10 Spur gear pair calculation according to DIN 3990 and other standards

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Chapter 10
Spur gear pair calculation according to DIN 3990 and other standards

    10.1   Start the calculation module
    10.2   Input of geometry data
    10.3   Input of tool data
    10.4   Input for the determination of allowances
    10.5   Representation of the tooth form
    10.6   The load capacity of the gearing
    10.7   Meshing interferences for external gears
    10.8   The message window
    10.9   The calculation results
    10.10   The documentation: The calculation report
    10.11   How to save the calculation
    10.12   The button ‘Redo’ and ‘Undo’
    10.13   The button ‘CAD’
    10.14   The button ‘Options’
    10.15   The internal gears
    10.16   Input of geometry data for internal gears
    10.17   Manufacturing process for internal gears
    10.18   Meshing interferences for internal gears
    10.19   Examples for internal gears in the eAssistant
    10.20   How to calculate the accurate tooth form of involute splines

10.1 Start the calculation module

Please login with your user name and your password. Select the module ‘Spur gear pair’ through the tree structure of the Project Manager by double-clicking on the module or clicking on the button ‘New calculation’.

The calculation module is opened in a new window.

Figure 10.1:

General overview

10.2 Input of geometry data

The eAssistant allows an easy and fast calculation of the geometry of spur gears according to DIN 3960, DIN 3961, DIN 3964, DIN 3967, DIN 3977 and DIN 868. The load capacity according to DIN 3990 is considered as well. The profile shift modification and the addendum chamfer are also integrated into the calculation. Find out more about the functions and possibilities for the calculation of spur gear pairs in the section ‘The input of geometry data’.

Please note: All important calculation results are determined and displayed immediately during the input of your values. Which means that after every input of data, the results are calculated again. You can also click on the button ‘Calculate’, then your inputs will be confirmed too. If you press the ‘ENTER’ or ‘Tabulator’ button of your keyboard or if you click on the next input field, your data will be also confirmed. Find futher information in the section 10.9 ‘The results’.

Figure 10.2:

The inputs in the geometry section

10.2.1 Normal module

The normal module $m n$ 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

10.2.2 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-carring 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.

10.2.3 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.

The 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.

10.2.4 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 $ad$ is modified automatically. The standard centre distance is an operand. If the sum of the profile shift coefficients = 0, $ad$ corresponds to the working centre distance $a$ .

10.2.5 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 determinded automatically.

Figure 10.3:

The standard centre distance and the working centre distance

In case of an overlarge profile shift modification, 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.

Example:

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 $β$ = $15∘$ . Now the standard centre distance and the working centre distance are determined automatically.

Figure 10.4:

The inputs

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

Figure 10.5:

The input ‘0’

Now the nearest integer value is used for the working centre distance.

Figure 10.6:

The integer value

10.2.6 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 10.16.1 ´The direction of helix angle’.

Figure 10.7:

Left and right-hand teeth

Example:

Select the option ‘left’ for gear 1:

Figure 10.8:

The option

That means: Gear 1 is left-handed, gear 2 is right-handed (external gearing).

Select the option ‘left’ for gear 2:

Figure 10.9:

The option

That means: Gear 2 is left-handed, gear 1 is right-handed (external gearing).

10.2.7 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 modification. Find more information about the profile shift coefficient in section 10.2.9 ‘Profile shift coefficient’. In section ‘Internal gearings’ you will get more information about the number of teeth for internal gearings.

10.2.8 Face width

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

Figure 10.10:

The face width

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


Standard values for the face width $b$ 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

10.2.9 Profile shift coefficient

Profile shift modifications 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 modifications 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 modification 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 modification 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.

Figure 10.11:

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

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.

Characteristics of the profile shift modification

A positive profile shift modification increases the tooth thickness, a negative profile shift modification decreases the tooth thickness.
With an increasing positive profile shift modification, the tip tooth 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 tip tooth thickness determines the limit for a very large profile shift modification, in particular for very small number of teeth.
The profile shift modification affects the operating pressure angle as well as the load capacity.
For a small number of teeth and with a negative profile shift modification, an undercut becomes a problem (see figure in section 10.2.9). 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 10.12:

The 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 10.13:

The balanced specific sliding

10.2.10 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 have 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 10.14:

The 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’.

10.2.11 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.

Figure 10.15:

Enable the input field

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.

