program 60-1164—compound epicyclic differential design ...covered by agma standard 6023-a88, and...

Program 60-1164—Compound Epicyclic Differential Design

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 geometric 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.) Therefore, there are three input/output elements in epicyclic gear differentials. The ring/sun ratio range for which these units can be designed with reasonable proportions is about 5:1 to 24:1. Below this range the planet gears become quite small and it becomes difficult to design the gears and planet bearings for reasonable life. Above this range 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 identical planets to assemble equally spaced around the center, the following relationship between the tooth numbers in the gears must be met: [(Nring∙N'pl-sun )+ (Nsun∙N'pl-ring)]/np = integer where: Nring = Number of teeth in ring gear N'pl-sun = Numerator of reduced planet ratio fraction Nsun = Number of teeth in sun gear N'pl-ring = Denominator of reduced planet ratio fraction np = Number of planet gears Npl-sun N'pl-sun reduced planet ratio fraction = ────── = ─────── Npl-ring N'pl-ring where: N'pl-sun and N'pl-ring have common factors removed.

UTS Integrated Gear Software

2

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/ß = integer ß = 360°/[(Nring∙N'pl-sun )+ (Nsun∙N'pl-ring)] where: ep = Angle between adjacent planet gears, deg ß = Least mesh angle, deg The model will calculate the planet interference outside diameter for the large planet. The large planet OD must be less than this diameter. The planet tooth tip clearance will be the amount the actual planet OD is less than the interference OD. The summation of radial loads on the sun and ring will be made to determine whether or not the radial loads on the sun and ring are balanced. The result is displayed as a “Yes” or “No” under “Radial Loads On Sun & Ring Balanced?”. It is not necessary (or even desirable) that PDring = PDsun+ PDpl-sun + PDpl-ring [where PD = Teeth/(Trans Pitch)]. 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 high φint 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.) The pitch line velocity is calculated and the minimum recommended AGMA quality class is determined is accordance with the ANSI/AGMA 6023-A88 Design Manual for Enclosed Epicyclic Gear Drives. The model contains a table of K factors and Unit Loads for use in estimating the size of the gears required to carry the necessary load. The K factor is a function of the compressive stress carried on the teeth and is proportional to the square of the stress. The Unit Load is a function of the bending stress in the root area of the gears and is directly proportional to the stress. Both factors are directly proportional to the

60-1164—Compound Epicyclic Differential Design

3

load. Different gear materials are, of course, capable of carrying different K and Unit Load factors for a given number of cycles. These factors are approximate because they do not contain many of the elements affecting the stresses on the gears. They are close enough, however, to allow us to get a “starting place” for our design with only the information at hand. It is essential, of course, to check our preliminary design with equations containing all the factors known to affect the operation and life of the gear set. The selection of the K factor and Unit Load is, of course, based on the material used for the gears and our best estimate of the load the gears will carry. The number of cycles the gears are required to run will also be part of the selection process. The first thing we need to determine is a “service factor” which adjusts the load to account for the extra load imposed on the gears from non-uniform torques produced by the driver and driven machines. A few selected service factors are contained in the table “SF”. The number of cycles we must run will be dealt with separately from the service factor and the service factors listed do not include adjustments for duration of service. Further information on service factors can be found in various AGMA standards pertaining to specific industries and applications. The service factors usually applied sometimes are not sufficient for critical drives running at high power and/or speed and must be used with caution. AGMA Standard 427, “AGMA Information Sheet, Systems Considerations for Critical Service Gear Drives” is an excellent source for information concerning the rating of these drives.


4

The table “KUL” contains K factors and Unit Loads for a number of materials and conditions for steel gears.

The table takes into account the class of gearing such as high speed, medium speed, etc. The accuracy to which the gears are made is also included along with the type of gear (spur, helical) and the heat treatment used.


