7 Spur Gear Pair Calculation According to DIN 3990 and Other Standards

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Chapter 7
Spur Gear Pair Calculation According to DIN 3990 and Other Standards

    7.1   Start the Calculation Module
    7.2   Input of Geometry Data
    7.3   Input of Tool Data
    7.4   Input for the Determination of Allowances
    7.5   Representation of Gear Tooth Form
    7.6   Calculation of Gear Load Capacity
    7.7   Input of Gear Tooth Profile and Gear Flank Modifications
    7.8   Meshing Interferences for External Gears
    7.9   Internal Gears
    7.10   Input of Geometry Data for Internal Gears
    7.11   Manufacturing Process for Internal Gears
    7.12   Meshing Interferences for Internal Gears
    7.13   eAssistant: Examples for Internal Gears
    7.14   Message Window
    7.15   Quick Info: Tooltip
    7.16   Calculation Results
    7.17   Documentation: Calculation Report
    7.18   How to Save the Calculation
    7.19   ‘Redo’ and ‘Undo’ Button
    7.20   The Button ‘CAD’
    7.21   The Button ‘Options’
    7.22   How to Calculate the Accurate Tooth Form of Involute Splines

7.1 Start the Calculation Module

Please login with your username and your password. Select the module ‘Cylindrical gear pair’ through the tree structure of the project manager by double-clicking on the module or clicking on the button ‘New calculation’.

Figure 7.1:

General overview

The eAssistant module allows an easy and fast calculation of the geometry of cylindrical gears according to DIN 3960, DIN 3961, DIN 3964, DIN 3967, DIN 3977 and DIN 868. The load capacity according to DIN 3990, ISO 6336 as well ISO/TR 13989 (scuffing) is considered as well. You can calculate external and internal spur and helical gears. The profile shift, the addendum chamfer and allowances are also integrated into the calculation.

7.2 Input of Geometry Data

All important calculation results will be calculated during every input and will be displayed in the result panel. A recalculation occurs after every data input. Any changes that are made to the user interface take effect immediately. Press the Enter key or move to the next input field to complete the input. Alternatively, use the Tab key to jump from field to field or click the ‘Calculate’ button after every input. Your entries will be also confirmed and the calculation results will displayed automatically. If the result exceeds certain values, the result will be marked red.

7.2.1 Normal Module

The normal module $mn$ is a basic parameter in the gear geometry and describes the size of a gear. Please note that the larger the module the larger the teeth. To represent the tooth size, the circular pitch and the diametral pitch are also used. The module is defined in $mm$ and is determined by the number of teeth. To modify the variety of the gearings, the module is standardised (see tables). The calculation with the eAssistant is possible with any modules including several decimal places.


Series of modules in mm according to DIN 780 series 1 (part 1)

0,05	0,06	0,08	0,10	0,12	0,16	0,20	0,25

0,3	0,4	0,5	0,6	0,7	0,8	0,9	1


Series of modules in mm according to DIN 782 series 1 (part 2)

1,25	1,5	2	2,5	3	4	5	6	8

10	12	16	20	25	32	40	50	60


Series of modules in mm according to DIN 780 series 2 (part 1)

0,055	0,07	0,09	0,11	0,14	0,18	0,22	0,28	0,35

0,45	0,55	0,65	0,75	0,85	0,95	1,125	1,375	1,75


Series of modules in mm according to DIN 780 series 2 (part 2)

2,25	2,75	3,5	4,5	5,5	7	9	11

14	18	22	28	36	45	55	70

7.2.2 Pressure Angle

The pressure angle is the angle between the line-of-action and the common tangent to the pitch circles. With an increasing distance from the base circle, the profile angles $αy$ increase too. The most common pressure angle now in use for spur gears is $20∘$ . This pressure angle is usually preferred due to its stronger tooth shape and reduced undercutting. The $25∘$ pressure angle has the highest load-carrying ability, but is more sensitive to center-distance variation and hence runs less quietly. The choice is dependent on the application. Open the program, a pressure angle of $20∘$ appears.

7.2.3 Helix Angle

For spur gears the helix angle is $β$ = $0∘$ , for helical gears the angle $β$ is up to $45∘$ due to the fact that the teeth for a helical gear are inclined by the angle. $45∘$ is also the maximum value that you can enter into the input field for the helix angle. For an external gearing a right-hand teeth and a left-hand teeth can only mesh correctly. For internal gearings pinion and gear must have the same direction.

Helical Gears

Helical gears are used to transmit power or motion between parallel shafts. Helical gears differ from spur gears in that they have teeth that are cut in the form of a helix on their pitch cylinders instead of parallel to the axis of rotation. As two teeth on the gear engage, it starts a contact on one end of the tooth which gradually spreads with the gear rotation, until the time when both the tooth are fully engaged. Finally, it recedes until the teeth break contact at a single point on the opposite side of the wheel. Thus force is taken up and released gradually. Helical gears offer a refinement over spur gears. The angled teeth engage more gradually than do spur gear teeth. This causes helical gears to run quieter and smoother than spur gears. Helical gears are used in areas requiring high speeds, large power transmission or where noise prevention is important.

7.2.4 Standard Centre Distance

The centre distance is the distance between the centre of the shaft of one gear to the centre of the shaft of the other gear. If you change the number of teeth, the standard centre distance $a d$ is modified automatically. The standard centre distance is an operand. If the sum of the profile shift coefficients = 0, $a d$ corresponds to the working centre distance $a$ .

7.2.5 Working Centre Distance

The working centre distance $a$ is the distance between the axes. In case of changing the normal module $mn$ , the working centre distance is determined automatically. If the profile shift is too large, the working centre distance can be modified manually at any time. If the standard centre distance and the working centre distance are equal, the profile shift coefficients will be set to the value ‘0’ automatically.

Figure 7.2:

Standard centre distance and the working centre distance

Enter the value ‘13’ for the number of teeth for gear 1 and the number of teeth ‘63’ for the gear 2, a ‘5’ normal module and a helix angle of $β$ = $15∘$ . The standard centre distance and the working centre distance are determined automatically.

Figure 7.3:

Inputs

Enter the value ‘0’ for the working centre distance into the input field and confirm with ‘Enter’ key or click on the ‘Calculate’ button.

Figure 7.4:

Input ‘0’

The standard centre distance and the working distance are equal.

Figure 7.5:

Centre distance

7.2.6 Direction of Helix Angle

Enter a value for the direction of the helix angle. When the gear is placed on a flat surface, the teeth of a left-hand gear lean to the left and the teeth of a right-hand gear lean to the right. It should be noted that a pair of helical gears on parallel shafts must have the same helix angle $β$ . However, the helix directions must be opposite, i.e., a left-hand mates with a right-hand helix. For an external gear pair the engaged gearings have different directions, internal gears have the same direction with the same helix angle. Find further information in the section 7.10.1 ‘Direction of Helix Angle’.

Figure 7.6:

Left and right-hand teeth

Select the option ‘left’ for gear 1. That means: Gear 1 is left-handed, gear 2 is right-handed (for external gears).

Figure 7.7:

Option

Select the option ‘left’ for gear 2. That means: Gear 2 is left-handed, gear 1 is right-handed (for external gears).

Figure 7.8:

Option

7.2.7 Number of Teeth

The number of teeth of a gear describes the number of the teeth on the full rim. The number of teeth is positive for external gears and negative for internal gears. Please note that the smaller the number of teeth the larger the influence of the profile shift. Find more information about the profile shift coefficient in section 7.2.9 ‘Profile Shift Coefficient’. In section ‘Internal gearings’ you will get more information about the number of teeth for internal gearings.

7.2.8 Facewidth

The facewidth $b$ is the length of the gear teeth as measured along a line parallel to the gear axis.

Figure 7.9:

Facewidth

Enter a value for the facewidth. The following table shows some additional information about the facewidth $b$ as well as minimum number of teeth $z$ .

Figure 7.10:

Enter the facewidth


Standard Values for the Facewidth $b$ and Minimum Number of Teeth $z$ ¹

Teeth, machine-cut	Gears on rigid shafts, that run in roller or excellent plain bearings, rigid substructure	$b ≤ 30 ...40 ⋅m$

	Gears in usual gear boxes, roller or plain bearings	$b ≤ 25 ⋅m$

	Gears on steel constructions, beams and suchlike	$b ≤ 15 ⋅m$

	Gears with excellent bearing in high duty gearings	$b ≤ 2 ⋅d1$

Teeth, cast roughly	Overhung gears	$b ≤ 10 ⋅m$

Gears with high circumferential velocity $(υ > 4m ∕s)$ and considerable power, when $ɛα > 1,5$		$z1 ≥ 16$

Gears with mean circumferential velocity $(υ = 0,8...4m∕s)$		$z1 ≥ 12$

Gears with low circumferential velocity $(υ < 0,8m∕s)$ or for low power for subordinated purposes		$z1 ≥ 10$

Basically external gearings		$z1 + z2 ≥ 24$

Basically internal gearings		$z2 ≥ z1 +10$

¹ from: Karl-Heinz Decker: Maschinenelemente: Gestaltung und Berechnung, 1992, p. 506, table 23.2

7.2.9 Profile Shift Coefficient

Profile shift can make spur gears or helical gears run more quietly and carry more load. If spacing errors of some magnitude are present, proper profile shift will give the teeth a little clearance at the first point of contact. If a pair of teeth are spaced too close together, there is a bump as the tooth comes into mesh. With the modification there is a little relief at the first point of contact. The profile shift affects the tooth form because the tool is shifted by the value $xm$ towards or away from the tip circle. The calculation of the tip diameter $da$ and root diameter $df$ includes the profile shift coefficient $x$ . According to DIN 3960 the profile shift is

positive when the profile reference line is shifted from the pitch circle to the tip circle,
negative when the profile reference line is shifted from the pitch circle to the root circle.

You can select the profile shift coefficients $x1$ and $x2$ . Please note that no meshing interferences occur. In case meshing interferences occur, you will get an appropriate message in the message window.

Figure 7.11:

Change the tooth form with the profile shift: number of teeth $z$ = 10; tooth 1: $x$ = 0,5; tooth 2: $x$ = 0; tooth 3: $x$ = -0,5

Characteristics of the Profile Shift

A positive profile shift increases the tooth thickness, a negative profile shift decreases the tooth thickness.
With an increasing positive profile shift, the tooth tip thickness and the root fillet become smaller, the axle load and the load capacity increase. This advantage occurs especially for a smaller number of teeth.
The minimum permitted tooth tip thickness determines the limit for a very large profile shift, in particular for very small number of teeth.
The profile shift affects the operating pressure angle as well as the load capacity.
For a small number of teeth and with a negative profile shift, an undercut becomes a problem (see figure in section 7.2.9). The undercut weakens the tooth root and a part of the tooth flank is cut off.

Here you get the possibility to dimension and optimize the profile shift coefficient. To optimize the profile shift coefficient, click on the calculator button.

Figure 7.12:

Profile shift coefficient

Enter either your own value for the profile shift coefficients into the input field or activate the option ‘Balanced specific sliding’. The coefficients will be modified. Enter either your own values for the profile shift coefficients or activate the option ‘Balanced specific sliding’. The factors are modified so that the specific sliding is balanced. The tooth flanks slide and roll on each other. The measure for the sliding velocity and the rubbing wear of the tooth flanks presents the relative sliding, the so-called sliding. The specific sliding is the ratio of the sliding velocity and radial velocity. The specific sliding shows which of the two gears could be damaged by the rubbing wear.

Figure 7.13:

Balanced specific sliding

7.2.10 Tip Diameter

The tip diameter $d a$ depends on the module and will be determined by the program automatically. If you change the profile shift, the tip diameter will change, too. There is the possibility to enable the tip circle using the ‘Lock’ button. Now you can add and modify the tip diameter very easily. Please note that the tip diameter has an influence on the modification of the tip diameter. Click on the button again to disable the input field. The value is determined again according to DIN.

Figure 7.14:

Tip diameter

In case you use a special tool, the tip diameter can be changed by a tool customization. Find out more about the tool data in the section ‘The input of tool data’.

7.2.11 Tip Diameter Allowance

The tip diameter allowance is determined according to DIN. Click on the ‘Lock’ button to enable the input field and enter your own value. If your values are out of range of the DIN, you will get an information in the message window. Click on the ‘Lock’ button and the input field is disabled again. The allowances are determined according to DIN.

Figure 7.15:

Enable the input field

7.2.12 Modification of Tip Diameter

The modification of the tip diameter $k$ is automatically determined by the program that a sufficient tip clearance is available. Click on the ‘Lock’ button to enable the input field and enter your own value. Such a modification of the tip diameter have an effect on the tip diameter.

7.2.13 Tip Clearance

Clearance $c$ is the distance between the root circle of a gear and the addendum circle of its mate. A certain clearance between the gears is necessary for a smooth operation.

Figure 7.16:

Tip clearance $c$

A distinction is made between two different kind of clearances. There is the tip clearance $c$ and the backlash $j$ . Standard gears have got a basic rack profile with a addendum coefficient $h = m a$ or a tool basic rack profile with $h=m fp$ . The dedendum coefficient $h f$ of the basic rack profile or the addendum coefficient $h ap$ of the tool basic rack profile has to be larger due to ensure that tip and root circle of the gears are not in contact.

Backlash $j$

If the gears are of standard tooth proportion design and operate on standard center distance, they would function ideally with neither backlash nor jamming. The general purpose of backlash is to prevent gears from jamming and making contact on both sides of their teeth simultaneously. Any error in machining which tends to increase the possibility of jamming makes it necessary to increase the amount of backlash. Consequently, the smaller the amount of backlash, the more accurate must be the machining of the gears. Runout of both gears, errors in profile, pitch, tooth thickness, helix angle and centre distance - all are factors to consider in the specification of the amount of backlash. In order to obtain the amount of backlash desired, it is necessary to change the tooth thickness or tooth space allowances (please see also section 7.4.8 ‘Backlash Normal Plane’).

7.2.14 Root Diameter and Allowances of Root

The root diameter $df$ depends upon the module, the profile shift and addendum coefficient of the basic rack profile. The root diameter is determined by the program. Therefore, the root diameter occurs as a result of the calculation. The allowances of root result from your calculation and will be determined automatically. The allowances depend upon the tooth thickness allowances. For instance, if you enter the value ‘0’ for a gear, then the allowances of root become ‘0’ for this gear as well.

Figure 7.17:

Root diameter and allowances

7.2.15 Inner and Outer Diameter

Here you can enter an inner diameter (for external gears) and outer diameter (for internal gears). It should be kept in mind that the inner diameter has to be smaller than the root diameter $df$ .

Figure 7.18:

Inner diameter

In case the inner diameter is larger than $df$ , then the program automatically corrects the value and enters the maximum value for the inner diameter. An appropriate message appears in the message window.

7.2.16 Web Width

The web width can be considered here. The web width is shown in the figure next to the input field. There is the possibility to modify the web width by using the ‘Lock’ button.

