Bevel and hypoid gearing - geometry [ISO 23509]

Table of Content:

Bevel and hypoid gearing - geometry [ISO 23509]

The calculation is designed for geometric designs and checks of bevel and hypoid gear with straight, oblique and curved teeth according the ISO 23590
The programme gives solutions to the following tasks:

  1. Preliminary design of the gear size

  2. Detail geometrical design for gear:
    - Straight
    - Oblique
    - Spiral
    - Zerol
    - Hypoid (Gleason, Oerlikon, Klingelnberg)

  3. Automatic design of a transmission with the minimum number of input requirements.

  4. Generation of standard 2D drawings.
  5. Precision 3D model of the bevel gearing.
  6. Precision 3D model of the hypoid gearing.
  7. Support for all CAD systems.

The calculations use procedures, algorithms and data from ISO 23509 and related standards AGMA ISO 23509 : A, ANSI, ISO, DIN, BS and specialized literature.

Hint: The comparative document "Choices of transmission" can be helpful when selecting a suitable transmission type.

User interface

 User interface.





Purchase, Price list

 Purchase, Price list.

Control, structure and syntax of calculations.

Information on the syntax and control of the calculation can be found in the document "Control, structure and syntax of calculations".

Information on the project.

Information on the purpose, use and control of the paragraph "Information on the project" can be found in the document  "Information on the project".

Theoretical foundations.

Estimated pinion / gear size (ISO 23509):

Estimation is for a gear with a shaft angle of 90°. For other axis angles, the estimation must be corrected.

Pinion torque T1

Pinion torque is a convenient criterion for approximate rating of bevel gears, requiring conversion from power to torque by the relation:

T1 = 9550 * P / n1

T1 ... pinion torque [N*m]
P ..... power [kW]
n1 ... pinion speed [/min]

Pinion outer pitch diameter de1

Pinion outer pitch diameter versus pinion torque, transmission ratio, material, gear type, load type...:

Pitting resistance

de1 = (0 + 15 * T1^0.325) / (u^0.35) * KM * SZF * PFG   [mm]

Bending strength

For T1 < 5000 [N*m]
de1 = (0 + 14 * T1^0.35) / (u^0.5) * LTF * SZF   [mm]

For T1 >=5000 [N*m]
de1 = (0 + 14 * T1^0.35) / (u^0.5) * u^0.14 * (T1/5000)^0.1 * LTF * SZF   [mm]

de1 ... Pinion outer pitch diameter [mm]
T1 .... Pinion torque  [N*m]
u ...... Transmission ratio
KM .... Material factor
SZF.... Straight and zerol bevel factor
PFG ... Precision-finished gears
LTF ... Load type factor

Preliminary hypoid pinion pitch diameter deplm1

For hypoid gears:

deplm1 = de1 - a / u

deplm1... Preliminary hypoid pinion pitch diameter [mm]
de1 ........ Pinion outer pitch diameter [mm] (larger value from previous calculation)
a ............ Hypoid offset [mm]
u ........... Transmission ratio


All used geometry formulas can be found in ISO 23509. Below is the geometry nomenclature.