5

The number of cycles needed is found by multiplying the speed (relative to the carrier) by the life required. (If the sun meshes with more than one planet this must be taken into account.) You may move the cursor to the appropriate location in the table and change the values for pinion cycles, service factor and ratio to suit the application. After solving the model the table will be updated to reflect the changed data. If you are using model 60-1164 for the first time you may wish to run the following example. The design of a compound epicyclic differential is a complex job and the example will take you through the design of the operating characteristics and mesh geometry along with preliminary sizing of the gears. The power flow across a speed range will also be considered. It is, of course, not necessary to follow this example in your design work, but the example will help you decide where to “jump in” until you are familiar with the model and design parameters.

Example Suppose we wish to design a spur gear differential planetary set with a sun gear speed of 1800 RPM and a ring gear speed of 150 RPM, to drive a conveyor system at speeds up to 150 RPM. The conveyor will run most of the time at 150 RPM. A hydraulic system will be connected between a constant speed electric motor and the carrier to control the speed of the conveyor. The motor will also drive the sun gear. (It is not meant to imply that this is a good design for a differential unit. Indeed, as you work this model you will encounter warnings that the output values are not covered by AGMA Standard 6023-A88, and other warnings about design problems. This example serves only to illustrate the model for this type of work.) The smallest number of teeth we wish to use is about 20. We will start with a normal diametral pitch of 1 and a normal pressure angle of 20 degrees for both the external and internal gear sets. (The pitch will be changed when we know more about the loads on the teeth.) We will make a reasonable guess for the ring gear teeth of 180 and use a sun gear of 20 teeth. We want an output speed on the ring gear of 150 RPM with a sun gear speed of 1800 RPM because we want the carrier locked (hydraulic power equal to zero) under these conditions.


6

Fig. 1A

Enter our initial conditions in the wizard input form, as shown in Figures 1A and 1B. If the sun and carrier speeds are input as positive the ring speed must be input as negative to indicate opposite rotation. (Click off the “Enable table?” checkbox and press Enter to clear the interactive table.) Report 1 shows the solved model.

Fig. 1B


7

Report 1

Model Title : Program 60-1164 Unit System: US

Description Value Unit Comment

MESSAGE FIELD

ERROR MESSAGE, internal mesh none

ERROR MESSAGE, external mesh none

ERROR MESSAGE, mesh - general

Prime factors greater than 100 unknown

NUMBER OF TEETH

Ring Gear Teeth 180

Ring Planet Teeth (Internal mesh) 69

Sun Planet Teeth (External mesh) 91

Sun Gear Teeth 20

Plot pitch diameters? 'y=yes Def: no n

CENTER DISTANCE AND PRESSURE ANGLES

Operating Center Distance 55.71430 in

Standard CD - External mesh 55.71430 in

Standard CD - Internal mesh 55.71430 in

Opr Press Angle - External mesh 20.0000 deg

Opr Press Angle - Internal mesh 20.0000 deg

PLANET SPACING

Least mesh angle (IDENTICAL planets 0.0202 deg


8



EQUALLY SPACED IDENTICAL PLANETS

p1 none

p2 #

p3 #

p4 #

Reduced planet ratio fraction 91/69

EXTERNAL MESH (SUN/PLANET)

Normal Diametral Pitch 1.000000 1/in

Normal Pressure Angle 20.000000 deg

Nominal helix Angle 0.000000 deg

Transverse Diametral Pitch 1.000000 1/in

Transverse Press Angle 20.000000 deg

Normal Module 25.400000 mm `

Transverse Module 25.400000 mm `

Axial pitch in

Opr Pitch Dia, planet 91.42860 in

Opr Pitch Dia, sun gear 20.00000 in

INTERNAL MESH (RING/PLANET)







9





Axial pitch in

Opr Pitch Dia, ring gear 180.00000 in


OPERATION AS GEAR UNIT OR DIFFERENTIAL

Enable table? 'e=enable 'c=clear c

Type of unit Star

Number of planets

ck_#

Effective planets (1 member floating)

Effective planets (All fixed)

Effective planets

Planet interference OD in

Radial Loads On Sun & Ring Balanced?

