Planetary Gearing with Spur and Helical Toothing

The calculation is designed for geometric and strength design and check of epicyclic gearing with straight and helical toothing. The application provides solutions for 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.
4. Calculation of complete geometric parameters (including corrected toothing).
5. Optimization of toothing by use of proper correction (balancing specific slips, minimizing specific slips, strength...).
6. Calculation of strength parameters, safety check.
7. Supplementary calculations (calculation of parameters of the existing gear, design of shafts, check dimensions).
8. Support of 2D and 3D CAD systems.
9. Drawings of an accurate tooth shape including data (X,Y coordinates).

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

List of standards: ISO 6336, ISO 1328, DIN 867, DIN 3960, DIN 3990, ISO 6336-5 and others.

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

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".

Theory.

Planetary gearings consist of a system of gearwheels and a carrier. The so-called suns are aligned with the carrier and the central axis of the gear mechanism. The planets are the gearwheels mounted in a pivoting way on the carrier and they mesh with the suns or with each other. The planets may have one, two or more gearings. Two or multi-speed planets have more constructional variants with wider possibilities; however, they are more complex and expensive to manufacture.

You can see an example of a simple planetary gearing with single-speed toothing of the planet below. This basic type of planetary gearing is also handled in complex in this program.

Simple planetary gearing (differential):

0 - Sun; 1 - Carrier; 2 - Ring gear; 3 - Planet.

If all three members of a simple planetary gearing (0, 1, 2) are free, the system is called a differential (2 degrees of freedom), which enables it to compose / decompose two moves into one. That is used e.g. for machine tools (composition) or for a car differential (move decomposition).

If one of the basic members (0 or 2) is connected to the frame, the system is called planetary gearing (1 degree of freedom), specifically a reductor in case of a drive outwards from the sun or a multiplicator in case of a drive outwards from the carrier. The system with the carrier connected to the frame is called a normal transmission or helical gearing.

 Planetary gearings may be arranged in various ways. The most frequent way includes the serial arrangement, where the total transmission ratio (efficiency) is determined by the product of partial transmission ratios (efficiency). The composed gearings often use the possibility to brake individual members, i.e. gear changing.

Advantages:

• Saving of space using an aligned arrangement of the driving and driven shafts
• Lower weight compared to a normal transmission
• High efficiency even at high transmitted outputs
• Low radial loading of centre member bearings
• Compact design

Disadvantages:

• • Complicated construction, higher requirements for production and assembly accuracy
• • Higher production costs
• • Some limiting conditions (complex assembly)

Use:

With respect to the above-specified advantages, the use of planetary gearings is popular in a wide range of applications (e.g. motor vehicle transmissions, building machinery, lifting equipment, marine transmissions, turbine reducers, etc.) The combination of planetary gearing with hydraulic or friction gearing is also common.

Applied formulas (calculation of gearing geometry).

Some of the most important formulas for calculation of the gearing geometry are specified below. In the formulas, indexes 1 and 2 are used for a pinion and a wheel (which is a pair: a sun and a planet, or a planet and a ring gear). For the internal gearing (ring gear), a negative value for the number of internal gear teeth is used, i.e. also a negative value for the centre distance and diameters.

In the case of planetary gearing, the individual gears are dependent on each other and the gearing must be handled as a whole, including the respective constraining conditions (see below).

Basic profile parameters: mn (module, DP for inch calculation), a (pressure angle), ha*, c*, rf* (cutting tool parameters)

Pinion and wheel parameters: z1, z2 (number of teeth of pinion and wheel), x1, x2 (addendum modification), b (helix angle), b (width of gearing)