10.2.12 Modification of the 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.

10.2.13 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 10.16:

The 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.

The 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 10.4.8 ‘Backlash normal plane’).

10.2.14 Root diameter

The root diameter $df$ depends upon the module, the profile shift modification 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.

10.2.15 Allowances of root

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.

10.2.16 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 10.17:

The 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.

Figure 10.18:

The message window

10.2.17 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.

The ‘Lock’ button is still disabled.

Figure 10.19:

The input field for the web width

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 face width. In case the web width is smaller than the face width, 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 10.20:

The web width

10.2.18 The chamfer

The chamfer can be considered.

Figure 10.21:

The chamfer

10.2.19 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.

Figure 10.22:

The 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 10.5 ‘The representation of the tooth form’). Click the button ‘Geometry’ and you get to the geometry input again.

10.3 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 10.23:

The input mask for tool data

10.3.1 The 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.

Figure 10.24:

The selection of tool

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 10.17 ‘The manufacturing process for internal gears’). With a rack, for instance, two gears can be meshed with any number of teeth. Therefore, one gear is considered as a tool which rolls in or cuts the gearing.

Figure 10.25:

The hob and gear shaper cutter

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 face width. 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 a unidirectional traverse at right angles. Both movements are relatively simple to effect, resulting in a very accurate process.

The field of application of the hob:

Recommended for gears with very wide face width
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.

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.

The 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 gears, chain gearings

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 10.26:

The constructed involute

The 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 10.27:

The tool

10.3.2 Basic rack 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 10.28:

The 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 10.3.4 ‘The protuberance’).

Figure 10.29:

The 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 dedenum coefficient are activated. Now you can modify the basic rack profile.

Figure 10.30:

The own input

The 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 10.31:

The button for the tool dimensioning

A new window is opened.

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 10.32:

The tool dimensioning

10.3.3 The 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’.

Figure 10.33:

The tip form

10.3.4 The 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.

The 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.

* * p*prp0 = (hprp0 --ρa0 ⋅(1--sin(αp)))⋅sin(αn --αp) + ρ*a0 ⋅(1 - cos(αn - αp)) cos(αp)

The following figure shows a representation.

Figure 10.34:

The 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.


The undercut for ground gears dependent upon module

Module $m$	$q$	$prp0$	$hap$	$ρao$

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

10.3.5 Machining allowance

You can consider an allowance for the tooth flank. The tool provides an allowance $q$ on the flank for the pre-cutting tool. The allowance is the smallest distance between the involutes and the pre-machining having the same root diameter. The following table provides the maximum machining allowances of the several methods:


The maximum machining allowances¹

An 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

¹ from: Linke, H.: Stirnradverzahnung Berechnung Werkstoffe Fertigung, Carl Hanser Verlag, München, Wien, 1996, p.638

The eAssistant provides the following allowances for the grinding of a gear:

Constant allowance with bottom of the tooth space

Protuberance: Cutter tooth profile is built up on the tip to provide an undercutnear the root of the gear being generated.

10.4 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 10.35:

The input of allowances

10.4.1 The gear quality

The choice of the right toothing quantity 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 are defined according to DIN standards arranged in ascending accuracy of 12 to 1. 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 10.36:

Tolerances according to the manufacturing process

Reference values for the selection of the quality, tolerances for gearings made of metal and plastics:


Toothing made of metal¹

$ν$	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 plastics¹

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 cutting¹

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

¹ from: Karl-Heinz Decker: Maschinenelemente: Gestaltung und Berechnung, 1992, p.512, table 23.3

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

Figure 10.37:

The listbox for the selection of quality

10.4.2 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 10.38:

The own input

10.4.3 Tooth thickness allowance

One of the most important criteria of gear quality is the specification and control of tooth thickness. As mentioned in 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 10.39:

The 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 10.4.8 ‘The backlash normal plane’).

Figure 10.40:

The tooth thickness allowances for gear 1 and gear 2

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 10.41:

The ‘Calculator’ button

A new window is opened.

Figure 10.42:

Calculation of tooth thickness allowances

Activate (checkmark) 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.

10.4.4 Tooth space allowance

The tooth space allowance $AW$ 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.

Figure 10.43:

The tooth space allowance for gear 1 and gear 2

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.

10.4.5 Measurement of the 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

A 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 10.44:

The 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).