Ring/Sun gear ratio 12.0000

Planet/Sun Ratio 4.5714

Ring/Planet Ratio 2.6250

ROTATION SPEED

Sun gear 1800.00000 rpm

Ring gear -150.00000 rpm

Carrier 0.00000 rpm


10



ROTATION SPEED RELATIVE TO CARRIER



Planet gear -393.75000 rpm

Carrier 0.00000 rpm

DRIVER/DRIVEN

Sun gear Driver

Ring gear Driven

Carrier

SUN/PLANET MESH

Relative pitch line velocity 9424.78 ft/min

Max Recommended Spacing Tolerance 0.00028 in

Min Recommended AGMA Quality Class Q14

RING/PLANET MESH

Relative pitch line velocity: 7068.58 ft/min



TORQUES ON ELEMENTS (NO LOSSES)

Sun gear 3501.39 lbf-in

Ring gear 42016.67 lbf-in

Carrier -45518.06 lbf-in


11



POWER (+ IN, - OUT) (NO LOSSES)

Sun gear 100.00 HP

Ring gear -100.00 HP

Carrier HP Ignore the speed warnings, as we do not have the unit sized as yet. Of course, we cannot have non-integer numbers of teeth for the planet gears. Blank the ring gear speed, enter the number of teeth for the planet gears as integers and solve again. Remember to click the “Refresh Values” button when you’ve completed all the input value changes in the wizard form. Report 2 is the solved model. Report 2



MESSAGE FIELD




Prime factors greater than 100 none

NUMBER OF TEETH

Ring Gear Teeth 180



12




Sun Gear Teeth 20








PLANET SPACING



p1 2

p2 3

p3 #

p4 #









13





Axial pitch in











Axial pitch in





Type of unit Star

Number of planets

ck_#



Effective planets


14








ROTATION SPEED



Carrier 0.00000 rpm





Carrier 0.00000 rpm

DRIVER/DRIVEN

Sun gear Driver

Ring gear Driven

Carrier

SUN/PLANET MESH





15



RING/PLANET MESH









Sun gear 100.00 HP


Carrier HP With 69 and 91 teeth in the planet gears and a center distance of 55.5 inches, φext and φint are both “standard” at 20 degrees. (The “Operating Center Distance” is defaulted to half way between the “standard” center distances for both meshes if the operating center distance is not entered. In this case all three are 55.5 inches.) We can assemble 2 or 3 planet gears with equal spacing. Suppose we want φext to be larger than the nominal pressure angle of 20 degrees. We need to reduce the number of planet teeth to run the external mesh on spread centers. For 68/90, 67/90, and 67/89 we can assemble only two planets equally spaced (Try it if you like). For 66/89, however, we can assemble three planets. Enter 66 and 89 for the planet teeth. After solving you should have Sheet 3. (We will adjust the final operating pressure angles later by adjusting the operating center distance.)


16

Report 3



MESSAGE FIELD





NUMBER OF TEETH

Ring Gear Teeth 180



Sun Gear Teeth 20








PLANET SPACING



17




p1 2

p2 3

p3 #

p4 #










Axial pitch in










18





Axial pitch in





Type of unit Star

Number of planets

ck_#



Effective planets






ROTATION SPEED



Carrier 0.00000 rpm




19





Carrier 0.00000 rpm

DRIVER/DRIVEN

Sun gear Driver

Ring gear Driven

Carrier

SUN/PLANET MESH




RING/PLANET MESH









Sun gear 100.00 HP


Carrier HP


20

The ring gear (conveyor) speed is about 148 RPM with this tooth combination. Enter 3 for the number of planets (see Figure 2). To check the hydraulic unit speed enter zero for ring gear speed and blank the carrier speed. Solving the model should produce the values shown in Report 4. Report 4