1. Transmission ratio
   i = z2 / z1
2. Module
   mt = mn / cos(b) ... Transverse
3. Circular pitch
   p = p • mn ... Circular pitch
   pt = p / cos(b) ... Transverse circular pitch
   ptb = pt • cos(at) ... Base circular pitch
4. Pressure angle
   at = arctg(tg(a) / cos(b)) ... Transverse pressure angle
   awn = arcinv(2 • (x1 + x2) / (z1 + z2) • tg(a) + inv(a)) ... Pressure angle at the pitch cylinder
   awt = arcinv(2 • (x1 + x2) / (z1 + z2) • tg(a) + inv(at)) ... Transverse pressure angle at the pitch cylinder
5. Base helix angle
   bb = arcsin(sin(b) • cos(a))
6. Diameter
   d = z • mt ... Reference diameter
   db = d • cos(at) ... Base diameter
   dw = d • cos(at) / cos(awt) ... Operating pitch diameter
   da = mn • (z / cos(b) + 2 • (ha* + x - DY) ... Tip diameter
   df = d - hf ... Root diameter
7. Center distance
   a = (z1 + z2) • mn / (2 • cos(b)) ... Center distance (pitch)
   av = a + (x1 + x2) • mn ... Center distance (production)
   aw = a • cos(at) / cos(atw)  ... Center distance (working)
   DY = (a - aw) / mn + (x1 + x2) ... Unit correction
8. Contact ratio
   e_a = ((da1^2 - db1^2)^0.5 + (da2^2 - db2^2)^0.5 - 2 • aw • sin(awt)) / (2 • p • cos(at) / cos(b)) ... Transverse contact ratio
   e_b = b / mn / p • sin(b) ... Transverse overlap ratio
   e_g = e_a + e_b ... Total overlap ratio

{02}

Designing and geometrical relations.

The formulas below use the following indexes.

For:

• Sun - 0
• Planet - 1
• Ring gear - 2

With respect to the possibilities of assembly and functioning of planetary gears, the geometry of the gearwheels cannot be chosen at random. In order to provide proper functioning, the following several conditions must be followed and observed.

Condition of alignment.

The planets of the planetary gearings gear with the suns, possibly with other planets. This calculation applies to the joint mesh of a planet with suns (planet, ring gear). As the planet and the ring gear have the same axis, the centre distance between the planet and both suns must be the same.

It applies for the generally corrected wheels that:
aw (0,1) = aw (1,2)
with aw (0,1)=mt • (z0+z1)/2 • COS(alfat)/COS(alfawt(0,1))
with aw (1,2)=mt • (z1+z2)/2 • COS(alfat)/COS(alfawt(1,2))

Note: In the program, a violation of this condition is indicated by the cells with calculated centre distance highlighted in red.

Condition of assembly.

For simple planets and uniform planet distribution, the following condition must be complied with:

g = (abs (z0) + abs (z2))/P
With:
g - must be a random integer
P - number of planets
z - number of teeth

Note: This condition need not always be achievable (e.g. if a specific transmission ratio is required). This condition may be bypassed by uneven distribution of the planets, which may result in greater demands on production, imbalance of the planet carrier, imbalance of inner forces and increased stress.

Condition of backlash between neighbouring planets.

This condition provides minimum backlash between the planets vmin (1-2 mm, 0.05 in).

Maximum planets P = int(asin((da1+vmin)/(aw • 2)))

Note: In the program, a violation of this condition is indicated by the cells with number of planets highlighted in red.

Applied formulas (calculation of stress and safety).

The calculations of stress and surface and bending safety are performed in accordance with the ISO 6336 standard. The basic formulas are specified below. Information on calculations of individual coefficients in the applied formulas can be found in the ISO 6336-1, 6336-2, 6336-3 standard.

Contact stress
sH0 = ZH • ZE • Ze • Zb • (Ft / (d1 • b) • (u + 1) / u)^0.5 ... Nominal contact stress
sH = ZB • sH0 • (KA • KV • KHb • KHa)^0.5 ≤ sHP ... Contact stress
sHG = sHlim • ZNT • ZL • ZV • ZR • ZW • ZX ... Pitting stress limit

Safety coefficient for surface durability
SH = sHG / sH

Tooth-root stress
sF0 = YFa • YSa • Ye • Yb • Ft / (b • mn) ... Nominal tooth-root stress
sF = sF0 • KA • KV • KFb • KFa ≤ sFP ... Tooth-root stress
sFG = sFlim • YST • YNT • Yd • YR • YX • YA • YT ... Tooth-root stress limit

Safety coefficient for bending durability
SF = sFG / sF

Note: The AGMA standard was not used for the strength calculation as it is simpler compared to ISO and includes fewer influences and input parameters (gearing is usually over designed for ANSI/AGMA). However, if the strength control according to ANSI/AGMA is required, it may be performed in the calculation module for internal and external spur toothing.