Figure 7.19:

Input field for the web width

The ‘Lock’ button is still disabled. Enter the values for the inner or outer diameter into the input field. Then the ‘Lock’ button is enabled and the web width gets the same value as the facewidth. In case the web width is smaller than the facewidth, then the gear body stiffness is affected due to the gear body coefficient $C R$ . The tooth spring stiffness changes which affects again the load capacity.

Figure 7.20:

Web width

7.2.17 Chamfer

The chamfer can be considered.

Figure 7.21:

Chamfer

7.2.18 Addendum Chamfer

The tooth ends of a gear are often rounded or chamfered. A chamfer is a small angled surface added on the end of a shaft along an edge. For the calculation you can consider the addendum chamfer. Meshing interferences can be removed by the addendum chamfer.

Figure 7.22:

Addendum chamfer

Please Note: If you define the geometry of the gear pair, you are able to look at the tooth form. Click on the button ‘Tooth form’ and select ‘Total view’ or ‘Detail view’ (find more information on the tooth form and its functions in section 7.5 ‘Representation of Tooth Form’). Click the button ‘Geometry’ and you get to the geometry input again.

7.3 Input of Tool Data

For the selection of the manufacturing process you have to consider the material, size of the gear, quantity, gear type (external or internal gears) and accuracy. The many methods of making gear teeth must be considered as well. The calculation program distinguishes between gear-tooth cutting and gear hobbing.

Figure 7.23:

Input mask for tool data

Please note: If you want to add some own notes, comments or a description, then use the comment line.

7.3.1 Tool

The most important manufacturing processes are gear hobbing and gear shaping. Select either the tool ‘Hob’ or ‘Gear shaper cutter’ by clicking the listbox. A ‘Constructed involute’ is also available. Basically, the selection of the tool depends on the gear type (external or internal gears). The external gears can be produced by cutting wherein the gear cutting tool is a hob. For internal gears a gear shaper cutter is used (see section 7.11 ‘Manufacturing Process for Internal Gears’).

Figure 7.24:

Selection of tool

Hob Generation

The hobbing is the most widely used method of cutting gear teeth. The hobbing process is quite advantageous in cutting gears with very wide facewidth. A very high degree of tooth-spacing accuracy can be obtained with hobbing. With regard to accuracy, hobbing is superior to the other cutting processes. A wide variety of sizes and kinds of hobbing machines are used. The rotating hob has a series of rack teeth arranged in a spiral around the outside of a cylinder, so it cuts several gear teeth at one time. To generate the full width of the gear, the hob slowly traverses the face of the gear as it rotates. Thus, the hob has a basic rotary motion and an unidirectional traverse at right angles. Both movements are relatively simple to effect, resulting in a very accurate process.

Field of Application of the Hob:

Recommended for gears with very wide facewidth
Recommended for external spur and helical gears up to module ‘40’ (Please keep in mind: it is an expensive tool for large modules)
Recommended for all basic rack profiles
The helix angle is arbitrary.

Figure 7.25:

Hob and gear shaper cutter

Gear Shaper Generation

The shaping process is a gear-cutting method in which the cutting tool is shaped like a pinion. If a gear is provided with cutting clearance and is hardened, it may be used as a generating tool in a gear shaper. The cutter reciprocates while it and the gear blank are rotated together at the angular-velocity ratio corresponding to the number of teeth on the cutter and the gear. The teeth on the gear cutter are appropriately relieved to form cutting edges on one face. Although the shaping process is not suitable for the direct cutting of ultra-precision gears and generally is not as highly rated as hobbing, it can produce precision quality gears. Usually it is a more rapid process than hobbing. Two outstanding features of shaping involve shouldered and internal gears. For internal gears, the shaping process is the only basic method of tooth generation.

Field of Application of the Gear Shaper Cutter:

Recommended for internal and external spur and helical gears
Racks
Special gearings, e.g., splined shaft connections, face or chain gears

Constructed Involute

In addition to the hob and the gear shaper cutter, you can also select the entry ‘Constructed involute’ as a tool. In case internal gears cannot be shaped with a gear shaper cutter, the tooth form calculation is still possible by using the constructed involute. This specifically applies for applications in the precision mechanics. This method allows a generation of the tooth form with a constant root fillet radius.

Figure 7.26:

Constructed Involute

Representation of Hob and Gear Shaper Cutter

The representation shows either the hob basic rack profile or the gear shaper cutter tooth profile. The radio buttons enable you to choose one of the graphical representation.

Figure 7.27:

Tool

7.3.2 Basic Rack Tooth Profile

To mesh two gears with each other, the parameters have to be coordinated. According to DIN 867 a rack is the basic rack profile. A gear with an infinite number of teeth will have straight lines for both the pitch and the base circles. The involute profile will be a straight line. The rack can be used to determine the basic parameters. Racks can be both spur and helical. A rack will mesh with all gears of the same pitch. The pressure angle and the gears pitch radius remain constant regardless of changes in the relative position of the gear and rack.

The following standard basic rack profiles are available for your calculation. Choose your profile from the listbox.

Figure 7.28:

Listbox for the basic rack profile

ISO 53 Profile A: is recommended for gears transmitting high torques
ISO 53 Profile B: is recommended for normal service
ISO 53 Profile C: is recommended for normal service, type C may be applied for manufacturing with some standard hobs.
ISO 53 Profile D: is recommended for high-precision gears transmitting high torques and consequently with tooth flanks finished by grinding or shaving. Care should be taken to avoid creating notches in the fillet during finishing which could create stress concentrations.
DIN 3972 Profile I
DIN 3972 Profile II
Profile 1 DIN 867
Profile 2 DIN 867
Profile 3 DIN 867
Profile 4 DIN 867

In addition to the standard basic rack profiles, you can also select a protuberance tool. When part of the involute profile of a gear tooth is cut away near its base, the tooth is said to be undercut. By using a protuberance tool an undercut near the root can be generated. Grinding notches at the tooth flank can be avoided during the grinding. That provides relief for subsequent finishing operations (see section 7.3.4 ‘Protuberance’).

Figure 7.29:

Selection of the protuberance tools

You can select the following profiles:

Prot 1.4- $6∘$ /0.085
Prot 1.5- $∘ 6$ 0.02
Prot 1.6- $6∘$ /0.02
Prot 1.4- $8∘$ /0.04
Prot 1.4- $∘ 8$ /0.066
Prot 1.4- $10∘$ /0.05
Prot 1.5- $10∘$ /0.02
Prot 1.6- $∘ 10$ /0.02
Prot 1.25- $14∘$ /0.024

Please Note: If you select ‘user defined input’, then the input fields for the edge radius, the addendum coefficient and the dedendum coefficient are activated. Now you can modify the basic rack profile.

Figure 7.30:

Own input

Modification of the Basic Rack Profile

In case you use special tools, the eAssistant offers an easy and comfortable solution. As mentioned above, the basic rack profile can be specified by the entry ‘user defined input’.

Figure 7.31:

Button for the tool dimensioning

Here you can change the tip and the root diameter for gear 1 and gear 2. Confirm your entries with the button ‘OK’. The listbox for the basic rack profiles displays then ‘user defined input’.

Figure 7.32:

Tool dimensioning

7.3.3 Tip Form

Select the entry ‘Gear shaper cutter’ and the listbox ‘Tip form’ is enabled. Then choose between ‘Full radius’ and ‘Radius with straight line’.

7.3.4 Protuberance

Undercut may be deliberately introduced to facilitate finishing operations. Undercut is the loss of profile in the vicinity of involute start at the base circle due to tool cutter action in generating teeth with low numbers of teeth. The protuberance cuts an undercut at the root of the gear tooth. The protuberance design is also used in some cases to permit the sides of gear teeth to be ground without having to grind the root fillet.

Corrections of the Tooth Form in the Calculation Program

Protuberance: Shaper cutters can be made with a protuberance at the tip. The protuberance cuts an undercut at the root of the gear tooth. This provides a desirable relief for a shaving tool.
Radius corners: The corners of cutter teeth are radiused and produce a controlled fillet in the root corners of the gear being generated. A smooth surface is created, notch effects are decreased and strength is increased.

The gearing tools for modified tooth forms may be used only in a specified range of number of teeth. The range of number of teeth itself is dependent upon the number of teeth of the working wheel and the allowed tolerances.

Determination of the Amount of the Protuberance from the Height of the Protuberance Flank

The following equation determines the amount of the protuberance. In case the height of the protuberance flank is given and not the amount of the protuberance, the amount of the protuberance may be calculated by this equation.

* (h*prp0 - ρ*a0 ⋅(1- sin(αp)))⋅sin(αn - αp) * pprp0 =---------------cos(α-)-------------- + ρa0 ⋅(1 - cos(αn - αp)) p

The following figure shows a representation:

Figure 7.33:

Height of the protuberance flank

To avoid grinding steps, a deviation in the tooth root area of the profile is a common and allowed method. Because of a grinding stock allowance, an undercut must be allowed. Hence, a larger tooth root thickness is necessary. The following table shows some determination of the undercut dependent upon the module.


Undercut for Ground Gears Dependent upon Module²

Module $m$	Allowance $q$	Protuberance $prp0$	Addendum Coefficient $hap$	Edge Radius $ρao$

2	0,160	0,260	2,900	0,500

2,5	0,170	0,280	3,625	0,625

3	0,180	0,300	4,350	0,750

4	0,200	0,340	5,800	1,000

5	0,220	0,380	7,250	1,250

6	0,240	0,420	8,700	1,500

7	0,260	0,460	10,150	1,7500

² from: Linke, H.: Stirnradverzahnung Berechnung Werkstoffe Fertigung, Carl Hanser Verlag, MÃ¼nchen, Wien, 2nd ed. 2010, p. 68, table 2.1/2

7.3.5 Machining Allowance

You can consider an allowance for the tooth flank. The tool provides an allowance $q$ on the flank and/or root for the pre-cutting tool. The allowance is the smallest distance between the involutes and the pre-machining having the same root diameter. In case you select the tool basic rack profile with protuberance, the allowance refers to the tooth flank. If the allowance of the tool basic rack profile is selected without protuberance, then tooth flank and tooth root get the allowance.

Figure 7.34:

The eAssistant provides the following allowances for the grinding of a gear: a) Constant allowance with bottom of the tooth space, b) Protuberance: Cutter tooth profile is built up on the tip to provide an undercut near the root of the gear being generated.


Maximum Machining Allowances³

Allowance per Tooth Flank	Manufacturing Process

$<$ 0,05 (0,10) mm	Finishing operation by cold rolling, gear shaving, honing, lapping

0,05 to 0,5 (1,5) mm	Grinding, profile grinding, (honing)

$>$ 0,5 mm, pre-cutting	Primary shaping, forming, cutting with geometrically determined edges except shaving, grinding and profile grinding in special cases

³ from: Linke, H.: Stirnradverzahnung Berechnung Werkstoffe Fertigung, Carl Hanser Verlag, MÃ¼nchen, Wien, 1996, p. 638

7.4 Input for the Determination of Allowances

A manufacturing of work-pieces with accurate nominal dimensions is impossible. Hence, a deviation from the nominal size has to be allowed. For a lot of applications the gear and the pinion of a pair must be independently manufactured and meshed without any modifications. That means, the parts have to be separately replaceable.

Figure 7.35:

Input of allowances

7.4.1 Gear Quality

The choice of the right toothing quality is determined by economical aspects depending upon the intended purpose and manufacturing process. In all fields of gearing, the control of gear accuracy is essential. Several classes or grades of accuracy can be set. 12 grades (12 to 1) are defined according to DIN standards. High accuracy grades can be set for a long-life, high speed gears. Lower accuracy grades will cover medium- or slow-speed grades. Accuracy grade ‘1’ describes the highest possible accuracy, ‘12’ a very low accuracy. The gear accuracy ‘1 to 4’ is mainly used for master gears, quality ‘5 to 12’ is used for gears (figure from: Niemann, G.: Maschinenelemente, Vol. 2, Getriebe allgemein, Zahnradgetriebe-Grundlagen, Stirnradgetriebe, 1989, p. 73, figure 21.4/1).

Figure 7.36:

Tolerances according to the manufacturing process

Select the appropriate quality between 1 and 12 by using the following listbox.

Figure 7.37:

Listbox for the selection of quality

The following table provides some reference values for the selection of the quality, tolerances for gearings made of metal and plastics:


Toothing Made of Metal⁴

$ν$	Machining	Quality	Tolerance Sequence
bis m/s	of Tooth Flanks	(Accuracy)	DIN 3967

0,8	cast, raw	12	2x30

0,8	rough-machined	11 or 10	29 or 28

2	finish milled	9	27

4	finish milled	8	26

8	fine finished	7	25

12	shaved or ground	6	24

20	precision-ground	5	23

40	precision-machined	4 or 3	22

60	precision-machined	3	22 or 21

Toothing Made of Injection Molding Plastics³

Application	d	Quality	Tolerance Sequence
	in mm	(Accuracy)	DIN 3967

Gearings with high requirements	to 10	9	27

Gearings with high requirements	10 to 50	10	28

Gearings with normal requirements	10 to 50	11	29

Gearings with low requirements	to 280	12	2 x 30

Toothing made of plastic manufactured by cutting³

Gearings with high requirements	to 10	8	25 to 27

Gearings with high requirements	10 to 50	9	26 to 28

Gearings with normal requirements	to 50	10	27,28

Gearings with normal requirements	50 to 125	11	27,28

Gearings with low requirements	to 280	12	28

⁴ from: Karl-Heinz Decker: Maschinenelemente: Gestaltung und Berechnung, 1992, p. 512, table 23.3

7.4.2 Backlash Allowance and Tolerance Sequence

The system for gearings is very similar to the DIN system of fits and tolerances. For the system of fits for gear transmissions letters are used to indicate the deviation from basic (nominal) size, a number defines the width. There are clearance fits for gearings, therefore, lower case characters ‘h’ to ‘a’ appear. If you select the entry ‘user defined input’, the input field for the tooth thickness allowances is enabled and you can define your individual values.

Figure 7.38:

Own input

7.4.3 Tooth Thickness Allowance

One of the most important criteria of gear quality is the specification and control of tooth thickness. The magnitude of tooth thickness and its tolerance is a direct measure of backlash when the gear is assembled with its mate. Dimensional changes, due to thermal expansion, do not allow a zero-backlash assembly. The tooth thickness allowance has to be determined that no jamming occurs. To prevent that jamming of gears during the operation, it is necessary to decrease tooth thickness by a minimum amount ( $A sne$ and $A sn$ ).

Figure 7.39:

Lower and upper tooth thickness allowances for gear 1 and gear 2

The tooth thickness allowances for teeth of external and internal gearings have to be negative. Then a backlash occurs (find more information on the backlash in section 7.4.8 ‘Backlash Normal Plane’).

The eAssistant offers the possibility to specify the tooth thickness allowances based on measured data or given test dimensions. Click on the ‘Calculator’ button.

Figure 7.40:

‘Calculator’ button

A new window is opened.