NUMBER OF TEETH

Ring Gear Teeth 180



Sun Gear Teeth 20




Fig. 2


21










Type of unit Planetary

Number of planets 3

ck_#

Effective planets (1 member floating) 3

Effective planets (All fixed) 2

Effective planets

Planet interference OD 96.56180 in

Radial Loads On Sun & Ring Balanced? Yes




ROTATION SPEED


Ring gear 0.00000 rpm

Carrier 137.02422 rpm


22







Carrier 0.00000 rpm

DRIVER/DRIVEN

Sun gear Driver

Ring gear Driven

Carrier Driven


Sun gear 100.00 HP The program has given us a warning message concerning the assembly of the planets. Vernier assembly will be required with these tooth counts and this number of planets. If the number of teeth in the ring and the number of teeth in the sun are both divisible by the number of equally spaced planets then the (identical) planets may have any number of teeth. If the ratio of the numbers of teeth on the planet shafts is an integer, then tooth marking for assembly is optional if the smaller of the two planet shaft gears is assembled first. If we have neither of these conditions, then vernier assembly is assured and a layout of the teeth will be required for tooth marking to make assembly possible. See ANSI/AGMA 6023. The carrier speed is about 137 RPM. The sun is always a driver and the ring and carrier are always driven. The torque and power is listed for all elements. The sun power is positive, indicating power input, and the ring and carrier power is negative,


23

indicating power out. (This should guide any studies of approach and recess action done on the teeth. It is not always apparent. Use the speeds relative to the carrier and relative power for load analysis.) The effective number of planets may be different from the actual number of planets, because of errors preventing the planets from sharing the load equally. When one or two members are allowed to float radially, the load sharing is better than when all members are constrained by bearings. In this case, if floating is utilized, the effective number of planets is 3, the same as the actual number of planets. (This is true only for 3 planets or less with float.) If all members are fixed, the effective number of planets is 2.44. To determine the load for an individual planet, the transmitted torque is divided by the effective number of planets. To size the unit we need to consider the allowable load on the teeth. We will use a floating design, so enter 3 for “Effective Planets”. We'll use a K-factor of 500 for the external mesh. (See table “KUL”, shown on page 4, for some help with K-factors). We will keep the face to diameter (aspect) ratio at about 1 (or less). It may take a few trials to accomplish this. We will start with a face width of 1.75 inches. Obtaining a solution for the pitch diameter of the sun will require iteration, so we will enter 1.75 inches as a guess for the operating pitch diameter of the sun to give the iterative solver some place to start. Enter 500 for the K-factor, 1.75 for the face width and 1.75 for the sun gear operating pitch diameter. Blank both normal pitches and the normal pressure angle for the internal mesh.


24

Sheet 1

You must enter the sun gear operating pitch diameter as a Guess value directly in the TK Solver Variable Sheet, as shown in Sheet 1. Enter the value in the input cell, then type a G in the status column or double-click the cell and pick Guess from the drop-down list. Report 5 shows the solved model. Report 5



