Helical Gear Reliability Prediction

Published Date: 02 Nov 2017

A gear is a wheel-like rotating machine component with a number of cut teeth that meshes with another toothed part in order to transmit torque. Two or more gears working in tandem are called a transmission and can produce a mechanical advantage through a gear ratio, see Figure 6.. Geared devices can be used to change the speed, torque, and/or direction of a power from a source. The most common situation is for a gear to mesh with another gear; however, a gear can also mesh with a non-rotating toothed part, for example a rack, thereby producing translation instead of rotation. When two gears of unequal number of teeth are combined, a mechanical advantage is produced, with both the rotational speeds and the torques of the two gears differing in a simple relationship. In two gear system the smaller gear is the pinion and larger one is the gear/wheel. In most applications the pinion is the driver, this arrangement results in reduction of the rotational speed but it increases torque.

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Figure 6. Simple gear and pinion assemble.

There are different methods of classifying gears. The most common method of classification is based on the formation of the gear teeth. An external gear is one with the teeth formed on the outer surface of a cylinder or cone. Conversely, an internal gear is one with the teeth formed on the inner surface of a cylinder or cone. It is also possible to classify gears based on the shape of the gear tooth. Some of these are: Spur gears where each tooth is straight and aligned parallel to the axis of rotation. Helical gears where the edges of the teeth are not parallel to the axis of rotation, but are set at an angle. Bevel gear is shaped like a right circular cone with most of its tip cut off. Figure Figure 6. shows some of these types of gears.

Figure 6. Common types of gears (ref. EngineersEdge).

An example of spiral bevel gear application, common in the lower gearbox (LGB) of a propulsion system, is shown in Figure 6..

Figure 6. Gears in propulsion systems.

Gear terminologies and definitions

In this section we will discuss parameters used to describe gear geometry and some common terminologies used in modern gear designs. Some of the gear geometry parameters are shown in Figure 6.. Figure 6. shows a meshing gear teeth and corresponding geometric parameters.

Figure 6. Geometric parameters of a gear/gear tooth (source RoyMech).

Some of the common gear design and manufacturing terminologies shown in Figure 6. and their definitions is given in Table 6 .

Table 6 Gear terminologies and definitions.

Addendum

The height by which a tooth projects beyond the pitch circle

Dedendum

The depth of a tooth space below the pitch circle. It is normally greater than the addendum of the mating gear to provide clearance.

Pitch circle

Circle on which tooth profile and tooth proportions are constructed. It is an imaginary circle that rolls without slipping with a pitch circle of the mating gear.

Backlash

The amount by which the width of the tooth space exceeds the thickness of the engaging tooth on the pitch circle

Circular pitch

The distance along the pitch circle between corresponding profiles of adjacent teeth

Tooth circular thickness

The length of the arc between the two sides of a gear tooth on the pitch circle

Contact ratio

The number of angular pitches through which a tooth surface rotates from the beginning to the end of contact

Diametral pitch

The ratio of the number of teeth to the pitch diameter of the gear.

Normal plane

The plane normal to the tooth surface at a pitch point and perpendicular to pitch plane

Pressure angle

Angle at pitch point between the line of pressure which is normal to tooth surface and the plane tangent to the pitch circle

Tip relief

An arbitrary modification of tooth profile whereby a small amount of material is removed near the tip of the gear tooth

Working depth

The depth of engagement of two gears (sum of two addendums)

Figure 6. Geometric parameters of a gear mesh (source RoyMech).

The meaning of the these parameters and their notation along with the measure of units are given in Table 6 .

Table 6 Gear geometric parameters and units of measure.

a

Operating centre distance, in mm

h

Tooth depth, in mm

b

Common face width, in mm

αn

Normal pressure angle at reference cylinder, in rad

B

Total face width of double helix gears, including gap, in mm

αt

Transverse pressure angle at reference cylinder, in rad

d

Reference diameter, in mm

αtw

Transverse pressure angle at working pitch cylinder, in rad

da

Tip diameter, in mm

β

Helix angle at reference cylinder, in rad

db

Base diameter, in mm

β b

Base helix angle, in rad.

df

Root/fillet diameter, in mm

Ɛa

Transverse contact ratio

dw

Working pitch diameter, in mm

Ɛb

Overlap ratio

x

Addendum modification coefficient

Ɛg

Total contact ratio

z

Number of teeth

ρao

Tip radius of the tool,

zn

Virtual number of teeth

ρF

Tooth root radius at the critical section, in mm

n

Rotational speed, in rpm

v

Linear speed at pitch diameter, in m/s.

U

Reduction ratio

mn

Normal module, in mm

Gear failure

The gear is said to be failed when it can no longer "efficiently" perform the operation for which it was designed. The most common modes of gear failure are wear, surface fatigue, plastic flow, and tooth breakage.

shows more detailed classification of gear failure modes along with the cause and effect of the damage.

Tooth breakage and fatigue failure of gears can lead to the catastrophic failure of equipments. Because of this, effective procedures and information to evaluate the load capacity and useful life of gears are needed. In this sense, ISO 6336 has introduced useful information to consider the bending and fatigue load capacity of gears. The procedure takes into account the pitting resistance (surface fatigue failure) and bending strength capacity (volumetric fatigue failure) of helical gears.