Efficiency and losses.

The losses in planetary gearing can be divided into losses due to idle run and losses due to loading. The losses due to idle run (lubrication, unloaded mesh, bearings) are difficult to specify analytically and in general, they are significantly lower than losses due to loading. The losses due to loading arise during power transmission and include:

Losses in gearing

The loss coefficient can be approximately described according to the formula:
Spur gearing: zz = 0.5 • f • p • e • (1/z1 +- 1/z2)
Helical gearing: zz = 0.25 / cos(b) • f • p • e • (1/z1 +- 1/z2)
with:
z1, z2 - number of teeth
f - friction coefficient (0.04 - 0.08)
e - mesh ratio
b - helix angle
Plus sign (+) for external gearing, (-) for internal gearing.

Losses in bearings

The loss output can be determined from the relation:
PVL = w • F • f • r
with:
w - angular velocity
F - resulting bearing loading (carrier, centrifugal force)
f - friction coefficient (0.001 - 0.005)
r - mean bearing radius

Note: The friction coefficient (for both gearing and bearing) is estimated in the calculation based on the selected accuracy level (gearing roughness) and used lubricant.

For the calculation of loss (efficiency) of the planet gearing, we will use the losses in the helical gearing (carrier stopped) with:

with:
zz0/z1 - loss coefficient sun - planet
zz1/z2 - loss coefficient planet - ring gear

z0,z2 - number of teeth of the sun and ring gear

For individual cases of output flow, it applies for the loss calculation:

Note: Detailed and accurate identification of losses in the planetary mechanism is demanding and depends on a detailed knowledge of construction, the used materials and operating conditions. Therefore, it is necessary to consider the values from the calculation as informative.

Calculation method.

The gearwheel transmissions are divided into:

Force gears - The gear designed mainly for output transfer and transformation, requires strength design/control (e.g. machinery drives, industrial gear boxes.).

Non-force gears - The gear with transferred torsional moment minimum compared to the gearwheel dimension does not require strength design/control (e.g. devices, control engineering.).

Design of force gear.

The task to design planetary gearing allows significant freedom in the choice of diameter and width parameters of gearwheels on the one hand, while on the other hand it is necessary to comply with a number of conditions (alignment, assembly...) in order to guarantee the gear functions. That is why it is appropriate to proceed in an iterative way and refine the solution and fine-tune the respective parameters gradually.

Fast (informative) design:

This procedure provides a quick view of the parameters of the designed gear. Although such designed gear is normally usable, you can improve its properties significantly by gradually optimizing the range of parameters. Use the following procedure in designing:
1. Set the performance parameters of the gearing (transferred output and speed). [1]
2. Choose the material for all gears, choose the load mode, operating and production parameters and the required safety coefficients. [2]
3. Perform automatic design -> Press the "Spur gearing"/"Helical gearing" button. [2.11]
4. Check the results.

Parameter optimizing:

Before optimizing the parameters, perform the "Fast (informative) design" described above first. Then proceed as follows:
1. If you want to use non-standard parameters of the tooth profile, set them in paragraph [3].
2. Set the gearwheel parameters (number of teeth, pressure angle and helix angle). [4.1-4.6]
3. Use the slider [4.7] to set the ratio of the sun width to its diameter, press the "Design gearing" button.
4. Check the dimensions of the designed gear in the schematic drawing. If the dimensions do not suit you, adjust the ratio of the pinion width and diameter and recalculate the gear [4.4].
5. In paragraph [5], fine-tune the centre distance, possibly sliding properties by changing the corrections.
6. Check and assess (compare with the Help) the dimensional and qualitative indicators. [6; 7; 8]
7. Check the safety coefficients. [9, 10]

Hint: You can change the gearing dimensions by the proper choice of material (possibly its finish).

Design of non-force gear.

