Program 60-1162—Compound Epicyclic Gear Design · PDF file60-1162—Compound...

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

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identical planets to assemble equally spaced around the center, the following relationship between the tooth numbers in the gears must be met: [(Nring·Npl-sun )+(Nsun·Npl-ring)]/np =integer where: Nring = Number of teeth in ring gear

Npl-sun = Number of teeth in sun gear planet Nsun = Number of teeth in sun gear

Npl-ring = Number of teeth in ring gear planet np = 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”.

ep/β = integer

β = 360° / [(Nring·Npl-sun )+(Nsun·Npl-ring)]

where: ep = Angle between adjacent planet gears, deg

β = Least mesh angle, deg

It is not necessary (or even desirable) that PDring = PDsun+PDpl-sun+PDpl-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, φext, and the planet/ring internal mesh, φint,will be equal to the nominal pressure angle of the systems. A higher operating pressure angle at the external mesh, φext, 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 highφint will tend to reduce the bending stress. If the ring gear rim thickness is 1.5 tooth depths or less, a low φint 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.

60-1162—Compound Epicyclic Gear Design (Parallel Axis)

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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


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Report 1

Model Title : Program 60-1162 Unit System: US

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 1.000000 1/in

Normal Pressure Angle 20.000000 deg

Nominal helix Angle 0.000000 deg


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Model Title : Program 60-1162 Unit System: US Transverse Pitch 1.0000 1/in

Transverse Press Angle 20.0000 deg

Normal Module 25.4 mm'

Transverse Module 25.4 mm'

Opr Pitch Dia, planet in

Opr Pitch Dia, sun gear in INTERNAL MESH (RING/PLANET)




Transverse Pitch 1.0000 1/in




Opr Pitch Dia, ring gear in

Opr Pitch Dia, planet in

The model has set the normal diametral pitch to 1, the normal 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 double-click the Status cell and pick “Guess” from the drop-down list that appears. This will


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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 meshm_x ERROR MESSAGE, external meshm_g ERROR MESSAGE, mesh - generalm_p Prime factors greater than 100

NUMBER OF TEETH:G 100 nr Ring Gear Teeth28 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 Ratiomi Ring/Planet Ratio

opr_cd in Operating Center Distancestd_cds in "Standard" CD - External meshstd_cdr in "Standard" CD - Internal meshtpa_x deg Opr Press Angle - External meshtpa_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: Carrierms_st Star Range (Normal = 5 to 24)

In: Sun Fixed: Carrier Out: Ringms_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 ofthe least mesh angle for assembly)

NUMBER OF EQUALLY SPACED PLANETSp1 (These are the 1st 4 up to 50 thatp2 will assemble without interference -p3 planets will assemble equally ifp4 (ring*sun planet)+(sun*ring planet)

divided by planets is an integer

EXTERNAL MESH (Sun/Planet):pn_s 1 1/in Normal Pitchnpa_s 20 deg Normal Pressure Angleha_s 0 deg Nominal helix Anglept_s 1 1/in Transverse Pitchtpa_s 20 deg Transverse Press Angle


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mod_n_s 25.4 mm Normal Modulemod_t_s 25.4 mm Transverse Moduleopr_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 none

ERROR MESSAGE, external mesh none

ERROR MESSAGE, mesh - general non_integer_teeth

Prime factors greater than 100 unknown NUMBER OF TEETH

Ring Gear Teeth 117


Sun Planet Teeth (External mesh) 69

Sun Gear Teeth 20

RATIOS


Model Title : Program 60-1162 Unit System: US Star Gear (Reverse rotation) 14.5000


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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.


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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.


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Report 3


MESSAGE FIELD



ERROR MESSAGE, mesh - general none

Prime factors greater than 100 none NUMBER OF TEETH

Ring Gear Teeth 118



Sun Gear Teeth 20

RATIOS


Star Gear (Reverse rotation) 14.5393



Ring/Planet Ratio 4.2143

Operating Center Distance 44.7500 in

Standard CD - External mesh 44.5000 in

Standard CD - Internal mesh 45.0000 in

Opr Press Angle - External mesh 20.8617 deg


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Model Title : Program 60-1162 Unit System: US Opr Press Angle - Internal mesh 19.1012 deg RATIO RANGE (SPEED REDUCTION GEARS)

Planetary Range (Normal = 6 to 25) normal

Star Range (Normal = 5 to 24) normal

Solar Range (Normal = 1.05 to 2.2) normal PLANET SPACING

Least mesh angle 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 # 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.


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Report 4


MESSAGE FIELD





Ring Gear Teeth 119



Sun Gear Teeth 20

RATIOS





Ring/Planet Ratio 4.2500 No Group





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Model Title : Program 60-1162 Unit System: US Opr Press Angle - External mesh 22.4904 deg

Opr Press Angle - Internal mesh 17.1683 deg RATIO RANGE (SPEED REDUCTION GEARS)









Number of equally spaced planets # EXTERNAL MESH (SUN/PLANET)








Opr Pitch Dia, planet 69.1591 in

Opr Pitch Dia, sun gear 20.3409 in


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Model Title : Program 60-1162 Unit System: US INTERNAL MESH (RING/PLANET)








Opr Pitch Dia, ring gear 117.0385 in

Opr Pitch Dia, planet 27.5385 in With this tooth combination we can assemble 3 planets equally spaced around the axis of the unit. With a normal 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 normal pitch to 10. Input 10 for the normal pitch of both meshes and solve. The completed data input form is shown in Figure 3, the solved model in Report 5.


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Fig. 3

Report 5


MESSAGE FIELD




Prime factors greater than 100 none


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Model Title : Program 60-1162 Unit System: US NUMBER OF TEETH

Ring Gear Teeth 119



Sun Gear Teeth 20

RATIOS










Opr Press Angle - Internal mesh 17.1683 deg RATIO RANGE (SPEED REDUCTION GEARS)






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Opr Pitch Dia, sun gear 2.0341 in INTERNAL MESH (RING/PLANET)








Opr Pitch Dia, ring gear 11.7038 in



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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


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Report 6


MESSAGE FIELD





Ring Gear Teeth 119



Sun Gear Teeth 20

RATIOS











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Model Title : Program 60-1162 Unit System: US Opr Press Angle - Internal mesh 20.0000 deg RATIO RANGE (SPEED REDUCTION GEARS)


















Opr Pitch Dia, sun gear 2.0682 in INTERNAL MESH (RING/PLANET)



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Model Title : Program 60-1162 Unit System: US 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 %


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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.

Program 60-1162—Compound Epicyclic Gear Design · PDF file60-1162—Compound...

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