Figure 10.45:

The number of teeth for the span measurement

By using the ‘Lock’ button you are able to activate the input field and you can enter your own input value for the number of teeth for the span measurement. If you click the button again, the previous input value appears.

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 $M dR$
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 even numberof teeth

Measurement over balls: External spur gear with odd numberof teeth

Measurement over balls: Internal spur gear with odd number ofteeth

The eAssistant already specifies the diameter of ball or pin for the test dimensions.

Figure 10.46:

The diameter of ball or pin

Enable the input field by clicking the ‘Lock’ button. Enter your own input value for the number of teeth for the span measurement. If you click on the button once again, the previous input value appears.

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’.

10.4.6 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 tooth thickness allowance, the tooth space tooth thickness and the backlashes are interdependent. Hence, if you change the centre distance, then the backlash is changed, too.

Figure 10.47:

The 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.

10.4.7 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 10.48:

The centre distance allowance

10.4.8 The 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
The radial backlash

Figure 10.49:

The 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.

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 allowances.

10.4.9 The backlash pitch diameter

The backlash pitch diameter $jt$ 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 10.50:

The backlash pitch diameter

10.4.10 The 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 10.51:

The radial backlash

10.5 Representation of the tooth form

A special highlight of this eAssistant calculation module is the presentation of the accurate tooth form with an animation and simulation of the tooth mesh. For the presentation you can select the lower, middle and upper allowances for the tooth thickness and centre distance.

Figure 10.52:

The tooth form

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 button ‘Tooth form’ and you get a general or detailed view of the tooth form. By clicking the buttons ‘Geometry’ or ‘Tool’, you can open the main input masks of the calculation program again.

Please note: Please note 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 ‘The button CAD’ contains some helpful information on this function 10.13.

10.5.1 Representation of the spur gear pair

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

Figure 10.53:

The spur gear pair

Please note: Please note 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 10.5.2 ‘The representation of the mesh’.

10.5.2 Representation of the mesh

Click on the ‘Detail view’ button.

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

Figure 10.54:

The 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.

10.5.3 Rotating angle

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

Figure 10.55:

The rotating angle

The rotation of the driving gear counter-clockwise

The rotation of the driving gear clockwise

10.5.4 The rotation

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

Figure 10.56:

The rotation

The continuous rotation of the driving gear counter-clockwise

The continuous rotation of the driving gear clockwise

The rotation is stopped.

10.5.5 Tooth thickness allowance

Click on the button ‘Detail view’.

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.

Figure 10.57:

The tooth thickness allowance in the main mask ‘Allowance’

For the representation of the tooth mesh, select the lower, middle and upper tooth thickness allowances for gear 1 and gear 2.

Figure 10.58:

The tooth thickness 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 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 middle tooth thickness allowance. At the first start of the tooth form, you get the middle tooth thickness allowance as a standard feature.

The representation of gear 1 including the lower, middle and upper tooth thickness allowance

The representation of gear 2 including the lower, middle and upper tooth thickness allowance

Please note: Enter different values which are larger or smaller than the upper and lower tooth thickness allowance. You will see that all previous tooth thickness allowances are used again.

10.5.6 Tip diameter allowance

Click on the button ‘Detail view’.

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.

Figure 10.59:

The tip diameter allowance in the main mask ‘Geometry’

For the representation of the tooth mesh, select the lower, middle and upper tip diameter allowances for gear 1 and gear 2.

Figure 10.60:

The tip diameter 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 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 middle tip diameter allowance. At the first start of the tooth form, you get the middle tip diameter allowance as a standard feature.

The representation of gear 1 including the lower, middle and upper tip diameter allowance

The representation of gear 2 including the lower, middle and upper tip diameter allowance

Please note: Enter different values which are larger or smaller than the upper and lower tip diameter allowance. You will see that all previous tooth thickness allowances are used again.

10.5.7 Centre distance allowance

Click on the button ‘Detail view’.

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.

Figure 10.61:

The centre distance allowance in the main mask ‘Allowances’

For the representation of the tooth mesh, select the lower, middle and upper centre distance allowances for gear 1 and gear 2.

Figure 10.62:

The centre distance 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 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 middle centre distance allowance. At the first start of the tooth form, you get the middle centre distance allowance as a standard feature.

