE FIELD ERROR MESSAGE, internal mesh ERROR MESSAGE, external mesh ERROR MESSAGE, mesh - general Prime factors greater than 100 15

UTS Integrated Gear Software NUMBER OF TEETH Ring Gear Teeth 119 Ring Planet Teeth (Internal mesh) 28 Sun Planet Teeth (External mesh) 68 Sun Gear Teeth 20 RATIOS Planetary Gear (Forward rotation) 15.4500 Star Gear (Reverse rotation) 14.4500 Solar Gear (Forward rotation) 1.0692 Planet/Sun Ratio 3.4000 Ring/Planet Ratio 4.2500 Operating Center Distance Standard CD - External mesh Standard CD - Internal mesh Opr Press Angle - External mesh Opr Press Angle - Internal mesh RATIO RANGE (SPEED REDUCTION GEARS) Planetary Range (Normal = 6 to 25) Star Range (Normal = 5 to 24) Solar Range (Normal = 1.05 to 2.2) PLANET SPACING Least mesh angle 4.4750 in 4.4000 in 4.5500 in 22.4904 deg 17.1683 deg 0.0416 deg 16

60-1162 Compound Epicyclic Gear Design (Parallel Axis) NUMBER OF EQUALLY SPACED PLANETS Number of equally spaced planets 2.0000 Number of equally spaced planets 3.0000 Number of equally spaced planets # Number of equally spaced planets # EXTERNAL MESH (SUN/PLANET) Normal Pitch 10.000000 1/in Normal Pressure Angle Nominal helix Angle Transverse Pitch Transverse Press Angle Normal Module Transverse Module Opr Pitch Dia, planet Opr Pitch Dia, sun gear INTERNAL MESH (RING/PLANET) Normal Pitch Normal Pressure Angle Nominal helix Angle Transverse Pitch Transverse Press Angle Normal Module Transverse Module Opr Pitch Dia, ring gear Opr Pitch Dia, planet 20.000000 deg 0.000000 deg 10.0000 1/in 20.0000 deg 2.54 mm' 2.54 mm' 6.9159 in 2.0341 in 10.000000 1/in 20.000000 deg 0.000000 deg 10.0000 1/in 20.0000 deg 2.54 mm' 2.54 mm' 11.7038 in 2.7538 in 17

UTS Integrated Gear Software With a center distance of 4.475 inches, φext is about 22.5 degrees and φint is about 17 degrees. (The Operating Center Distance is defaulted to the mid-point between the standard center distances for the two meshes if the operating center distance is not entered.) If we wish we can use this center distance. If we want to increase φext and φint all we need do is increase the operating center distance. Let's bring φint up to nominal. Change the operating center distance to the standard center distance for the internal mesh. Input 4.55 for the operating center distance and solve. Figure 4 is the completed data input form and Report 6 shows the result. Fig. 4 18

60-1162 Compound Epicyclic Gear Design (Parallel Axis) Report 6 MESSAGE FIELD ERROR MESSAGE, internal mesh ERROR MESSAGE, external mesh ERROR MESSAGE, mesh - general Prime factors greater than 100 NUMBER OF TEETH Ring Gear Teeth 119 Ring Planet Teeth (Internal mesh) 28 Sun Planet Teeth (External mesh) 68 Sun Gear Teeth 20 RATIOS Planetary Gear (Forward rotation) 15.4500 Star Gear (Reverse rotation) 14.4500 Solar Gear (Forward rotation) 1.0692 Planet/Sun Ratio 3.4000 Ring/Planet Ratio 4.2500 Operating Center Distance Standard CD - External mesh Standard CD - Internal mesh Opr Press Angle - External mesh 4.5500 in 4.4000 in 4.5500 in 24.6718 deg 19

UTS Integrated Gear Software Opr Press Angle - Internal mesh RATIO RANGE (SPEED REDUCTION GEARS) Planetary Range (Normal = 6 to 25) Star Range (Normal = 5 to 24) Solar Range (Normal = 1.05 to 2.2) PLANET SPACING Least mesh angle NUMBER OF EQUALLY SPACED PLANETS 20.0000 deg 0.0416 deg Number of equally spaced planets 2.0000 Number of equally spaced planets 3.0000 Number of equally spaced planets # Number of equally spaced planets # EXTERNAL MESH (SUN/PLANET) Normal Pitch Normal Pressure Angle Nominal helix Angle Transverse Pitch Transverse Press Angle Normal Module Transverse Module Opr Pitch Dia, planet Opr Pitch Dia, sun gear INTERNAL MESH (RING/PLANET) Normal Pitch 10.000000 1/in 20.000000 deg 0.000000 deg 10.0000 1/in 20.0000 deg 2.54 mm' 2.54 mm' 7.0318 in 2.0682 in 10.000000 1/in 20

60-1162 Compound Epicyclic Gear Design (Parallel Axis) Normal Pressure Angle 20.000000 deg Nominal helix Angle 0.000000 deg Transverse Pitch 10.0000 1/in Transverse Press Angle 20.0000 deg Normal Module 2.54 mm' Transverse Module 2.54 mm' Opr Pitch Dia, ring gear 11.9000 in Opr Pitch Dia, planet 2.8000 in APPROXIMATE GEAR MESH EFFICIENCY External mesh loss (gears only) 0.500 % Internal mesh loss (gears only) 0.295 % Planetary efficiency 99.26 % Star efficiency 99.20 % Solar efficiency 99.95 % RELATIVE POWER (% OF INPUT) Relative power planetary 93.53 % Relative power star 100.00 % Relative power solar 6.47 % REACTION TORQUE (% OF INPUT) Reaction torque Planetary 1445.00 % Reaction torque star 1545.00 % Reaction torque solar 6.92 % OUTPUT TORQUE (% OF INPUT) Output torque Planetary 1545.00 % Output torque star 1445.00 % Output torque solar 106.92 % 21

UTS Integrated Gear Software φext is now about 24.7 degrees and φint is 20 degrees, which is nominal. The higher pressure angle at the external mesh will result in lower bending and compressive stress and, since the number of load cycles imposed on the external mesh is higher than the internal mesh, this might be an advantage. This completes the solution, and all design data for the geometry of the gear set is solved for in the model. Note that there are no error or caution messages in the error message block. Of course, this is not the only solution to this design problem. The model was solved progressively to obtain this solution. With the backsolving capability of TK Solver, you might wish to investigate other solutions. To consider K-factor and unit loads at the same time as the unit geometry, see UTS gear model 60-1164. NOTE: The relative power of a true epicyclic unit is often misunderstood. The input, output and reaction torques of any gear unit must balance. The carrier of a planetary or solar reduction unit is rotating in the same direction as the input number. Therefore, the mesh velocity of the teeth is less than in a non-eipcyclic gear. The power at the teeth is a product of load and linear velocity. Since the linear velocity is less than rotation speed multiplied by pitch radius, the relative power is less than the shaft transmitted power. When relative power is used in calculations, the relative speed must also be used. 22