Program 60-125—Crossed Axis Involute Helical Gears ... · PDF fileProgram...

Program 60-125—Crossed Axis Involute Helical Gears

Introduction Crossed axis involute helical gears are involute helicoid gears that are mounted on axes that do not intersect and are not parallel. There is no difference between a helical gear used in a crossed axis application and a helical gear used in a parallel axis application. The gears are merely mounted in a different configuration. Crossed axis involute gears are also called spiral gears in some of the literature. If there are only a few teeth in one of the gears the gear set is sometimes called a worm gear set. This is not a correct definition of a worm gear set, but this usage is established in the field and is difficult to change. A worm gear set usually has one or both members enveloping the other. UTS literature also describes crossed axis helicals as “non-enveloping” worm gears so that these gears will be recognized in the field. The contact between crossed axis gears is theoretical point contact, while contact between worm gears and parallel axis gears is theoretical line contact. This condition restricts the load capacity of the crossed axis gears compared with the worm gears or parallel axis gears because of much higher contact stress. While the capacity of crossed axis gears is limited, they are the easiest gears to manufacture and assemble. Relatively large changes in center distance, shaft angle and axial assembly position have little or no effect on the smoothness and capacity of the gear set. The contact between parallel axis gears is on a plane of contact internally tangent to the base cylinders of the gears. A plane cannot be tangent to a pair of cylinders that are not parallel. Therefore, contact takes place between crossed axis gears on a line which is internally tangent to the base cylinders of the gears. (This contact line does not, generally, pass through the “line of centers” or common normal to the axes. The contact line will pass through the line of centers only if the gears and CD are made to “standard” proportions.) If the contact ratio of the gears is greater than two then the load is shared between two points instead of being carried on one and the load capacity is doubled. The designer should try to obtain a contact ratio greater than two. (With crossed axis gears a contact ratio of 1.9 is little better than a contact ratio of 1.2.) This is more easily accomplished using low pressure angles. If a contact ratio over two cannot be obtained due to geometrical constraints then a high pressure angle should be used to increase the radii of curvature of the contacting flanks and reduce the contact stress. (The tooth bending stress is usually of little concern since crossed axis gears have a much higher bending capacity than flank contact capacity.)

UTS Integrated Gear Software

2

Much of the literature on crossed axis gears is rather incomplete (and some of it is wrong). Reference 1 is a relatively new book by Professor J.R. Colbourne of the University of Alberta, Canada, in Edmonton. The section on the geometry of crossed helical gears is quite complete and very thorough and is highly recommended for engineers responsible for design and manufacturing of these gears. This model can be used for the design of crossed helical gears produced by hobbing, shaping and forming. For formed gears the shape of the root fillet is that produced by a basic straight sided rack. For hobbed or shaped gears the tooth shape is that produced by these tools. A plot of the teeth of the gears in mesh is available in the model. The plot is of the virtual spur gears and so is representative of the teeth in the “normal” plane. The model checks many conditions to be sure that the gear set will function as intended. (Gear load is not checked; it must be judged against the load table for the material to be used.)

60-125—Crossed Axis Involute Helical Gears

3

Examples Example 1 Example 1 is a crossed axis gear set with a 22-tooth driver and a 55-tooth driven. The normal diametral pitch is 10 and the normal pressure angle is 14.5 degrees. We will make both gears with a 45 degree helix angle, because the sliding in the mesh will be minimum if the helix angle on both gears is half the shaft angle. Both gears will be hobbed (non-topping) without any post processing. The gear material is to be carburized steel. The expected coefficient of friction is about 0.06 with the lubricant being used. The housing and the gears have about the same coefficient of thermal expansion and the gears are not expected to run more than about 100 deg F hotter than the housing so we can use the backlash recommended by the model. The gears will be mounted on shafts that are at right angles. The driver will carry 25 HP at 1800 RPM. Open a new analysis in 60-125. We will use all the default values furnished by the model except the Young’s modulus for the driven gear. Select “Profile shift and center distance” and “Base backlash & driver normal tooth thickness” when these choices appear. We will enter load data. The form rounds off the operating center distance; the entered value should be 5.44472. The completed data input form is shown in Figure 1A. We will configure a plot: accept the default roll angle, enter 3 for the number of teeth and 1 for the driver tooth number. The solved model’s input and output values are shown in Report 1A.


4

Fig. 1A


5

Report 1A

Model Title : Program 60-125 Unit System: US NORMAL PLANE

Diametral_Pitch 10.000000 1/in

Nominal Pressure Angle 14.500000 deg

Module 2.540000 mm `

Circular_Pitch 0.3142 in

Base Pitch 0.304 in

COMMON

Shaft Angle (+ or -) 90.0000 deg

Operating_center distance 5.445 in

Standard_center distance 5.445 in

Length of contact 0.7029 in

Contact ratio (SAP > TIF) 2.3110

Contact_below finished involute? No

Gear_ratio 2.5000

DRIVER

Number of teeth 22

Hobbed ('hob), Shaped ('shp) hob

Outside Diameter 3.3112 in

Start_Tip Modification NA in

Roll_at_start of tip modification NA deg


6

Model Title : Program 60-125 Unit System: US Normal_OD tip relief NA in

Normal_circular OD tip relief NA in

Transverse_circular_OD tip relief NA in

Effective_outside diameter 3.3112 in

Normal_tooth_thickness_at_EOD 0.0958 in

Normal_EOD tip relief NA in

Pointed tooth diameter (No tip mod) 3.5407 in

Reference PD 3.1113 in

Finished normal tooth thickness 0.1550 in

Total normal circular finish stock on tooth 0.0000 in thickness

Helix angle (+ Right, - Left) 45.000000 deg

Lead_angle (+ Right, - Left) 45.0000 deg

Minimum face width (bi-rotation) 0.5232 in

Transverse_diametral pitch 7.0711 1/in `

Transverse_circular pitch 0.444 in

Transverse_module 3.5921 mm `

Transverse_pressure angle 20.0895 deg

Transverse_tooth thickness 0.2192 in

Transverse_base pitch 0.417 in

Base_helix angle 43.20288 deg

Axial pitch 0.444 in

Lead 9.7743 in

Inv/fillet intersection dia (TIF) 2.9562 in

Roll_at_inv/fil intersection dia 8.7967 deg


7

Model Title : Program 60-125 Unit System: US Root diameter 2.8533 in

Whole depth of tooth 0.2289 in

Minimum_fillet radius 0.0331 in

Base_diameter 2.9220 in

DRIVEN

Number of teeth 55





Normal_OD tip relief NA in















8

Model Title : Program 60-125 Unit System: US Transverse_circular pitch 0.444 in







Lead 24.4359 in



Root diameter 7.5203 in




OPERATING DATA

Separation of pitch planes 0.000 in

Contact_path to line of centers 0.0000 in

Working depth 0.1999 in

OPERATING DATA NORMAL PLANE

Base_backlash 0.0040 in

Diametral_pitch 10.0000 1/in

Circular_pitch 0.314 in

Pressure_angle 14.5000 deg


9

Model Title : Program 60-125 Unit System: US OPERATING DATA DRIVER

Pitch_diameter 3.1113 in

Normal_Tooth Thickness 0.1550 in

Helix_Angle (+ Right, - Left) 45.0000 deg

Lead_Angle (+ Right, - Left) 45.0000 deg

Transverse_Diametral Pitch 7.0711 1/in `

Transverse_Circular Pitch 0.444 in

Transverse_Module 3.5921 mm `

Transverse_Pressure Angle 20.0895 deg

Transverse_Tooth Thickness 0.2192 in

Angular_backlash 0.215 deg

Start of active profile (SAP) 2.9702 in

Normal_tooth thickness at SAP 0.1796 in

Normal_space width at SAP 0.1272 in

Root_clearance 0.0290 in

OPERATING DATA DRIVEN










10

Model Title : Program 60-125 Unit System: US Transverse_Tooth Thickness 0.2192 in