Figure 7.41:

Calculation of tooth thickness allowances

Activate gear 1 and gear 2 and enter the input values. Confirm with the button ‘OK’. The ‘Lock’ button next to the input field for the tooth space allowances is enabled. Now you can change the tooth space allowances.

7.4.4 Tooth Space Allowance

The tooth space allowance $A W$ is the difference between the actual dimension and the nominal dimension of the span measurement $W k$ . The actual measurement of the span measurement gets smaller for external gears by negative allowances for a zero-backlash assembly. The upper and lower tooth space allowance are displayed as well. For an own input of the tooth thickness allowances, the tooth space allowances can be defined as well. The ‘Lock’ button next to the input field of the tooth space allowances is enabled. Therefore, you can change the tooth space allowances.

Figure 7.42:

Tooth space allowance for gear 1 and gear 2

7.4.5 Measurement of Tooth Thickness

The tooth thickness of a gear may be measured directly with calipers or it may be determined indirectly by diameter pins. The sizing of gears may be controlled by double-flank composite checks and centre distance settings corresponding to maximum and minimum tooth thickness specifications. Different measurement methods are used:

At pitch circle (chordal),
Span measurement across several teeth,
Measurement over pins or balls that are placed in diametrically opposed tooth spaces,
Check of the centre distance allowance with zero-backlash engagement by using a master gear in a flank roll tester.

In the following you get some information on the widely used measurement methods:

Span measurement $W k$
Measurement by diameter over balls or pins, the measurement by using balls and pins

Span Measurement across Several Teeth

Span measurement $Wk$ is the measurement of the distance across several teeth in a normal plane. As long as the measuring device has parallel measuring surfaces that contact on an unmodified portion of the involute, the measurement will be along a line tangent to the base cylinder. It is a widely used method for gauging the tooth thickness by using the span measurement. The tooth thickness of spur or helical gears is often measured with calipers. An advantage is that the dimensions can be influenced during the manufacturing.

Figure 7.43:

Span measurement

The calculation program determines the number of teeth for the span measurement (number of teeth across the span measurement has to be gauged). By using the ‘Lock’ button you are able to activate the input field and you can enter your own input value. If you click the button again, the previous input value appears.

Figure 7.44:

Number of teeth for the span measurement

Tooth Thickness Measurement by Diameter over Pins or Balls

The tooth thickness is often checked by measurement over pins $M dR$ or balls $M dK$ . The pins or balls are placed in diametrically opposed tooth spaces (even number of teeth) or nearest to it (odd number of teeth). Measurement over pins is the measurement of the distance taken over a pin positioned in a tooth space and a reference surface. The reference surface may be the reference axis of the gear, a datum surface or either one or two pins positioned in the tooth space or spaces opposite the first. The measurement over pins is only used for spur gears and external helical gears. For the measurement values a distinction is made between:

Measurement over balls $MdK$
Measurement over pins $MdR$
Measurement over pins for a spur gear
Measurement over pins for external helical gears with even number of teeth
Measurement over pins for external helical gears with odd number of teeth

For an external gear the measurement over balls $MdK$ is the largest outer measure. The two balls are placed in diametrically opposed tooth spaces. The balls have to be in the same plane perpendicular to a gear axis. For an internal gear (see figure: ‘Internal spur gear with odd number of teeth’) the measurement over balls is the smallest inner measure between the balls. $DM$ is the diameter of ball or pin. The internal gear is generally checked for tooth thickness with measuring pins, like the external gear. However, the measurement is made between the pins instead of over pins.

Measurement over balls: External spur gear with evennumber of teeth

Measurement over balls: External spur gear with oddnumber of teeth

Measurement over balls: Internal spur gear with oddnumber of teeth

The eAssistant already specifies the diameter of ball or pin for the test dimensions. Enable the input field by clicking the ‘Lock’ button. Enter your own input value for the diameter. If you click on the button once again, the previous input value appears.

Figure 7.45:

Diameter of ball or pin

Please Note: In the calculation report you can find all results for the span measurement or measurement over balls and pins in section ‘Test dimensions’.

7.4.6 Tolerance Field for Centre Distance

The general purpose of backlash is to prevent gears from jamming and making contact on both sides of their teeth simultaneously. The center distance and the gear fits have an important influence on the backlash. The gear fit selection defines the tolerances of the centre distance with the backlash. The gear fit selection provides only one tolerance field. The allowances are indicated for the ‘JS’ field. These conform to the ISO basic tolerances. The backlash is dependent upon the tooth thickness allowances and the tooth space allowances. Hence, if you change the centre distance, then the backlash is changed, too.

Figure 7.46:

Tolerance field for the centre distance

Select the option ‘user defined input’ from the listbox. Now you are able to enter your own centre distance allowances. Confirm your entries with Enter. The backlashes are automatically determined.

7.4.7 Centre Distance Allowance

The centre distance allowance $Aa$ is the allowed deviation of the centre distance from the nominal centre distance. The allowances are indicated with $±$ to get no improper major allowances from the nominal centre distances with gears having several axes.

Figure 7.47:

Centre distance allowance

7.4.8 Backlash Normal Plane

A gear fit has to be determined, so that two gears can be meshed. For that, a proper backlash must be provided for the mesh to avoid jamming of the gears. The eAssistant offers three different backlashes: the backlash normal plane, the backlash pitch diameter and the radial backlash.

Figure 7.48:

Backlash normal plane

Besides errors in manufacturing and assembling, the variation in backlash will depend considerably on the tooth thickness tolerances and centre distance of the gears. The DIN system represents a standard centre distance and provides the backlash by changing the tooth thickness. The backlash between the meshing teeth adjusts the deviations of the tooth thicknesses, centre distance and tooth form using the tooth thickness $A sni$ and tooth space allowances $A sne$ . The lowest tooth thickness allowance $A sni$ indicates the maximum backlash, the upper tooth thickness allowance indicates the minimum backlash $A sne$ . In addition to the tooth thickness allowance and centre distance allowance, errors in profile and pitch are also factors to consider in the specification of the amount of backlash.

Please note: The backlash depends also on thermal expansions, deformation of elementes and displacement of casing. These impacts must be considered for the determination of the tooth thickness.

7.4.9 Backlash Pitch Diameter

The backlash pitch diameter $j t$ refers to the backlash at the pitch circle. The backlash pitch diameter may be the length of the pitch circle arc in which the gear rotate against its mating gear.

Figure 7.49:

Backlash pitch diameter

7.4.10 Radial Backlash

The radial backlash is the difference of the centre distance between the working condition and zero-backlash engagement. The radial backlash $j r$ matters especially for very small modules (m $<$ 0,6 mm).

Figure 7.50:

Radial backlash

7.5 Representation of Gear Tooth Form

A special highlight of this calculation module is the presentation of the accurate gear tooth form with an animation and simulation of the tooth mesh. For the presentation you can select the lower, upper and mean allowances for the tooth thickness, tip diameter and centre distance. When you define the geometry for the gear pair, then you can have a look at the tooth form at any time. Click on the ‘Tooth form’ button and you get a general or detailed view of the tooth form. By clicking the ‘Geometry’ or ‘Tool’ buttons, you can open the main input masks of the calculation module again.

Figure 7.51:

Tooth form

Please Note: Please keep in mind that all values are later taken over to the DXF output and CAD generation. In case you change the tooth thickness allowance or the centre distance allowance in the tooth form mask, then the last modification is taken over to the DXF output. The Section 7.20 ‘CAD button’ contains some helpful information on this function.

Figure 7.52:

DXF output

7.5.1 Representation of Cylindrical Gear Pair

Click on the ‘Tooth form’ button to represent the gear tooth form.

Figure 7.53:

Cylindrical gear pair

Please Note: Please keep in mind that you can check the backlash and the mesh ratio only in the presentation of the mesh. The gear mesh will be discussed in Section 7.5.2 ‘Representation of Mesh’.

7.5.2 Representation of Mesh

Click on the ‘Detail view’ button. You get a larger representation of the gear tooth form. Now you can see the detailed tooth mesh. Click on the ‘Total view’ button to obtain an entire view of the cylindrical gear pair.

Figure 7.54:

Detail view of the mesh

Please Note: The representation of the tooth mesh allows you to look at the tooth thickness allowances, the tip diameter and centre distance allowances as well the tooth mesh and to check the influence of these values. The tooth form mask provides various functions. Find a short description of these functions in the following section.

7.5.3 Rotating Angle

Enter an rotating angle for the rotation of the spur gear pair.

Figure 7.55:

Rotating angle

Rotation of the driving gear counter-clockwise

Rotation of the driving gear clockwise

7.5.4 Rotation

When you click on one of the two arrows, a continuous rotation of the spur gears occurs.

Figure 7.56:

Rotation

The continuous rotation of the driving gear counter-clockwise

The continuous rotation of the driving gear clockwise

The rotation is stopped.

7.5.5 Tooth Thickness Allowance

Click on the ‘Detail view’ button and the tooth mesh is represented in detail. Now you can change the tooth thickness allowance, that is already given in the main mask for the ‘Allowances’, within the tolerance limit. All changes are displayed immediately. For the representation of the tooth mesh, select the lower, upper and mean tooth thickness allowances for gear 1 and gear 2.

Figure 7.57:

Tooth thickness allowance in the main mask ‘Allowance’

The both arrows indicate the lower and upper allowance. The active input is grayed out and disabled. Click on the left arrow and you will get the representation for the lower tooth thickness allowance. The right arrow shows the representation for the upper tooth thickness allowance. The middle button displays the mean tooth thickness allowance. At the first start of the tooth form, you will get the mean tooth thickness allowance as a standard feature.

The tooth thickness allowances can be defined between the lower and upper allowance.

Figure 7.58:

Tooth thickness allowance

Please Note: In case you have specified the lower and upper tooth thickness allowance for gear 1 and gear 2 in the input mask ‘Allowances’ by using the calculator button, then the manually defined values appear here as lower and upper tooth thickness allowance.

7.5.6 Tip Diameter Allowance

Click the ‘Detail view’ button and the tooth mesh is represented in detail. Now you can change the tip diameter allowance, that is already given in the main mask for the ‘Allowances’, within the tolerance limit. All changes are displayed immediately. For the representation of the tooth mesh, select the lower, upper and mean tip diameter allowances for gear 1 and gear 2.

Figure 7.59:

Tip diameter allowance in the main mask ‘Geometry’

The both arrows indicate the lower and upper allowance. The active input is grayed out and disabled. Click on the left arrow and you will get the representation for the lower tip diameter allowance. The right arrow shows the representation for the upper tip diameter allowance. The middle button displays the mean tip diameter allowance. At the first start of the tooth form, you will get the mean tip diameter allowance as a standard feature.

Figure 7.60:

Tip diameter allowance

Please Note: In case you have specified the lower and upper tip diameter allowance for gear 1 and gear 2 in the input mask ‘Geometry’ by using the ‘Lock’ button, then the manually defined values appear here as lower and upper tip diameter allowance.

7.5.7 Centre Distance Allowance

Click on the ‘Detail view’ button and the tooth mesh is represented in detail. Now you can change the centre distance allowance, that is already given in the main mask for the ‘Allowances’, within the tolerance limit. All changes are displayed immediately. You can check the operation of the gears by using various centre distance settings. For the representation of the tooth mesh, select the lower, upper and mean centre distance allowances for gear 1 and gear 2.

Figure 7.61:

Centre distance allowance in the main mask ‘Allowances’

The both arrows indicate the lower and upper allowance. The active input is grayed out and disabled. Click on the left arrow and you will get the representation for the lower centre distance allowance. The right arrow shows the representation for the upper centre distance allowance. The middle button displays the mean centre distance allowance. At the first start of the tooth form, you will get the mean centre distance allowance as a standard feature.

Figure 7.62:

Centre distance allowances

Please Note: In case you have specified the lower and upper centre distance allowance in the input mask ‘Allowances’ by using the ‘User defined input’ option, then the manually defined values appear here as lower and upper centre distance allowance.

7.6 Calculation of Gear Load Capacity

Gears fail by tooth breakage, pitting as well as by scuffing. The strength is determined by the loads, the geometry of gearing as well as selected materials. The calculation of the load capacity is about the proof of the following strength factors that result from the above-mentioned forms of damage:

Load capacity of the tooth root (safety against failure of the toothing due tooth breakage)
Load capacity of the tooth flank (safety against failure of the toothing due to pitting)
Scuffing load capacity (safety against failure of the toothing due to scuffing)

Figure 7.63:

Load capacity

Load Capacity of the Tooth Root - Tooth Breakage

Tooth breakage is a fatigue failure. Pitting, scuffing or wear may weaken the tooth so that it breaks. The slow progress of the fracture apparently causes the metal to break like brittle material. A tear or grinding notch may cause a tooth breakage. Gear tooth fractures ordinarily start in the root fillet. The tooth breakage can destroy an entire gearing and leads to a failure of the gearing. Sometimes a new tooth will break as a result of severe overload or a serious defect in the tooth structure. According to DIN 3990, an operation with a reduced load is possible after a tooth breakage, if just a small portion of a tooth broke off and the other parts of the gearing are intact.

For a high load capacity of the tooth root, the following methods are advantageous: positive profile shift (for small number of teeth), usage of hardened and tempered or case-hardened materials with larger load capacity of the tooth root, larger root fillet, larger module

Load Capacity of the Tooth Flank - Pitting of Gear Teeth

Pitting is a fatigue failure and is characterized by little bits of metal breaking out of the surface and thereby leaving small holes or pits, so that oil seeps into the pits. This is caused by high tooth loads leading to excessive surface stress, a high local temperature due to high rubbing speeds or inadequate lubrication. The cracking of the surface develops, spreads and ultimately results in small bits breaking out of the tooth surface. But it is often possible to get some years of service out of gears that have pitted rather extensively.

For a high load capacity of the tooth flank, the following methods are advantageous: large number of teeth, positive profile shift (for small number of teeth), higher pressure angle, large hardness of tooth flank, nitriding, more viscous oil

Scuffing Load Capacity

Scuffing is a surface destruction and it can be caused by a lubrication failure. Tears and scratches appear on the rubbing surface of the teeth. This form of damage is called ‘scuffing’. The terms of ‘scuffing’ and ‘scoring’ are used interchangeably. Scuffing is an important form of damage leading to component replacements in lubricated mechanical systems. Compared with tooth breakage and pitting, it is not a fatigue failure, it can come very quickly. A short overload can lead to scuffing and the gearing fails. Scuffing is apt to occur when the gears are first put into operation because the teeth have not sufficient operating time to develop smooth surfaces. Due to the scuffing, the temperature, the forces and the noise increase, the gear teeth finally break off. The following factors may influence the occurrence of scuffing:

Gear material
Lubrication
Surface condition of tooth flanks
Sliding velocity
Load
Impurities in a lubricant

After the occurrence of scuffing, high-speed gears apt to additional dynamic forces that cause usually pitting or tooth breakage. The high surface temperature may cause a breakdown of the lubricating film. The following factors support scuffing:

High loads
Kind of lubrication: Non-alloy oil protects less against scuffing than E.P. oil (extreme pressure)
High oil temperature
Rough oil surface
Low gear quality: Larger contact ratio and tooth alignment errors may cause local stresses by impacts and unbalanced carrying.

