Bevel gearing with straight, oblique and curved teeth.

Table of Content:

Bevel gearing with straight, oblique and curved teeth.

The calculation is designed for geometric and strength designs and checks of tapered toothing with straight, oblique and curved teeth. The programme gives solutions to the following tasks:

  1. Calculation of helical and straight toothing.
  2. Automatic design of a transmission with the minimum number of input requirements.
  3. Design for entered coefficients of safety (static, dynamic).
  4. Calculation of complete geometric parameters (including corrected toothing).
  5. Calculation of strength parameters, safety check.
  6. Supplementary calculations (calculation of parameters of the existing gear, temperature rise, design of shafts)
  7. Support of 2D and 3D CAD systems.

The calculations use procedures, algorithms and data from standards ANSI, ISO, DIN, BS and specialized literature.

List of standards: DIN 3971, DIN 3991 Kegelradern 1-4, ISO 6336 1-3, DIN 3965 Toleranzen für Kegelradverzahnungen 1-4, ISO 1328, DIN 3990, ANSI B6.1-1968, AGMA 2001-C95, AGMA 908-B89/95, AGMA 2003-A86/88, AGMA 2005-B88 and others.

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.

Rolling gears with tapered wheels can be used for the formation of kinematic and force bound between concurrent shafts (mostly at angles of axes Σ=90°). According to the course of the teeth wheels with straight, oblique and curved teeth are distinguished. Compared with cylindrical wheels, tapered wheel are more demanding in view of production and installation. Production of such toothing needs specialised tools and machines and it is more difficult to achieve the desired level of accuracy. In case of on-the-fly-seating the danger of deformation is increased; this also means worse mesh conditions (above all, for wheels with straight teeth). Wheels with oblique and curved teeth are used for higher speeds, higher loads and higher gear ratios (up to i=10).

Geometry of tapered wheels.

Geometry of a gearing consists of a pair of truncated cones (foot and head cones) and a pitch cone between them.

Division by positions of the foot and head cones

Type I – Head and foot conical surfaces have a common top.
Type II – The foot cone top is shifted to create a constant width of the teeth gap.
Type III – Constant height of teeth. Surfaces of all cones are parallel.

Division by shapes of guiding curves of teeth

A – Straight teeth, B – Oblique teeth, C – Circular teeth, D – Circular teeth ("Zerol"),
E – Paloid teeth, F – Eloid teeth

Table of division of bevel gearing

Guiding straight line

Name Height of the tooth Dimensions, notes

Radial straight line

Straight toothing variable met-standardised, a=20°, 15°, 14.5°, 17.5°, b=bm=0°
Gearings with low requirements, higher noise level, lower circumference speeds v=2-3 m/sec (6-10ft/sec).

Oblique straight line

Oblique toothing variable met-standardised, a=20°, 15°, 14.5°, 17.5°, b=bm=20°-40° (po 5°)

Higher circumference speeds, lower noise level, higher loading, longer life, lower sensitivity to inaccuracies and deformations, higher gear ratio can be used (<10)


Circular arch

variable; heat, pitch and foot cones do not have any common top mmn-standardised, amn=20°, 17.5°, 14.5°, bm=30°-45°
mostly 35°)
mmn-standardised, amn=20°, 17.5°, 14.5°, bm=0°
constant mmn-standardised, amn=20°, 17.5°, 14.5°, bm=25°-45°
4. Evolvent
Paloid toothing
constant mmn-standardised, amn=20°, 17.5°,  bm=30°-38°
Eloid toothing
constant mmn-standardised, amn=17.5°,  bm=30°-50°
Cyclopaloid toothing
Klingelnberg (Ger.)
constant mmn-standardised, amn=20°, 17.5°,  bm=0°-45°



Process of calculation.