PLOT CONFIGURATION

Mark inv/fil intersections? y

Mark mod/inv intersections? y

Number of teeth on plot 1

Driver_contact roll angle of deg

Driver tooth number 1

ROLL ANGLES DRIVER

Start_of_active profile 10.4489 deg

Actual_Start of active profile 10.4489 deg

Actual_End of active profile 30.5424 deg

Effective outside diameter 30.5424 deg

ROLL ANGLES DRIVEN





HOB DRIVER

Hob type n


11

Model Title : Program 60-125 Unit System: US Flank angle 14.5000 deg

Tip to Reference Line 0.1250 in

Tooth thickness at Reference Line 0.1571 in

Reference_Line to Start Mod Ramp _ in

Pressure Angle of Mod Ramp _ deg

Tip_radius 0.0300 in

Protuberance 0.0000 in

Protuberance_angle from flank _ deg

Protuberance_pressure angle _ deg

Tip_to_flank/prot intersection _ in

Ref_Line to Hob SAP 0.0880 in

Normal_Space Width at Hob SAP 0.1116 in

HOB DRIVEN

Hob type n

Flank angle 14.5000 deg












12

Model Title : Program 60-125 Unit System: US Normal_Space Width at Hob SAP 0.1074 in

SHAPER DRIVER

Number_of_Teeth _

Outside_Diameter _ in

Normal_Tooth_Thickness _ in

Tip_Radius - Normal Plane _ in

Protuberance - Normal Plane _ in

Center distance with shaper cutter NA in

Start_of_active_profile diameter in

SHAPER DRIVEN

Number_of_Teeth _

Outside_Diameter _ in

Normal_Tooth_Thickness _ in

Tip_Radius - Normal Plane _ in

Protuberance - Normal Plane _ in

Center distance with shaper cutter NA in


FORMED DRIVER

Flank Angle _ deg

Tip to Reference Line _ in

Tooth Thickness at Reference Line _ in

Tip_radius _ in

Radial_tip chamfer _ in

Normal_tip radius _ in


13

Model Title : Program 60-125 Unit System: US Normal_tip_relief exponent _

FORMED DRIVEN

Flank Angle _ deg

Tip to Reference Line _ in

Tooth Thickness at Reference Line _ in

Tip_radius _ in

Radial_tip chamfer _ in

Normal_tip radius _ in

Normal_tip_relief exponent _

LOADING

Maximum sliding velocity 2130.1 ft/min

Approx_coefficient of friction 0.060

Approx_efficiency 89 %

Approx_efficiency backdriving 89 %

Anti-backdrive safety factor _

Imposed normal tooth load 821.951 lbf

Overload factor 1.00

Total_normal tooth load 410.976 lbf

LOADING AT_CENTER OF CONTACT INTERVAL

Angle between contact curvatures 27.935 deg

Specific compressive stress, s 189335 psi

Load/compressive stress factor, Cc 897.9 psi

LOADING DRIVER

Power 25.0000 HP


14

Model Title : Program 60-125 Unit System: US Rotational speed 1800.0 rpm

Torque 875.347 lbf-in

Young`s modulus 30000000.00 psi

Contact_curvature inclination angle 13.760 deg

Tangential force 562.695 lbf

Separating force 205.800 lbf

Axial_force 562.695 lbf

Overturning moment 873.103 lbf-in

LOADING DRIVEN

Rotational speed 720.0 rpm







Overturning moment 2190.641 lbf-in Figure 1A is a plot of the teeth in mesh at the first point of contact on the driver (driver start of active profile, SAP). The contact ratio is over two so that there are either 2 or 3 teeth in contact at all times. (Note that the efficiency is the same no matter which gear is driving. This is due to both helix angles being equal to one half the shaft angle.)


15

Fig. 1B

The Specific compressive stress is about 189,000 psi and the Load/compressive stress factor is about 900 psi. Sheet 1B is the approximate load-stress table from the model. The approximate allowable Load/compressive stress for carburized steel is 1058 psi and our design is a little below this. Note that hardened steel “wears in” very little and a run-in will not increase the capacity of the set. (The factors in the table are approximate and from several sources. It is suggested that the table be updated and changed as you accumulate more accurate data.)


16

Sheet 1B


17

Next we will check the operating conditions of the gear set when the shaft angle is off 1 degree. We will assume that the tolerance on the shaft angle is +/-1 degree. First we will check conditions at 91 degrees. This time we will enter normal tooth thickness for both driver and driven, and enter 91 degrees for the shaft angle. The completed data input form is shown in Figure 1C and the solved model in Report 1B. Fig. 1C


18

Report 1B

Model Title : Program 60-125 Unit System: US

NORMAL PLANE





Base Pitch 0.304 in

COMMON







Gear_ratio 2.5000

DRIVER

Number of teeth 22






19






















Lead 9.7743 in




20





DRIVEN

Number of teeth 55




















21








Lead 24.4359 in







OPERATING DATA

Separation of pitch planes -0.096 in

Contact_path to line of centers -0.1611 in








22


























23







PLOT CONFIGURATION






ROLL ANGLES DRIVER





ROLL ANGLES DRIVEN





HOB DRIVER

Hob type n


24

Model Title : Program 60-125 Unit System: US Flank angle 14.5000 deg












HOB DRIVEN

Hob type n













25

Model Title : Program 60-125 Unit System: US Normal_Space Width at Hob SAP 0.1074 in

LOADING













LOADING DRIVER

Power 25.0000 HP










26

Model Title : Program 60-125 Unit System: US LOADING DRIVEN








Overturning moment 2192.014 lbf-in The backlash has dropped from 0.004 inch to 0.0012 inch and the contact ratio has dropped only slightly. Next we will enter 89 degrees for the shaft and check the set at the low end of the shaft angle tolerance. Report 1C is the solved model. (Because only is one input is changed, the completed input form is not shown.)