Bevel gear nomenclature - Axial plane

  1. back angle

  2. back cone angle

  3. back cone distance

  4. clearance - c

  5. crown point

  6. crown to back

  7. dedendum angle - Thetaf1, Thetaf2

  8. face angle - deltaa1, deltaa2

  9. face width - b

  10. front angle

  11. mean cone distance - Rm

  12. mean point

  13. mounting distance

  14. outer cone distance - Re

  15. outside diameter - dae1, dae2

  16. pitch angle - delta1, delta2

  17. pitch cone apex

  18. crown to crossing point - txo1, txo2

  19. outer pitch diameter - de1, de2

  20. root angle - deltaf1, deltaf2

  21. shaft angle - Sigma

  22. equivalent pitch radius

  23. mean pitch diameter - dm1, dm2

  24. pinion

  25. wheel

Bevel gear nomenclature — Mean transverse section

  1. whole depth - hm

  2. pitch point

  3. clearance - c

  4. circular thickness

  5. circular pitch

  6. chordal addendum

  7. chordal thickness

  8. backlash

  9. working depth - hmw

  10. addendum - ham

  11. dedendum - hfm

  12. equivalent pitch radius

Hypoid - nomenclature

  1. face apex beyond crossing point - tzF1

  2. root apex beyond crossing point - tzR1

  3. pitch apex beyond crossing point - tz1

  4. crown to crossing point - txo1, txo2

  5. front crown to crossing point - txi1

  6. outside diameter - dae1, dae2

  7. outer pitch diameter - de1, de2

  8. shaft angle - Sigma

  9. root angle - deltaf1, deltaf2

  10. face angle of blank - deltaa1, deltaa2

  11. wheel face width - b2

  12. hypoid offset - a

  13. mounting distance

  14. pitch angle - delta2

  15. outer cone distance - Re

  16. pinion face width - b1

Bevel gear tooth tapers

  1. depth

  2. slot width

  3. thickness

  4. space width

Root line tilt

  1. pitch cone apex

Bevel gear depthwise tapers

A. Standard depth taper
B. Constant and modified slot width
C. Uniform depth

  1. mean whole depth

  2. mean addendum

  3. mean dedendum

Machining Principle.

Two basic principles of machining are primarily used for the calculation and generation of the 3D models of the gear wheels.

A. For face hobbed gears

The curvature is an extended epicycloid (locus), which is created by rolling the tool on the basic circle of the epicycloid.

B. For face milling gears

The curvature radius is equal to the radius of the tool.


Input values for selected methode




Method 0

Method 1

Method 2

Method 3



Shaft angle







Hypoid offset





z1, z2

z1, z2

Number of teeth







Mean pitch diameter of wheel







Outer pitch diameter of wheel







Wheel face width







Mean spiral angle of pinion







Mean spiral angle of wheel







Cutter radius







Number of blade groups
(only face hobbing)






Process of calculation.

This calculation is intended primarily for geometry design. The calculation or a gear size design, respectively, is only indicative and needs to be checked with respect to the relevant standard or the documentation supplied by the manufacturer of the specific machining centre.

Gear dimension design:

  1. Enter the power and operating parameters of transmission (transmitted power, rotation speed, type of gears, material, type of load, ...). [1]

  2. Enter the gear ratio or the number of teeth, respectively; the angle of shaft axes angle; the shaft offset; and the toothing width and angle [2]

  3. Specify the tooth profile parameters. [3]

  4. Based on the type of toothing, select the relevant method of calculation 0, 1, 2, or 3. [4,5,6,7]

  5. Using the relevant method, enter the input geometric parameter or run iteration of the calculation.

  6. Check the results. Use the graphic representation in paragraph [9.0].

  7. If necessary, load the 2D drawing into the 2D CAD system, or generate data for creation of a precision 3D model.

Options of basic input parameters. [1]

Enter basic input parameters of the designed gearing in this paragraph.

1.1 Calculation units.

Use the drop-down menu to choose the required system of units for calculation. After the units are switched, all values will immediately be re-calculated automatically.

1.2 Transferred power.

Enter the power to the driven gear. Usual values are in the range 2 - 500 kW / 3-700 HP, in extreme cases up to 4000 kW / 6000 HP.

1.3 Speed (Pinion / Gear).

Enter the speed of the driven gear. Extreme speed can reach 50 000 rpm. The speed of the driven gear is calculated using the number of teeth of both gears.

1.4 Torsional moment (Pinion / Gear).

This is the result of the calculation and cannot be entered.

1.5 Gearing type

This calculation allows users to solve a choice of gearing types. Select the type of toothing from the list. By making this selection, you also select a range of coefficients which affect the estimate of the gear assembly size and the coefficients for geometry calculations.

  1. Standard straight toothing - Methode 0

  2. Standard oblique toothing - Methode 0

  3. Spiral bevel gears (non hypoids) - Methode 0

  4. Zerol bevels - Methode 0

  5. Hypoid spiral bevel gears (Gleason) - Methode 1

  6. Hypoid spiral bevel gears (Oerlikon) - Methode 2

  7. Hypoid spiral bevel gears (Klingelnberg) - Methode 3

Note: To calculate the geometry, always use the relevant paragraph marked as Method 0, 1, 2 or 3.

1.7 Material of the pinion / gear

For materials other than case-hardened steel at 55 minimum HRC, the pinion outer pitch diameter, as given in Figure B.2 ISO 23509, is to be multiplied by the material factor KM

1.9 Precision-finished gears.

"Precision finish", refers to a machine finishing operation which includes grinding, skiving and hard cut finishing.

1.10 Load type.

Statically loaded gears should be designed for bending strength rather than pitting resistance For statically loaded gears which are subject to vibration, the pinion outer pitch diameter, as given in Figure B.2 ISO 23509, is to be multiplied by 0,70. For statically loaded gears which are not subject to vibration, the pinion outer pitch diameter, as given in Figure B.2, B.3 ISO 23509, is to be multiplied by 0,60.