Geared transmissions can be divided into:

Power gearing - In case of gearing designed, above all, for a power transmission and transformation, it is necessary to perform a strength design/check (for example, drives of machines, industrial gearboxes, etc.).
Non-power gearing - In case of gearing with minimum transferred torsional moment with respect to the size of the gears, it is not necessary to perform any strength design/check (for example, instruments, regulation devices, etc.).

Note: This calculation is designed for designs of toothing with straight and oblique teeth. It can also be used for orientation in case of wheels with curved teeth. For exact calculation of wheels with curved teeth it is necessary to use the calculation instructions (software) that are supplied by the producer of the respective machining equipment.

Design of power gearing.

The design of the gearing cannot be solved directly and allows considerable freedom in options of diameter and width parameters of toothed wheels. Therefore it is necessary to proceed iteratively and make the design more and more exact and tune the monitored parameters.

Fast (orientation) design:

This procedure gives a quick view of the parameters of the designed gearing. Although gearing designed in this way can be used, step-by-step optimising of several parameters can substantially improve the properties of the designed gearing. Proceed in the design as follows:

  1. Enter power parameters of the gearing (transferred power, speed, desired gear ratio). [1]
  2. Select material of the pinion and wheel, loading mode, operational and production parameters and safety coefficients. [2]
  3. Use the button for "Automatic design" (select oblique or straight teeth). [2]
  4. Check the results.

Optimising parameters:

Before optimising parameters, first execute the "Fast (orientation) design", as described above, and then proceed as follows:

  1. Select the type of toothing and parameters of the tooth profile. [3]
  2. Set up parameters of wheels (angle of axes, number of teeth, angle of meshing, slope of teeth). [4.1,4.2,4.3, 4.4]
  3. Using the slider [4.4] adjust the ratio between the width of the pinion and its diameter, then press the button " Navrhnout ozubení - Design the gearing".
  4. Check dimensions of the designed gearing in the schematic illustration. In case the dimensions do not meet your requirements, adjust the ratio width/diameter of the pinion and recalculate the gearing [4.4].
  5. Parameters of the gearing can be further improved in par. [5] using changes of corrections.
  6. Check and assess (compare with Help) the dimensional and qualitative indexes [6; 7; 8].
  7. Check safety coefficients [9, 10].
Hint: Dimensions of the gearing can be changed significantly using a change of material (or its surface treatment).

Design of non-power gearing.

When designing non-power gearing, it is not necessary to solve and check any strength parameters. Directly choose, therefore, a suitable number of teeth and the module [4.1, 4.7] and check dimensions of the designed gearing.

Hint: When designing non-power gearing, choose a suitably low transferred power.

Options of basic input parameters. [1]

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

1.1 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.

1.2 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.

Hint In case you need to calculate the transmission ratio and know input and output speeds, press the button to the right of the input field and perform the respective calculation in the chapter (section) of supplements.

1.3 Torsional moment (Pinion / Gear).

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

Hint: In case you need to obtain the transferred power from the torsional moment and speed, press the button on the right and perform the respective calculation in the chapter of supplements.

1.4 Transmission ratio.

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.

1.5 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".

Options of material, loading conditions, operational and production parameters. [2]

When designing power gearing, enter other supplementary operational and production input parameters in this paragraph. Try to be as accurate as possible when selecting and entering these parameters as each of them may dramatically affect the properties of the designed gearing.

2.1, 2.2 Material of the pinion/gear.

The option is performed, above all, according to the following aspects:

  1. Strength
  2. Price of the material and its heat treatment
  3. Workability
  4. Hardenability
  5. Degree of loading
  6. Dimensions of the gear
  7. Seriality of production

Usually the principle that the pinion has to be harder than the gear (20-60 HB) is followed, whilst the difference in hardnesses increases with increasing hardness of the gear and the transmission ratio. For quick orientation, the materials are divided into 8 groups marked with the letters A to H. Perform selection of the material in the pop-up list separately for the pinion and for the gear. In case you need more detailed information on the chosen material, proceed to the sheet "Material".