For a high scuffing load capacity, the following methods are advantageous: E.P. oils (oil that contains chemical additives), a careful running-in period of the gearing, low sliding velocity due to tip relief and a smaller module

Please note: There are two different types of scuffing - cold and hot scuffing. Both types describe a damage on the flank. The scuffing problem is not limited to high-speed gears. Scuffing can also occur on slow-speed gears. The slow-speed scuffing is called cold scuffing and the high-speed hot scuffing. Cold scuffing is not often observed. Hence, all further comments and information refer to hot scuffing.

7.6.1 Activate Load Capacity

The calculation of load capacity of cylindrical gears is standardized according to DIN 3990 and ISO 6336. The standards give different methods to calculate the load capacity in DIN 3990 and ISO 6336. The eAssistant provides all calculations according to DIN 3990 Method B and ISO 6336 Method B. Hence, you can check the load capacity of tooth root and tooth flank as well as the scuffing fast and easily. The scuffing safeties are determined according to the integral and flash temperature method. The material properties, the endurance, face load factor, application factor as well as the kind of lubrication and the selected lubrication are taken into consideration for the calculation. There are extended input options to influence the number of load changes or the roughness. A grinding notch can be integrated into the calculation and the mode of operation can be selected.

Click on the ‘Load capacity’ button to get to the calculation mask. You will notice that all input fields or listboxes are disabled. When you select the entry ‘DIN 3990 Method B’ or ‘ISO 6336 Method B’ from the listbox ‘Calculation method’, all input fields are enabled. In case you do not need the calculation for load capacity, the calculation can be disabled. Thus, the size of the calculation report becomes smaller.

Figure 7.64:

Activate the calculation for load capacity

7.6.2 Inputs for Load Capacity - General Factors

Comment

You can add a description or a short comment to gear 1 and gear 2.

Figure 7.65:

Add a comment

Material Selection

Select an appropriate material directly from the listbox or click on the button ‘Material’. Eventually, the material database opens.

Figure 7.66:

Listbox ‘Material’

The material database provides some detailed information on the several kinds of material. If the listbox is active, the two arrow keys ‘Up’ and ‘Down’ of your keyboard allows you to search through the database, so you can compare the different values with each other.

Figure 7.67:

Material database

In order for gears to achieve their intended performance, life and reliability, the selection of a suitable material is very important. Steel is the most common material that is used for gears. There are a number of steels used for gears, ranging from plain carbon steels through the highly alloyed steels from low to high carbon contents. The choice will depend upon a number of factors, including size, service and design. For pinion and gear, the same hardened and tempered steel may be used. It has to be kept in mind that unhardened gears with equal hardness should not be meshed with each other because scuffing is apt to occur. A hardened or nidrided gear $HRC > 50$ smoothes the tooth flanks of the hardened and tempered mating gear, reduces the form deviations and increases the load capacity of the tooth flank. For a mating of hardened gears, no hardness difference is necessary. The final selection of the material should be based upon an understanding of the material properties and application requirements.

Hardening and tempering differs from hardening by annealing at high temperatures. The temperature range for hardening and tempering ranges from $400∘$ to $700∘$ C while after hardening, parts are annealed at a low temperature. On the other hand, a distinction is made between the material. For hardening, steel contains more than 0,6 to 0,7% of carbon, for hardening and tempering less than 0.6% of carbon. However, there is no well-defined limit between hardening and tempering and hardening.

Kind of Material

Steel casting: Steel casting belongs to the ferrous metals that include carbon (up to max. 2 %) and are poured into sand molds to produce several components. Due to a higher melting temperature, steel casting is more difficult to cast than cast iron. Steel casting is cheaper than ground or forged gears.

Steel: Steel is the most common material and is used for medium and high-loaded gears.

Nidrided steel: Nitriding is adding nitrogen to solid iron-base alloys by heating the steel in contact with ammonia gas or other suitable nitrogenous material. This process is used to harden the surface of gears.

Case-hardened steel: Case-hardened steel is a quality and high-grade steel with low carbon content. Case-hardened steel is usually formed by diffusing carbon (carburization), nitrogen (nitriding) into the outer layer of the steel at high temperature and then heat treating the surface layer to the desired hardness. When the steel is cooled rapidly by quenching, the higher carbon content on the outer surface becomes hard while the core remains soft and tough.

Blackheart malleable cast iron (pearlitic structure): Malleable cast iron is a heat-treated iron carbon alloy. Two groups of malleable cast iron are specified, whiteheart and blackheart cast iron. Blackheart malleable cast iron is used for parts with a complex shape, in which a high durability, shock resistance and good machining are important. Malleable cast iron is used for smaller dimensions and has got a higher strength and toughness than steel castings.

Cast iron with spheroidal graphite (pearlitic structure, bainitic structure, ferritic structure): Cast iron usually refers to gray cast iron but identifies a large group of ferrous alloys that contain more than 2 % of carbon. It is extremely rare that the maximum carbon content is higher than 4.5 %. Cast iron is a low-priced material. However, cast iron has less toughness and ductility than steel. Cast iron with spheroidal graphite can be used for parts with higher vibration stress.

Heat-treated steel: Hardening and tempering is a heat-treating technique for steels by quenching from the hardness temperature and annealing at a high temperature so that the toughness is increased significantly. At the same time, a higher elastic limit is reached. Annealing temperatures and times differ for different materials and with properties desired, steel is usually held for several hours at about $400∘$ C to $700∘$ C. Some steels have to be cooled very quickly (Annealing: in order to achieve the intended properties of work pieces (e.g., desired strength or toughness), reheating of the work pieces to certain temperatures is necessary.).

Gray cast iron: Gray cast iron is used for complex shapes and offers low cost and an easy machinability. It provides excellent damping properties but it is a disadvantage that the load capacity is very low.

Please Note: When you select the option ‘User defined input’, then all inputs and options are enabled and you can specify your individual material very easily. Your inputs will be saved to the calculation file.

Please be advised that changing the material will delete your defined inputs and you have to enter the inputs again.

Figure 7.68:

Own input of a material

Application Factor $KA$

The application factor $KA$ evaluates the external dynamic forces that affect the gearing. These additional forces are largely dependent on the characteristics of the driving and driven machines as well as the masses and stiffness of the system, including shafts and couplings used in service. Because scuffing is not a fatigue failure, the application factor shall consider the stronger influence of several load peaks during the calculation of the scuffing load capacity. Several load peaks affect directly only the flank temperature. Because of that, the same application factor $KA$ can be used for the calculation of the scuffing load capacity as well as of the load capacity of the tooth root and tooth flank. The application factor is determined by experience. An application factor of ‘1.0’ is best thought of a perfectly smooth operation. The following table gives some values according to DIN 3990.


Application Factors $KA$ According to DIN 3990-1: 1987-12⁵

Working Characteristics	Working Characteristics of the Driven Machine

of the Driving Machine	Uniform	Light Shocks	Moderate Shocks	Heavy Shocks

Uniform	1,0	1,25	1,5	1,75

Light Shocks	1,1	1,35	1,6	1,85

Moderate Shocks	1,25	1,5	1,75	2,0

Heavy Shocks	1,5	1,75	2,0	2,25 or higher

⁵ from: DIN 3990 Part 1, December 1987, p. 55, table: A1

Working Characteristics of the Driving Machine

Uniform: e.g., electric motor, steam or gas turbine (small, rarely occurring starting torques)
Light Shocks: e.g., electric motor, steam or gas turbine (large, frequently occurring starting torques)
Moderate Shocks: e.g., multiple cylinder internal combustion engines
Heavy Shocks: e.g., single cylinder internal combustion engines

Working Characteristics of the Driven Machines

Uniform: e.g., steady load current generator, uniformly loaded conveyor belt or platform conveyor, worm conveyor, light lifts, packing machinery, feed drives for machine tools, ventilators, centrifuges, centrifugal pumps, agitators and mixers for light liquids or uniform density materials, shears, presses ...
Light Shocks: e.g., heavy lifts, crane slewing gear, industrial and mine ventilator, centrifugal pumps, agitators and mixers for viscous liquids or substances of non-uniform density, multi-cylinder piston pumps ...
Moderate Shocks: e.g., rubber extruders, continuously mixers for rubber and plastics, wood-working machine, lifting gear, single cylinder piston pumps ...
Heavy Shocks: e.g., excavators (bucket wheel drives), rubber kneaders, foundry machines, brick presses, peeling machines, rotary drills ...

Please Note: You will find a ‘Question mark’ button next to the input field. Click on this button and the above-mentioned table opens. The ‘Question mark’ button is an additional feature and provides further information. You will find this button next to several input fields.

Figure 7.69:

The question mark button

Face Load Factor $KH β$

The face load factor takes into account the effects of the non-uniform distribution of load over the gear facewidth on the surface stress $KH β$ , on the tooth root stress $KF β$ and on the scuffing $KBβ$ . The face load factor is determined according to DIN 3990, Part 1 Method B.

Figure 7.70:

Face load factor for the surface pressure

When you start the calculation module, the value ‘1.25’ is entered into the input field. In case you already use a defined face load factor, you can save the certain factor to a template file. Then the calculation module starts with the individual face load factor. When you click on the calculator symbol, the input mask for the face load factor opens. In the top input field ‘Face coeff.’ you can find the default value ‘1.25’. You will notice that the lower input fields and listboxes are disabled. By using the ‘OK’ button you can take over the default value to the main mask. There is a listbox next to the input field for the face load factor. When you open the listbox, the entry ‘DIN 3990 T1 Method B’ appears.

Figure 7.71:

Listbox with the selection of DIN

As soon as you select this entry from the listbox, the remaining input fields and listboxes are enabled. The face load factor is determined automatically but you still cannot take over the value to the main mask. In order to take over the calculated value, you have to add further inputs from the input mask for the face load factor. When the button ‘OK’ is activated, then the determined face load factor can be confirmed with the button ‘OK’.

Please Note: However, there is the possibility to take over the value, determined according to DIN, to the main mask without changing the extensive settings. When you click on the calculator button next to the face load factor, the above-mentioned input mask opens. The face load factor $KH β$ is displayed in the input field. Open the adjacent listbox and select the entry ‘DIN 3990 T1 method B’. The face load factor is calculated but the button ‘OK’ is still disabled.

Figure 7.72:

Face load factor

Open the listbox again and select the entry ‘User defined’. Now the ‘OK‘ button is enabled and you can take over the face load factor.

Figure 7.73:

Take over the face load factor

Mesh Misalignment $F βx$

The path of teeth is marked by the path of tooth traces. The tooth trace is the section of a tooth flank with the reference surface. The mesh misalignment $F βx$ considers all influences of manufacturing, assembly and deformation that may intensify and compensate each other.

Figure 7.74:

Tooth trace

The mesh misalignment is determined according to DIN 3990, part 1 method C. Using this method, portions of the mesh misalignment are considered caused by a deformation of pinion and pinion shaft as well as manufacturing inaccuracies. $F βx$ consists of $f sh$ and $f ma$ . $f sh$ is the mesh misalignment due to bending and torsion of the pinion and pinion shaft, therefore it is a mesh misalignment due to deformation. The mesh misalignment $f ma$ is a misalignment due to manufacturing inaccuracies and is dependent upon the gear accuracy and the facewidth of the gear.

Figure 7.75:

Mesh misalignment

Please Note: Select the entry DIN 3990 method B from the listbox for the face load factor, then the factor is determined according to DIN. The selection and input fields are enabled. User-defined inputs for the mesh misalignment are also possible.

Figure 7.76:

User-defined selection

Position of Tooth Contact Pattern

The tooth contact pattern gives some insight into the required geometry and accuracy of gears. While rolling off each other, a tooth flank will not come into contact with every point of its mating flank. A tooth contact pattern is a representation of contact surfaces of two engaged tooth flanks of gear pair. Under operating conditions, an even load distribution over the facewidth and tooth depth is to be accomplished. For a contact pattern, a thin layer of a marking compound is applied to the flanks. After that, the gear pair is rotated as long as the tooth contact pattern appears. Then the gears are visually inspected to check the tooth contact pattern which is indicated by a light wear pattern on the mating tooth surfaces. The optimization of the contact pattern plays an important role for improving smoothness and quietness of operation.

Figure 7.77:

Select contact pattern

Click on the ‘Question mark’ button and you will get a representation of the contact pattern according to DIN 3990, Part 1.

Figure 7.78:

Open contact pattern

Figure 7.79:

Contact pattern according to DIN 3990, Part 1, 1987

Figure 7.80:

Contact pattern according to DIN 3990, Part 1, 1987

Pinion Corrections

Errors in manufacturing and elastic deformations that may influence the load capacity can be adjusted by using intentional deviations from the involute (modification of the tooth depth) and theoretical tooth trace (modification of the facewidth). Lead crowning and end relief are the most important pinion corrections and are advantageous for a good load distribution over the facewidth of a gear. Due to lead crowning or end relief, a non-uniform load distribution can be reduced. The calculation program allows you to select one of the above-mentioned pinion corrections from the listbox.

Figure 7.81:

Selection of the pinion correction

Lead Crowning

Lead crowning is a common modification that results in the flank of each gear tooth having a slight outward bulge in its center area. A crowned tooth becomes gradually thinner towards the end of the teeth. The purpose of lead crowning is to ensure that manufacturing inaccuracies and deformations are adjusted under load and that the tooth ends are relieved. In general, lead crowning $Cc$ is carried out symmetrically to the centre of the facewidth.

Figure 7.82:

Lead crowning

End Relief

Due to mesh misalignments, an overloading of the tooth ends occurs. Therefore, this kind of pinion correction is used to protect the tooth ends against overloading. Generally, the size of the relief at both sides of the tooth flank is equal. If crown shaving and crown grinding are not possible, then end relief is recommended.

Figure 7.83:

End relief

Pinion Arrangement - Stiffening Effect

DIN 3990 describes the stiffening effect as follows:

When $d∕d ≥ 1,15 1sh$ , then stiffening is assumed; when $d ∕d < 1,15 1 sh$ , there is no stiffening; furthermore, scarcely any or no stiffening at all is to be expected when a pinion slides on a shaft and feather key or a similar fitting, nor when normally shrink fitted (DIN 3990, part 1, edition December 1987, Beuth Verlag GmbH Berlin, figure 6.8, p. 33).

Figure 7.84:

Pinion arrangement

Transmitted Power - Power Distribution for the Dimensioning of the Face Load Factor $kHβ$

The transmitted power $k$ is the percentage of the power which will be transmitted through the pinion tooth mesh, in the ratio of the full power which is transmitted through the pinion shaft. For example: The power input on a shaft is 10 kW. 60% is transmitted through the tooth mesh and the remaining 40% is transmitted to the end of the shaft. Now you have to define 6 kW for the pinion to dimension the gearing. To determine the face load factor, you have to enter 60% of the transmitted power because the stronger deformation of the shaft due to the full torque transmission (10 kW) is taken into consideration.