27

Report 1C


NORMAL PLANE





Base Pitch 0.304 in

COMMON







Gear_ratio 2.5000

DRIVER

Number of teeth 22





28

Model Title : Program 60-125 Unit System: US Roll_at_start of tip modification NA deg






















Lead 9.7743 in



29

Model Title : Program 60-125 Unit System: US Roll_at_inv/fil intersection dia 8.7967 deg





DRIVEN

Number of teeth 55



















30

Model Title : Program 60-125 Unit System: US Transverse_diametral pitch 7.0711 1/in `








Lead 24.4359 in







OPERATING DATA










31


























32







PLOT CONFIGURATION






ROLL ANGLES DRIVER





ROLL ANGLES DRIVEN






33

Model Title : Program 60-125 Unit System: US HOB DRIVER

Hob type n













HOB DRIVEN

Hob type n









34

Model Title : Program 60-125 Unit System: US Protuberance_angle from flank _ deg






LOADING













LOADING DRIVER

Power 25.0000 HP




35

Model Title : Program 60-125 Unit System: US Young`s modulus 30000000.00 psi






LOADING DRIVEN









The backlash under this condition is only 0.0005 inch. If these conditions are at the extreme minimum effective center distance (worst case of tolerances or thermal changes, for example) this may be satisfactory, but it could be a problem if there is any chance of further reduction in backlash. The contact ratio is still well over two.

Next we should check the other end of the tolerance and thermal range to find the maximum backlash and minimum contact ratio (see UTS Model 60-146). Keep in mind that if the contact ratio drops below two we will lose half our capacity. Figure 1D and 1E are the completed data input form and a portion of the Plot Configuration tab, respectively. (Again, the form rounds the standard center distance; enter 5.44772.) Report 1D is the solved model for maximum effective CD after these changes have been made.


36

Fig. 1D


37

Fig. 1E

Report 1D


NORMAL PLANE





Base Pitch 0.304 in

COMMON




38

Model Title : Program 60-125 Unit System: US Standard_center distance 5.445 in




Gear_ratio 2.5000

DRIVER

Number of teeth 22



















39









Lead 9.7743 in







DRIVEN

Number of teeth 55









40

Model Title : Program 60-125 Unit System: US Effective_outside diameter 7.9751 in


















Lead 24.4359 in







41

Model Title : Program 60-125 Unit System: US Base_diameter 7.3049 in

OPERATING DATA









OPERATING DATA DRIVER















42

Model Title : Program 60-125 Unit System: US Root_clearance 0.0364 in
















PLOT CONFIGURATION




Driver_contact roll angle of 10.9379 deg


ROLL ANGLES DRIVER



43

Model Title : Program 60-125 Unit System: US Actual_Start of active profile 10.9378 deg



ROLL ANGLES DRIVEN





HOB DRIVER

Hob type n














44


HOB DRIVEN

Hob type n













LOADING










45

Model Title : Program 60-125 Unit System: US LOADING AT_CENTER OF CONTACT INTERVAL




LOADING DRIVER

Power 25.0000 HP









LOADING DRIVEN










46

The backlash is 0.0084 inch and the contact ratio is still above two. Figure 1F is a plot of the teeth at the first point of contact for this condition. Fig. 1F

This set of conditions should also be checked at 89 and 91 degree shaft angles to find any effect on the contact ratio at maximum effective center distance.


47

Example 2 Example 2 is a crossed axis gear set with a 1 tooth driver and a 36 tooth driven. This type of set is often called a worm gear set even though neither gear envelops the other. The driver will be steel and the driven will be acetal (Young’s Modulus about 400,000 psi). The dynamic coefficient of friction between steel and acetal is about 0.15. Both gears will be formed. (See UTS Model 60-410 for help with tools and molds for formed gears.) The shaft angle will be 80 degrees. This enables us to try a design with a 80 degree helix angle (10 degree lead angle) on the driver and to use a spur gear for the driven. (If the plastic gear is a spur gear we can use wire electric discharge machining to make the gear mold directly.) The normal diametral pitch will be 24 and we will use a 14.5 degree pressure angle to try to obtain a contact ratio over two. The required load is 0.5 lbf-in torque on the driver at 4,200 RPM. The gears will be lubricated (at least initially) and are expected to run only slightly above room temperature. (The thermal conduction of plastics is very poor but in this case we have a steel driver to carry the heat away quickly.) Because both gears are to be manufactured by forming, it is necessary to specify tooth tip radii for rolling or molding. First, however, we will design the gears without the tip radii and then add them later. Because the plastic is a much weaker material than the steel, we will also adjust the tooth thickness of the gears at the same time as we add the tip radii. Figure 2A is the completed data input form for this model. (The form rounds the operating center distance; you should enter .87191.) We will configure a plot; a portion of the completed Plot Configuration tab is shown in Figure 2B. Report 2A is the solved model.


48

Fig. 2A

Fig. 2B


49

Report 2A


NORMAL PLANE





Base Pitch 0.127 in

COMMON







Gear_ratio 36.0000

DRIVER

Number of teeth 1

Hobbed ('hob), Shaped ('shp) frm




50























Lead 0.1329 in



51






DRIVEN

Number of teeth 36
















Lead_angle (+ Right, - Left) _ deg



52








Axial pitch in

Lead in







OPERATING DATA










53




















Lead_Angle (+ Right, - Left) _ deg






54







PLOT CONFIGURATION






ROLL ANGLES DRIVER





ROLL ANGLES DRIVEN






55


FORMED DRIVER

Flank Angle 14.5000 deg


Tooth Thickness at Reference Line 0.0654 in


Radial_tip chamfer 0.0000 in

Normal_tip radius 0.0000 in


FORMED DRIVEN








LOADING








56

Model Title : Program 60-125 Unit System: US Overload factor 1.00






LOADING DRIVER

Power 0.0333 HP


Torque 0.500 lbf-in







LOADING DRIVEN










57

The Load/compressive stress factor is about 1,140 psi. The load-stress table gives us an allowable factor of about 1,150 psi after an extensive run in. (The capacity increases because the hard driver wears into the plastic gear and the set is then more like a single enveloping worm set than a crossed axis set. The “point” contact has become a “line” contact. For this reason a “hunting” ratio should never be used when there is more than one tooth in the harder gear.)

If an extensive run-in cannot be assured, then this design is not adequate, as the allowable Load/compressive stress factor is only about 191 psi at initial contact and 290 psi after a short run-in. Note that the efficiency with the “worm” driving is about 54% and with the gear driving about 15%. This set would backdrive and could not be used in a one-way drive without an additional mechanism to prevent reverse rotation.

Figure 2C is a plot of the gears in mesh at the first point of contact.

Fig. 2C

Now we will add the radii on the tooth tips. Instead of rounding the corners of the gears from the present outside diameters (and reducing the contact ratio) we will add material to the tooth tips while keeping the new effective outside diameters at the present outside diameters. At the same time we will thin the steel driver teeth and thicken the plastic driven teeth in the ratio of about 35% and 65% of the normal circular pitch. (This is a “rule of thumb” and seems to work pretty well.) Don't forget to change the basic rack tooth thickness opposite to the change in gear tooth thickness to keep the root diameters about the same.


58

After a little experimenting we have the model shown in Figure 2D and Report 2B. Fig. 2D


59

Report 2B


NORMAL PLANE





Base Pitch 0.127 in

COMMON







Gear_ratio 36.0000

DRIVER

Number of teeth 1






60






















Lead 0.1329 in




61





DRIVEN

Number of teeth 36




















62







Axial pitch in

Lead in







OPERATING DATA










63


























64







PLOT CONFIGURATION






ROLL ANGLES DRIVER





ROLL ANGLES DRIVEN





FORMED DRIVER



65

Model Title : Program 60-125 Unit System: US Tip to Reference Line 0.0554 in






FORMED DRIVEN








LOADING










66






LOADING DRIVER

Power 0.0333 HP


Torque 0.500 lbf-in







LOADING DRIVEN










67

Figure 2E is the Plot Configuration tab with inputs that produce the plot shown in Figure 2F. It is a plot of our new gears in mesh at the first point of contact. Fig. 2E

Fig. 2F

After taking tolerances, thermal and moisture absorption into account (see UTS gear model 60-146) we might have the model shown in Figure 2G and Report 2C.