  1. Low-loaded gears, piece production, small-lot production, smaller dimensions
  2. Low-loaded gears, piece production, small-lot production, greater dimensions
  3. Medium-loaded gears, small-lot production, smaller dimensions
  4. Medium-loaded gears, small-lot production, great dimensions
  5. Considerably loaded gears, lot production, smaller dimensions
  6. Heavily-loaded gears, lot production, greater dimensions
  7. Extremely loaded gears
  8. High speed gears

Materials A,B,C and D, so-called. soft gears - The toothing is produced after heat treatment; these gears are characterized by good running-in, do not have any special requirements for accuracy or stiffness of support if at least one gear is made of the chosen material.

Materials E,F,G and H, so-called. hard gears - Higher production costs (hardening +100%, case hardening +200%, nitriding +150%). Heat treatment is performed after production of toothing. Complicated achievement of the necessary accuracy. Costly completion operations (grinding, lapping) are often necessary after heat treatment.

Own material values - In case you wish to use a material for production of toothing that is not included in the delivered table of materials, it is necessary to enter some data on this material. Proceed to the sheet "Materials". The first 5 rows in the table of materials are reserved for definition of your own materials. Enter the name of the material in the column designed for names of materials (it will be displayed in the selection sheet) and fill in successively all parameters in the row (white fields). After filling in the fields, go back to the sheet "Calculation", choose the newly defined material and continue in the calculation.

Warning: The material table includes options for the used materials. As the strength values of the material depend very much on the semi-product dimensions, the method of heat treatment and particularly the supplier, it is necessary to consider the values in the material table as orientation ones. It is recommended to consult the particular and accurate values with your technologist and supplier or take them from particular material sheets.

2.3 Loading of the gearbox, driving machine - examples.

Setting of these coefficients substantially affects the calculation of safety coefficients. Therefore, try to enter as accurate a specification as possible when selecting the type of loading. Examples of driving machines:

  1. Continuous: electric motor, steam turbine, gas turbine
  2. With light shocks: hydraulic motor, steam turbine, gas turbine
  3. With medium shocks: multi-cylinder internal combustion engine
  4. With heavy shocks: single-cylinder internal combustion engine

2.4 Loading of gearbox, driven machine - examples:

Setting these parameters substantially affects the calculation of safety coefficients. Therefore, try to enter as accurate a specification as possible when selecting the type of loading. Examples of driven machines:

  1. Continuous: generator, conveyor (belt, plate, worm), light lift, gearing of a machine tool traverse, fan, turbocharger, turbo compressor, mixer for materials with a constant density
  2. With light shocks: generator, gear pump, rotary pump
  3. With medium shocks: main drive of a machine tool, heavy lift, crane swivel, mine fan, mixer for materials with variable density, multi-cylinder piston pump, feed pump
  4. With big shocks: press, shears, rubber calendar, rolling mill, vane excavator, heavy centrifuge, heavy feeding pump, drilling set, briquetting press, kneading machine

2.5 Type of gearing mounting.

Adjusting this parameter influences the calculations of the safety coefficient. The type of seating defines the coefficient of irregularity of loading arising, above all, from deflections of the shafts. Select the type of seating according to the definition and illustration.

  1. Both wheels seated on-the-fly: It is gearing whose wheels are on-the-fly-seated. The shaft is fixed (built-in) on one side of the wheel only.
  2. One wheel seated on-the-fly: It is gearing with one wheel on-the-fly-seated and the is seated between bearings
  3. Gearing seated on both ends: It is gearing with wheels seated between bearings. The shaft is fixed (built-in) on both sides of the wheel.

Type 1: Rigid case, rigid shafts, robust roller or tapered roller bearings.
Type 2: Less rigid case, longer shafts, ball bearings.