1.12 Accuracy grade - ISO1328

It only affects the design of the backlash.

Preliminary / Approximate design of geometrical parameters. [2]

In this paragraph, select the gear ratio, numbers of teeth, angle of axes, offset of axes, etc. Then you obtain a preliminary design of the size and shape of toothing.

- The entered number of teeth and the angle of axes are used for all the methods of the geometry calculations.
- The diameters, the axes offsets, the width of toothing, and the angle of toothing are then offered in a specific paragraph (Methods 0 to 3) as the recommended values (minimum and maximum).

2.1 Transmission ratio / from table

The optimum transmission ratio varies in the range 1-5. In extreme cases this ratio can reach up to 10. The transmission ratio can be entered in the left input field using the keyboard. The right pop-up list contains recommended values of the transmission ratio and when selecting a value from this list, the chosen value is added to the field on the left automatically.

2.2 Recommended (minimum) number of teeth pinion / wheel

The recommended (and minimum) number of pinion teeth is based on the ISO 23590 standard. The recommended number of the gear teeth is then calculated based on the gear ratio entered on the previous line.

2.3 Number of teeth pinion / wheel

Enter the number of pinion teeth. The number of gear teeth is calculated based on the required gear ratio [2.1]. After checking the checkbox on the right, it is possible to enter the number of the gear teeth directly.

2.4 Actual transmission ratio / deviation.

As the actual transmission ratio is the ratio of the number of teeth of both gears (integers), the actual transmission ratio will be mostly different from the desired (entered) one. The value of the "Actual transmission ratio" is displayed on the left; the percentage deviation from the desired transmission ratio is displayed on the right. This deviation for the transmission ratio should be in the range:
i = 1 to 4.5 ........... 2.5%
i is greater than 4.5... 4.0%

Hint: In case you need to design gearing with a transmission ratio as accurate as possible or need to distribute the transmission ratio among more gears in the gearbox, use "Calculation of the transmission ratio".

2.5 Angle of shaft axes.

Enter the angle of axes of individual wheels (mostly 90°). The calculation also enables options of other values. The case when the angle of the pitch cone exceeds 90° is indicated by a red cell. (This creates conical gearing that cannot be produced on usual machines).

2.6 Hypoid offset / max. value (25% of de2)

In most cases, the hypoid offset is determined by the application. Pinion offset is designated as being positive or negative.

Picture illustrates positive and negative pinion offsets as seen from the wheel apex.

A. Left-hand pinion mated with right-hand wheel
B. Right-hand pinion mated with left-hand wheel

It is recommended that positive pinion offset be used because of the increasing diameter of the pinion, higher face contact ratio, pitting and bending load capacity. ln general, due to lengthwise sliding, the offset should not exceed 25 % of the wheel outer pitch diameter and, for heavy-duty applications, it should be limited to 12,5 % of the wheel pitch diameter.

2.7 Pitch diameter (outer)

Based on power and geometry inputs, the outer pitch diameter is estimated.

2.8 Outer transverse module / Outer transverse DP

Since tooling for bevel gears is not standardized according to module, it is not necessary that the module be an integer. However, it is possible in calculation to calculate the wheel size according to the required module.

2.11 Face width / max. recommended value

After clearing the button the value of the toothing width can be entered. Ticking the button automatically selects the maximum value.
Generally, the face width is 30% of the cone distance Re2 or 10*met, whichever is less. However, design parameters may require values to be larger or smaller. For zerol bevel gears, the face width should be multiplied by 0.83 and should not exceed 25 % of the Re2.
For shaft angles less than 90° the face width larger than given may be used. For shaft angles greater than 90° the face width smaller than given should be used.
The hypoid pinion face width is generally greater than the face width of the wheel.

2.12 Mean spiral angle

Common design practice suggests that the spiral angle be selected to give a face contact ratio of approximately 2.0. For high-speed applications and maximum smoothness and quietness, face contact ratios greater than 2.0 are suggested, but face contact ratios less than 2.0 are allowed.

For hypoid sets, the pinion spiral angle may be calculated using Formula:

betam1 = 25 + (z2 / z1)^0.2 + 90 * a / de2

z1, z2 .... Number of teeth
a ........... Hypoid offset [mm]
de2 ....... Gear outer pitch diameter [mm]

Initial data for tooth profile parameters (used in all calculation methodes) [3]

Enter basic input parameters for tooth profile in this paragraph.