68

Fig. 2G


69

Report 2C


NORMAL PLANE





Base Pitch 0.127 in

COMMON







Gear_ratio 36.0000

DRIVER

Number of teeth 1





70























Lead 0.1329 in



71






DRIVEN

Number of teeth 36



















72








Axial pitch in

Lead in







OPERATING DATA










73


























74







PLOT CONFIGURATION






ROLL ANGLES DRIVER





ROLL ANGLES DRIVEN






75

Model Title : Program 60-125 Unit System: US FORMED DRIVER








FORMED DRIVEN








LOADING









76

Model Title : Program 60-125 Unit System: US Total_normal tooth load 24.790 lbf





LOADING DRIVER

Power 0.0333 HP


Torque 0.500 lbf-in







LOADING DRIVEN










77

We have a serious problem under these conditions. The contact ratio has dropped below 2 and all the load must be carried on one contact “point”. The Load/compressive stress factor has gone from about 1,140 psi to about 1,970 psi. This is much more than the material can stand; this design would not be adequate. Reducing the tolerance, thermal and moisture effects may be enough to save the design. Otherwise, a new approach may have to be taken.


78

Figure 2H is the Plot Configuration tab that produces the plot in Figure 2I, a plot of the gears in mesh at the first point of contact. Fig. 2H

Fig. 2I

The design of successful plastic gears is very difficult because of the high thermal coefficients, moisture absorption and tolerances required by plastics. The problem is compounded by the requirement for radii at all corners.


79

Glossary Angle between contact curvatures

This is the angle between the tooth generators at the center of the contact interval. This angle is used in the calculation of the contact stress. Angular backlash

This is the angle through which the gear can be rotated when the other gear is held stationary. Approx coefficient of friction

The coefficient of friction between the teeth depends upon many factors, such as materials, lubricant and sliding velocity. The default coeffecient of friction is for a hardened steel driver and a bronze driven. Since the sliding conditions are similar to single enveloping worm gears, the default values are taken from AGMA standard 6034-B92, Practice for Enclosed Cylindrical Wormgear Speed Reducers and Gear Motors. See the References below. If the materials are not steel and bronze the coefficient of friction should be entered for the materials used. Approximate efficiency

The efficiency of the gear set is dependent upon the coefficient of friction and the sliding velocity. The efficiency is calculated from information in Analytical Mechanics of Gears by Earle Buckingham. See the References below. At center of contact interval

This location is half way between the outside diameters of the gears along the path of contact. It is at this point that the compressive stress between the teeth is calculated. Axial force

This is the force normal to the plane of rotation and normal to the line of centers produced by the torque on the gear.


80

Axial pitch

The axial pitch is the distance from tooth flank to tooth flank in the axial direction. The axial pitch of the gear is a constant at any diameter greater than or equal to the base diameter and is equal to the lead divided by the number of teeth. It is negative for left hand gears. Unless the helix angles are the same, the driver and driven gears will have different axial pitches. Base backlash

The base backlash is the distance along the path of contact between the unloaded flanks of the gear teeth. If there is a separation of the pitch planes the backlash would be between the unloaded tooth flanks along a line of action on the other side of the line of centers from the loaded tooth flanks. This backlash is the theoretical backlash based on the tooth thickness of the gears, the operating center distance, the helix angles of the gears and the shaft angle. The actual operating backlash will be the same only if an adjustment is made to account for the difference between effective and measured tooth thickness during manufacture of the gears. Base diameter

The base diameter is the diameter of the base cylinder from which the involute tooth flank is generated. The involute of a circle may be defined as the locus of the end of a line as it is unwound from the circle. In the case of involute gears, the surface of the tooth would be described by the edge of a sheet that is unwound from the base cylinder. For a spur gear the edge of the sheet would be parallel to the axis of the cylinder. For a helical gear the edge of the sheet would be cut at the base helix angle from the axis of the base cylinder. Base helix angle

The base helix angle is the angle of the tooth flank on the base diameter. It is measured from the axis of rotation of the gear. The algebraic sign will be positive for right hand gears and negative for left hand gears.


81

Basic rack form for formed gears

A gear that is to be formed by a process such as stamping or molding should be designed using a basic rack form for generating the root fillet area. If this is not done, fillet interference with the mating gear tooth tip is possible. If a suitable basic rack form is used the possibility of interference is reduced. If a tip radius is used on the mating gear that extends above the standard addendum height for the basic rack the mesh should be checked for secondary contact between the mating tooth tip and the fillet area. This may occur out of the involute contact zone and off the involute line of action. Center distance with shaper cutter

This is the cutting center distance between the shaper cutter and the gear to produce the required shaped tooth thickness. Check for contact below finished flank

The model compares the diameter at the involute/fillet intersection (TIF) with the start of active profile (SAP) diameter. If the SAP is below the TIF then the contact ratio calculated using the path of contact bounded by the outside diameters is not correct. The contact ratio will be less than the calculated value. If the TIF cannot be lowered by changing the tooling or tooth form, then the outside diameter of the mating gear should be reduced (unless the condition is caused by required operation at a minimum center distance tolerance condition). The contact ratio will also be reduced if either of the face widths is less than the minimum required or there is an error in the assembled axial position of the gears with respect to the line of centers. Of course, the model cannot detect these conditions. Contact curvature inclination angle

This angle is between the involute surface generator and the tangent to the helix angle at the center of the contact interval. It is used to calculate the angle between the generators of the teeth at the contact point.


82

Contact path to line of centers

The involute flanks of crossed helical involute gears contact on a path of contact which is a straight line internally tangent to the base cylinders of the gears. When there is a separation of the pitch planes the path of contact does NOT pass through the line of centers. Only if the gear set is “standard” as to dimensions and shaft angle will the path of contact pass through the line of centers. If the center distance is not equal to the sum of the reference pitch radii and/or the shaft angle is not equal to the sum of the nominal helix angles there will be a separation of the pitch planes and the path of contact will not pass through the line of centers. The contact path to line of centers is the distance from the line of centers to the contact path. Contact ratio

The contact ratio is the active length of the line of action divided by the normal base pitch. It may be defined as the “average number of teeth in contact”. The equations used to calculate the contact ratio do not detect undercut, Start of Active Profile below the True Involute Form or too narrow a face width. If the SAP is below the TIF on either gear or the face widths are too narrow (or the gears are improperly assembled) then the contact ratio will be reduced. The gear set must have a profile contact ratio greater than one to be a conjugate set and for good load capacity the contact ratio should be above two if possible. If a contact ratio between one and two is necessary then a high operating normal pressure angle should be considered to reduce the contact stress. NOTE: The contact ratio should be checked at its minimum value to insure that it is

not too low due to tolerances on the center distance and gears. Driven

The driven is the gear which is under torque applied in the direction opposite to the gear rotation. Driver

The driver is the gear which is under torque applied in the same direction as the gear rotation.