Note: Type of application for AGMA
Type1: Automotive, aircraft and high-quality commercial
Type2: General commercial

2.6 Accuracy grade.

When choosing the degree of accuracy of the designed gearing, it is necessary to take into account the operating conditions, functionality and production feasibility. The design should be based on:

The accuracy of toothing is chosen to the necessary extent only, because achievement of a high degree of accuracy is costly, difficult and conditioned by higher demands for technological equipment.

Table of surface roughness and maximum peripheral speed

Degree of accuracy ISO 1328 3 4 5 6 7 8 9 10 11
Degree of accuracy DIN 3965 2 3 4 5 6 7 8 9 10
Degree of accuracy AGMA 13 12 11 10 9 8 7 6 5
Max. surface roughness Ra max [nm] 0.1-0.2 0.4 0.8 1.6 1.6 3.2 6.3 12.5 25
Max. peripheral speed [m/s] helical teeth 50 40 30 20 12 8 5 3 3

Max. peripheral speed for straight teeth is <5 [m/s]

Orientation values for options of the degree of accuracy according to the specification.


Degree of accuracy


Degree of accuracy


Control gears 2 - 4 13-12
Measuring instruments 3 - 6 13-10
Turbine reducers 3 - 5 13-11
Aviation reducers 3 - 6 13-10
Machine tools 3 - 7 13-9
Aviation engines 5 - 6 11-10
High speed gearboxes 5 - 6 11-10
Passenger cars 6 - 7 10-9
Industrial gearboxes 7 - 8 9-8
Light ship engines 7 9
Rolling mills, locomotives 8 - 9 8-7
Heavy ship engines, tractors 8 - 9 8-7
Building and agricultural machines 8 - 10 8-6
Textile machines 7 - 9 9-7

2.7 Coefficient of one-off overloading.

The coefficient gives the ratio between the maximum (start-up) and nominal torque of the driving machine. The coefficient substantially affects the calculation of the safety coefficient with one-off overloading (start-up) of the gearing. The coefficient can be found in the catalogue of the producer of the driving unit.

Recommended values:

Three-phase asynchronous electric motor ... 2-3

2.8 Desired service life.

The parameter specifies the desired service life in hours. Orientation values in hours are given in the table.


Household machines, seldom used devices 2000
Electric hand tools, machines for short-term runs 5000
Machines for 8-hour operation 20000
Machines for 16-hour operation 40000
Machines for continuous operation 80000
Machines for continuous operation with log service life 150000

2.9 Coefficient of safety (contact/bend).

The recommended values of the safety coefficient vary within the range:

Hint: Use recommendations in the help to estimate the safety coefficient.

2.10 Automatic design.

Decide whether you wish to design straight or oblique toothing. You can use the following recommendations for your option:

In case of the "Automatic design", adjustment of parameters of the gearing is based on the entered power and operational parameters [1.0; 2.0] and generally applicable recommendations. However, manual optimising enables a design of the gearing with better parameters (weight, size) and/or modifications of dimensions based on own design requirements.

Warning: The "Automatic design" may change the parameters that have already been changed in other paragraphs; therefore use the "Automatic design", above all, for preliminary determination of the gearing parameters.

Parameters of the tooth profile. Type of toothing. [3]

In this paragraph select a type of toothing and parameters of the tooth profile. The calculation is designed, above all, for the design of conical toothing with straight teeth (straight, oblique), nevertheless, it is possible to use it for an orientation design of toothing with curved teeth.

3.1 Guiding curve of the toothing (Type of toothing).

Select a type of cone toothing. The calculation is designed primarily for the design of conical toothing with straight and oblique teeth (I/A, I/B). For orientation it is possible to use it also for wheels with curved teeth (C, D, E, F). For exact calculation of wheels with curved teeth the calculation instructions (software) that are supplied by the producer of the respective machining equipment must be used.

3.2 Addendum - Coefficient of height of the tooth head.

The coefficient is adjusted automatically according to the selected type of toothing [3.1].In case of the need to adjust own values, tick the box [3.1].