Figure 7.85:

Transmitted power

Reference Gear

The inputs for the power, speed and torque apply for the appropriate gear that is selected in the listbox. For the other gear, speed and torque are determined from the reference gear.

Power and Torque

The power, torque and speed are dependent upon each other. Click on the adjacent button ‘T/P’ to switch between the input for the torque and the input for the power. When you click on the ‘TP’ button, then you can enter either the torque or the power. The values are converted. The description of the input field changes accordingly into ‘Torque’ or ‘Power’.

Kind of Lubrication and Lubricant Selection

Lubrication serves several purposes but its basic and most important function is to protect the sliding and rolling tooth surfaces from seizing, wear and friction. The friction of the tooth flank is responsible for flank wear, gear heating and gear noise. A reduced flank friction improves the efficiency that is dependent on the tooth load, circumferential velocity, gear quality and the surface condition of the tooth flanks. In order that the gearing should work properly, the selection of a lubricant is an important choice. A liquid lubricant is a good choice and can be easily introduced between the contacting surfaces. In addition, a lubrication has to reduce frictional heat and has to protect the surfaces against corrosion. The bearings and clutches in a gearing require also an appropriate lubricant. Therefore, the lubricant has to be suitable as well. Oil and greases are the most common lubricants. The compounding of oils provides a combination and generation of various properties. Oil offers a wider range of operating speeds than greases. They are easier to handle and are most effective. Special E.P. (extreme pressure) oils have been developed for slow-speed, highly-loaded vehicle gears. These oils develop chemical compounds on the contacting gear-tooth surfaces. Grease is a combination of liquid and solids. Grease has the advantage of remaining in place and not spreading as oil. It can provide a lubricant film at heavily loads and at low speeds.

Figure 7.86:

Open the selection of a lubricant

Liquid lubricants may be characterized in many different ways. Viscosity is one very important property of a lubricant and determines the oils lubricating efficiency.

For the selection of liquid lubricants applies: the smaller circumferential velocity and larger the contact pressure as well as the roughness of tooth flanks, the higher the viscosity. A higher viscosity will result in a higher hydrodynamic load capacity and an increased scuffing load limit where scratching and scuffing of the tooth flanks occur (Muhs/Wittel/Jannasch/Voßiek: Roloff/Matek Maschinenelemente, 17th revised edition, published by Vieweg, Wiesbaden 2005).

Please Note: If the viscosity is too low, the oil film will not be sufficiently formed and if the viscosity is too high, the viscosity resistance will also be high and cause temperature rise. For higher speed, a lower viscosity oil should be used and for heavy loads, a higher viscosity oil should be used.

Gears that are running primarily in a gearbox are lubricated with oil. A distinction is made between oil splash lubrication and oil injection lubrication.

Oil splash lubrication: The oil splash lubrication is an easy, reliable and reasonable lubrication system. It is a type of lubrication used in enclosed gear drives. In splash lubrication, the gear tooth dips into a tray of lubricant and transfers the lubricant to the meshing gear as it rotates. As a result, oil reaches all of the places where it is needed. The oil splash lubrication can be used for average speed applications.

Oil injection lubrication: With the oil injection lubrication, the oil can be filtered, cooled and checked and the oil is directly fed to the bearings. The amount of oil can be controlled according to the heat dissipation requirements. The gearbox is used as an oil tank reservoir from which several units can be supplied. The oil is sprayed directly by a pump injector into the mating surfaces.

Grease lubrication: The selection of the grease is dependent upon the circumferential velocity, the kind of application and the service temperature. A grease lubrication requires low maintenance and protects against contamination. Grease lubrication is suitable for any gear system that is opened or enclosed, so long as it runs at low speed. The grease should have a suitable viscosity with good fluidity especially in a enclosed gear unit.

Click on the ‘Lubricant’ button and open database. The extensive database contains the lubricants including all detailed information about the oils and greases (e.g., density, viscosity, load stage of FZG test). You can find out more about the FZG test in Section 7.6.3 ‘Extensive Input Options for Scuffing Load Capacity’. Select ‘User defined input’ from the listbox to define your individual lubricant.

Figure 7.87:

Lubricant selection

7.6.3 Extended Input Options for Load Capacity of Tooth Root and Tooth Flank

The main mask of the load capacity provides the ‘Tooth root/flank’ button, click on that button and the extended input options appear.

Figure 7.88:

Selection of extended input options

If you do not change any inputs in the following mask, then the default input values are used.

Figure 7.89:

Extended input options for tooth root and tooth flank

Roughness

The surface roughness of the tooth flanks influences the load capacity of the tooth flanks. The average roughness $R z$ is the arithemitc average of five individually measured roughness values. The input of the roughness occurs for root and flank of pinion and gear. The right choice of the surface roughness is determined by economical aspects depending upon the intended purpose and manufacturing process. A fine surface can be very expensive because of the high manufacturing costs. A surface that is too rough may not fulfill the required functionality.

Grinding Notch

A grinding notch may significantly reduce the fatigue strength and a tooth breakage can occur due to a grinding notch. Shot-peening can be used to increase the fatigue strength of gears that are damaged by a grinding notch. A careful grinding of the notch is basically suitable.

Figure 7.90:

Grinding notch

Hardening Depth Root/Flank

The hardening depth is significantly for the pitting load capacity and is determined by the depth of surface layer heated to hardening temperature, the hardenability of the material and the effect of the quenching method.

Case-hardening: Steels get their specific features by case-hardening. This combined heat treatment process consists of the following subprocesses:

Carburizing, i.e., using carbon for the surface
Hardening, i.e., heat treatment to achieve a hardened and wear-resistant surface
Annealing (stress relief)

Please Note: The calculation module determines the optimal hardening depth automatically, but the hardening depth can be defined also individually. If the individual hardening depth is smaller than the optimal hardening depth, then the fatigue strength is reduced accordingly. The determination of the optimal hardening depth and reduction of fatigue strength with reduced hardening depth is based on: „Tobie, Thomas: Zur Grübchen- und Zahnfußtragfähigkeit einsatzgehärteter Zahnräder, Dissertation Technische Universität München (Lehrstuhl für Maschinenelemente, Forschungsstelle für Zahnräder und Getriebebau) 2001, Section. 10.3: Eingliederung der Versuchsergebnisse in das Rechenverfahren nach DIN 3990“.

Figure 7.91:

Hardening depth

Technology Factor $Y T$

The technology factor $Y T$ considers the change of the strength of the tooth root by machining process.

σ = σ Y Flim Flim0 T

$σ Flim0$

Fatigue strength of the tooth root from material data

$σFlim$

Fatigue strength of the tooth root with influence of the technology factor

$YT$	Technology factor (see following table)


Technology Factor $YT$ According to Linke⁶

Kind of Manufacturing of the Tooth Root	Technology Factor $YT$

Shot peening:	1,2 bis 1,4
Applies for case-hardened or carbonitrided gears; not ground in the hardened layer

Rolling:	1,3 bis 1,5
Applies for flame and induction hardened gears; not ground in the hardened layer

Grinding:	General: 0,7
Applies for case-hardened or carbonitrided gears	for CBN grinding wheel: 1

Shape cutting:	1
Does not apply for ground gears

⁶ from: Linke, H.: Stirnradverzahnung Berechnung Werkstoffe Fertigung, Carl Hanser Verlag Muenchen Wien, 1996, p. 320, table 6.5/6

Click the ‘Question mark’ button to open the tables for the technology factor and mode of operation factor.

Figure 7.92:

Open the tables

Mode of Operation Factor $YA$

The fatigue strength of the tooth root $σFlim$ is corrected with the influence of the mode of operation.

σFlim = σFlim0YA

$σ Flim0$

Fatigue strength of the tooth root from material data

$σFlim$

Fatigue strength of the tooth root with influence of the mode of operation factor

$YA$	Mode of operation factor (see following table)

The following reference values can be used for the mode of operation factor $YA$ . DIN 3990 specifies identical values for swelling and alternating load.


Mode of operation factor $YA$ according to Linke⁷

Mode of Operation	Mode of Operation Factor $Y A$	Direction of Load

Swelling	1

Alternating	0,7

Oscillating	0,85 - 0,15 $lgNrev --6--$ (for $6 1 ≤ Nrev ≤ 10$ ) 0,7 (for $6 Nrev > 10$ )

Please Note: $Nrev$ = Number of load direction changes during operation time

⁷ from: Linke, H.: Stirnradverzahnung Berechnung Werkstoffe Fertigung, Carl Hanser Verlag Muenchen Wien, 2nd edition 2010, p. 321, table 6.5/7

Dynamic Coefficient $KV$

The dynamic coefficient $KV$ considers additional inner dynamic forces. Inner dynamic forces are caused by mesh alignments, lead crowning, deformation of teeth, the housing, shafts and gear bodies as well as oscillation of the wheel masses. As the circumferential velocity of the gear rim increases, the dynamic forces increase. The forces decrease with an increasing load of the teeth. Click the ‘Lock’ button to change the dynamic coefficient.

Transverse Coefficient $KH α$

The transverse coefficients account for the effect of the non-uniform distribution of transverse load between several pairs of simultaneously contacting gear teeth on the surface pressure ( $KH α$ ), stress leading to scuffing ( $KBα$ ) and loading of the tooth root ( $KF α$ ). Click the ‘Lock’ button to change the transverse coefficient.

Mesh Load Factor $K γ$

The mesh load factor takes into account an uneven distribution of the total circumferential force for gearings with multiple transmission paths or for planetary gear trains with more than three planets. For transmission paths, the total circumferential force is distributed to several mesh. For gearings without transmission paths, the value is set at ‘1.0’.

Carried Width

When the facewidth of pinion and gear is not equal, then a maximum overhang of ‘1 x m’ at each tooth end is assumed as a carried width. Unhardened portions of surface-hardened gear tooth flanks (including transition zone) consider only 50 % as the carried width. However, if it is foreseen that because of crowning or end relief

the contact does not extend to the end of face, then the smaller facewidth shall be used for both pinion and gear. Click the ‘Lock’ button to change the carried width.

Permit Pitting

In specific cases, the development of pits on the gear flank is allowed. Use this option to permit several pits. In general, initial pitting is considered normal and is not a cause for concern. In particular, case-hardened and nitrided gears usually has the tendency to pit near the tooth root and lead eventually to fatigue breakage. Here an individual assessment is necessary. In some cases (aerospace industry), pits are absolutely not permitted. For turbo transmissions, pits may lead to oscillations and increased additional dynamic forces.

Figure 7.93:

Permit pitting

7.6.4 Extended Input Options for Scuffing Load Capacity

The scuffing load capacity offers different extended input options. Click on the ‘Scuffing’ button and the extended input options appear.

Figure 7.94:

Extended input options ‘Scuffing’

Thermal Contact Coefficient $BM$

The thermal contact coefficient $BM$ is required for the determination of the flash factor. The flash factor considers the influence of the material properties of gear and pinion on the flash temperature.

Relative Structure Factor $X WrelT$

The relative structure factor $X WrelT$ is primarily intended to take into account influence of the material properties on the scuffing load capacity and is determined by:

XW XW relT = X---- W T

$XW$	The lower table provides the empirically determined relative structure factor.

$XWT$

The relative structure factor for the test gears that are used for the determination of the scuffing

temperature. $XW T$ = 1,0 for the FZG gear test.


Structure Factor $XW$ ⁸

Material/Heat Treatment	Structure Factor $XW$

Through-hardened steel	1,00

Phosphated steel	1,25

Copper plated steel	1,50

Bath and gas nitrided steel	1,50

Hardened carburized steel, with austenite content:
- less than average	1,15
- average	1,00
- greater than average	0,85

Austenitic steel (stainless steel)	0,45

⁸ from: Linke, H.: Stirnradverzahnung Berechnung Werkstoffe Fertigung, Carl Hanser Verlag Muenchen Wien, 1996, p. 367, table 6.5/16

Load Stage of Standard FZG Gear Test

Because scuffing is not a fatigue failure, a standard FZG gear test was developed to determine the scuffing load capacity of a lubricant under certain operating conditions. The gear test, known as FZG gear test (Institute for Machine Elements Gear Research Center, University Munich, Germany), is a standardized method according to DIN 51354. At the FZG, the different influences on scuffing are extensively investigated. The test is performed on a standard FZG test machine using standard test gears. Standardized, case-hardened and ground spur gears with a large one-side profile shift are used. The load is increased gradually on a FZG gear test rig with defined technical parameters. There are 12 load stages and the gears are inspected for scuffing after every load stage. Finally, the load stage is determined where scuffing of the gear teeth occurs and where the flank area is damaged by scratches. The higher the load stages, the better the industrial gear lubricants resistance to scuffing.

Figure 7.95:

Selection of load stage

Type of Profile Modification

For high-duty gearings, it is possible to change the theoretical involute. Using the listbox to define the type of profile modification. You can select the following options:

without profile modification
for high-duty gearing
for uniform mesh

Figure 7.96:

Profile modification

The force distribution factor $X Γ$ evaluates the influence of the force distribution over several pairs of meshing teeth. A polygon-like shape over the line of action represents the progress of the force distribution factor. The values of the points A and E depend upon the type of profile modification. According to DIN 3990 (see Part 4, p. 17), the force distribution factor is as follows:

Force Distribution Factor Without Profile Modification and With Profile Modification for High-Duty Gears

Figure 7.97:

Without profile modification

Figure 7.98:

For high-duty gears (the pinion drives)

Force Distribution Factor With Profile Modification for High-Duty Gears and for Uniform Mesh

Figure 7.99:

For high duty gears (the gear drives)

Figure 7.100:

For uniform mesh

Contact Temperature Along the Path of Contact

The contact temperature varies along the path of contact due to the progress of the flash temperature.

Contact Temperature Without Profile Modification

Figure 7.101:

Contact temperature without profile modification

Contact Temperature With Profile Modification

Figure 7.102:

Contact temperature with profile modification

Contact Temperature for Uniform Mesh

Figure 7.103:

Contact temperature for uniform mesh

Flash Temperature and Integral Temperature Method

High surface temperatures due to high loads and sliding speeds can cause a lubricant film breakdown. Because of that, there are two calculation methods in DIN 3990 that are based on different criteria for the development of a damage. The eAssistant provides both the integral temperature method and flash temperature method:

Flash temperature method defines a variable contact temperature along the path of contact.
Integral temperature method defines a weighted average of the surface temperature along the path of contact.

Flash Temperature Method

The flash temperature is the temperature at which a gear-tooth surface is calculated to be hot enough to destroy the oil film and allow instantaneous welding at the contact point. The contact temperature $ϑB$ in any point of contact $Y$ results from the sum of the bulk temperature $ϑM$ and the flash temperature $ϑfla$ :

ϑB = ϑM + ϑfla

According to the flash temperature method, there is no scuffing as long as the contact temperature $ϑ B$ (as the sum of bulk temperature $ϑ M$ and flash temperature $ϑ fla$ ) does not exceed the scuffing contact temperature in all points of contact. The scuffing temperature $ϑ S$ to be a characteristic value for the material-lubricant-material system of a gear pair, to be determined by gear tests with the same material-lubricant-material system.