3.1 Nominal design pressure angle drive side / coast side

There are three normal pressure angles that are to be considered.
- Nominal design pressure angle alfad, is the start value for the calculation. It may be half of the sum of pressure angles or different on drive and coast side.
- Generated pressure angle alfan, is the pressure angle of the generating gear; alfan can be found on the tooth flank in the mean normal section.
- Effective pressure angle, alfae, is a calculated value.
The most commonly used design pressure angle for bevel gears is 20°. This pressure angle affects the gear design in a number of ways. Lower generated pressure angles increase the transverse contact ratio, reduce the axial and separating forces and increase the toplands and slot widths. The converse is true for higher pressure angles. Based on the requirements of the application, the engineer may decide to choose higher or lower design pressure angles. Lower effective pressure angles increase the risk of undercut.
For hypoid gears, it could be reasonable to have unequal generated pressure angles on the coast and drive sides, in order to balance the mesh conditions. If full balance of the mesh conditions is recommended, the influence factor of limit pressure angle, falfalim, is set to "1". Then the limit pressure angle, alfalim, is added to the design pressure angle alfad, on the drive side and subtracted on the coast side in order to obtain the generated normal pressure angle, alfan.
As for bevel (non-hypoid] gears, the limit pressure always is equal to zero, the nominal design pressure angles have the same values as the generated pressure angles. If the effective pressure angles have the same values, the mesh conditions on coast and drive side are equal.
Straight bevels
To avoid undercut, use a nominal design pressure angle of 20° or higher for pinions with 14 to 16 teeth and 25° for pinions with 12 or 13 teeth.
Zerol bevels
On zero bevels 22.5° and 25° nominal design pressure angles are used for low tooth numbers, high ratios, or both, to prevent undercut. Use a 22.5° nominal design pressure angle for pinions with 14 to 16 teeth and a 25° nominal design pressure angle for pinions with 13 teeth.
Spiral bevels
To avoid undercut a 20° design pressure angle or higher for pinions with 12 teeth or fewer teeth may be used.
To balance the mesh conditions on coast and drive side, the influence factor of limit pressure angle should be falfalim = “1”. For the use of standard cutting tools, the value of falfalim may be different from "1". The nominal design pressure angles 18° or 20° may be used for light-duty drives; higher pressure angles such as 22.5° and 25° for heavy-duty drives.

3.4 Data type selection (I or II)

Two alternative sets of input data, those used by American Gear Manufacturers Association AGMA - (data type ll) and those used in European Standards - (data type l) be available to calculate one or more of the four design methods shown.

After selecting the type, the related values will also be calculated for the second set of input data and displayed in the right column. The data in the right column are then used for the calculation. In the green column, the recommended values resulting from the previous data (toothing type, gear ratio, number of teeth) are shown.

3.5 Profile shift coefficient

Recommended values for gearings with the angle of axes 90°:
Gear ratio / xhm1

3.6 Basic crown gear addendum factor

In common cases, the addendum factor khap is set to khap = 1.00 and the dedendum factor khfp is set to khfp = 1.25.

3.7 Basic crown gear deddendum factor

In common cases, the addendum factor khap is set to khap = 1.00 and the dedendum factor khfp is set to khfp = 1.25.

3.8 Thickness modification coefficient (theoretical)

Adjust the value of the unit change of the tooth thickness here. Recommended values for gearings with the angle of axes 90°:
Gear ratio / xsmn

3.9 Mean addendum factor of wheel

This factor apportions the working depth between the pinion and wheel addendums. The pinion addendum is usually longer than the wheel addendum, except when the numbers of teeth are equal. Longer addendums are used on the pinion to avoid undercut. Suggested values for shaft angles Sum = 90° for cham are on the right. Other values based on sliding velocity, topland or point width limits, or matching strength between two members, can be used. Clause8 from ISO 23509 gives the limits for the mean addendum factor to prevent undercut on pinion and wheel.

3.10 Depth factor

Normally, a depth factor kd = 2.000 is used to calculate mean working depth hmw, but it can be varied to suit design and other requirements. The value on right gives the suggested depth factors based on pinion tooth numbers.

3.11 Clearence factor

While the clearance is constant along the entire length of the tooth, the calculation is made at mean point. Normally, the value of 0.125 is used for the clearance factor kc, but it can be varied to suit the design and other requirements.

3.12 Circular thickness factor

The mean normal circular thickness is calculated at the mean point. Values of kt based on balanced bending stress are on the right. Other values of kt may be used if a different strength balance is desired.