Recommended values:

The coefficient can be changed within a wide range and depends on the desired parameters of toothing, way of production, production machine and tool. Detailed information can be found in the instructions for the production machine or specialised literature.

3.3 Unit head clearance.

See 3.2

3.4, 3.5 Coefficient of foot radius.

Its size depends on the unit head clearance. The standard value is rf*=0.38. The recommended values are given above input cells. When the box is ticked, the recommended values are transferred automatically to the input cells.

Recommended values:

In case there are no particular reasons for the option of a non-standard value, do not change the default values.

Design of a module (Diametral Pitch) and geometry of toothing. [4]

This paragraph can be used to design the geometry of a gearing. The geometry design substantially influences many parameters such as functionality, safety, life and price.

4.1 Number of teeth.

Enter the number of teeth of the pinion. Calculation of the number of teeth of the wheel is based on the desired gear ratio. Finding the optimum number of teeth is not an unambiguous task and cannot be solved directly. The number of teeth influences mesh conditions, noise, effectiveness and production costs. Therefore the number of teeth is usually selected and made more accurate according to qualitative and strength indexes.

Generally, the rule says that increasing the number of teeth leads to:

Recommended values for gearing with angles of axes 90°:

Gear ratio

 Range z1
1 18-40
1.12 18-38
1.25 17-36
1.6 16-34
2 15-30
2.5 13-26
3 12-23
4 10-18
5 9-14
6 8-11

Lower values are chosen for hardened wheels with curved teeth, higher values are chosen for straight non-hardened teeth.

Hint: In case the number of teeth of the pinion and wheel is known and the gear ratio must be calculated, press the button to the right of the input field and execute the respective calculation in the chapter of supplements.

4.2 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).

4.3 Pressure angle.

This determined parameters of the basic profile, the value 20° being mostly chosen. A change in the pressure angle a can influence the functional and strength properties.

When choosing the pressure angle there may be chosen the transverse pressure angle (used for straight toothing) or normal pressure angle (used for curved teeth).

The letter "X" marks the basic circles.

Increasing the pressure angle leads to:

Selection of the values:

Recommended values:

In case of no requirements for the designed gearing, it is recommended to use 20°.

4.4 Base helix angle.

Toothing with base helix angle = 0 (straight toothing) is used rarely, only for less loaded gearing up to a circumference speed of approx. 5 m/s (10 ft/s). Wheels with oblique or curved teeth are used for higher speeds. Values between 20° and 40° (usually per 5°) are chosen for straight oblique teeth.

4.5 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 characterised 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

4.6 Width of toothing to the surface straight line of the cone (b/Re).

Using the slider adjust the value of the dimensionless coefficient that shows the ratio between the toothing width and surface straight line of the cone [4.7].

4.7 Width of toothing to the surface straight line of the cone (b/Re).

This parameter can be used to design the module size and thus the basic geometric parameters of the wheel.

The recommended maximum value is:

Low to medium loaded gearings: 0.2 - 0.3
Highly loaded gearings: 0.3 - 0.35

Adjusting this parameter can be performed using the slider that is located on row [4.6]. After adjusting this parameter press the button "Design gearing". This procedure executes a design of toothing that meets the requirements of the desired safety [2.9] and other input parameters.

In case you tick the button on row [4.9], the maximum possible value of the toothing width to the surface straight line Re and to the tangential module on the outer perimeter met is chosen automatically.

After "Design gearing" is executed, check the dimensions (widths and diameters of the wheels and their weights). In case the result is not satisfactory, modify the input parameters of the gearing and repeat the "Design gearing".

Hint 1: In case it is not possible to approximate the desired dimensions of the gearing using changes of this parameter, try to modify the number of teeth of the pinion, angle of slope of the teeth or select another material.

4.8 Module of toothing / Diametral Pitch.

This is the most important parameter that determines the teeth site and thus the size of the gearing. In general it is applicable that a smaller module can be used for a higher number of teeth (higher value P with inch-version of the calculation) and vice versa. In the right pop-up list you can find standardised values of the module / (Diametral Pitch with inch-version of the calculation) and the value selected from this list is automatically entered into the field on the left.