The strength parameters do not require handling or controlling when designing a non-force gear. Therefore, choose the proper number of planets [4.1], teeth [4.3] and module [4.9] directly and check the dimensions of the designed gearing.

Hint: When designing the non-force gear, choose appropriately low transferred power.

Options of basic input parameters. [1]

In this paragraph, set the basic input parameters of the designed gearing.

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 Transmission type (input/output).

Choose the transmission type. The first part on the drop-down menu line specifies which member of the planetary gearing will be the input (you can choose the input power and speed). The second part (after the arrow) specifies the output member. After switching, the speed inputs are also adjusted [1.4] so that the input member speed is non-zero.

1.3 Transferred power.

Set the power on the input gearing member. The normal values range from 0.1 - 3000 kW / 0.14-4200 HP, and up to 65000 kW /100000 HP in extreme cases.
Press the button on the right to perform the calculation for the maximum possible power for the current gearing parameters.

1.4 Speed.

Set the speed of the gearing input member. The highest speed may be up to 150,000 rpm. The checked checkbox after the input cells indicates locking of the respective planetary gearing member. After unchecking, the mechanism acts as a differential => The speed for two gearing members can be selected.
The output member speed (in bold) is a function of the number of teeth of individual gearwheels. However, as the number of teeth cannot be designed randomly, it is appropriate to handle the required output speed in detail in paragraph [4.0].

Note: Use the following line [1.5] to perform the preliminary design of the number of teeth in order to achieve the requested speed of the output member.

1.5 Requested speed.

Set the requested output member speed. It must be within the range specified in the green cell. After pressing the button on the right, the number of teeth will be designed and selected in order to achieve the requested speed. The number of teeth and speed can be handled in detail in paragraph [14.0].

Note: The speed range is specified for the commonly used number of teeth. A more accurate calculation enabling you to use extreme values is provided in chapter [14.0].

1.6 Torsional moment.

It is a result of the calculation and cannot be set.

1.7 Speed (planet in planet carrier).

It specifies the planet speed in the carrier. It is important for calculating the capacity of the planet bearing, which is often a critical place of the gearing.

1.8 Transmission ratio z1/z0, z2/z1, (z2/z0).

It is the transmission ratio between individual gearing members. The third value is important - the transmission ratio (z2/z0), which indicates the value of the helical gearing used in further calculations.

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.

2.2, 2.3, 2.4 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.

2.5 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

2.6 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

2.7 Type of gearing mounting.

Setting this parameter affects the calculation of the safety coefficient. The type of mounting defines the load unevenness coefficient resulting mainly from the shaft deflections. Use the following definition and picture to choose/estimate the type of mounting.

A. Double-sided symmetrically supported gearing: It is gearing with the gearwheels mounted symmetrically between the bearings (the distances between the bearings and wheel edge are the same).
B. Double-sided non-symmetrically supported gearing: It is gearing with the wheels mounted asymmetrically between the bearings (the distances between the bearing and wheel edge are different).
C. Overhung gearing: It is gearing with the wheels overhung. The shaft is attached (fixed) from only one side of the gearwheel.

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

Note: The construction variants of planetary gearing are significantly richer than for normal spun gearing. In addition, in case of planetary gearing, the forces are composed and act on the sun as well as the ring gear more favourably so the choice of type of mounting will be most affected by the planet mounting (see the picture).

2.8 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:

• Peripheral speed, transferred power.
• Operating conditions, desired service life and reliability.
• requirements for kinematics accuracy, noise, vibrations.

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

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

2.9 Desired service life.

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

2.10 Coefficient of safety (contact/bend).

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

• Coefficient of safety in contact SH = 1.1 - 1.3
• Coefficient of safety in bend SF = 1.3 - 1.6

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

2.11 Automatic design.

Decide whether you wish to design straight or helical toothing. The following recommendations can be used for your option:

• Straight teeth - Suitable for slow speed and heavily loaded gearing, zero axial forces, higher weight.
• Helical toothing - Suitable for high speed gearing; characterized by lower noise, higher loading capacity, necessity to catch axial forces.