Results section [4,5,6,7]

The following paragraphs describe the above-mentioned methods of calculation corresponding to the selected type of toothing [1.5]. Expand the corresponding method and specify the input parameters of toothing. The recommended values based on the draft design (see par. 1.0, 2.0, and 3.0) are shown on the right side. When using the methods 1, 2 and 3, it is usually required to perform an iteration to finish the calculation of the geometric results.

Using the '<<' button located at the beginning of the paragraph on the right [4.1, 5.1, 6.1, 7.1], fill in the values from the preliminary draft (paragraphs [1.0], [2.0], [3.0]).

4.1, 5.1, 7.1 Outer pitch diameter (wheel)

Choose the value de2 (dm2). It should be larger than the value on the right based on the preliminary design.

6.1 Mean pitch diameter (wheel)

Choose the value de2 (dm2). It should be larger than the value on the right based on the preliminary design.

4.2, 5.2, 6.2, 7.2 Face width (gear)

Choose the value b2. It should be smaller than the value on the right, which depends on the input parameters specified in this paragraph.

4.3, 5.3, 6.3, 7.3 Mean spiral angle

Choose the value betam2 (betam1). It should be greater than the value on the right side shown in the brackets for achievement of the mesh coefficient eb =2.0, paragraph [2.12]. The permitted range is given in the angle brackets '<...>'.

5.4, 6.4, 7.4 Hypoid offset / max. value (25% of de2)

Select the value a. It should be in the range of values on the right side. It it dependent on the entered input parameters in this paragraph. See paragraph [2.6]

4.4, 5.5, 6.5, 7.5 Outer normal backlash, Outer transverse backlash

Suggested minimum values of the outer backlash are on the right. It will be noted that the backlash allowance is proportional to the module. Two ranges of values are given: one for ISO accuracy grades 4 to 7, the other for ISO accuracy grades 8 to 12, according to ISO 1328-1. See [1.12]. The brackets contain the recommended mean value.

4.5, 5.6, 6.6, 7.6 Cutter radius

Since bevel gear design and manufacture are functions of the cutter radius and, for face hobbed gears, also the number of blade groups, There is a list of standard cutters. The recommended range is given in the angle brackets '<...>'.

Nominal cutter radii rc0 and blade groups z0

4.6, 5.8, 6.8, 7.8 Depthwise taper, (adendum and dedendum angles)

Select the tooth profile that corresponds to the type of toothing and your requirements. After making your selection, the angles Thetaa2 and Thetaf2 will be proposed accordingly, which will be used in the calculation. Upon checking the checkbox, custom values may be entered.

4.23, 5.48, 6.53, 7.38 Mean normal module, Outer transverse module

If you need to achieve the exact mmn (met) module, enter the required value and push the "<” button. The de2 (dm2) value will be calculated. In most cases, it will be necessary to run the iteration again and repeat the calculation (Methods 1, 2, and 3).

Force conditions (forces acting on the toothing). [8]

In loaded gearing there appear some forces that are transferred to the machine structure. Knowledge of these forces is fundamental for correct dimensioning of the equipment. The orientation of the forces can be seen in the following illustration. Row [8.4, 8.6] shows the amounts of the forces if the sense of rotation of the gearing agrees with the directions in the illustration, row [8.5, 8.7] shows the amounts of forces if the directions are opposite. In case the amount of the force is negative, it acts in the opposite direction than as illustrated. Illustration A shows the left direction of the pitch, illustration B the right direction.

8.1 Selection of source for forces calculation

To calculate the forces, use the same method as was used for the geometry solution. The geometrical values obtained from the method selected including the input data (see par. 1 - 3) shall be used for the calculation of forces.

8.2 Direction of the teeth pitch (pinion)

According to the direction of the teeth pitch, right and left wheels are distinguished. The teeth of wheels in the mesh must have opposite directions of curving. The gearing as a unit is characterized by the direction of curving of the pinion teeth.

In case of gearing with oblique and curved teeth, a rotary movement mostly in one direction is desired. The direction of teeth curving is then chosen to push the wheels out of the mesh by axial forces (the teeth enter the mesh by their thicker ends on the outer surface of the wheels).

The illustration shows the direction of the pinion teeth pitch:
A - Left
B - Right

Graphic output, CAD systems.

Information on options of 2D and 3D graphic outputs and information on cooperation with 2D and 3D CAD systems can be found in the document "Graphic output, CAD systems".

In addition to the standard 2D output drawing, this calculation also includes ready background data for the creation of a precision 3D model of the bevel, or hypoid gearing suitable for manufacturing, printing, or testing.