The module can be entered using an option between the normal module "mmn" (toothing with curved teeth) and tangential module "met" (straight and oblique toothing) by the respective setting in the selection list.

Hint: The design of the correct size of the module is quite a complicated task. Therefore it is recommended to use a procedure for designs of toothing based on the ratio between the toothing width to the surface straight line of the cone [4.6, 4.7].

4.9 Face width.

After clearing the button the value of the face width can be entered. Ticking the button automatically selects the maximum value.

4.10 Approximate weight of the gearing.

It is calculated as the weight of solid wheels (without lightening or holes, see the illustration). It can be used for fast orientation during the design procedure.

4.11 Minimum safety coefficient.

This row always gives the lower of the coefficients for the pinion and wheel. The first column shows the safety coefficient for fatigue in the contact, the other column shows the safety coefficient for bending fatigue.

Correction of toothing. [5]

Radial (height) and tangential (perimetral) displacement of the cutters in the course of the production process can change the geometric, kinematic and strength characteristics of the toothing. The radial (height) displacement is determined by the coefficient x, the tangential (perimetral) displacement is determined by the coefficient xt. Conical gearings are produced mostly as toothing VN, thus x = x1 = -x2 and xt = xt1 = -xt2

Both values can be adjusted in this paragraph.

Correction of toothing can lead to:

Hint: Detailed information on possibilities and ways of correction can be found in specialised literature.

5.1 Correction type.

Select one of the recommended types of correction in the selection list. The recommended values x1 and xt are in row [5.2].

5.3 Permissible undercutting of teeth.

The practice permits slight undercutting of teeth. The given value is the minimum (limit) which leads to permissible undercutting of teeth. The correction value should not be lower except in some special cases.

5.4 Preventing undercutting of teeth.

It is the minimum correction value that can be used to avoid undercutting of teeth.

5.5 Pinion addendum modification coefficient setting.

This slider is designed for fast changes in the correction. In case the field to the right of the slider is ticked, movements of the slider control the amount of correction x. This function can be used at the moment when you wish to optimise some of the qualitative or strength parameters of toothing; the most important of them are given in rows [5.8-5.11].

5.6 Addendum modification coefficient Pinion / Gear.

Here you can find the value of pinion displacement x1 and wheel displacement x2. In case you wish to enter a unit displacement of the pinion using the keyboard, clear the box in row [5.5].

Recommended values for gearings with the angle of axes 90°:
Gear ratio
1 0
1.12 0.10
1.25 0.19
1.6 0.27
2 0.33
2.5 0.38
3 0.40
4 0.43
5 0.44
6 0.45

5.7 Unit change of the tooth thickness.

Adjust the value of the unit change of the tooth thickness here.

Recommended values for gearings with the angle of axes 90°:
Gear ratio
1 0
1.12 0.010
1.25 0.018
1.6 0.024
2 0.030
2.5 0.039
3 0.048
4 0.065
5 0.082
6 0.100

Qualitative indexes.

It is advisable to monitor the behaviour of these indexes when changing the corrections. Exceeding critical values is indicated by a change in the digit colour.

5.8 Total mesh index.

For a detailed explanation, see [8.1] and [8.2]

5.9 Unit tooth thickness on the tip diameter.

It is a dimensionless parameter (ratio of the tooth thickness and module) and depends, above all, on the tooth shape. It is influenced by the following parameters:

Recommended values

Usually it is 0.25 - 0.4. It is higher for low values of unit displacement and hardened wheels. A lower value than the recommended one is indicated by a red text, exceeding the limit of the tooth sharpness is indicated by a red field.

5.10, 5.11 Safety coefficient for contact and bending fatigue.

For detailed information, see [10].