According to ISO 6336, gear failures can be classified in two major categories. One is tooth breakage failure and other is surface failure. Each of these failure modes has a different causes and driving conditions as shown in

. In gear failures as provided in Table 1, tooth breakage failure largely occurs due to misalignment problems that results in excessive surface contact fatigue. These problems can easily be reduced or eliminated by adopting the correct gear mounting procedures during assembly. However, gear surface failures normally occur due to critical operational parameters such as external loading, lubrication and operating speed [3-5]. These operational parameters are actually stimulating the generation of a complex wear mechanism on the gear surface which ultimately leads a surface failure during gear operation.

Gear failure modes

The distinguishing characteristic of materials associate with the lost of resistance under the action of repeated or fluctuating stresses is called fatigue failure. The study of fatigue failure is not an exact and absolute science, of which precise results can be obtained. The prediction of fatigue fracture is very often approximate and relative, with many components of the statistical calculation, and there are a great many factors to be considered, even for very simple load cases. In this sense the determination of the fatigue limit for materials with industrial purposes demands a great variety of test to define the magnitude of fatigue limit reported at a specific number of cycles. This argument works equally for any other mode of gear failure, for example, gear tooth bending. For this reason, it is important to distinguish all types of failure modes in order to determine the reliability of gears. Some of the common failure modes in gears operating under any service condition is described in the next section.

Wearing is the removal of metal, worn away normally in a uniform manner from the contacting surface of the rear teeth. Specific types of gear wear include abrasive wear, corrosive wear, and scoring. Abrasive wear is caused by an accumulation of abrasive particles in the lubrication system. Corrosive wear is caused by the presence of water or chemical additives in the lubrication oil resulting in deterioration of the gear surface from chemical action. Scoring type of wear is caused by failure of the lubricant film due to overheating resulting in metal-to-metal contact and tearing of the surface metal.

Surface fatigue is the failure of gear as a result of repeated surface or even subsurface stresses that are beyond the endurance limit of the gear material. Surface fatigue generally results in the removal of metal and formation of cavities on the gear tooth surface. This pitting can be caused by the gear tooth surfaces not properly conforming to each other due to lack of proper alignment. Spalling is similar to pitting except that the pits are relatively larger, shallower, and very irregular in shape. Spalling is usually caused by excessive contact stress on the gear tooth surface.

Plastic flow is generally due to the cold working of the gear tooth surfaces, caused by heavy loads and rolling/sliding action of the gear mesh. The result of high contact stress level in the gear mesh is the yielding of the surface/subsurface material and thus gear tooth deformation.

Gear tooth breakage is a failure caused by the fracture of a whole tooth or substantial portion of the tooth. Gear overload or cyclic stress of the gear tooth beyond the endurance limit of the gear material causes bending fatigue and eventually a crack originating in the root section and then propagating and tearing away of the tooth or part of the tooth. Gear overload is generally caused by a bearing failure of any kind, system dynamic loading, or contaminants entering gear mesh area.

Table 6 Gear Failure Modes, Causes, and Effects (Ref. ISO 6336-2, [R1]).

Failure Mode

Failure Cause(s)

Failure Effect(s)

Pitting (Macro/Micro)

Cyclic contact stress transmitted through lubrication film

Local contact of asperities produced by inadequate lubrication film caused by:

Excess load/temperature, lower oil viscosity, lower operating speed, water saturation in the oil.

Tooth surface damage

Tooth root fillet cracking

Tooth bending fatigue

Surface contact fatigue

Tooth failure

Tooth shear

Fracture

Tooth failure

Scuffing

Lubrication breakdown

Wear and then eventual tooth failure

Plastic deformation

Excessive loading

Surface yielding

Surface damage resulting in:

Vibration

Noise

Tooth failure

Spalling

Fatigue

Mating surface deterioration

Welding

Galling

Tooth failure

Tooth bending fatigue

Surface contact fatigue

Tooth failure

Contact fatigue

Surface contact fatigue

Tooth failure

Thermal fatigue

Incorrect heat treatment during production

Tooth failure

Abrasive wear

Contaminants in gear mesh area and lubrication system

Tooth scoring

Gear vibration/Noise

Gear stress calculation

In reliability prediction of gears a number of parameters play significant role and the stress applied to the gear system is one of these parameters. In gear design there are a number of force components that are considered for optimal life of gear under a prescribed loading condition. The two most significant force components are bending stress and contact stress. The next two sections give a general introduction on tooth bending and contact stresses.

Gear tooth bending stress

Wilfred Lewis in 1893 provides a formula for estimating the bending stress in a gear tooth. He modeled a gear tooth taking the full load at its tip as simple cantilever beam. If we substitute a gear tooth for the rectangular beam, we can find the critical bending stress point at the root fillet of the gear tooth. This bending stress is given by the formula:

Where, is the root bending stress (N/m2), is transmitted tangential load (Newton), is face width (meters), is the module (m), is the Lewis form Factor. The Lewis form Factor is a function of number of teeth, pressure angle and depth of the gear tooth.