The representation of the lower, middle and upper centre distance allowance

Please note: Enter different values which are larger or smaller than the upper and lower centre distance allowance. You will see that all previous centre distance allowances are used again.

10.6 The load capacity of the gearing

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)

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 modification (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 modification (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 EP oil (extreme pressure)
High oil temperature
Rough oil surface
Low toothing quantity: 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: EP 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

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.

The calculation of the load capacity of spur gears is standardized according to DIN 3990. In DIN 3990, there are different methods for the determination of the load capacity. The eAssistant provides all calculations according to DIN 3990 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 coefficient, 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.

Figure 10.63:

The load capacity

10.6.1 Activate the load capacity

Click on the button ‘Load capacity’ 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’ from the listbox ‘Calculation method’, all input fields are enabled. In case you do not need the calculation for the load capacity, the calculation can be deactivated. Thus, the size of the calculation report becomes smaller.

Figure 10.64:

Activate the calculation for the load capacity

10.6.2 The inputs for the load capacity - General factors

Comment

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

Figure 10.65:

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 10.66:

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 10.67:

The material database

Please note: When you are in the main mask of the load capacity and the listbox ‘Material’ is active, then you can use the two arrow keys ‘Up’ and ‘Down’ to display the different safeties in the result panel.

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.

The difference between hardening and tempering and hardening

Hardening and tempering differs from hardening by annealing at high temperatures. The temperature range for hardening and tempering ranges from 400^∘C 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 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), an 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 activated and you can specify your individual material very easily. Your inputs will be saved in the calculation file. But in case you select another material from the listbox, then your defined inputs get lost. You have to enter these inputs again.

Figure 10.68:

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 $K A$ according to DIN 3990-1: 1987-12

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

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 ...

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.

Face coefficient $KH β$

The face coefficient $KH β$ evaluates non-uniform load distribution across the face width due to manufacturing inaccuracies and elastic deformations. The effect of the non-uniform load distribution is considered by the face coefficients $KHβ$ for the surface pressure, $KFβ$ for the stress, $KBβ$ for scuffing. The face load factor is determined according to DIN 3990, part 1 method B.

Figure 10.69:

The face coefficient for the surface pressure

When you start the eAssistant, the value ‘1.25’ is entered into the input field. In case you already use a defined face coefficient, you can save the certain factor as a template file. Then the program starts with the individual face coefficient. Find further information on the topic in section 4.17 ‘The template file’.

When you click on the calculator symbol, the input mask for the face coefficient opens. In the top input field ‘Face coeff.’ you can find the default value of ‘1.25’. You will notice that the lower input fields and listboxes are disabled. By using the button ‘OK’ you can take over the default value to the main mask.

There is a listbox next to the input field for the face coefficient. When you open the listbox, the entry ‘DIN 3990 T1 method B’ appears.

Figure 10.70:

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 coefficient 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 coefficient. When the button ‘OK’ is activated, then the determined face coefficient 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 coefficient, the above-mentioned input mask opens. The face coefficient $K Hβ$ is displayed in the input field. Open the adjacent listbox and select the entry ‘DIN 3990 T1 method B’. The face coefficient is calculated but the button ‘OK’ is still disabled.

Figure 10.71:

Face coefficient

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

Figure 10.72:

Take over the face coefficient

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.

Figure 10.73:

Tooth trace

The mesh misalignment $F βx$ considers all influences of manufacturing, assembly and deformation that may intensify and compensate each other. 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 face width of the gear.

Figure 10.74:

Mesh misalignment

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

Figure 10.75:

User-defined selection

Position of tooth contact pattern

A tooth contact pattern - 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 face width and tooth depth is to be accomplished. For rigid gearings, the size and the position of the tooth contact pattern under a light load can be used alternatively. Tooth contact patterns serve to assess both the toothing geometry of an individual gear and the state of two engaged gears under a light load. For a contact pattern, a thin layer of a marking compound is applied on four to six 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 contact pattern that is indicated by a light wear pattern on the mating tooth surfaces.

Click on the question mark button and you get several contact pattern according to DIN 3990, part 1.

Figure 10.76:

Part of a contact pattern

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 face width). Crowning and end relief are the most important pinion corrections and are advantageous for a good load distribution over the face width of a gear. Due to 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 10.77:

Selection of the pinion correction

Lead crowning: 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 crowning is to ensure that manufacturing inaccuracies and deformations are adjusted under load and that the tooth ends are relieved. In general, the lead crowning $Cc$ is carried out symmetrically to the centre of the face width.