83

Driver Power

The driver power is the power applied to the driving gear. The amount of power available at the driven gear depends on the efficiency of the gear unit. Depending on the sliding velocity, among other things, some crossed axis helical gear sets have a relativly low efficiency. (The lowest sliding velocity will occur when the gear operating helix angles are each one half of the shaft angle.) Effective outside diameter

This diameter will be the same as the physical outside diameter (exclusive of tip chamfers or tip radii) unless the mating gear has undercut in the root fillet area that removes part of the active portion of the involute flank. If there is a tip chamfer (but not tip relief) or tip radius on the gear and the mate is not undercut enough to reduce the contact, this is the largest diameter on the tooth that engages the mating tooth on the path of contact. If there is undercut the active portion of the line of action may be set by the undercut diameter instead of the outside diameter of the mate. This condition is undesirable, as a portion of the tooth near the OD is not used and may as well be removed. Excessive undercut reduces the contact ratio, and every effort should be made to alter the design of the gears or generating tools to eliminate it. Finished normal tooth thickness at Ref PD

This is the “circular” thickness of the tooth at the reference pitch diameter after any finishing operation measured in the plane which is normal to the reference helix angle. Flank angle

This is the angle of the straight sided portion of the rack or hob tooth measured from a normal to the pitch line of the rack or hob. The only pitch requirement for a rack or hob to produce a given gear is that the normal base pitch of the tool and gear be the same. Under some conditions it is desirable that the flank angle of a rack or hob be different than the nominal pressure angle of the gear. For example, the flank angle can be less than the pressure angle of the gear as long as the linear pitch is adjusted to keep the normal base pitch the same. When this is done with a hob it is called a “short lead” hob. This type of rack or hob will reduce the generating diameter and lower the undercut break-out point or reduce the undercut.


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Formed gear normal tip radius

A tip radius on a formed gear is defined and measured in the normal plane. The default value is zero. Formed gear normal tip relief exponent

The shape of the tip relief deviation from the involute can be controlled with the tip relief exponent. The exponent controls the rate of drop away from the involute with the radial distance away from the start of tip relief diameter. For example, if the exponent is one the drop from the involute would be linear. If the exponent is two the drop would be parabolic. The value of the exponent must be between one and five. The default value is 3/2. Formed gear tip chamfer

The tip chamfer is measured on a radial through the gear center from the outside diameter of the gear exclusive of any tip modification. The chamfer will be at approximately a 45 degree angle with the radial to the outside diameter. The default value is zero. Gear production method

• The model allows for gears to be produced by three different methods. • The gears may be generated by a rack type cutting tool such as a hob or rack

type shaper cutter. The cutter may be of three types: 1) A non-topping cutter with or without protuberance to leave undercut for a

finishing operation such as shaving or grinding. 2) A semi-topping cutter with a ramp in the cutter root used for producing a

“chamfer” at the tooth tip. This cutter may also have protuberance. 3) A tip relief cutter which has a higher flank angle for part of the flank toward

the cutter root. This cutter is used to obtain tip relief on the gear. This cutter may also have protuberance but is seldom used as a finishing operation would remove part or all of the tip relief.

(For any of these cutters enter 'hob.) • A generating tool shaped like a gear may be used and is called a shaper

cutter. This tool may have protuberance to leave undercut for a finishing operation such as shaving or grinding. To use this cutter enter 'shp.

• The gear may be formed by methods such as molding or milling with a tool shaped like the tooth space. In this case a basic rack is used to generate the shape of the tooth space and root fillet. This method is used to produce gear sets that are less likely to have fillet interference than gears using circular arcs and straight lines to form the root fillet. The gear may have a tip chamfer or a tip radius. Tip relief may be applied if desired. (Enter 'frm if the gear is formed.)


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Gear ratio

The gear ratio is the number of teeth in the driven gear divided by the number of teeth in the driver. It may be greater than or equal to one (reduction gear) or less than one (speed increaser). For crossed axis helical gears the ratio is NOT, in general, the ratio of operating pitch diameters and base circles. Helix angle

This is the nominal helix angle for the gear at the reference pitch diameter. It is measured from the axis of rotation of the gear. In this model a right hand helix angle is considered positive and a left hand helix angle is considered negative. The equations in the model follow a signed convention and the algebraic signs of the helix angles must be entered accordingly. The reference pitch diameter may be found by dividing the number of teeth by the nominal transverse pitch. The nominal transverse pitch is found by multiplying the nominal normal pitch by the cosine of the nominal helix angle. The gear will have a different helix angle at all diameters from the base diameter to the outside diameter. The lead of the gear is the axial advance of a tooth in one turn about the gear axis of rotation. The lead of the gear is a constant at any diameter greater than or equal to the base diameter. Hob type

Gears may be generated by a rack type tool such as a hob or rack type shaper cutter. The cutter may be of three types: Non-Top: A non-topping cutter with or without protuberance to leave undercut for a

finishing operation such as shaving or grinding. (Enter 'n) Semi-Top: A semi-topping cutter with a ramp in the cutter root used for producing a

“chamfer” at the tooth tip. This cutter may also have protuberance. (Enter 's)

Tip-Relief: A tip relief cutter which has a higher flank angle for part of the flank toward the cutter root. This cutter is used to obtain tip relief on the gear. This cutter may also have protuberance but this is seldom used as a finishing operation would remove part or all of the tip relief. (Enter 'r)


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Imposed normal tooth load

This is the total load imposed on all gear teeth in mesh in a direction normal to the tooth flanks along the path of contact by the torque applied to the driver. Involute/fillet intersection dia (TIF)

This is the diameter at the intersection between the involute profile and the fillet portion of the tooth. On generated gears this may be a smooth tangent point or, in the case of undercut gears, the point of intersection of the involute and undercut trochoidal fillet. It is the lowest involute portion of the tooth and is sometimes called the true involute form (TIF) diameter. (This use of the term “true involute form diameter” should not be confused with the “required true involute form diameter” found in gear specifications as a required limit when the gear is inspected.) Lead

The lead is the axial advance of a tooth in one turn about the gear axis of rotation. It equal to the axial pitch times the number of gear teeth and is negative for left hand gears. The lead of the gear is a constant at any diameter greater than or equal to the base diameter. Lead angle

This is the nominal lead angle for the gear at the reference pitch diameter. It is measured from a normal to the axis of rotation of the gear. In this model a right hand lead angle is considered positive and a left hand lead angle is considered negative. The equations in the model follow a signed convention and the algebraic signs of the lead angles must be entered accordingly. The gear will have a different lead angle at all diameters from the base diameter to the outside diameter. The lead of the gear is the axial advance of a tooth in one turn about the gear axis of rotation. The lead of the gear is a constant at any diameter greater than or equal to the base diameter. Length of contact

The length of contact is the active length of the path of contact. (The path of contact is an internal tangent line between the base cylinders of the gears. The position of the path of contact is set by the shaft angle, the gear pressure and helix angles, and the center distance.)