MESSAGE FIELD

ERROR MESSAGE, internal mesh

ERROR MESSAGE, external mesh




25

Title : Program 60-1164 Unit System: US


NUMBER OF TEETH

Ring Gear Teeth 180



Sun Gear Teeth 20




Standard CD - External mesh in

Standard CD - Internal mesh in

Opr Press Angle - External mesh deg

Opr Press Angle - Internal mesh deg

PLANET SPACING



p1

p2

p3

p4



Normal Diametral Pitch 1/in


26





Transverse Diametral Pitch 1/in


Normal Module mm `

Transverse Module mm `

Axial pitch in




Normal Diametral Pitch 1/in

Normal Pressure Angle deg


Transverse Diametral Pitch 1/in

Transverse Press Angle deg

Normal Module mm `

Transverse Module mm `

Axial pitch in






Number of planets 3


27



ck_#



Effective planets 3






ROTATION SPEED








Carrier 0.00000 rpm

DRIVER/DRIVEN

Sun gear Driver

Ring gear Driven

Carrier Driven


28



SUN/PLANET MESH


Max Recommended Spacing Tolerance in

Min Recommended AGMA Quality Class

RING/PLANET MESH


Max Recommended Spacing Tolerance in

Min Recommended AGMA Quality Class






Sun gear 100.00 HP

Ring gear HP

Carrier -100.00 HP

PER PLANET GEAR

Relative power @ ext mesh (no loss) 30.79580 HP

Relative power @ int mesh (no loss) -30.79580 HP

Tooth tangential load at sun 1291.40 lbf

Tooth tangential load at ring 1821.31 lbf

Face width - sun/planet 1.75000 in

K factor - sun/planet 500.00 psi


29



Unit load - sun/planet psi

Helical (Axial) contact ratio

Face width - planet/ring in

K factor - planet/ring psi

Unit load - planet/ring psi


Centripetal acceleration on planet 2.62676 G's

APPROXIMATE EFFICIENCY

Ext Friction Coefficient 0

Sun/Planet Power Loss 0.92 HP

Int Friction Coefficient 0

Ring/Planet Power Loss 0.46 HP

Total Power Loss (Gear Losses Only) 1.39 HP

Approx Efficiency 98.6 % With a K-factor of 500 and a 1.75-inch face width the operating pitch diameter of the sun is about 1.8 inches. The aspect ratio is a little less than one. Next we wish to establish the normal pitch. Set the standard center distance for the external mesh to the operating center distance to find an approximate pitch. (Do not type in the value shown here, because it has been rounded off for display purposes. Instead, copy the output value for the operating center distance, and paste it as an input value for the standard center distance. See the partial Variable Sheet in Sheet 2.) After solving, the pitch is seen to be about 11.065, as shown in Report 6.


30

Sheet 2

Report 6



NUMBER OF TEETH

Ring Gear Teeth 180



Sun Gear Teeth 20


Operating Center Distance in


Standard CD - Internal mesh in

Opr Press Angle - External mesh deg


31



Opr Press Angle - Internal mesh deg

PLANET SPACING



p1

p2

p3

p4










Axial pitch in

Opr Pitch Dia, planet in

Opr Pitch Dia, sun gear in




32





Number of planets 3

Effective planets 3

ROTATION SPEED




Sun gear 100.00 HP

PER PLANET GEAR


K factor - sun/planet 500.00 psi We will use 10 normal pitch as 11 pitch is not usually “standard”. We will also use 10 pitch for the internal mesh as a starting point. Blank the standard center distance for the external mesh along with the face width of the sun mesh. Enter 10 for the normal pitch of both meshes and 20 for the normal pressure angle of the internal mesh. Report 7 shows the solved model.


33

Report 7



MESSAGE FIELD



ERROR MESSAGE, mesh - general vernier assembly


NUMBER OF TEETH

Ring Gear Teeth 180



Sun Gear Teeth 20








PLANET SPACING



34




p1 2

p2 3

p3 #

p4 #










Axial pitch in










35





Axial pitch in






Number of planets 3

ck_#



Effective planets 3






ROTATION SPEED





36







Carrier 0.00000 rpm

DRIVER/DRIVEN

Sun gear Driver

Ring gear Driven

Carrier Driven

SUN/PLANET MESH




RING/PLANET MESH









37




Sun gear 100.00 HP

Ring gear HP

Carrier -100.00 HP

PER PLANET GEAR







Unit load - sun/planet 8352.41 psi


Face width - planet/ring in

K factor - planet/ring psi

Unit load - planet/ring psi



APPROXIMATE EFFICIENCY

Ext Friction Coefficient 0

Sun/Planet Power Loss 0.92 HP

Int Friction Coefficient 0

Ring/Planet Power Loss 0.46 HP

Total Power Loss (Gear Losses Only) 1.39 HP

Approx Efficiency 98.6 %


38

Let us suppose that we have a limit of about 6000 psi unit load for the external mesh. (See table “KUL” for suggested unit loads.) The unit load with a 1.366-inch face is over 8300 psi. Blank the K-factor and enter 6000 for the unit load for the external mesh. (This will increase the face width and we could, instead, use a coarser pitch.) At the same time we will obtain a face width for the internal mesh. We will use a K-factor for the internal mesh of 180 psi as we anticipate using a machineable hardness steel for the ring gear. Enter 180 for the internal mesh K-factor. Solve for Report 8. Report 8



NUMBER OF TEETH

Ring Gear Teeth 180



Sun Gear Teeth 20









Axial pitch in



39












Axial pitch in






Number of planets 3

ck_#



Effective planets 3






40




ROTATION SPEED








Carrier 0.00000 rpm

DRIVER/DRIVEN

Sun gear Driver

Ring gear Driven

Carrier Driven

SUN/PLANET MESH




RING/PLANET MESH





41








Sun gear 100.00 HP

Ring gear HP

Carrier -100.00 HP

PER PLANET GEAR









Face width - planet/ring 0.19730 in

K factor - planet/ring 800.00 psi

Unit load - planet/ring 81540.17 psi




42

We now have a face width for the sun/planet mesh of about 1.9 inches and the aspect ratio is still less than one. The unit load at the ring is over 18000 psi. With a ring soft enough to machine the allowable unit load limit might be about 4500 psi. To reduce the unit load on the internal mesh we will change the number of teeth and the pitch.