Figure 10.78:

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 10.79:

End relief

Pinion arrangement - The stiffening effect

DIN 3990 describes the stiffening effect as follows:

When $d1∕dsh ≥ 1,15$ , then stiffening is assumed; when $d1∕dsh < 1,15$ , 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 10.80:

Pinion arrangement according to DIN 3990

Figure 10.81:

Pinion arrangement according to DIN 3990

Transmitted power - Power distribution for the dimensioning of the face coefficient $kH β$

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% are transmitted through the tooth mesh and the remaining 40% are 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 coefficient, 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.

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 selection of a lubricant

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 effeciency 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 EP (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.

Liquid lubricants may be characterized in many different ways. Viscosity is one very important property of a lubricant and determines the oil´s 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 Maschinenlemente, 17th revised edition, published by Vieweg, Wiesbaden 2005.) 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.

Figure 10.82:

Open the selection of a lubricant

Select the kind of lubrication from the listbox.

Figure 10.83:

Selection of the lubricant

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.

Subsequently, the corresponding database is queried and the lubricants are displayed.

Click on the button ‘Lubricant’ and open database. The extensive database contains the lubricants including all detailed information about the oils and greases (e.g., density, viscosity, force stage of the FZG test). You can find out more about the FZG test in section ‘Extensive input options for the scuffing load capacity’.

10.6.3 Extended input options for the load capacity of tooth root and tooth flank

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

Figure 10.84:

Selection of the extended input options

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

Figure 10.85:

The extended input options for tooth root and tooth flank

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 10.86:

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: The 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 eAssistant 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. However, as an alternative to the calculation of the optimal hardening depth, the following table shows some guideline values for amounts of individual hardening depth:


Hardening depth after case-hardening (e.g., flame and induction hardening)
for gears according to Linke¹

Normal module $m n$ in mm	Hardening depth $Rht$ in mm

3 to 3,5	0,7 + 0,4

4 to 4,5	0,9 + 0,4

5 to 6	1,2 + 0,5

7 to 8	1,6 + 0,6

9 to 12	2,0 + 0,8

14 to 16	2,7 + 1,0

18 to 22	3,5 + 1,4

25 to 28	4,5 + 1,8

¹ from: Linke, H.: Stirnradverzahnung Berechnung Werkstoffe Fertigung, Carl Hanser Verlag München, Wien, 1996, S. 520, Tab.: 7.4/8

Technology factor $YT$

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


Technology factor $YT$ according to Linke¹

Kind of manufacturing of the tooth root	Technology factor $Y T$

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

¹ from: Linke, H.: Stirnradverzahnung Berechnung Werkstoffe Fertigung, Carl Hanser Verlag München Wien, 1996, p.320, table: 6.5/6

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

$σFlim0$

Fatigue strength of the tooth root from material data

$σ Flim$

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

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


Mode of operation factor $YA$ according to Linke¹

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

¹ from: Linke, H.: Stirnradverzahnung Berechnung Werkstoffe Fertigung, Carl Hanser Verlag München Wien, 1996, p.321, table: 6.5/7

Dynamic coefficient $K V$

The dynamic coefficient $K V$ 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.

Transverse coefficients

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 ( $KFα$ ).

Mesh load factor

The mesh load factor takes into account an uneven distribution of the total circumferential force for gearings with 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 face width 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 face width shall be used for both pinion and gear.

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.

10.6.4 Extended input options for the scuffing load capacity

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

Figure 10.87:

Selection of extended input options

Figure 10.88:

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 $XW relT$

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

X = XW--- W relT XW 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. For the FZG gear test $XW T$ = 1,0 applies.


Structure factor $XW$ ¹ ¹

Material/Heat treatment	Structure factor $XW$

Hardened and tempered steel	1,00

Phosphated steel	1,25

Coppered Steel	1,50

Liquid nitrided and gas-nitrided steel	1,50

Case-hardened steel
- with low austenite content	1,15
- with normal austenite content	1,00
- with high austenite content	0,85

Austenitic steel (stainless steel)	0,45

¹ from: Linke, H.: Stirnradverzahnung Berechnung Werkstoffe Fertigung, Carl Hanser Verlag München Wien, 1996, S. 367, Tab.: 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 modification 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 lubricant´s resistance to scuffing.