Note: The 3D model is created as a simulation of the basic machining process. They do not contain any optimizations. Therefore, for gearing that requires optimization (stress, noise, surface treatment, etc.), choose the appropriate software or a supplier that can optimize the calculated parameters according to ISO 23509 accordingly to your requirements.

9.0 Graphic output, CAD systems.

1. In the "Output of a 2D drawing into" list, choose the target CAD system (target application) to which the picture should be generated, or a "DXF File" to convert the drawing into a .DXF file.

2. In the "Scale of 2D drawing" list, set up the drawing scale. The drawing is always created in the scale 1:1. The scale allows you to set only certain parameters of the drawing, such as the size of the text or overlapping of the axes.

3. If necessary, set up another control elements as well. Most calculations also include other setting options, which depend on the calculation and type of the plotted object. Explanation of these supplementary options can be found in the help for the respective calculation.

4. Start plotting using the button with the icon of the desired drawing.

Hint: In most cases, it is quite sufficient to choose the "Automatic" scale, which is set up with regards to the size of the plotted objects.
Note1: The CAD system (target application) must be started before generating the drawing. If it is not started or if an error appears in communication between the calculation and the target program, it is possible to save the drawing as a file in .DXF format.
Note2: If you use the local language setting of your keyboard, use the same keyboard setting in the calculation and in the target program as well (for trouble-free communication using the "SendKeys" command).

9.4 Definition of the gear wheel dimensions.

The functional dimensions are based on the calculation. However, in this section you can define the wheel dimensions in a manner that allows them to also meet other requirements. This mainly concerns the assembly dimensions (mdi, mde). After ticking the check box in [9.5], you can enter the preferred gap sizes a, b. Upon changing these values, the parameters of the assembly distance also change (mdi, mde), as do the diameters (dmdi, dmde). On the right side of the calculated values [9.7–9.10], it is possible to enter the required dimension value, and upon pressing the button '<=1' or '<=2' the gap values a, b are adjusted for the pinion or gear wheel such that the mdi, mde, dmdi, and dmde values correspond to the required value.

Caution: The calculation does not check the limit values; nonsensical input data will give nonsensical results.

The mutual relationship of the entered values is clear from the figure.

Note: The settings of these dimensions affect not only the 2D drawing but also the generated basis for the 3D model.
Note: The dimensions tzF, tz, and tzR in the calculation are geometrical parameters.

9.11 Text description (Information for BOM).

Locate the text description in the 2D drawing by pressing the button “Draw”. The text can be edited after the tick off box has been activated.
If it is supported by the respective module for entering models into a 3D CAD system, the contents of individual rows is entered into user attributes of the model and these can be used when generating a BOM. (Details can be found in Help for connection to the respective 3D CAD system.)

9.18 Table of parameters.

A series of calculations (gearing, springs, etc.) enables entering of the respective table with text information on the calculated object into the drawing. The table can be selected from the respective list (in case the calculation enables more types to be entered). Drawing of the table can be activated by pressing the button “Draw a table”.

9.19 Generation of the basis for the 3D models.

Defining a 3D model is not a completely trivial issue, and defining the parametric model for an arbitrary CAD system is complicated. For this reason, we have prepared an option to generate the background data for the creation of a precision non-parametric model, in principle for an any 3D CAD system.

The creation of a precision model thus consists of three simple operations.

  1. Creating the wheel without gearing (simple rotation of the wheel profile).

  2. Downloading the profiles of the gaps between the teeth at defined intervals, their interconnection and deduction of the created volume from the created wheel.

  3. Generating a rotary field of gaps and thereby creating of a precision gear wheel.

General procedure:

  1. Set the output parameters [9.20–9.31] of the generated dimensions suitable for your 3D CAD system (examples and details in the help file and the examples).

  2. In point [9.33] set the rotation (mirroring) of the wheel profile chosen in the figure so that upon insertion into your selected surface in your CAD system, the profile is rotated around the positive z-half-axis.

  3. Generate the profile of the selected wheel (First button).

  4. In the CAD system, insert the profile into the selected plane and create the wheel semi-product by rotating around the z-axis.

  5. Generate the cross sections of the tooth gaps (Second button).

  6. Gradually insert the generated cross sections into the model.
    For wheels with spur gearing, two cross sections are sufficient for 1–9.
    For wheels with bevels or curved teeth or for hypoid wheels, it is usually sufficient to use cross sections 1-3-5-7-9.
    For higher precision, it is possible to use all 9 cross sections.