Basic dimensions of gearing. [6]

This paragraph gives a well arranged listing of all basic dimensional parameters of toothing. For clarity you can find here an illustration of the most important dimensional parameters. It is recommended to use specialized literature for more a detailed explanation of individual parameters.

Marking of dimensions according to ISO (DIN)

Marking of dimensions according to ANSI (AGMA)

Virtual spur gear toothing. [7]

Each conical wheel (straight, oblique) can be assigned to a virtual cylindrical wheel with straight teeth, whose profile is in practice the same as the normal profile of a conical wheel in its mean central section. Parameters of this comparative wheel can be found in this paragraph.

Qualitative indexes of a gearing. [8]

This includes the parameters which inform us of the quality of the designed toothing. It is advisable to compare them with the recommended values.

8.1 Transverse contact ratio / overlap ratio.

For smooth meshing of gears, it is necessary that the other pair of teeth enters in meshing before the first pair is released. The contact ratio in the face plane says how many teeth are in meshing simultaneously. With the value ea=1 this corresponds to a limit case when only one pair of teeth is in meshing at the given moment. With the value ea=2, there are two teeth in meshing simultaneously. In case the value is between 1< ea<2, the meshing will include partly one pair of teeth and partly two pairs. The parameter depends on a number of effects. (increases with the number of teeth, decreases with the pressure angle at the pitch cylinder aw). The transverse overlap ratio is applicable in the case of helical gearing (angle b>0) and then the meshing angle is evaluated eg [8.2](sum ea a eb).

Recommended value:

According to the complexity of the gearing, this parameter should not be lower than 1.1 to 1.2.

8.2 Total contact ratio.

This is the sum of transverse contact and overlap ratios.

Recommended value:

This is specified using the same recommendations as ea in case of spur gearing. This means that eg must always be higher than 1.2.

8.3 Resonance speed.

This is the speed at which the angle speed is the same as the proper angle vibration frequency of the gearing.This causes undesired resonance effects.

8.4 Resonance ratio.

This is the ratio of pinion speed and "Critical speed".

In case the designed gearing works in the range of critical speed (N ~ 1), the resonance ratio N is indicated by a red number. In such cases, modifications of the designed gearing (changes of numbers of teeth) or consultations with a specialist are recommended.

8.5 Approximate weight of gearing.

It is roughly calculated as the weight of the solid wheel (without holes for shafts or lightening holes) according to the illustration with the real dimensions of the toothing in par. [4.0]. It can be used for fast orientation in the design.

8.6 Efficiency of the gearing.

Exact determination of the coefficient is difficult. Therefore, an approximate calculation based on the number of teeth, mesh coefficient, angle beta and friction coefficient is used. The choice of friction coefficient is based on the chosen level of toothing accuracy [2.6] within the range 0.04-0.08

Coefficients for safety calculation. [9]

Calculation according to DIN 3991.

The standard DIN 3991 defines 4 levels (A, B, C, D) of complexity of the determination of the coefficient used for calculating safety coefficients. The coefficients are mostly determined in this calculation using the methodology B or C (exceptionally D). Detailed information and formulas for determination of the respective coefficients can be found in the appropriate standard.


The life coefficient [9.18, 9.29] – According to DIN 3991 the value of the contact and bending fatigue for the given number of cycles is used directly. This calculation uses the basic value of the fatigue limit multiplied by the respective life coefficient calculated from the basic number of loading cycles, exponent of the Wohler’s curve and the actual number of cycles.

According to DIN 3991 the coefficient of alternate loading [9.27] and coefficient of production technology [9.28] are not considered here. Therefore, the values of these coefficients are set to 1.0.

Calculation according to AGMA.

The methodology according to standards AGMA 2003-A86/88, AGMA 2005-B88 is used for calculating the safety coefficients in the calculation of toothing in inches.