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 10.89:

Profile modification

Contact temperature along the path of contact

Figure 10.90:

Contact temperature with zero-deviation involute profile (no profile modification)

Figure 10.91:

Contact temperature for profile modification

Figure 10.92:

Contact temperature for an uniform mesh

The integral temperature method and flash 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 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:

The integral temperature method defines a weighted average of the surface temperature along the path of contact.
The flash temperature method defines a variable contact temperature along the path of contact.

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:

SintS = ϑSint-≥ SSmin ϑint

$ϑintS$

Scuffing integral temperature

$ϑint$

Integral temperature

DIN 3990 provides some reference values from practical experience:

$SintS<1,0$

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.

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 $ϑ B$ in any point of contact $Y$ results from the sum of the mass temperature $ϑ M$ and the flash temperature $ϑ fla$ :

ϑ = ϑ + ϑ B M fla

According to the flash temperature method, there is no scuffing as long as the contact temperature $ϑB$ (as the sum of mass temperature $ϑM$ and flash temperature $ϑfla$ ) does not exceed the scuffing contact temperature in all points of contact. The scuffing contact temperature $ϑS$ is determined from a material-lubricant-material system in gear tests and is transferred to the gear pair.

Figure 10.93:

Contact temperature along the path of contact

The letters 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 values of the contact temperature along the path of contact

$ϑoil$

Temperature of the lubricating oil before mesh

$ϑS$	Scuffing temperature

The factor of safety $SBmin$ 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 $SBmin ≈ 3$ .

10.7 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 modification as well as a very small tip clearance may cause meshing interferences. Interference takes place between the tips of the teeth 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 modification,
other manufacturing tools

When internal gearsets have a too small difference between the number of teeth in the pinion and the number of teeth in the gear (smaller than ‘10’), there meshing interferences may occur. Meshing interferences do not occur very often for external gearings.

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

The condition for a smooth and continuous tooth action is:

εγ = εα + εβ > 1

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 10.94:

The total contact ratio

10.7.2 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.

Please note: Opposed to external gearings, meshing interferences occur more often for internal gearings. The section 10.18 discusses this issue.

10.8 The message window

The calculation module contains a message window. You get all information, warnings and hints. The information will also appear in the calculation report later. After the completion of your calculation, click on the ‘Report’ button to create a calculation report (see section 10.10 ‘The documentation: The calculation report’).

Figure 10.95:

The message window

10.9 The calculation results

All important calculation results, such as the safety foot root ot flank, are determined and displayed immediately during the input of values.

Figure 10.96:

The calculation results

Enter the values and the result is determined and displayed immediately. Which means that after every input of data, the results are calculated again. Move between the fields using the Tab key of your keyboard or use the mouse to click in the next input field. Your inputs will be confirmed. Press ‘Enter’ or the ‘Calculate’ button, your input will be confirmed as well.

10.10 The documentation: The calculation report

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

Figure 10.97:

The button ‘Report’

The calculation report is opened.

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.

Figure 10.98:

The calculation report

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 10.99:

Save, print and PDF features

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 button ‘Save’.
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 dialog box for saving the report.

10.11 How to save the calculation

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

Figure 10.100:

The ‘Save’ button

If you have activated the option ‘Enable file save local’ in the Project Manager and the option ‘Local’ in the calculation module, a standard Windows dialog for saving the file on your workstation appears.

Please note: You must not forget that the calculation module has to be closed to activate the option ‘Enable file save local.’

Figure 10.101:

Windows dialog for saving the file

In case you have not activated this option, a new window is opened and you can save the calculation on the eAssistant server.

Figure 10.102:

Save the calculation

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 ‘File’.

10.12 The button ‘Redo’ and ‘Undo’

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

Figure 10.103:

Button „Vorwärts“ und „Zurück“

10.13 The button ‘CAD’

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

Figure 10.104:

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 and intelligently. 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: You have to activate the option ‘Enable file save local’ to allow the generation of CAD data.

In the following you get some information on the DXF output and the eAssistant CAD plugin as well.

10.13.1 The DXF output for an 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 10.105:

The DXF output

A new window is opened.

Figure 10.106:

The settings for the 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

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

Now you can save the DXF file on 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.