  7. Interlink the entered cross sections with the LOFT operation and deduct the created element from the wheel profile.

  8. Create rotary fields around the z-axis from the gap between the teeth (number of copies = number of teeth).

  9. Insert the wheel into the system, and use the dimensions a, mdi,or mde from [9.7 or 9.8] to locate it correctly.

  10. Some CAD systems may also require the profile paths (Third Button) to correctly link the profiles.

Caution: The settings and procedures for selected CAD systems are given in the help file.

Settings for some CAD systems, remarks.


Autodesk Inventor

B: 1.1-1.2; C: 15-25; D: Yes; E: 5; F: No; G: XLSX; [9.22] Wheel profile rotation 270, Mirror No.


Wheel creation: 2D Sketch - XZ plane, import points from xlsx, connect points by line, rotate profile around z-axis
Tooth creation: 3D Sketch - Import profile points from xlsx, connect points by curve, close the open profile manually by line, repeat for profiles 1-9, join profiles by the LOFT operation (subtract from wheel).



B: 1.1-1.2; C: 15-25; D: Yes; E: 5; F: Yes; G: DXF for wheel profile, G: SLDCRV for profiles and control curves; [9.22] Wheel profile rotation 90, Mirror Yes


Wheel creation: 2D Sketch - XZ plane, import dxf (select units correctly when importing), connect points by line, rotate profile around z-axis
Tooth creation: 3D Sketch
a) Import profile points from SLDCRV, connect points by curve, repeat for sections 1-9.
b) Control curve import (Profile Path) from SLDCRV. Select one or more control curves.
c) Join profiles by the LOFT operation (subtract from wheel). Select profiles and control curve(s).


B: 1.1-1.2; C: 15-25; D: Yes; E: 5; F: Yes; G: DXF for wheel profile, G: XLSX for profiles and control curves; [9.22] Wheel Profile Rotation 270, Mirror No.


Wheel creation: Open DXF from file xx_Pinion-cone.dxf (Options ...-> Import 2D geometry, Part Sketch, Right (YZ)), rotate profile around z-axis
Tooth creation:
a) Surfacing => Curve by table, importing profiles from XLSX (Parameters: Curve Fit = Smoothing Off, Curve End Conditions = Closed, Natural), repeat for sections 1-9.
b) Home => Lofted Cutout



B: 1.1-1.2; C: 15-25; D: Yes; E: 5; F: Yes; G: IBL [9.22] Wheel Profile Rotation 270, Mirror No.


Wheel creation: load a wheel profile from a file xx_Pinion-cone.ibl, rotation around the z axis
Tooth creation:
a) Create a blend element from the xx_All_InOne.ibl file to create a single dental space.
b) Creating a rotary field by multiplying the blend element.


To generate curves, it is necessary to use a third-party tool capable of reading the points and connecting them into a curve. Catia does not allow direct reading of the curve points.
It is naturally possible to manually copy the individual cross-sections from the generated files to the definition tables of the curves.

Note: For other CAD systems, the gear wheel creation procedure is analogous.

9.20 Generation / Direction of the teeth pitch (pinion).

Choose whether the data should be generated for the pinion or gear wheel. The selection of the type of orientation of the gearing applies to the pinion; the wheel is oriented in the opposite direction.

9.21 Pre-fixes of the names of saved files / Saving remarks.

It is possible to set the value at 01–10. When saving the files necessary for the creation of the 3D model, a name is offered by default, for instance, '01_Pinion'. This is suitable, for instance, for the creation of various versions of the model.

If the save remark switch setting is 'Yes', a text file containing the complete input parameters of the gearing is generated together with the profiles.

Caution: The program overwrites any existing files with identical names without warning.

9.22 Rotation / mirroring of the gear wheel profile in the coordinate system.

Set the rotation (mirroring) of the wheel profile (First button) to be rotated around the z-axis. By rotating, it is possible to correctly position the wheel profile in the coordinate system of your CAD system. A graphic representation is in figure [9.34].

Values for some CAD systems:

Inventor - 270
Solidworks - 90
Solidedge - 270
Creo - 270

9.23 Root fillet radius of tooth rf (pinion / wheel).

The radius is given in multiples of the module. For hypoid gearing and gearing with larger angle of contact alpha, it is suitable to start with a value of 0.1 and increase this value according to requirements after checking the model. DIN uses the values 0.25 to 0.38, ANSI uses the value 0.3. In principle, it is possible to select an arbitrary value in the approximate range of 0.02 to 0.4.