Figure 7.104:

Progress of contact temperature along the path of contact

Please note: Points A to E mark the important points from the beginning to the end of the mesh.

The safety against scuffing $SB$ is determined according to the flash temperature method:

-ϑS---ϑoil-- SB = ϑBmax - ϑoil ≥ SBmin

$ϑBmax$

Maximum contact temperature along the path of contact

$ϑoil$

Oil temperature before reaching the mesh

$ϑS$	Scuffing temperature

The safety factor $SBmin$ is dependent on whether the gearing is put into operation after a good running-in period. With a careful running-in period, there is no scuffing damage up to $SBmin ≈ 1$ . Without a running-in period, there is no scuffing up to $SBmin ≈ 3$ (according to Linke).

Integral Temperature Method

According to the integral temperature method, scuffing occurs when the integral temperature exceeds the scuffing integral temperature. The scuffing integral temperature is assumed as a characteristic value for the material-lubricant-material system of a gear pair and is determined from gear tests. The scuffing safety according the integral temperature method $SintS$ is calculated as follows:

ϑintS SintS = ϑ----≥ SSmin int

$ϑintS$

Scuffing integral temperature

$ϑint$

Integral temperature

As uncertainties and inaccuracies in the assumptions cannot be excluded, it is necessary to introduce a safety factor. According to Linke, the following reference values can be used:

$SintS<1,0$

In all probability, scuffing damages are expected to occur.

$1,0≤SintS≤2,0$

For a careful running-in period of the gearing, good contact pattern and real assumed

loads, there are no scuffing damages to be expected.

$SintS>2,0$

There is no risk of scuffing.

7.6.5 Input Options for Load Capacity According to ISO 6336 Method B

In addition to DIN 3990 Method B, it is possible to calculate the load capacity of tooth root and tooth flank according to ISO 6336 (2008) Method B. Select the ISO 6336 Method B from the listbox and the input fields are enabled.

Figure 7.105:

Activate load capacity according to ISO 6336 Method B

There is a strong similarity between the DIN 3990 standard and the ISO 6336 standard and most parts of the load capacity calculation according to ISO 6336 correspond to DIN 3990. In fact, the ISO 6336 evolved from the DIN 3990 standard. There are factors that influence the tooth root stress and tooth flank strength, but the differences are minor and the influence on the safety of root, flank and scuffing is very low. One large difference is the calculation of the helix angle factor $Z β$ as well as the calculation of the long life factors ( $Z NT$ and $Y NT$ ) for the tooth root stress. Another difference between DIN 3990 and ISO 6336 is the critical stress point on the root fillet. ISO 6336 uses the tangency point of a $60∘$ angle as the most critically stressed point on the root fillet for internal gears. The DIN 3990 standard uses the tangency point of a $30∘$ angle as the critical stress point on the root fillet for external and internal gears as the basis for the calculation of the load capacity.

$60∘$ Tangent for Internal Gears

The tooth form factor $Y F$ is the form factor, which represents the influence on nominal tooth root stress of the tooth form with load applied at the outer point of single pair tooth contact. According to DIN 3990, the form factor for external and internal gears is calculated at the tooth root at the point of the $30∘$ tangent. This method appeared to be inaccurate, especially for internal gears. ISO 6336 uses the tangency point of a $60∘$ angle as the most critically stressed point on the root fillet for internal gears. This method is more precise and leads to higher safeties for the tooth root.

Figure 7.106:

$60∘$ tangent for internal gears

Life Factors $YNT$ and $ZNT$

The long life factor $YNT$ accounts the higher tooth root stress and the long life factor $ZNT$ accounts the higher contact stress including static stress, which may be tolerable for a limited life (number of load cycles). The factors mainly depend on the quality of the material, heat treatment, number of load cycles as well as notch sensitivity, surface conditions and gear dimensions. The life factors $YNT$ and $ZNT$ can be read from the following graph for the static and reference stress as a function of material and heat treatment.

Figure 7.107:

DIN 3990 and ISO 6336: Life factors

Long life factor $YNT$ :
With optimum lubrication, material and manufacturing $YNT$ = 1,0 may be used for the number of load cycles $NL=3⋅106$ . For static stresses $NL ≤ 103$ , the long life factor is 2,5.

Long life factor $ZNT$ :
With optimum lubrication, material and manufacturing $ZNT$ = 1,0 may be used for the number of load cycles $NL=5⋅107$ . For static stresses $NL ≤ 105$ , the long life factor is 1,6.

There are significant differences in the calculation of the life factors $YNT$ and $ZNT$ when comparing DIN 3990 and ISO 6336. Values appropriate to the relevant number of load cycles, $NL$ , are indicated by the S-N curve (also known as a Woehler curve). The S-N approach is different. The long life factor for the load capacity of the tooth root approaches 1.0 for the range of long life (depending on the material, usually at $3⋅106$ load cycles). In DIN 3990, the life factor remains 1.0 for higher number of load cycles. According to ISO 6336, the factor for materials decreases from 1.0 to 0.85 at $1010$ load cycles. Only after that, the factor remains 0.85 at $1010$ load cycles. For gears in the long life range, the calculations according to ISO 6336 result in significantly smaller safeties (15 % lower) for tooth root and tooth flank. This also applies for the long life factor for flanks. Use the ‘Lock’ button to modify the long life factors $YNT$ and $ZNT$ . The input fields are enabled and you can define your own value for the factors. Please remember to keep the modified input field open or the default values will be used again.

Figure 7.108:

Input of life factors

Face Load Factor $KH β$

Another difference between ISO and DIN is the determination of the mesh stiffness $cγ$ . The mesh stiffness is needed for the calculation of the face load factor $KH β$ . The factor $KH β$ takes into account uneven distribution of load over the facewidth due to mesh misalignment caused by inaccuracies in manufacture and elastic deformations. For the calculation of the face load factor, ISO 6336 uses a mesh stiffness $cγ$ that is reduced by 15 % in comparison to the mean stiffness the DIN 3990 use for the calculation of $KH β$ . This results in slightly lower face load factors.

Rim Thickness Factor

If the rim thickness is too thin and not sufficient to provide full support for the tooth root (e.g., for planetary gear trains or internal gears), then this can be accounted for by the rim thickness factor $YB$ . $YB$ is expressed as a function of the tooth depth for external gears and as a function of the normal module for internal gears. The load capacity decreases with a gear rim thickness $sR < 1,2⋅ht$ for external gears or $2,8 ⋅mn$ , for internal gears $sR< 3,5 ⋅mn$ (ISO 6336 Edition 2006 - Was ist neu?: Dr.-Ing. R. Heß, Dipl.-Ing. B. Kisters, A. Friedr. Flender AG, Bocholt, Tagungsbeitrag Dresdener Maschinenelemente Kolloquium 2009).

Helix Angle Factor $Z β$

Another difference is the determination of the helix angle factor $Zβ$ . Independent of the influence of the helix angle on the length of path of contact, this factor accounts for the influence of the helix angle on the load capacity of the tooth flanks, allowing for such variables as the distribution of load along the lines of contact. $Zβ$ is dependent only on the helix angle, $β$ . The given formulae for the determination of the helix angle factor are different in DIN 3990 and ISO 6336. DIN 3990 gives the following formular to calculate the helix angle factor ( $β$ is the helix angle at reference circle):

Z = ∘cos-β- β

In ISO 6336, the helix angle factor is defined as follows:

Zβ = √-1---- cosβ

Figure 7.109:

Comparison of helix angle factor in DIN and ISO

Work Hardening Factor $ZW$

The work hardening factor $ZW$ has been revised in ISO 6336. This factor is used to take into consideration the Hertzian pressure which serves as a basis for the calculation of the load capacity of the tooth flanks. The work hardening factor accounts for the increase in surface durability due to meshing a steel gear with a hardened or substantially harder pinion with smooth tooth flanks. In DIN 3990 and the previous version of ISO 6336, the work hardening factor was dependent solely on the flank hardness of the softer gear.

Surface Roughness

The increase in the surface durability of the soft gear depends not only on any work hardening of this gear, but also on other influences such as flank surface roughness. In addition, the influence of the surface roughness is addressed in ISO 6336. Tooth flank curvature, pitch line velocity and lubricant viscosity are taken into account in the calculation. The work hardening factor is reduced for gears with hard, rough surface. Gear teeth with rough surfaces may wear a softer mating teeth. Wear of the surface is not covered by ISO 6336. Especially for rough pinion surfaces, values of $ZW$ ¡ 1 may be evaluated. As in this range effects of wear can limit the surface durability, $ZW$ is fixed at $ZW$ = 1,0 (ISO 6336 Edition 2006 - Was ist neu?: Dr.-Ing. R. Heß, Dipl.-Ing. B. Kisters, A. Friedr. Flender AG, Bocholt, Tagungsbeitrag Dresdener Maschinenelemente Kolloquium 2009).

7.6.6 Scuffing Load Capacity According to ISO/TR 13989

ISO 6336 does not provide a calculation method for scuffing. For the safety against scuffing, the Technical Report ISO/TR 13989 shall be preferably used. ISO/TR 13989 is a Technical Report (March 2000). This document is not to be regarded as an ‘International Standard’. It is proposed for provisional application so that information and experience of its use in practice may be gathered. The scuffing load capacity is calculated according to ISO/TR 13989 Part 1 (Flash temperature method) and Part 2 (Integral temperature method) as soon as you select ‘ISO 6336 Method B’.

Figure 7.110:

Scuffing load capacity according to ISO/TR 13989

Thermal Contact Coefficient $BM$

Lubricant Factor $XL$

The lubricant factor $XL$ depends on the the type of lubricant. You can select the following types:

Mineral oils $(XL$ = 1,0
Water soluble polyglycols $XL$ = 0,6
Non-water soluble polyglycols $XL$ = 0,7
Polyalfaolefins $XL$ = 0,8
Phosphate esters $XL$ = 1,3
Traction fluids $XL$ = 1,5

Figure 7.111:

Type of lubricant

Relative Structure Factor $X WrelT$

The relative structure factor $X WrelT$ is primarily intended to take into account influence of the material properties on the scuffing load capacity and is determined by:

XW XW relT = X---- W T

$XW$	The lower table provides the empirically determined relative structure factor.

$XWT$

The relative structure factor of test gears that are used for the determination of the scuffing

temperature. $XW T$ = 1,0 for the FZG gear test.


Structural Factor $XW$ ⁹

Material/Heat Treatment	Structural Factor $XW$

Through-hardened steel	1,00

Phosphated steel	1,25

Copper plated steel	1,50

Bath and gas nitrided steel	1,50

Hardened carburized steel, with austenite content:
- less than average	1,15
- average (10 % to 20 %)	1,00
- greater than average	0,85

Austenitic steel (stainless steel)	0,45

⁹ from: Linke, H.: Stirnradverzahnung Berechnung Werkstoffe Fertigung, Carl Hanser Verlag München Wien, 1996, p. 367, table 6.5/16

Load Stage of Standard FZG Gear Test

Because scuffing is not a fatigue failure, a standard FZG gear test was developed to determine the scuffing load capacity of a lubricant under certain operating conditions. The gear test, known as FZG gear test (Institute for Machine Elements Gear Research Center, University Munich, Germany), is a standardized method according to DIN 51354. At the FZG, the different influences on scuffing are extensively investigated. The test is performed on a standard FZG test machine using standard test gears. Standardized, case-hardened and ground gears with a large one-side profile shift are used. The load is increased gradually on a FZG gear test rig with defined technical parameters. There are 12 load stages and the gears are inspected for scuffing after every load stage. Finally, the load stage is determined where scuffing of the gear teeth occurs and where the flank area is damaged by scratches. The higher the load stages, the better the industrial gear lubricants resistance to scuffing. Click the ‘Lock’ button in order to select a load stage.

Figure 7.112:

Selection of load stage

Flash Temperature Method and Integral Temperature Method

High surface temperatures due to high loads and slidings speeds can cause a lubricant film breakdown. Because of that, there are two calculation methods in ISO/TR 13989 that are based on different criteria for the development of a damage. The eAssistant provides both the integral temperature method and flash temperature method:

Flash temperature method defines a variable contact temperature along the path of contact.
Integral temperature method defines a weighted average of the surface temperature along the path of contact.

Flash Temperature Method

ϑB = ϑM + ϑfla

Figure 7.113:

Progress of contact temperature along the path of contact

Please note: Points A to E mark the important points from the beginning to the end of the mesh.

The safety against scuffing $SB$ is determined according to the flash temperature method:

-ϑS---ϑoil-- SB = ϑBmax - ϑoil ≥ SBmin

$ϑBmax$

Maximum contact temperature along the path of contact

$ϑoil$

Oil temperature before reaching the mesh

$ϑS$	Scuffing temperature

Integral Temperature Method

ϑintS SintS = ϑ----≥ SSmin int

$ϑintS$

Scuffing integral temperature

$ϑint$

Integral temperature

As uncertainties and inaccuracies in the assumptions cannot be excluded, it is necessary to introduce a safety factor. According to Linke, the following reference values can be used:

$SintS<1,0$

In all probability, scuffing damages are expected to occur.

$1,0≤SintS≤2,0$

For a careful running-in period of the gearing, good contact pattern and real assumed

loads, there are no scuffing damages to be expected.

$SintS>2,0$

There is no risk of scuffing.

7.7 Input of Gear Tooth Profile and Gear Flank Modifications

Manufacturing errors, misalignment in the assembly of the gears as well as displacement of the gears under load lead to non-uniform load carrying across the facewidth and move a significant concentration of load to the tooth edges. Intentional deviations from the involute profile (profile modification) and from the theoretical tooth trace (flank modification) are used to minimize the manufacturing inaccuracies and elastic deformations, to improve the running behavior and to reduce the noise of the gear pair.

Figure 7.114:

Input of profile and flank modifications

There are different types of tooth corrections. Short and long linear or short and long circular tip and root relief are typical tooth profile modifications. Lead crowning and end relief are common flank modifications. These corrections tend to give better load distribution over the facewidth and can reduce the effects of misalignment. Using profile or flank modification requires an appropriate degree of gear accuracy. The minimum required gear manufacturing accuracy is DIN quality 7 or better. Design details should be based on a careful estimate of the deformations and manufacturing deviations of the gearing.

7.7.1 Gear Tooth Profile Modification

In case of gear tooth profile modification, parts of the involute profile are changed to reduce the load in that area. Profile modifications help gears to run more quietly and to regulate transmission errors.

Figure 7.115:

Gear tooth profile modification

Tip Relief and Root Relief

Tip relief is a modification whereby material is removed at the tips of the gear tooth. In case of root relief, a small amount of material is removed near the root of the gear tooth. The modified tooth profile merges as continuously as possible into the theoretical tooth profile. Profile crowning can be seen as a combination of tip and root relief.