Blank the number of teeth and the normal pitch for the internal mesh. Enter 2 inches for the internal mesh face width and 4500 for the internal mesh unit load. Blank the K-factor for the internal mesh. These changes are shown in Figures 4A and 4B

In the TK Solver Variable Sheet, change the operating center distance and the tangential load at the ring to input values. Type “I” in the Status column for these variables, or double-click in the Status cell and pick “Input” from the drop-down list, as shown in Sheet 3. After solving

Fig. 4A

Fig. 4B


43

you should have the numbers in Report 9. Sheet 3

Report 9



MESSAGE FIELD



ERROR MESSAGE, mesh - general non integer teeth


NUMBER OF TEETH

Ring Gear Teeth 98



Sun Gear Teeth 20



44









PLANET SPACING



p1 none

p2 #

p3 #

p4 #










Axial pitch in


45













Axial pitch in






Number of planets 3

ck_# Chk Spc



Effective planets 3


Radial Loads On Sun & Ring Balanced? No



46





ROTATION SPEED








Carrier 0.00000 rpm

DRIVER/DRIVEN

Sun gear Driver

Ring gear Driven

Carrier Driven

SUN/PLANET MESH




RING/PLANET MESH





47








Sun gear 100.00 HP

Ring gear HP

Carrier -100.00 HP

PER PLANET GEAR













Centripetal acceleration on planet 2.97310 G's Let's change to 5 normal pitch for the ring. Enter 5 for the pitch and blank the tangential load at the ring and the ring face width. Report 10 is the solved model.


48

Report 10



MESSAGE FIELD





NUMBER OF TEETH

Ring Gear Teeth 102



Sun Gear Teeth 20








PLANET SPACING



49




p1 none

p2 #

p3 #

p4 #










Axial pitch in










50





Axial pitch in






Number of planets 3

ck_# Chk Spc



Effective planets 3






ROTATION SPEED





51







Carrier 0.00000 rpm


Sun gear 100.00 HP

PER PLANET GEAR


Unit load - planet/ring 4500.00 psi With this pitch and number of teeth the output speed is too far from our original design point. Set the carrier speed to 137 RPM and solve again. (Report 11.) Report 11



MESSAGE FIELD





52




NUMBER OF TEETH

Ring Gear Teeth 88



Sun Gear Teeth 20








PLANET SPACING



p1 none

p2 #

p3 #

p4 #



53











Axial pitch in











Axial pitch in




54






Number of planets 3

ck_# Chk Spc



Effective planets 3






ROTATION SPEED





Sun gear 100.00 HP

PER PLANET GEAR




55

Enter 88 and 32 teeth for the external mesh, blank the carrier speed, and solve. (Report 12.) Report 12



MESSAGE FIELD





NUMBER OF TEETH

Ring Gear Teeth 88



Sun Gear Teeth 20









56



PLANET SPACING



p1 2

p2 3

p3 #

p4 #










Axial pitch in








57







Axial pitch in






Number of planets 3

ck_#



Effective planets 3






ROTATION SPEED




58





Sun gear 100.00 HP

PER PLANET GEAR


Unit load - planet/ring 4500.00 psi Everything looks OK. Let's set the operating center distance to the standard center distance for the internal mesh. This will set the operating pressure angle for the internal mesh to nominal and increase the operating pressure angle for the external mesh above nominal. Enter 5.6 for the operating center distance. We will make the face widths 2 inches for the external and 1.875 inches for the internal meshes. Enter these values and solve to obtain Sheet 13. Report 13