9.24 Coefficient of extension of the profile path.

This states the increase of the gap between the profiles, which forms the volume deducted from the wheel base (the face width must be exceeded). In normal cases, 1.1 is suitable; for a non-standard profile it is possible to increase this to 2.0.

Caution: Use only the extension of the profile path that is absolutely necessary. For high values, it is possible to get outside the definition range of the path.

9.25 Number of points of tooth flank.

It states how many points will be generated to define the tooth side curve. Normally, the suitable value is 20. For special cases (limitations of the CAD system, precision, etc.) it is possible to use values 12 to 200.

Caution: The calculation internally adds several points in such a manner that the created involute exceeds the outer diameter.

9.26 Add a profile superstructure.

Above the generated profile of the gap between the teeth, another piece is added, which can be suitable for better selection of the curves, or whose presence is necessary to correctly interpret the curves.

9.27 The number of curvature angle points of the superstructure.

It states the number of points to be used for the curvature of the connection and end of the superstructure. In most cases, five is sufficient.

9.28 Close the generated profile.

Some CAD systems require a closed profile, while some require a profile that is manually opened or closed. The switch connects/disconnects the first and last points of the profile.

9.29 Output of curves generated to the format.

Various CAD systems support importing points that form the curves in various formats. The following formats are available:

9.30 Marking of the points on the tooth curve (only dxf).

It inserts a small circle into each point of the curve. Suitable in a case where the CAD system is capable of downloading only the DXF format and it is necessary to create a profile manually, point by point.

9.31 Generating the number of cross sections.

The program is capable of generating individual cross sections that define the intermediate tooth gap (see above). The number of the cross sections is automatically added to the file name. In most cases, it is best to generate all 9 cross sections in a single step by selecting 'All'. The cross sections are generated and numbered from the outer diameter.

The selection 'All in one' allows the generation of all cross sections in one file of format:

*.XLSX (suitable for Catia, etc.)



- first shape x,y,z



- second shape x,y,z





*.IBL (for Creo)



Begin section

Begin curve

- first shape x,y,z

Begin section

Begin curve

- second shape x,y,z


9.32 Generate number profile path.

Some CAD systems require the directrix (directrices) in addition to the connected profiles. Enter the sequence number of the curve and press the button.


The numbering of the directrices is shown in the figure and passes through the corresponding point of the profiles.

If you enter 0, a curve is generated that lies on the surface of the pitch cone and runs through the centre of each profile.

9.33 Summary list of parameters.

The parameters of the gearing that fully define the geometry and important dimensions are entered in short form in the line.

9.34 Graphs and figures.

A. Top left.

The graph in the top left shows how the wheel profile will be generated. It is possible to turn and mirror it in such a manner that it suits the requirement (after input to the CAD system) that the wheel profile must be rotated according to the positive z-half-axis. The individual profiles of the inter-tooth gaps are defined in relation to this half-axis. Settings can be changed in the selection lists in the graph.

B. Bottom left.

The figure shows the sequence of steps necessary to generate the wheel.

C. Right side.

The graph shows the geometry of contact curves of the teeth in the pitch plane.

The wheel and the gear tooth curve path are shown in blue.
The pinion and the pinion tooth curve path are shown in green.
Black is used to indicate the machine tool, rolling circle.
Red is used to indicate the creative path of the tool that is shared by the pinion and wheel.

The button '=0' is used to set the gearing to the zero position. The buttons '+' and '-' are used to turn the wheel and pinion in the gear ratio.

For offset of the axes a=0 (method 0), the pinion and wheel merge into a green circle.

Setting calculations, change the language.

Information on setting of calculation parameters and setting of the language can be found in the document "Setting calculations, change the language".

Workbook modifications (calculation).

General information on how to modify and extend calculation workbooks is mentioned in the document "Workbook (calculation) modifications".

List of Standards, literature list:

ISO 23509: 2016

- Bevel and hypoid gear geometry

- Géométrie des engrenages coniques et hypoides

- Geometrie kuželového a hypoidního ozubení

ISO 23509:2016 specifies the geometry of bevel gears. The term bevel gears is used to mean straight, spiral, zerol bevel and hypoid gear designs. If the text pertains to one or more, but not all, of these, the specific forms are identified. ISO 23509:2016 is intended for use by an experienced gear designer capable of selecting reasonable values for the factors based on his knowledge and background. It is not intended for use by the engineering public at large.

AGMA ISO 23509 : A

American Gear Manufacturers Association

Equivalent Standard(s) Relationship - ISO 23509 : 2016 Identical

ISO 10300-1 : 2014


ISO 10300-3 : 2014


ISO 10300-2 : 2014