10.13.2 eAssistant CAD plugin for several CAD systems

Considering tolerances, addendum chamfer and profile modification, the geometry for spur gear pairs can be calculated up to the accurate tooth form. External and internal gears as well as cylindrical and helical gears can be created automatically as a 3D part.

Accomplish a calculation and click on the button ‘CAD $⇒$ SolidKiss_nG Interface’.

Figure 10.107:

SolidKiss_nG

Now open the CAD system (e.g., SolidWorks, Solid Edge, Autodesk Inventor or Catia). The plugin enables you to open all eAssistant modules directly through the CAD menu. At the push of the button, the part can be created as a 3D part on the basis of the calculated data.

The CAD plugin has to be installed on your system.

A simple mouse-click allows you to add all necessary manufacturing data of a gear wheel to the drawing. The data is diplayed as a table. The appearance and size of that table are individually configurable.

Please note: Please visit our web site www.eAssistant.de to find detailed information on the CAD plugin.

10.14 The button ‘Options’

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

Figure 10.108:

The button ‘Options’

A new window is opened.

Here you can change now the following settings:

Minimum safety tooth root
Minimum safety flank
Minimum safety scuffing (integral)
Minimum safety 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 report

10.15 The 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. 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.

Figure 10.109:

Internal gear in the eAssistant

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.

10.16 Input of geometry data for internal gears

Please note: 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 10.110:

The internal gearing

10.16.1 Direction of the 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.

10.16.2 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 10.111:

An 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 10.112:

The helix angle

10.16.3 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.

Figure 10.113:

The input of a negative number of teeth

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

10.16.4 Centre distance

The working centre distance is always negative.

Figure 10.114:

The negative centre distance

Please note: 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 10.115:

A negative diameter

10.16.5 Profile shift modification

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 modification 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 geatings: the profile shift modification is positive, $x⋅m>0 n$ , when the profile reference line is shifted from the pitch circle to the pitch circle and negative, $x⋅m< 0 n$ , when the profile reference line is shifted from the pitch circle to the root circle.

Figure 10.116:

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

A positive profile shift modification 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.

Please note: A positive profile shift modification may be disadvantageous for internal gears. It is comparable with a negative profile shift modification for external gears.

A negative profile shift modification 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 space width 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.

Please note: A negative profile shift modification may be advantageous for internal gears. In this case, it is comparable with a positive profile shift modification for external gears.

For external and internal gear pairs the impacts of positive and negative profile shift modification are similar.

10.17 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 10.117:

The 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.

10.18 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 (The tooth root meshing interference on the gear shaper cutter)
Tooth crest meshing interference
Feed meshing interference
Radial assembly interference

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

10.18.1 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.

10.18.2 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.

10.18.3 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 appears. The term of the mesh and the load capacity are decreased.

10.18.4 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.

The meshing interference can be avoided by changing the number of teeth and by a negativeprofile shift modification.

10.18.5 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.

If the sum of the profile shift modification is decreased, feed meshing interferences can beavoided.

10.18.6 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. This meshing interference may occur for a small difference between the number of teeth.

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

Figure 10.118:

The radial assembly interference

10.19 Examples for internal gears in the eAssistant

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.

10.19.1 Extremely small number of teeth (pinion)

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

Figure 10.119:

Example*

Figure 10.120:

eAssistant: A small number of teeth

10.19.2 Standard tooth profile

$z1$ = 20; $z2$ = -28

Figure 10.121:

Example*

Figure 10.122:

eAssistant: A standard tooth profile

10.19.3 Small difference of number of teeth

A large gear transmission ratio with an extremely small difference of number of teeth: $z 1$ = 29; $z 2$ = -30. Meshing interferences can be avoided by a large modification of tip diameter.

Figure 10.123:

Example*

Figure 10.124:

eAssistant: A small difference of number of teeth

* Examples taken from: K. Roth: Zahnradtechnik: Band I, Stirnradverzahnungen - Geometrische Grundlagen (1989, p.198)

10.20 How to calculate the accurate tooth form of involute splines by using the spur gear pair calculation module

10.20.1 Select basic data for the involute spline

Please login with your user name 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 10.125:

Involute splines

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

Figure 10.126:

Profilgeometrieauswahl

10.20.2 Modify the 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 10.127:

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

Please enter the tool data as specified for gear 1.