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The length of contact should be determined only by the outside diameters of the gears. If either gear is undercut high enough on the involute flank, then the length of contact will be determined by the undercut diameter instead of the outside diameter of the mate. If either gear is too narrow (or mounted too far from the line of centers), the length of contact will be determined by the end of the gear instead of the OD of the mate. Both of these conditions should be avoided. Load/compressive stress factor

A factor is needed to relate the load and compressive stress after the gear set has run in for a period of time. A careful run-in will increase the load capacity for some materials by increasing the area under load. In effect, the wear during run in changes the crossed axis helical gears to enveloping worm gears. The load/compressive stress factor equation is from Analytical Mechanics of Gears by Earle Buckingham. See the References below. Maximum sliding velocity

This is the maximum value of the velocity of sliding between the teeth. It occurs at the tooth tip of one of the gears. For gear sets with high shaft angles the sliding is mostly axial. As the shaft angle is reduced the radial component of the sliding becomes more pronounced. Minimum face width (bi-rotation)

This is the minimum face width required to insure that, under normal conditions of assembly, the active length of the path of contact will not be shortened by the edge of the tooth face. It has been calculated so that both directions of rotation are accommodated. The face has been increased by 0.1/(Normal Diametral Pitch) or 0.1*(Normal Module) at each end to allow for possible axial position errors in assembly.


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Nominal transverse diametral pitch

This is the nominal transverse diametral pitch of the gear. Unless the helix angles are the same, the driver and driven gears will have different transverse diametral pitches. The nominal transverse diametral pitch is the number of teeth per inch of reference pitch diameter. This value, along with the nominal transverse pressure angle, sets the transverse base pitch of the gear. The nominal transverse diametral pitch is defined in the transverse plane. The transverse plane is a plane set normal to the axis of rotation of the gear. This is the plane in which most gear element calculations are done and is the plane in which the elements of a gear tooth are inspected. The flanks of the teeth are involutes of the base circle in this plane. Normal base pitch

The normal base pitch is the distance between same handed tooth flanks along a tangent line to the base circle. It is also the distance on the basic rack from one straight flank to the next normal to the flanks. The only requirements for conjugate tooth action are that the normal base pitches of the gears be the same and the contact ratio be greater than 1. Normal circular pitch

This is the nominal normal circular pitch of the system. This value, along with the nominal normal pressure angle, sets the normal base pitch of the system. The only requirements for conjugate tooth action are that the normal base pitches of the gears be the same and the contact ratio be greater than 1. The normal circular pitch, normal diametral pitch and the normal pressure angle are convenient numbers used in the “standards” to specify the normal base pitch. The nominal normal “circular” pitch is the distance on the basic rack, with a flank angle equal to the nominal pressure angle, from tooth to tooth along a line parallel to the tooth tips. The circular pitch is equal to pi divided by the diametral pitch. (The normal base pitch is the distance from tooth flank to tooth flank along a tangent line to the base circle. It is also the distance on the basic rack from one straight flank to the next normal to the flanks.)


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Normal circular tip relief at the outside diameter

This is the amount of tip relief measured at the outside diameter in the normal plane. (If the tooth has a formed tip chamfer or tip radius the tip relief is measured from the projected involute as if there were no chamfer or radius.) Normal diametral pitch

This is the nominal normal diametral pitch of the system. This value, along with the nominal normal pressure angle, sets the normal base pitch of the system. The only requirements for conjugate tooth action are that the normal base pitches of the gears be the same and the contact ratio be greater than 1. The normal diametral pitch and the normal pressure angle are convenient numbers used in the “standards” to specify the normal base pitch. (The normal base pitch is the distance from tooth flank to tooth flank along a tangent line to the base circle. It is also the distance on the basic rack from one straight flank to the next normal to the flanks.) Normal module

This is the nominal normal module of the system. This value, along with the nominal normal pressure angle, sets the normal base pitch of the system. The only requirements for conjugate tooth action are that the normal base pitches of the gears be the same and the total contact ratio be greater than one. The normal module and the normal pressure angle are convenient numbers used in the “standards” to specify the normal base pitch. (The normal base pitch is the distance from tooth flank to tooth flank along a tangent line to the base circle. It is also the distance on the basic rack from one straight flank to the next normal to the flanks.) Normal Plane

The normal plane is a plane set normal to the reference helix angle at the reference pitch diameter. This is the plane in which a rack type tool that generates the gear is specified. In spur gears, of course, the normal plane is the same as the transverse plane or plane of rotation. The involute flanks of the teeth are not involutes of the base circle in this plane but are involutes of a base ellipse defined by the intersection of the normal plane with the base cylinder. Normal space width at SAP

This is the “circular” space width at the start of active profile diameter measured in the plane which is normal to the helix angle at the SAP diameter.


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Normal tip relief at effective OD

This is the amount of tip relief measured normal to the involute profile at the effective outside diameter. This is the tip relief seen at the first point of contact (driven tip relief) and the last point of contact (driver tip relief). This relief should match the required tip relief for the first and last points of contact. Normal tip relief at the outside diameter

This is the amount of tip relief measured normal to the involute profile at the outside diameter. (If the tooth has a formed tip chamfer or tip radius the tip relief is measured from the projected involute as if there were no chamfer or radius.) It is an input value for formed gears if tip relief is desired. Normal tooth thickness at SAP

This is the “circular” thickness of the tooth at the start of active profile diameter measured in the plane which is normal to the helix angle at the SAP diameter. Normal Tooth Thickness at effective OD

This is the “circular” thickness of the tooth at the effective outside diameter measured in the plane which is normal to the helix angle at the effective outside diameter. The tooth thickness at the effective outside diameter should, in general, be more than than 0.3/(Normal Diametral Pitch) or 0.3*(Normal Module) for case hardened gears. If the gear is case hardened a smaller thickness may cause a very brittle tooth tip that is subject to breakage under shock loads. (If the included angle between the normal to the involute and the tangent to the effective outside diameter is at least 90 degrees or if the gear has a tip chamfer or tip radius this “rule of thumb” may not apply.) Number of teeth in shaper cutter

This is the number of teeth in the shaper cutter. For disc-type cutters some shaper cutter manufacturers have standard cutters and cutter blanks. If a standard cutter or blank can be used it may save cost and lead time to specify the number of teeth required to take advantage of standard stock.


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Number of teeth on plot

This is the number of teeth in each gear shown on the plot of the driver, driven or mesh. Since the plot contains at least one helical gear the number of teeth plotted is restricted as the virtual spur gear is used. The restrictions on the number of teeth plotted are as follows:

Contact ratio less than one: One tooth plotted Contact ratio between one & two: Max of two teeth plotted Contact ratio between two & three: Max of three teeth plotted

Operating center distance

This is the shortest distance between the centers of rotation of the gears after mounting. It may vary with such factors as tolerance of the housing and temperature. Operating helix angle

This is the helix angle of the gear at the operating pitch diameter. It is measured from the axis of rotation of the gear. The operating helix angle is set by the base helix angles of both gears along with the shaft angle. If the angle is right hand it is positive and if left hand it is negative. The algebraic sum of the operating helix angles must be equal to the shaft angle. Operating lead angle

This is the lead angle of the gear at the operating pitch diameter. It is measured from a normal to the axis of rotation of the gear. The operating lead angle is set by the base helix angles of both gears, along with the shaft angle. If the angle is right hand it is positive; if left hand, it is negative. Operating normal circular pitch

This is the operating circular pitch of the system measured in the normal direction at the operating pitch diameters. Operating normal diametral pitch

This is the operating normal diametral pitch of the system measured in the normal direction at the operating pitch diameters.