With the "Automatic design" the setting of parameters of the gearing is based on the entered power and operational parameters [1.0; 2.0] and on generally applicable recommendations. Manual optimizing can mostly provide toothing with better parameters (weight, dimensions) or enable modifications of the dimensions based on your own constructional requirements.

Warning: "Automatic design" can modify the parameters which were modified already in other paragraphs. Therefore, use the "Automatic design", above all, for preliminary determination of gearing parameters.

Parameters of the tooth profile. [3]

Set the parameters of machining tool and tooth tip relief in this paragraph. The parameters affect most of dimensions of the toothing, tooth shape and the following strength parameters, stiffness, durability, noise, efficiency and others. If you don't know the accurate parameters of the production tool, use the standardized type from the list box on line [3.1], namely:

1. DIN 867 (a=20deg, ha0=1.25, hf0=1.0, ra0=0.38, d0=0, anp=0deg, ca=0.25) for calculation in SI units and
3. ANSI B6.1 (a=20deg, ha0=1.25, hf0=1.0, ra0=0.3, d0=0deg, anp=0, ca=0.35) for calculation in inches.

External toothing.

You can define two types of the tool in the form, with protuberance (A) and without protuberance (B). If you define a tool without protuberance, set the protuberance dimension d0=0. Set the tool dimensions according to the dimensions in the picture in the modulus multiples "value" x "modulus" (calculation in SI units) or as a quotient of "value"/"Diametral Pitch" (inch calculation). Choose a pressure angle in paragraph [4]. The base of tooth can be either bevelled or rounded. Therefore choose only one way.

The diagram shows the shape of a tool tooth for wheel/pinion. If you change tool dimensions, press the respective button which will provide redrawing in accordance with the current set values.

An accurate shape of tooth and toothed wheel, check of interferences, etc. is described in the paragraph dealing with graphic output and CAD systems.

Internal toothing.

Internal toothing is made, in overwhelming majority of cases, by cutting using a circular tool. For the purposes of this calculation we will consider a tool with basic parameters identical with the designed toothing (an0=an, b0=b, mn0=mn). However, angle b cannot be chosen at random when manufacturing the internal toothing, as it is necessary to proceed from the properties of the machine tool and available tools and it is appropriate to consult this selection with a technologist.

You can see an example of such tool in the picture. The current status of tool sharpening equates its unit correction x0. Resharpening of the tool results in the change of correction and subsequently in the change of tool hub diameter. If the value for correction x0 is unknown, just measure the current hub diameter and use the change of correction x0 [3.13] to adjust the hub diameter da0 [3.14] to a required value.

3.11 Unit tooth tip relief.

Unit tooth tip relief "ca" affects the diameter of addendum circle. Normally ca=0.25 is chosen which guarantees preventing interference for normally used corrections. If the accurate tool parameters are known, it is possible to choose smaller c*, namely 0.15 to 0.1, and thus achieve increase of profile bite coefficient. Interference can and should be checked on a detailed drawing, see the paragraph dealing with graphic output and CAD systems. Line [3.10] gives minimum tooth tip relief which can be achieved using the selected tool. The selection of smaller tool tip relief is indicated by red color of input field. Button "<" will transfer the minimum value into the input field. Minimum unit tooth tip relief can be reduced by increasing the height of the tool base.

Note: The planet has two unit head clearances (sun => planet and planet => ring gear) while both are dependent and only one of them may be specified. That can be done by checking the option on the right.

Design of a module and geometry of toothing. [4]

The geometry of the gearing can be designed in this paragraph. The design of the geometry substantially affects a number of other parameters such as functionality, safety, durability or price.

4.1 Number of planets.

Choose the number of planets. Usually 2 to 6 planets are used; 3 planets are used most often. The green box shows the maximum number of planets possible for the respective number of teeth of the sun and ring gear. If the assembly condition is not complied with, the cell number turns red.

4.2, 4.3 Number of teeth.

Use line [4.3] to set the number of sun and ring gear teeth. The number of planet teeth is calculated automatically. As the number of teeth in the planetary gearing cannot be chosen at random (see the theoretical part), erroneous or improper combinations of the number of teeth are signalized by a red number. For the sake of simplifying, line [4.2] contains the buttons that enable you to increase/decrease the number of teeth while the number of teeth and addendum modifications are calculated in order to keep the assembly conditions.