Figure 7.116:

Tip and root relief

There are different types that can be chosen for the profile modification including short and long modification. A simple type of profile modification is the linear tip relief on pinion and gear. Linear tip and root relief can be applied to one or both gears. The listbox allows to select linear and circular profile modification.

Figure 7.117:

Selection of tip relief

Linear and Circular Tip and Root Relief

Figure 7.118:

Linear tip relief

Figure 7.119:

Linear root relief

Figure 7.120:

Circular tip relief

Figure 7.121:

Circular root relief

In case you know the relief length, you can enter this value into the input field. In case the length is unknown, select ‘short’ or ‘long’ from the listbox. The eAssistant determines the relief length automatically. If a diameter is given on the drawing, it is possible to define the diameter. Click the ‘d/l’ button on the right side to switch between the input for diameter or length.

Figure 7.122:

Diameter and length

You can define the transition from the modified to the unmodified range. For circular relief, the input of the transition start and transition end is not required. When you enable the checkbox ‘Use theoretical length of path of contact’, then the theoretical path of contact of a gear with a rack is used.

Figure 7.123:

Use theoretical length of path of contact

Profile Crowning

Profile crowning can be seen as a combination of tip relief and root relief. Material is removed from the tip and from the root of the tooth.

Figure 7.124:

Profile modification

You can select symmetric profile crowning from the listbox and add a value for $C ha$ to the input field.

Figure 7.125:

Symmetric modification

7.7.2 Gear Flank Modifications

Gear flank modifications are intentional deviations from the theoretical tooth trace. Lead crowning and end relief are typical gear flank modifications and are advantageous for a better load distribution across the facewidth of the gear. Both can compensate for misalignment so that the stresses do not rise at the tooth ends.

Figure 7.126:

Flank modification

End Relief

Due to mesh misalignments, an overloading of the tooth ends occurs. Therefore, end relief is used to protect the tooth ends from overloading. Usually, the relief applied is the same at both ends of the teeth. In case crown shaving and crown grinding are not possible, then end relief is recommended.

Figure 7.127:

End relief (double-sided)

End relief can be applied to both tooth ends or to the left or right end of the tooth.

Figure 7.128:

Selection of end relief

End Relief: Double-sided, Left and Right Side

Figure 7.129:

End relief (left side)

Figure 7.130:

End relief (right side)

Figure 7.131:

End relief (left side)

According to DIN 3990 Part 1, the height of end relief is calculated as follows:

For through-hardened gears: $Ce ≈ Fβxcv$ plus a manufacturing tolerance of 5 to 10 $μm$ . Thus, by analogy with $Fβxcv$ (initial equivalent misalignment) in DIN 3990, $Ce$ should be approximately:

Ce = fsh +1,5 ⋅fHβ

For surface hardened and nitrided gears: $C ≈ 0,5⋅F e βxcv$ plus a manufacturing tolerance of 5 bis 10 $μm$ .
$C e$ should be approximately:

C = 0,5⋅(f + 1,5 ⋅f ) e sh Hβ

When the gears are of such stiff construction that $fsh$ can for all practical purposes be neglected or when the helices have been modified to compensate deformation, the following is appropriate:

Ce = fHβ

60 % to 70 % of the above values is appropriate for very accurate and reliable gears with high tangential velocities.

According to DIN 3990 Part 1, the width (or length) of end relief can be determined as follows:

Figure 7.132:

End relief

For approximately constant loading and higher tangential velocities:

$le$ = smaller of the values $0,1 ⋅b$ or $1⋅m$

The following is appropriate for variable loading, low and average speeds:

bred = (0,5 to 0,7)⋅b

Lead Crowning

Lead crowning is a common way of crowning is the so-called lead crowning. This type is employed in order to compensate for manufacturing deviations and load-induced deformations of the gears and in particular to relieve the tooth-endloading.

Figure 7.133:

Lead crowning

Gears are usually crowned symmetrically about the mid-facewidth and the tooth center is slightly thicker than the tooth edges. The tooth flanks of the gear have a slight outward bulge in its center area. It is possible to select symmetrical and asymmetrical lead crowning from the listbox.

Figure 7.134:

Symmetrical crowning

Figure 7.135:

Asymmetrical crowning

According to DIN 3990 Part 1, the height of lead crowning can be determined as follows:

Cb = 0,5 ⋅(fsh + 1,5 ⋅fHβ)

When the gears are of such stiff construction that $fsh$ can for all practical purposes be neglected or when the helices have been modified to compensate for deformation at mid-face width, the following value can be substituted:

Cb = fHβ

Subject to the restriction $10 ≤ Cb ≤ 25 μm$ plus a manufacturing tolerance of about 5 $μm$ , 60 % to 70 % of the above values are adequate for extremely accurate and reliable high speed gears.

Figure 7.136:

Height of lead crowning

7.8 Meshing Interferences for External Gears

If parts of the flank of gear and mating gear mesh outside of the path of contact or if the contact ratio is $εγ < 1$ , then meshing interferences may occur. A large profile shift as well as a very small tip clearance may cause meshing interferences. Interference takes place between the tip of the tooth of the gear and root fillet area of the mating tooth. In some cases, the interference may be eliminated by decreasing the addendum of only one gear teeth. Due to meshing interferences, operating noise, gear failure (e.g., tooth breakage) and an increased wear can occur. In case of a basic rack profile, meshing interferences can be manipulated or removed by the following:

Tip relief,
Addendum chamfer,
Changing of profile shift,
Other manufacturing tools

7.8.1 Meshing Interferences Due to Low Contact Ratio

To assure smooth continuous tooth action, a pair of teeth must already have come into engagement. Especially for spur gear pairs a low contact ratio can appear:

the profile shift is too large and a small number of teeth,
the tip relief is too large,
an undercut occurs due to insufficient profile shift and small number of teeth.

The condition for a smooth and continuous tooth action is:

ε = ε + ε > 1 γ α β

The result panel displays the total contact ratio. In case the condition $εγ = εα + εβ > 1$ is not fulfilled, the total contact ratio will be marked in red. Furthermore, you will get an appropriate warning in the message window.

Figure 7.137:

Total contact ratio

7.8.2 Meshing Interferences Due to No Involute Area

For external gearings it is evident that interference is first encountered by the addendum of the gear teeth digging into the mating-pinion tooth flank. The accurate gear tooth form is helpful to check meshing interferences considering gear tolerances. The calculation module warns as soon as meshing interferences occur. Opposed to external gearings, meshing interferences occur more often for internal gearings. The Section 7.12 ‘Meshing Interferences for Internal Gears’ discusses this issue.

7.9 Internal Gears

The eAssistant provides the calculation of internal gears. A special feature of spur and helical gears is their capability of being made in an internal form, in which an internal gear mates with an ordinary external gear. An internal involute gear has either spur or helical teeth cut on the inside of a ring.

Figure 7.138:

Internal gear in the eAssistant

Its most common use is in a planetray gear train. The external gear must not be larger than about two-thirds the pitch diameter of the internal gear when full-depth $20∘$ pressure angle teeth are used. The axes on which the gears are mounted must be parallel.

General advantages of internal gears:

The centre distance is less than for external gears and makes it desirable in some applications where space is very limited.
Good surface endurance due to a convex profile surface working against a concave surface. The teeth of an internal gearing are relatively thick and strong. Hence, a low tooth root stress occurs.

General disadvantages of internal gears:

The assembly has to be considered. Due to a small difference between the number of teeth in the pinion and gear, internal gears will not assemble radially, but axially.
Fewer types of machine tools can produce internal gearings, usually a special tooling is required.
Low velocity ratios are unsuitable and in many cases impossible because of interferences. Interferences for internal gears occur far more frequently than for external gearings.
The use of rack-type tools is not possible for internal gearings. Only a few number of teeth provides defined features. Hence, a check regarding meshing interferences is necessary.

7.10 Input of Geometry Data for Internal Gears

Some inputs for the internal gears differ from the inputs for external gears. Nevertheless, internal gears can be calculated very fast. The following provides the most important changes for the input of internal gears.

Figure 7.139:

Internal gearing

7.10.1 Direction of Helix Angle

For an external gearing a right-hand teeth and a left-hand teeth can only mesh correctly. An internal gear has the same helix angle in degrees and the same hand its mating pinion. A right-hand pinion meshes with a right-hand gear and vice-versa.

Figure 7.140:

Direction of helix angle

7.10.2 Internal Helical Gears

Internal gears may be either spur or helical. Internal helical gearings have their advantages and disadvantages just like external helical gearings.

Figure 7.141:

Internal helical gear created in 3D

General advantages over internal spur gears are:

The gear pair offers a more quietly operation,
Gears are capable transmitting heavier loads,
A smaller number of teeth is possible than for internal spur gears.

General disadvantages are:

Axial loads cannot be avoided,
much more difficult to produce than internal spur gears

For the creation of an internal helical gear, only the helix angle $β$ has to be considered.

Figure 7.142:

Helix angle

7.10.3 Number of Teeth

Because the internal gear is reversed relative to the external gear, the tooth parts are also reversed relative to an ordinary external gear. Tooth proportions and standards are the same as for external gears except that the addendum of the gear is reduced to avoid trimming of the teeth in the fabrication process. The number of teeth is negative for internal gears. The tip, reference and root diameter are negative as well.

Please Note: Please note that you can enter a negative number of teeth only for gear 2.

Figure 7.143:

Input of a negative number of teeth

7.10.4 Centre Distance

The working centre distance is always negative. As soon as you enter a negative number of teeth, the centre of distance becomes negative as well. The diameters of internal gear pairs are negative. The eAssistant modifies that during the input of a negative number of teeth for gear 2.

Figure 7.144:

Negative centre distance

7.10.5 Profile Shift

For an internal gear the tooth tip is enlarged by shifting towards the tooth centre and the tooth root is enlarged by shifting away from the tooth centre. The internal gear is reversed relative to the external gear. A profile shift of an internal gear is positive in direction to the tooth tip and negative in direction to the tooth root. It applies for both internal and external gearings:

The profile shift is positive, $x⋅mn > 0$ , when the profile reference line is shifted from the pitch circle to the pitch circle.
The profile shift is negative, $x ⋅mn < 0$ , when the profile reference line is shifted from the pitch circle to the root circle.

A positive profile shift has the following influences:

The tooth root becomes thinner, the dedendum $hf$ gets smaller, the addendum $ha$ gets larger. Due to a thick and strong tooth root, there is no danger of tooth root breakage.
The tip circle and the root diameter increase, but get smaller according to the absolute value. Thereby, a smaller internal gear is developed.

The positive profile shift may be disadvantageous for internal gears. It is comparable with a negative profile shift for external gears. A negative profile shift has the following influences:

The tip circle and root diameter become smaller, but get larger according to the absolute value. A larger internal gear is developed.
The spacewidth at the tooth root gets smaller. For a smaller number of teeth there is a risk that a pointed tooth tip occurs, the risk of notch effects increases.

Figure 7.145:

Changing the tooth form using profile shift: Number of teeth $z$ = -50; tooth 1: $x$ = -1.5; tooth 2: $x$ = 0; tooth 3: $x$ = +0.5

Please Note: A negative profile shift may be advantageous for internal gears. In this case, it is comparable with a positive profile shift for external gears. For external and internal gear pairs the impacts of positive and negative profile shift are similar.

Figure 7.146:

Input of profile shift

7.11 Manufacturing Process for Internal Gears

For internal gears, the shaping process is the only basic method of tooth generation. Internal gears cannot be hobbed. Only in some very special cases rack-type tools can be used. They can be shaped, milled or cast. In small sizes they can be broached. Both helical and spur internals can be finished by shaving, grinding, lapping or burnishing. In case the gear shaper cutter itself is generated by using a rack tool, then the mesh of the gear flanks is limited by the proper tooth tip of the gear rack.

Figure 7.147:

Selection of the tool for gear 2

An internal gear mates with an ordinary external gear and the number of teeth of the external gear must be less than that of the gear to be cut for the internal gear. A rack profile can be a basic rack profile for internal gears. But the basic rack profile cannot be used for generating internal gears. Internal gears are produced by a gear shaper cutter. The number of teeth of the gear shaper cutter must be, according to the amount, smaller than the number of teeth of the internal gear. The shaping is a continuous process. The cutting tool is a spur shaper cutter. During the machining, tool and gear roll on each other. A feed motion occurs.

7.12 Meshing Interferences for Internal Gears

The gear mesh of an internal gear pair can be much more difficult than for external gears. Interferences for internal gears occur far more frequently than for external gearings. In case a meshing interference takes place, a warning is displayed in the message window. The following meshing interferences can appear in the calculation module:

Tooth root meshing interference on the pinion
Tooth root meshing interference on the internal gear
Generation meshing interference (tooth root meshing interference on the gear shaper cutter)
Tooth crest meshing interference
Feed meshing interference
Radial assembly interference

Please note: Meshing interferences may be eliminated or minimized by tip easing on the internal gear or on the pinion by increasing the pressure angle or helix angle.

7.12.1 Tooth Root Meshing Interference on the Pinion

When the tooth tip of the internal gear interferes the root fillet radius, then a tooth root meshing interference on the pinion occurs.

7.12.2 Tooth Root Meshing Interference on the Internal Gear

When the tooth tip of the pinion interferes the root fillet radius of the internal gear, then a tooth root meshing interference occurs.

7.12.3 Generation Meshing Interference

When shaper cutter and internal gear are in mesh, the generation meshing interference occurs due to tool cutter action in generating teeth with low numbers of teeth. Because of this interference there is a loss of the involute profile at the tooth tip. The term of the mesh and the load capacity are decreased.

7.12.4 Tooth Crest Meshing Interference

The tooth crest meshing interference may occur when the tooth crests of pinion and internal gear overlap during the hobbing process outside of the plane of action. For number of teeth differences of $|z2|- z1 < 10$ this meshing interference may occur frequently. For the generation of internal gears with gear shaper cutter, tooth crest meshing interference appears.

Please Note: Meshing interference can be avoided by changing the number of teeth and by a negative profile shift.

7.12.5 Feed Meshing Interference

If the chosen gear shaper cutter is too large and the teeth of the internal gear are cut off in the feed direction, a feed meshing interference occurs.

Please Note: If the sum of the profile shift is decreased, feed meshing interferences can be avoided. It is also possible to adjust the number of teeth of the gear shaper cutter.

7.12.6 Radial Assembly Interference

When internal gearsets have a too small difference between the number of teeth in the pinion and the number of teeth in the gear, there may be interference between the tips of the teeth. The interference is most apt to occur as the pinion is moved radially into mesh with the gear. It is possible to get around the radial-interference difficulty by assembling the set by an axial movement of the pinion.

Please Note: A radial assembly interference can be removed by decreasing the profile shift coefficients and addendum coefficient of pinion and internal gear.