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Operating normal pressure angle

This is the normal pressure angle at the operating pitch diameters. Operating normal tooth thickness

This is the “circular” thickness of the tooth at the operating pitch diameter measured in the plane which is normal to the operating helix angle. Operating pitch diameter

The operating pitch diameter is the diameter on the gear where the helix angle of the gear is the operating helix angle. The operating helix angle is set by the base helix angles of both gears, along with the shaft angle. Operating transverse circular pitch

This is the distance between same handed tooth flanks along the operating pitch diameter. It is equal to pi divided by the operating transverse diametral pitch. Operating transverse diametral pitch

This is the operating transverse diametral pitch of the gear measured at the operating pitch diameter. It is the number of teeth per inch of operating pitch diameter. This value, along with the operating transverse pressure angle, is consistent with the transverse base pitch of the gear. Operating transverse module

This is the transverse module of the gear at the operating pitch diameter. It is millimeters of operating pitch diameter per tooth. Operating transverse pressure angle

This is the transverse pressure angle at the operating pitch diameter. This pressure angle, combined with the operating transverse diametral pitch, must be consistant with the transverse base pitch. The gear will have a different pitch and pressure angle at any diameter larger than the base diameter. The transverse base pitch will remain constant. Operating transverse tooth thickness

This is the circular thickness of the tooth measured along the operating pitch diameter in the transverse plane which is normal to the axis of rotation of the gear.


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Outside diameter

This is the physical outside diameter of the gear blank. In this model it is not produced by generating tools but would be sized (or specified for formed gears) before generating. Overload factor

This factor accounts for any overload imposed on the gears other than the load due to the driver torque. The default value is 1. Overturning moment

This is the moment produced by the axial force on the gear teeth. The bearings must react this moment as an additional radial force on each bearing. The model calculates the value of the overturning moment when contact is at the center of the contact interval. Pointed tooth diameter

This is the diameter at which the teeth would come to a point. The tooth thickness would be reduced to zero. The effect of tip relief is not included and only the projected involute flank is considered. It is, of course, necessary to keep the actual outside diameter far enough below this diameter to obtain a satisfactory tooth top land width. Pressure angle

This is the normal pressure angle at the reference pitch diameter. It is the pressure angle that, combined with the normal nominal diametral pitch, determines the normal base pitch. The gear will have a different pitch and nominal pressure angle at any given diameter. The normal base pitch will remain constant. (The transverse base pitch will also remain constant.) The normal nominal pressure angle will be equal to the operating normal pressure angle if the center distance is “standard”. The flank angle of hobs or rack type shapers need not be equal to the normal nominal pressure angle. If the flank angle is not equal to the normal nominal pressure angle the cutter is “off lead”. Only the normal base pitch of the gear and cutter must match. Pressure Angle of Mod Ramp

This is the flank angle of the modification ramp for a semi-topping or tip relief hob. It is measured from a normal to the reference line.


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Protuberance

If a gear is to be finished after hobbing or shaping by processes such as shaving or grinding a protuberance type hob or shaper cutter is often used. The protuberance is a “wide spot” near the tip of the cutter tooth. This will produce undercut in the gear root fillet area for clearance of the finishing tool. Protuberance angle from flank

When protuberance is provided on a hob the protuberance radius is blended back into the main flank with a straight sided ramp. The protuberance angle from flank is the acute angle of the ramp measured from the main flank. Protuberance pressure angle

When protuberance is provided on a hob, the protuberance radius is blended back into the main flank with a straight sided ramp. The protuberance pressure angle is the pressure angle of the ramp measured from a normal to the hob reference line. Reference Line to Start Mod Ramp

This is the distance from the hob reference line to the beginning of the modification ramp toward the root of the hob tooth. This ramp produces the tooth tip “chamfer” when semi-topping hobs are used and the tip relief when tip relief hobs are used. Reference pitch diameter

This is the reference diameter obtained by dividing the number of teeth by the nominal transverse diametral pitch. All reference data is measured at this diameter unless specified otherwise. Roll angle at effective OD

The roll angle at the effective OD is the angle through which the geometric involute generator has turned in the transverse plane from the base cylinder to the effective outside diameter. In many cases this value is used in setting and reading involute checking equipment. Roll angle at inv/fillet intersection (TIF)

The roll angle at the involute/fillet intersection diameter is the angle through which the geometric involute generator (in the transverse plane) has turned from the base cylinder to the TIF diameter of the gear. In many cases this value is used in setting and reading involute checking equipment.


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Roll angle at SAP

The roll angle at the start of active profile is the angle through which the geometric involute generator has turned in the transverse plane from the base cylinder to the start of active profile diameter. In many cases this value is used in setting and reading involute checking equipment. This angle must, of course, be positive or interference will result; there is no involute profile below the base circle. An often suggested minimum value is about 5 degrees. Below this angle the radius of curvature of the involute is very small, making it difficut to manufacture and inspect the gear. In addition, the sliding velocity between the teeth in the transverse plane becomes quite high. Roll angle at start of tip modification

This is the roll angle at the start of the tooth tip modification. Root clearance

The root clearance is the clearance between the root diameter of a gear and the outside diameter of the mating gear. The value usually used to avoid interference is about 0.25/(Normal Diametral Pitch) or 0.25*(Normal module). It should be noted that interference may still exist between the tips of the mating gear teeth and the curved fillet even though there is root clearance. The best way to check for this condition is to observe the plot of the teeth in various positions at the minimum center distance and maximum root diameters and ODs. Root diameter

This is the diameter at the root of the tooth. This may be a portion of a circle for a flat root tooth or the lowest point on a trochoid for a full fillet tooth. Rotational speed

The rotational speeds of the driver and driven gears are inversely proportional to the number of teeth. In general, the speeds are NOT proportional to the operating pitch diameters or base diameters. Separating force

This is the force in the direction of the line of centers produced by the torque on the gear.


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Separation of pitch planes

Crossed helical gears may be analyzed by considering each gear to be in mesh with the basic normal rack. In the general case, each of the gears will mesh with the rack with a different pitch plane. The separation of pitch planes is the distance between the pitch plane of the driver and driven gears. The pitch planes are perpendicular to the line of centers and, therefore, the separation of the pitch planes is measured along the line of centers. When there is a separation of the pitch planes the path of contact does NOT pass through the line of centers. Only if the gear set is “standard” as to dimensions and shaft angle will the pitch planes coincide. If the center distance is not equal to the sum of the reference pitch radii and/or the shaft angle is not equal to the sum of the nominal helix angles there will be a separation of the pitch planes. Shaft angle

This is the angle between the driver and driven shafts or axes of rotation. It can be measured about the common normal to the shaft axes. The value of the shaft angle is approximately equal to the sum of the nominal helix angles of the gears. (It is exactly equal to the sum of the operating helix angles of the gears.) In this model right hand gears have a positive helix angle and left hand gears have a negative helix angle. Therefore, since the limits on the gear helix angles are 0 (spur gears) and less than 90 degrees, the shaft angle must be between -180 degrees and +180 degrees, excluding 0. (If the shaft angle is 0 then the set is a parallel axis set and this model does not apply.) Shaper cutter normal tip radius

The radius on the tip of the shaper cutter tooth is measured in the normal plane of the cutter. Shaper cutter normal tooth thickness

This is the normal circular arc thickness of the shaper cutter teeth at the cutter reference pitch diameter. Shaper cutter outside diameter

The shaper cutter outside diameter is controlled by the amount of root clearance required.