Hint: If the red numbers signal a problem in assembly, use the buttons on line [4.2] to fine-tune the number of teeth in order to maintain the assembly.

The number of planet teeth can be changed in a narrow range from the optimum value (+-2). Perform the change by selecting from the list on line [4.2]. After the change, the values of the addendum modification are calculated automatically in order to comply with the condition of alignment.

Note: The change in the number of planet teeth is important e.g. if we want to avoid a commensurable number of teeth for the sun and planet, or to improve the qualitative indicators (change in unit correction).

In general, the principle applies that increasing the number of teeth (in case of the same distance of axes) results in:

• • increase in surface capacity (contact, sticking, wear)
• • improvement in mesh ratio
• • reduction in bending capacity
• • reduction in production costs

Recommended values:

In general, a higher number of teeth is recommended for helical gearing and higher performances.

4.5 Normal pressure angle.

This angle determines parameters of the basic profile and is standardized to an angle of 20°. Changes in the pressure angle a/F affect functional and strength properties. Changes in the meshing angle, however, require non-standard production tools. In case there is no special need to use another meshing angle, use the value of 20°.

The letter "X" marks the basic circle.

Increasing the meshing angle allows:

• reduction in the danger of undercutting and interference
• to reduce slipping speeds
• increased loading capacity in contact, seizure and wear
• increased rigidity of the toothing
• increased noise and radial forces

Option of values

• Straight toothing with increased loading capacity requirement - 25 to 28 degrees
• Helical toothing - up to 25 degrees
• Gearing with a special requirement for quietness - 15 to 17.5 degrees

Recommended values:

In case you do not have any special requirements for the designed gearing, it is recommended to use 20°.

4.6 Helix angle.

The gearing with helix angle = 0 (spur gearing) is used for slow-motion and heavily stressed gearings.
The gearing with helix angle > 0 (helical gearing) is used for fast-motion gearings, it has lower noisiness and better load capacity, and enables a lower number of teeth without undercutting.

Recommended values

The beta angle is usually chosen from the following line of 6,8,10,12,15,20, 25,30,35,40 degrees.

Note: In the production of internal toothing (of the ring gear) it is impossible to choose the angle b randomly. It is necessary to apply the properties of the machine tool and the tools available and it is best to consult the choice with the technologist.

4.7 Setting the ratio of the width of the sun to its diameter.

Use the slider to set the value for the dimensionless coefficient, which defines the ratio of the width of the sun to its diameter [4.8].

4.8 Ratio of the width of the sun to its diameter.

This parameter is used to design the size of the module as well as the gearwheel's basic geometrical parameters (width, diameter). The recommended maximum value is given in the right column and depends on the selected gearwheel material, on the method of gearwheel mounting and on the gearing transmission ratio. Use the slider located on line [4.7] to set this parameter. After setting the parameter, press the "Design gearing" button. That way you will design the gearing conforming to the required safety [2.10] and other input parameters.

After the "Gearing design" is over, check the dimensions (gearwheel widths and diameters, weight). If you are not satisfied with the result, adjust the parameter of the pinion width to diameter [4.7, 4.8] and repeat the "Gearing design".

Upon the start of "Gearing design" the modules (DP) are gradually placed into the calculation from the table, the width of the gearing is calculated and the minimum module (DP), which still conforms to the strength conditions, is defined.

Recommended values:

Lower values - narrower gearwheel design, larger module, spur gearing
Higher values - wider gearwheel design, lower module, helical gearing

Note: Exceeding the recommended range is indicated by a change in number colour. It is possible to use values lower than the recommended ones without problem. Values higher than the recommended ones should be consulted with a specialist.

Hint: If you cannot approach the required gearwheel dimensions using the change in this parameter, try to change the number of teeth, helix angle or choose a different material.

4.9, 4.10 Module / Diametral Pitch.

This is the most important parameter that defines the size of the tooth, i.e. of the gearing. In general, it applies that for a higher number of teeth, a smaller module can be used (higher P value for inch version of calculation) and vice versa. In the right drop-down menu, there are standardized module values / Diametral Pitch and the value selected from this menu is added automatically to the box on the left.
Use the selection on the right to switch between the option to set the module or Diametral Pitch.