MESSAGE FIELD






59



NUMBER OF TEETH

Ring Gear Teeth 88



Sun Gear Teeth 20








PLANET SPACING



p1 2

p2 3

p3 #

p4 #





60









Axial pitch in











Axial pitch in






Number of planets 3


61



ck_#



Effective planets 3






ROTATION SPEED








Carrier 0.00000 rpm

DRIVER/DRIVEN

Sun gear Driver

Ring gear Driven

Carrier Driven


62



SUN/PLANET MESH




RING/PLANET MESH









Sun gear 100.00 HP

Ring gear HP

Carrier -100.00 HP

PER PLANET GEAR








63









Centripetal acceleration on planet 2.94098 G's This completes the solution and all design data for the geometry of the epicyclic differential gear set is solved for in the model. An AGMA quality of at least Q8 is recommended for the sun and planet and Q7 for the ring. Note that there are no error or caution messages in the error message block except the warning concerning vernier assembly. Of course, this is not the only solution to this design problem. The model was solved progressively to obtain this solution. With TK Solver’s multidirectional solving ability, you might wish to investigate other solutions. This model contains a table with which the relationship among speed, torque and power can be quickly explored. The table uses the geometric data from the Variable Sheet but is independent of the speed, torque and power data on the Variable Sheet. To use the table, first click off the checkbox for “Enable table?”, press Return, and solve. Then go to the TK Solver model and select the table “iTable” from the Table Sheet or from the drop-down table list in the Toolbar. Put the same input data as in the data input form in the table, just to compare results. Your screen should look like Sheet 4. Solve and you should have the data in Sheet 5.


64

Sheet 4

Sheet 5


65

Now let’s change the speed of the ring gear. Change the ring gear speed to -75 RPM and solve. Your table should match Sheet 6. Sheet 6

Note that the table indicates whether an element is driver or driven. The algebraic signs of the power for each element also indicates the direction of power flow. This data is helpful in assessing the implications of approach and recess action. The model contains a plot routine that gives a visual check on the pitch diameters. Check the checkbox for “Plot pitch diameters?” and solve to obtain the plot “PD's”. See Figure 5.


66

Fig. 5

A table of coordinates is available for the planet bearing locations. Sheet 7 is the table “planets”.


67

Sheet 7

In many situations it can be helpful to investigate the behavior of the elements of a differential over a range of operating conditions. For example, suppose that we wish to obtain a plot of the power carried by the motor, the hydraulic unit and the conveyor over a speed range of 0 to 150 RPM of the conveyor. Suppose that conveyor power is known to conform to this equation: Horsepower = 100(RPM/150)(1.1) The maximum speed for the conveyor is 150 RPM where the power is 100 HP. In terms of the variables used in the model this equation would be written: Pr=-100*(Nr/-150)^1.1 where: Pr = the power removed from the ring by the conveyor Nr = the ring gear RPM (The ring gear RPM is negative indicating opposite direction from the motor and the power is negative indicating power out of the differential) To set up the model the first step is to enter the equation on the Rule Sheet. Go to the bottom of the Rule Sheet and type in the equation. (Sheet 14).


68

Sheet 8

Next go to the wizard form and check off the checkbox for “Enable table?” and “Plot pitch diameters?” to avoid waiting for the solution of the table and plot as we proceed. To check the equation we entered on the Rule Sheet, enter 1800 for sun gear speed (electric motor) and —150 for ring gear speed (maximum conveyor speed). Blank the sun gear power. Report 14 shows the model after solving. Report 14