Note: Most coefficients are calculated additionally and found from information defined in paragraphs [1, 2, 4 and 5] so that the user does not receive any inquiries that cannot be answered by him. In case you are an expert in the field of strength checks of toothed wheels, you can rewrite the formulas for the determination of individual coefficients using your own numerical values.
Hint: A detailed description of the functions of individual coefficients, how they are calculated and their limitations can be found in the appropriate standard DIN/ISO/AGMA or in specialised literature.

Safety coefficients. [10]

Two basic strength calculations are usually carried out, namely for bend and for contact. The following safety coefficients are calculated in this calculation:

As initial values of the safety coefficient you can use:

Safety coefficients can then be modified according to general recommendations for options of safety coefficients and according to your own experience.

Force conditions (forces acting on the toothing). [11]

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 [11.3, 11.4] shows the amounts of the forces if the sense of rotation of the gearing agrees with the directions in the illustration, row [11.5, 11.6] 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. The pitch direction can be chosen in row [4.5].

11.7 Peripheral speed on the pitch diameter.

It is another important qualitative parameter that affects the desired accuracy of the gearing [2.6] and the manner of lubrication (Lubrication of wheels). The maximum recommended speed for the chosen level of accuracy is shown in the green cell on the right.

11.8 Specific load / Unit load.

It is another qualitative index that is used for the calculation "Indexes of irregularity of loading of the tooth".

Note: Not for AGMA

Parameters of the chosen material. [12]

This paragraph lists material characteristics of the pinion and gear materials.

Hint: Your own material values can be entered in the sheet "Material".

Power, warming-up, gearbox surface. [13]

This paragraph enables an orientation calculation of the lost heat and gearbox surface necessary for dissipation of this heat.For purposes of the calculation, fill in the first three input parameters:

13.1 Ambient air temperature.

13.2 Maximum oil temperature.

Temperature of the oil in the gearbox should be in a range from 50 to 80 °C; a lower temperature should be found in smaller modules. More exact determination of temperature depends on the chosen construction and used materials. Higher temperatures bring a danger of lower backlash and the gearing could seize.

13.3 Coefficient of heat dissipation.

This depends on the construction and ambient environment of the gearbox. Initially, it is possible to choose:
for ISO/DIN:

  • 8 to 11 [W/m2/K] for small closed rooms
  • 14 to 17 [W/m2/K] for well ventilated halls


  • 1.4 to 1.9 [BTU/sq.ft/h/F] for small closed rooms
  • 2.5 to 3.0 [BTU/sq.ft/h/F] for well ventilated halls

13.4 Power losses.

This depends on the total transferred power and efficiency of the gearing.

13.5 Gearbox surface.

This parameter gives the minimum surface of the gearbox necessary for dissipation of power losses and maintaining the desired oil temperature.

Preliminary design of shaft diameters (steel). [14]

Diameters of shafts (steel) that correspond to the desired load (transferred power, speed) are designed in this paragraph. These values are for orientation only; the final design should be made using a more exact calculation.

Auxiliary calculations. [15]

Auxiliary calculations are available in this paragraph. When entering values, use the same units as in the main calculation. To transfer the entered and calculated values to the main calculation, press the button "OK".

Lubrication of gears.

Use the following table for your decision on the manner of lubrication of the gearing.

Type of lubrication Peripheral speed in
[m/s] [ft/min]
Oil-bath lubrication < 12 < 2400
Pressure spray lubrication > 12 > 2400
Oil-mist lubrication > 60 > 12000

Graphic output, CAD systems [16].

Supplements- This calculation:

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".

16.4 Radius of the cutting tool (for a 3D model)

This parameter determined the radius of the cutting tool in the production of circular teeth. It must be used only as a model in the 3D CAD system (if the respective model supports detailed toothing).

16.5, 16.6 Amount of the inner / outer offset.

These parameters set the offset amount with the toothed wheel, see the illustration.

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".

Supplements - This calculation:

Material list - Method of heat treatment
1...Non-treated thermally, annealed normalizationally
3...Cemented, hardened, surface hardened