Note: Designing the correct size of module is a rather complicated task. Therefore, we recommend using the procedure for the design of gearing based on the ratio of the pinion width to diameter [4.5].

4.13 Face width.

The width of the gearing of individual gearwheels is measured using a pitch cylinder. After the button on the right is pressed, the values conforming to the selected ratio yd from line [4.7, 4.8] are completed.

Recommended values:

Depend on the selected material and type of gearing construction [2.1, 2.2, 2.3, 2.5]. See the previous line for the recommended value range.

4.14 Working face width.

This is a common width of both gears on rolling cylinders. If the gears are not in offset positions (Fig. 4.1), it is mostly the width of the gear. This width is used for strength checks of toothing.
In case the check box in this row is enabled, the "Working width of toothing" is filled out automatically with the lower value of the width of toothing from the previous row [4.9]

4.17 Approximate weight of the gearing.

This is calculated as the weight of full cylinders (without weight removal and holes). It serves for quick orientation during the design.

Note: For internal toothing, the weight of wheel is counted as a tube with thickness equal to tooth depth.

4.18 Minimum coefficient of safety.

The line always gives the lowest of the coefficients for individual gearwheels. The first column specifies the safety coefficient for surface durability, the second column gives the safety coefficient for bending durability.

4.19 Movement of gears (step and current angle).

As it is not always easy to imagine the mutual movement of all gearwheels (especially for a differential movement), it is possible to simulate the movement of individual gearing members. Specify the step which the input member is to use to move and use the slider to change the input member gear angle.

4.20 Normal backlash.

This is the perpendicular (shortest) distance between non-working sides of teeth. A backlash is necessary to create a continuous layer of lubricant on sides of teeth and to overcome production inaccuracies, deformations and thermal expansion of individual elements of the mechanism. Very small clearances are required in gearing of control systems and instruments and if it is not possible to eliminate it, gearing with automatic take up of backlash is usually used. Great backlash must be chosen with heavily loaded gearing (thermal expansion) and high-speed gearing (hydraulic resistance and shocks with pushing of oil off the inter-tooth gaps.

Recommended values:

In practice, the recommended values are chosen empirically and you can follow the recommended values in row [4.21].

After entering the backlash, the working axis distance [6.10] is modified so that the entered backlash is created.

Note: After the change in requested normal backlash, the condition of mutual alignment is violated, therefore, it is necessary to recalculate the coefficients of the addendum modification, see [5.0].

Correction of toothing. [5]

The correction of external gearing or internal gearing proper may be used to affect a range of parameters. First of all, it is necessary to ensure functioning, and then the performance or strength parameters may be improved. In case of planetary gearing, the situation is more complicated. The correction of individual wheels cannot be changed "randomly". First of all, the alignment must be provided, which means that the centre distance of the sun and the planet must be the same as the centre distance of the planet and the ring gear. That means that the corrections depend on each other and if e.g. the correction of the planet is changed, the correction of the sun and ring gear must also be changed in order to maintain the condition of alignment. In this paragraph, you can choose/change the correction of individual gearwheels while the program controls the gearing parameters and informs you in case of an erroneous or out-of-line setting.

While changing the correction, you can also check the most important qualitative parameters, e.g. mesh ratio, specific slip and safety.

Pictures in the calculation.

On the left there is a detail of a machine tool (the machining process may be simulated). The accurate shape of the tooth is black and the accurate shape of the machine tool is green. On the right there is a detail of the mutual position of a pitch, tip, root and base diameter in the mesh point (dashed - root diameter, dot-and dash - pitch diameter, solid- tip diameter).

5.1 Types of corrections.

Line [5.2-5.4] specifies some minimum values of corrections in order to achieve the selected parameters of individual gears.

5.2 Permissible undercutting of teeth.

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

This is the minimum value of a correction which can be used without admissible (minor, tolerable) undercutting of teeth.

5.3 Preventing undercutting of teeth.

5.4 Prevents tapering of teeth.