MESSAGE FIELD






69



NUMBER OF TEETH

Ring Gear Teeth 88



Sun Gear Teeth 20








PLANET SPACING



p1 2

p2 3

p3 #

p4 #





70









Axial pitch in











Axial pitch in





Type of unit Diff

Number of planets 3


71



ck_#



Effective planets 3






ROTATION SPEED



Carrier -2.69122 rpm





Carrier 0.00000 rpm

DRIVER/DRIVEN

Sun gear Driver

Ring gear Driven

Carrier Driver


72



SUN/PLANET MESH




RING/PLANET MESH









Sun gear 98.06 HP


Carrier 1.94 HP

PER PLANET GEAR








73









Centripetal acceleration on planet 0.00115 G's According to the conveyor speed/power equation, after solving you should have -100 for the ring gear horsepower. Let’s check the horsepower carried by the elements of the system as the conveyor is operated over a speed range of 0 to 150 RPM. We will use TK Solver's list-solving capability to obtain the information we want. The conveyor speed, which is the same as the ring gear speed, varies from 0 to 150 RPM. In the TK Solver Variable Sheet, place the cursor in the Status column for ring gear speed, Nr, and type “L”, or double-click in the Status cell and pick “List” from the drop-down list. This will establish Nr as a list. We also need to label the sun gear power, Ps, ring gear power, Pr, and the carrier power, Pc, as lists. (The carrier power is the same as the hydraulic power as the hydraulic unit is connected to the carrier.) And because we would also like to plot the motor power, we need to add another equation to the Rule Sheet. The motor power is the sum of the hydraulic (carrier) power and the sun gear power. In terms of our variables, the equation is: Pm=Pc+Ps. Move to the bottom of the Rule Sheet and add this equation. (Sheet 9)


74

Sheet 9

The new variable “Pm” will appear at the bottom of the Variable Sheet. Label “Pm” as a list. Sheet 10 shows the Variable Sheet in condensed form to show the list variables. Sheet 10

All of the lists have been established but, of course, none of them contains any data. We wish Nr, the ring gear or conveyor speed, to be an input list. If there is an entry in the input column, then Nr will be an input list.


75

The value on the Variable Sheet will not be used in list solving (unless it is also in the list) but only instructs TK that Nr is an input list. The other lists will be output lists as there is no entry in the input columns. The next job is to enter the values we desire in the list for Nr. Go to the list subsheet for Nr. Instead of typing in the RPM values we need, we will use the automatic list fill feature. Click the List Fill button in the TK Solver Toolbar. Accept the “Linear” and “Fill by Step” defaults. Enter 0 for the first value, -3 for the step size, and-150 for the last value. The list will fill with values from 0 to -150 by a step size of -3. Sheet 11 shows the list in condensed form. Return to the Variable Sheet and list solve by pressing F10 or the Iterative Solver. The values from the list Nr will be used, one by one, for input. The model will solve and the solutions for the other variables will be placed in their respective lists. Now that we have lists of conveyor speeds, Nr, and the corresponding power for the elements of the differential, we can quickly have a plot of these values. Go to the Plot Sheet and label a plot “Power”. Select “Line Chart” and type in the title “Power vs Conveyor RPM”. See Sheet 12.

Sheet 11


76

Sheet 12

“Dive” into the plot subsheet for “Power”: go to the Plot Sheet, select “power,” and click the right mouse button. Set “Display Zero Axes:” to X-axis, label the X and Y axes, enter Nr as the “X-Axis List” and the other lists as “Y-Axis” lists. This is all that is necessary. The screen and the settings are shown in Sheet 13. Sheet 13

To generate the plot, press F7, or click the Plot button in the TK Solver Toolbar. Your plot should look like Figure 6.


77

Fig. 6

We can see from the plot that power circulates through the differential and back through the hydraulic unit to the motor for conveyor speeds up to about 147 RPM. At higher speeds, power flows from the motor through the hydraulic unit to the differential. (This is because of the carrier speed of 147 instead of 150 RPM with the ring at 0 RPM.) The maximum power the hydraulic unit must handle is about 70 HP at about 10 RPM. If a table is useful, it can be quickly made using the Table Sheet. Note: The relative power in a differential is often misunderstood. The input and output torques of any gear unit must balance. The carrier of a differential is rotating. Therefore, the meshing velocities of the teeth are different from a non-differential gear. The power carried by the teeth is a product of load and linear velocity. Because the linear velocity is different from rotation speed multiplied by pitch radius, the relative power is different from the shaft transmitted power. The relative power should be used in load calculations and, of course, the relative speed must then also be used.

program 60-1164—compound epicyclic differential design ...covered by agma standard 6023-a88, and...

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