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Specific compressive stress

The compressive stess between the contacting tooth flanks is calculated at the center of the contact interval. The stress depends on:

1) Load between the teeth (The load is modified by the contact ratio.) 2) The modulus of elasticity of each of the gear materials 3) The radius of curvature of the teeth at the contact point (The curvature at

the center of the contact interval is controlled by the pressure angle of the teeth.) The equations from Roark`s Formulas for Stress and Strain, Table 33, Case 4, page 652. See the References below.

“Standard” center distance

The “standard” center distance is the center distance obtained by adding the standard (or reference) pitch radii of the gears. (The standard pitch diameters are found by dividing the number of teeth in each gear by the nominal transverse pitch.) It is the distance at which the reference pitch diameters are tangent. If a gear set is mounted at the “standard” center distance all reference data and operating data is the same. Most crossed helical gears are designed to operate on or close to “standard” center distance. Small deviations from standard in no way compromise the design and there are, at times, advantages to using non-standard center distance. Non-standard gears may be made with the same tooling as “standard” gears with no increase in cost. Start of active profile

This is the lowest diameter on the involute tooth flank that is contacted by the mating tooth. This diameter must, of course, be above the base diameter or interference will result unless the gear is undercut enough to prevent it. The inspection form diameter is an inspection diameter smaller than the SAP diamater to insure that the true involute form is low enough on the gear. The inspection form diameter is sometimes called the “required TIF (true involute form) diameter”. A rule of thumb is to keep the required TIF below the start of active profile by about 0.025/(Normal Dia Pitch) or 0.025*(Normal module) per side. This, of course, can be done only if the involute/fillet diameter is low enough.


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Starting diameter for tooth tip modifications

This is the diameter on the gear where tip modification starts. The tip modification is produced by a rack type gererating tool or is formed. It is an input value for formed gears. It includes: 1) Tip relief (including “chamfers”) produced by a tip relief or a semi-topping cutter. 2) Tip relief produced by forming. It does not include chamfers or tip radii produced by forming. Tangential force

This is the force in the plane of rotation normal to the line of centers produced by the torque on the gear. Thickness at Reference Line

This is the normal tooth thickness of the rack or hob tooth at the reference line on the rack or hob. The distance from the tip to the reference line is arbitrary with involute tools with straight sided flanks. Any number of distances could be used as long as the correct tooth thickness at this distance is specified. Two different tools cannot be directly compared unless the distance from the tool tip to the reference line is the same for both or has been adjusted for. Tip to flank/protuberance intersection

When protuberance is provided on a hob the protuberance radius is blended back into the main flank with a straight sided ramp. The tip to flank/protuberance intersection is the distance from the hob tip to the intersection of the protuberance ramp and the main flank. Tip radius

This is the radius (in the normal plane) on the tip of the rack or hob tooth. If you have entered a tip to reference line dimension and tooth thickness (along with protuberance for a hob) and you wish a full tip radius on the tool, enter a large value for the tip radius. The model will calculate the largest possible tip radius and display the value.


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Tip to Reference Line

This is the distance from the rack or hob tip to the reference line on the tool. The distance from the tool tip to the reference line is arbitrary with involute tools with straight sided flanks. Any number of distances could be used as long as the correct tooth thickness at this distance is specified. Two different tools cannot be directly compared unless the distance from the tip to the reference line is the same for both or has been adjusted for. Torque

The torque is the twisting effort applied to the gears about their axes of rotation. The difference between the driver and driven torque is determined by the numbers of teeth on the gears and the efficiency of the drive. Total normal finish stock on tooth thickness

This is the difference in normal tooth thickness at the reference pitch diameter between the pre-finish operation and the finished tooth. The value must be zero for a formed gear. Total normal tooth load

This is the load on one tooth contact from the driver torque and the overload modified by the contact ratio. (The product of the driver torque load and overload factor is divided by the minimum number of teeth in contact.) Transverse base pitch

The transverse base pitch is the distance from tooth flank to tooth flank in the transverse plane along a tangent line to the base circle. Transverse circular pitch

This is the nominal transverse circular pitch of the gear. It is the distance between same handed tooth flanks along the reference pitch diameter. Unless the helix angles are the same the driver and driven gears will have different transverse circular pitches. This value, along with the nominal transverse pressure angle, sets the transverse base pitch of the gear.


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Transverse circular tip relief at the outside diameter

This is the amount of tip relief measured at the outside diameter in the transverse plane. (If the tooth has a formed tip chamfer or tip radius the tip relief is measured from the projected involute as if there were no chamfer or radius.) Transverse module

This is the nominal transverse module of the gear. It is millimeters of reference pitch diameter per tooth. Unless the helix angles are the same the driver and driven gears will have different transverse modules. Transverse pressure angle

This is the transverse pressure angle at the reference pitch diameter found by dividing the number of teeth by the transverse pitch. It is also the pressure angle that, combined with the transverse diametral pitch, determines the transverse base pitch. The gear will have a different transverse pitch and pressure angle at any given diameter. The transverse base pitch will remain constant. Unless the helix angles are the same the driver and driven gears will have different transverse pressure angles. Transverse tooth thickness at Ref PD

This is the circular thickness of the tooth measured along the reference pitch diameter in the transverse plane which is normal to the axis of rotation of the gear. Working depth

The working depth is the distance along the line of centers between the overlapping outside diameters of the driver and driven. DO NOT use the working depth to calculate the start of active profile diameters on the gears. These diameters are set by the intersections of the outside diameters with the path of contact (if there is no undercut and the face widths are sufficient). Young`s modulus

This is Young`s modulus of elasticity for the gear material. The default value for the driver is 30 million psi (207 million kPa) for steel. The default value for the driven is 17 million psi (117 million kPa) for aluminum bronze.


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References: 1. Colbourne, J.R., The Geometry of Involute Gears. Springer-Verlag, New York, 1987 2. Buckingham, Earle, Analytical Mechanics of Gears. McGraw-Hill, New

York,1949, republished by Dover, New York, 1962 3. Young, Warren C.: Roark’s Formulas for Stress and Strain, 6th Edition, (Table 33, Case 4, page 652). McGraw-Hill, New York, 1989 4. AGMA 6034-B92, Practice for Enclosed Cylindrical Wormgear Speed Reducers and Gearmotors. Data extracted with the permission of the publisher, the American Gear Manufacturers Association, 1500 King Street, Suite 201, Alexandria, VA 22314 5. AGMA 2001-B88, Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth. Data extracted with the permission of the publisher, the American Gear Manufacturers Association, 1500 King Street, Suite 201, Alexandria, VA 22314

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