Figure 10.128:

The inputs for the tool

10.20.3 Enter the data for the involute spline

Now enter the data for the involute spline, that you have previously taken from the involute spline module, into the geometry mask of the gear module. Click on the button ‘Geometry’.

Figure 10.129:

The geometry

Enter the normal module, the pressure angle and the number of teeth.

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.

The calculation module ‘Involute splines’ provides the addendum modification $x ⋅m$ . Therefore, you have to enter for the profile shift coefficient $x$ for gear 1 $xm∕m = x$ (involute spline: $x$ is positive).

Figure 10.130:

The inputs

10.20.4 Define the tooth thickness allowances

Finally, you have to define the tooth thickness tolerances. Click on the button ‘Allowances’. Here an individual input is possible as well.

Figure 10.131:

The allowances

10.20.5 Accurate tooth form

Have a look at the accurate tooth form by clicking the button ‘Tooth form’.

Figure 10.132:

The tooth form

Click on the button ‘Detail view’. Here the accurate tooth form is graphically represented and you can select the tooth thickness allowances (lower, middle and upper) and the tip diameter allowances (lower, middle, upper). Now you are able to export the tooth form via the button ‘CAD’ to a 3D CAD system. Create also an DXF output via the menu item ‘CAD $⇒$ DXF Output’.

Figure 10.133:

The 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 4.17 ‘The template file’.

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Chapter 10Spur gear pair calculation according to DIN 3990 and other standards

10.1 Start the calculation module

10.2 Input of geometry data

10.2.1 Normal module

10.2.2 Pressure angle

10.2.3 Helix angle

10.2.4 Standard centre distance

10.2.5 Working centre distance

10.2.6 Direction of helix angle

10.2.7 Number of teeth

10.2.8 Face width

10.2.9 Profile shift coefficient

Characteristics of the profile shift modification

10.2.10 Tip diameter

10.2.11 Tip diameter allowance

10.2.12 Modification of the tip diameter

10.2.13 Tip clearance

10.2.14 Root diameter

10.2.15 Allowances of root

10.2.16 Inner and outer diameter

10.2.17 Web width

10.2.18 The chamfer

10.2.19 Addendum chamfer

10.3 Input of tool data

10.3.1 The tool

Hob generation

The field of application of the hob:

Gear shaper generation

The field of application of the gear shaper cutter:

Constructed involute

10.3.2 Basic rack profile

10.3.3 The tip form

10.3.4 The protuberance

Corrections of the tooth form in the calculation program

10.3.5 Machining allowance

10.4 Input for the determination of allowances

10.4.1 The gear quality

10.4.2 Backlash allowance and tolerance sequence

10.4.3 Tooth thickness allowance

10.4.4 Tooth space allowance

10.4.5 Measurement of the tooth thickness

A span measurement across several teeth

Tooth thickness measurement by diameter over pins or balls

10.4.6 Tolerance field for centre distance

10.4.7 Centre distance allowance

10.4.8 The backlash normal plane

10.4.9 The backlash pitch diameter

10.4.10 The radial backlash

10.5 Representation of the tooth form

10.5.1 Representation of the spur gear pair

10.5.2 Representation of the mesh

10.5.3 Rotating angle

10.5.4 The rotation

10.5.5 Tooth thickness allowance

10.5.6 Tip diameter allowance

10.5.7 Centre distance allowance

10.6 The load capacity of the gearing

Load capacity of the tooth root - Tooth breakage

Load capacity of the tooth flank - Pitting of gear teeth

Scuffing load capacity

10.6.1 Activate the load capacity

10.6.2 The inputs for the 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 coefficient

Mesh misalignment

Position of tooth contact pattern

Pinion corrections

Pinion arrangement - The stiffening effect

Transmitted power - Power distribution for the dimensioning of the face coefficient

Reference gear

Power and torque

Kind of lubrication and selection of a lubricant

10.6.3 Extended input options for the load capacity of tooth root and tooth flank

Grinding notch

Hardening depth root/flank

Chapter 10
Spur gear pair calculation according to DIN 3990 and other standards

Application factor $KA$

Face coefficient $KH β$

Mesh misalignment $F βx$

Transmitted power - Power distribution for the dimensioning of the face coefficient $kH β$

Technology factor $YT$

Mode of operation factor $YA$

Dynamic coefficient $K V$

Thermal contact coefficient $BM$

Relative structure factor $XW relT$