Figure 7.148:

Radial assembly interference

7.13 eAssistant: Examples for Internal Gears

For the pairing of external and internal gears the number of teeth of $z1 ≥ 14$ and $|z2| ≥ 40$ is used, if possible with a number of teeth difference of more than 6 to 10 teeth. However, there are applications where the number of teeth $z1$ is only ‘3’. The following three examples show that these cases are possible by using the eAssistant (Examples taken from: K. Roth: Zahnradtechnik: Vol. I, Stirnradverzahnungen - Geometrische Grundlagen (1989, p. 198))

7.13.1 Extremely Small Number of Teeth (Pinion)

Large gear transmission ratio with an extremely small number of teeth (pinion): $z1$ = 3; $z2$ = -28

Figure 7.149:

Example

Figure 7.150:

Gear Engineer: Small number of teeth

7.13.2 Standard Tooth Profile

$z 1$ = 20; $z 2$ = -28

Figure 7.151:

Example

Figure 7.152:

Gear Engineer: Standard tooth profile

7.13.3 Small Difference of Number of Teeth

A large gear transmission ratio with an extremely small difference of number of teeth: $z1$ = 29; $z2$ = -30. Meshing interferences can be avoided by a large modification of the tip diameter or negative profile shift.

Figure 7.153:

Example

Figure 7.154:

Gear Engineer: Small difference of number of teeth

7.14 Message Window

The calculation module provides a message window. This message window displays detailed information, helpful hints or warnings about problems. One of the main benefits of the program is that the software provides suggestions for correcting errors during the data input. If you check the message window carefully for any errors or warnings and follow the hints, you are able to find a solution to quickly resolve calculation problems.

Figure 7.155:

Message window

7.15 Quick Info: Tooltip

The quick info tooltip provides additional information about all input fields and buttons. Move the mouse pointer over the input field or button, then you will get the additional information. This information will be displayed in the quick info line.

Figure 7.156:

Quick info line

7.16 Calculation Results

All results will be calculated during every input and will be displayed in the result panel. A recalculation occurs after every data input. Any changes that are made to the user interface take effect immediately. In case a minimum safety is not fulfilled, the result will be marked red. Press the Enter key or move to the next input field to complete the input. Alternatively, use the Tab key to jump from field to field or click the ‘Calculate’ button after every input. Your entries will be also confirmed and the calculation results will displayed automatically.

Figure 7.157:

Calculation results

7.17 Documentation: Calculation Report

After the completion of your calculation, you can create a calculation report. Click on the ‘Report’ button.

Figure 7.158:

The button ‘Report’

The calculation report contains a table of contents. You can navigate through the report via the table of contents that provides links to the input values, results and figures. The report is available in HTML and PDF format. Calculation reports, saved in HTML format, can be opened in a web browser or in Word for Windows.

Figure 7.159:

Calculation report

You may also print or save the calculation report:

To save the report in the HTML format, please select ‘File’ $⇒$ ‘Save as’ from your browser menu bar. Select the file type ‘Webpage complete’, then just click on the ‘Save’ button.
If you click on the symbol ‘Print’, then you can print the report very easily.
If you click on the symbol ‘PDF’, then the report appears in the PDF format. If you right-click on the PDF symbol, you should see the ‘Save Target As’ option. Click on that option and you will see the Windows save dialog.

7.18 How to Save the Calculation

When the calculation is finished, you can save it to your computer or to the eAssistant server. Click on the button ‘Save’.

Figure 7.160:

‘Save’ button

Before you can save the calculation to your computer, you need to activate the checkbox ‘Enable save data local’ in the project manager and the option ‘Local’ in the calculation module. A standard Windows dialog for saving files will appear. Now you will be able to save the calculation to your computer.

Figure 7.161:

Windows dialog for saving the file

In case you do not activate the option in order to save your files locally, then a new window is opened and you can save the calculation to the eAssistant server. Please enter a name into the input field ‘Filename’ and click on the button ‘Save’. Then click on the button ‘Refresh’ in the project manager. Your saved calculation file is displayed in the window ‘Files’.

Figure 7.162:

Save the calculation

7.19 ‘Redo’ and ‘Undo’ Button

The ‘Undo’ button allows you to reset your inputs to an older state. The ‘Redo’ button reverses the undo.

Figure 7.163:

The button ‘Redo’ and ‘Undo’

7.20 The Button ‘CAD’

The top menu bar of the eAssistant provides the button ‘CAD’.

Figure 7.164:

Button „CAD“

The eAssistant plugin for various CAD systems (e.g., SolidWorks, Solid Edge, Autodesk Inventor and Catia) enables you to combine calculation and design very easily. On the basis of the eAssistant calculation, you can generate spur gears in a 2D DXF format or create as a 3D part within seconds.

Please note: In order to allow the generation of CAD data, you need to activate the option ‘Enable file save local’ in the project manager.

7.20.1 DXF Output for Accurate Tooth Form

Click on the menu item ‘CAD $⇒$ DXF Output’. Now you are able to create the accurate tooth form of any involute gearing in the 2D DXF format. Use the various setting for the DXF output.

Figure 7.165:

DXF output

For the DXF output the following options are possible:

DXF output for gear 1 or gear 2
DXF output with points
DXF output with lines
DXF output with poly lines
Number of teeth
Input of a required layer name where the contour should be placed
Save the DXF file including the header

Figure 7.166:

Settings for the DXF output

When you have defined all settings, then click on the button ‘OK’. A standard Windows dialog is opened to save the file.

Figure 7.167:

Saving the DXF file

Now you can save the DXF file to your computer. Enter a name for the file and click on the button ‘Save’. It is not necessary to specify the file extension. The file is identified automatically.

7.20.2 eAssistant CAD Plugin

The eAssistant plugin for various CAD systems (e.g., SolidWorks, Solid Edge, Autodesk Inventor and Catia) enables you to combine calculation and design very easily and fast. Based on your eAssistant calculation, you can generate spur and helical gears as a 3D part within seconds. A single menu pick in the eAssistant software transfers the eAssistant calculation data to the CAD system. Based on these parameters, the automatic creation of a 3D parametric model starts in the CAD system. Allowances, addendum chamfer, profile shift and profile modifications (tip and root relief) are taken into consideration.

Figure 7.168:

CAD plugin

The CAD model stores all features and dimensions as design parameters. The eAssistant calculation is linked and associated to the part and can be opened at any time throughout the entire design phase. This is also possible if one part contains different calculations. Click the button ‘CAD’ and select the CAD plugin. Open the CAD system and start the generation by clicking the integrated button ‘eAssistant’. Do not forget to activate the option ‘Enable save data local’ in the project manager.

Please note: First you need to download and install the right CAD plugin for your CAD system. The plugin is available on our web site www.eAssistant.eu. After installation, an integrated button called ‘eAssistant’ appears in the CAD system.

The eAssistant CAD plugin also supports an automatic creation of 2D detail drawings for manufacturing. With just one click, the design table with all manufacturing details can be placed on the sheet. There is no need to manually add all design table parameters to the drawing.

Please note: For further information, please visit our web site www.eAssistant.eu or read the CAD plugin manual.

7.20.3 Manufacturing Data

The button ‘CAD $⇒$ Manufacturing data’ allows to save the manufacturing data to a text file.

Figure 7.169:

Output of manufacturing data

7.21 The Button ‘Options’

Click on the button ‘Options’ in the top menu bar of the eAssistant to change some general settings.

Minimum safety tooth root / tooth flank
Minimum safety scuffing (integral) / scuffing (flash)
Factor for minimal gear ring thickness: the factor can be specified by the user
Chord of tooth root thickness analog FVA: This option has only effect on the calculation with protuberance tools
Number of decimal places for the calculation report

Figure 7.170:

The options

7.22 How to Calculate the Accurate Tooth Form of Involute Splines by Using the Spur Gear Pair Calculation Module

7.22.1 Select Basic Data for Involute Spline

Please login with your username and your password. Select the module ‘Involute splines’ through the tree structure of the Project Manager by double-clicking on the module or clicking on the button ‘New calculation’. Now select the basic data for the geometry according to DIN 5480. In case you have the DIN standard, then have a look at the standard. Click on the button ‘Selection’ and you get to the profile geometry selection.

Figure 7.171:

Involute splines

The profile geometry selection provides you the basic data for the involute spline.

Figure 7.172:

Profile geometry selection

7.22.2 Modify Basic Rack Profile

Now close the involute spline module and open the calculation module ‘Spur gear pair’. Click on the button ‘Tool’ and modify the basic rack profile.

Figure 7.173:

Tool

Please enter all tool data for the involute spline for gear 1 and the tool data for the gear 2 (internal gearing with a negative number of teeth). Select for gear 1 the entry ‘user defined input’ from the listbox ‘Basic rack profile’ and enter the following data:

For hobbing according to DIN 5480:

Edge radius = 0,16
Addendum coeff. = 0,6
Dedendum coeff. = 0,45

For shaping according to DIN 5480:

Edge radius = 0,16
Addendum coeff. = 0,65
Dedendum coeff. = 0,45

For broaching according to DIN 5480:

Edge radius = 0,16
Addendum coeff. = 0,55
Dedendum coeff. = 0,45

Select the tool ‘Gear shaper cutter’ from the listbox for gear 2 and change the ‘Basic rack profile’ to ‘user defined input’ as well. Please enter the tool data as specified for gear 1.

Please Note: For the tip form you have to select ‘Radius with straight line’.

Figure 7.174:

Inputs for the tool

Please note: In case the calculation of the gear tooth form does not work properly, please select ‘Constructed involute’ as a tool for gear 2.

7.22.3 Enter Data for Involute Spline

Now enter the data for the involute spline into the geometry mask of the gear module. Click on the ‘Geometry’ button. Begin to enter the normal module, the pressure angle and the number of teeth.

Figure 7.175:

Geometry

Please Note: The pressure angle for involute splines is $30∘$ according to DIN 5480. The number of teeth for gear 1 is positive (for the shaft) and for gear 2 (for the hub) negative. If the modification tip is not set automatically to the value ‘0’, please click on the ‘Lock’ button and enter the value ‘0’ for both gear 1 and gear 2. The tip allowances can be defined as well by clicking on the ‘Lock’ symbol.

For the profile shift coefficient $x*$ for gear 1 you have to enter $x*m∕m = x*$ (Involute spline: $x*$ is often positive).

Figure 7.176:

Inputs

7.22.4 Define Tooth Thickness Allowances

Finally, you have to define the tooth thickness tolerances. Click the ‘Allowances’ button. An individual input is also possible.

Figure 7.177:

Allowances

7.22.5 Accurate Tooth Form

Have a look at the accurate tooth form by clicking the ‘Tooth form’ button. Click on the ‘Detail view’ button.

Figure 7.178:

Tooth form

Here the accurate tooth form is graphically represented and you can select the tooth thickness allowances (lower, upper and mean allowances) and the tip diameter allowances (lower, upper and mean allowances). Create also an DXF output via the menu item ‘CAD $⇒$ DXF Output’.

Figure 7.179:

Tooth form

Please note: We recommend you to define a template file (e.g., for the tool data). Therefore, it is not necessary to enter the tool data again at every start. That saves both time and work. All you have to do is to define a template. If you now open the calculation module, the module starts with your individual values (e.g., a pressure angle of $30∘$ ). Find further information in the section 4.19 ‘Template File’.

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Chapter 7Spur Gear Pair Calculation According to DIN 3990 and Other Standards

7.1 Start the Calculation Module

7.2 Input of Geometry Data

7.2.1 Normal Module

7.2.2 Pressure Angle

7.2.3 Helix Angle

Helical Gears

7.2.4 Standard Centre Distance

7.2.5 Working Centre Distance

7.2.6 Direction of Helix Angle

7.2.7 Number of Teeth

7.2.8 Facewidth

7.2.9 Profile Shift Coefficient

Characteristics of the Profile Shift

7.2.10 Tip Diameter

7.2.11 Tip Diameter Allowance

7.2.12 Modification of Tip Diameter

7.2.13 Tip Clearance

Backlash

7.2.14 Root Diameter and Allowances of Root

7.2.15 Inner and Outer Diameter

7.2.16 Web Width

7.2.17 Chamfer

7.2.18 Addendum Chamfer

7.3 Input of Tool Data

7.3.1 Tool

Hob Generation

Field of Application of the Hob:

Gear Shaper Generation

Field of Application of the Gear Shaper Cutter:

Constructed Involute

Representation of Hob and Gear Shaper Cutter

7.3.2 Basic Rack Tooth Profile

Modification of the Basic Rack Profile

7.3.3 Tip Form

7.3.4 Protuberance

Corrections of the Tooth Form in the Calculation Program

Determination of the Amount of the Protuberance from the Height of the Protuberance Flank

7.3.5 Machining Allowance

7.4 Input for the Determination of Allowances

7.4.1 Gear Quality

7.4.2 Backlash Allowance and Tolerance Sequence

7.4.3 Tooth Thickness Allowance

7.4.4 Tooth Space Allowance

7.4.5 Measurement of Tooth Thickness

Span Measurement across Several Teeth

Tooth Thickness Measurement by Diameter over Pins or Balls

7.4.6 Tolerance Field for Centre Distance

7.4.7 Centre Distance Allowance

7.4.8 Backlash Normal Plane

7.4.9 Backlash Pitch Diameter

7.4.10 Radial Backlash

7.5 Representation of Gear Tooth Form

7.5.1 Representation of Cylindrical Gear Pair

7.5.2 Representation of Mesh

7.5.3 Rotating Angle

7.5.4 Rotation

7.5.5 Tooth Thickness Allowance

7.5.6 Tip Diameter Allowance

7.5.7 Centre Distance Allowance

7.6 Calculation of Gear Load Capacity

Load Capacity of the Tooth Root - Tooth Breakage

Load Capacity of the Tooth Flank - Pitting of Gear Teeth

Scuffing Load Capacity

7.6.1 Activate Load Capacity

7.6.2 Inputs for Load Capacity - General Factors

Comment

Material Selection

Kind of Material

Application Factor

Working Characteristics of the Driving Machine

Working Characteristics of the Driven Machines

Face Load Factor

Mesh Misalignment

Position of Tooth Contact Pattern

Pinion Corrections

Lead Crowning

End Relief

Pinion Arrangement - Stiffening Effect

Transmitted Power - Power Distribution for the Dimensioning of the Face Load Factor

Chapter 7
Spur Gear Pair Calculation According to DIN 3990 and Other Standards

Backlash $j$

Application Factor $KA$

Face Load Factor $KH β$

Mesh Misalignment $F βx$

Transmitted Power - Power Distribution for the Dimensioning of the Face Load Factor $kHβ$

Technology Factor $Y T$

Mode of Operation Factor $YA$

Dynamic Coefficient $KV$

Transverse Coefficient $KH α$

Mesh Load Factor $K γ$

Thermal Contact Coefficient $BM$

Relative Structure Factor $X WrelT$

$60∘$ Tangent for Internal Gears

Life Factors $YNT$ and $ZNT$

Face Load Factor $KH β$

Helix Angle Factor $Z β$

Work Hardening Factor $ZW$

Thermal Contact Coefficient $BM$

Lubricant Factor $XL$

Relative Structure Factor $X WrelT$