It is fact that, when teeth mesh, the load is delivered to the teeth with some degree of impact. Furthermore, the working condition of the gear, like wheel and pinion size, the speed of the shaft, the properties of the lubricant used, and vibrations from nearby components or machines have significant influence on the transmitted tangential load and thus the bending stress. For example, If only the effect of the impact is considered, it is necessary to introduce a velocity factor in the Lewis formula to account for the impact. Now the Lewis equation becomes,

Where, is the velocity factor. A similar change has to be made in the Lewis formula to account for all factors mentioned earlier.

Gear tooth contact stress

The contact stresses on the surface of matting gear teeth are usually determined by formula derived from Hertzian theory of non-adhesive elastic contact by H. Hertz’s in 1882. Hertz determined the width of the contact band and the stress pattern when various geometric shapes were loaded against each other. The Hertz formula for the width of the band of contact can be applied to, for example, spur gears quite easily. This can be done by considering that the contact condition of gears are equivalent to those of cylinders having the same radius of curvature at the point of contact as the gear have, see [R4].

Assuming and are the respective radius of curvature of an involute curves at the contact point, then the Hertz equation for contact stresses in the teeth becomes

where and are the pitch radius of the pinion and gear respectively, is a contact factor, and is the pressure angle. The Hertz equation for contact stress also needs to be modified depending on the contribution of the gear design, operational and environmental factors towards the tangential force.

Damage accumulation

Gears subjected to any kind of operating condition experience different types of failure mode, see section 1.1 on gear failure modes. Furthermore, each type of stresses result in different type of gear failure. In reliability analysis of gears it is then important to devise a method for combining the damage caused by different sources or failure modes. The most simple and yet practical method for accumulation of gear damages caused by different failure modes is the Palmgren-Miner rule described below.

Palmgren-Miner Rule

Suppose a body can tolerate only a certain amount of damage, D. If this body experiences damages from N sources, then we might expect that failure will occur if

There are two areas of interest in gear failure analysis where this linear damage accumulation concept can be used. The first one is for the accumulation of gear damage from different sources like surface fatigue, tooth bending stresses, scuffing damages, scoring, etc. We can also use this linear damage concept in a fatigue setting by considering the situation where the gear is subjected to cycles at alternating stress , cycles at alternating stress , … , cycles at alternating stress . Note that this is usually the case for gears under dynamic loading conditions. From the S-N curve shown in Figure 6., it is possible to calculate the number of cycles to failure for any stress level of interest.

Figure 6. An example of S-N curve, see [R2].

It is reasonable in this case to let the fractional damage at stress level simply be , so that the Palmgren-Miner rule would say that either or both gear tooth pitting and bending failures occur when

where is the number of cycles the gear is under stress level and is the total number of cycles the gear can survive before failure at stress level , or in other words is the remaining number of cycles before failure.

One of the limitations of the linear damage rule is that it does not consider the effect of sequence of multiple loading scenarios. For example, in a two-stress-level fatigue test in which a higher load is followed by a lewer load, the sum of the cycle ratios is less than 1. However, if a lower load is followed by a higher load, the summation of the cycle ratios is greater than 1.

Standards for gear life rating

There are many gear tooth and gearbox rating standards existing in the world. For a given gearbox, the rating system that is used can give very different answers in the amount of power that can be transmitted. If a user is not specific or does not have a basic understanding of the different rating system, the reliability of the gearbox can be dramatically affected. The intent of this section is to compare the recent API, AGMA, ISO, DIN, and DNV gear standards regarding gear and gearbox design and manufacturing.

Currently, ISO and AGMA standards are getting more popularity in industries. Regarding gear design, specially spur gears, DIN 3990 series and its modified release ISO 6336 series are partially equivalent to AGMA 2101-D04, according to EuroTrans, see Table 6 . Most of these standards give separate guidelines for both spur and helical gears; however ISO 6336 gives single general guidelines for both spur and helical gears only with minor changes in formulas and calculation procedures.

ISO 6336 consists of several parts dedicated to both spur and helical gears. ISO 6336-1 provide basic principles, introduction and general influence factors. ISO 6336-2 covers calculation of surface durability or pitting. ISO 6336-3 covers calculation of tooth bending strength. Part-2 and Part-3 are used to predict the two prominent failure modes in gears, according to ISO 6336. Part-5, ISO 6336-5, covers strength and quality of gear materials. The last part, ISO 6336-6, covers calculation of service life of gears under variable loading conditions. Table 6 shows equivalence of these ISO parts with the equivalent DIN and AGMA standards see [R1].

ISO 10300 on the other hand has three parts dedicated only to bevel gears. ISO 10300-1: introduction and general influence factors, ISO 10300-2: calculation of surface durability (pitting), and ISO 10300-3: calculation of tooth root strength. All the three parts of ISO 10300 use similar gear rating procedure as in the first three parts of ISO 6336, except changes in formulas to account for the conical shape of bevel gears.

Table 6 Equivalence between ISO Standards and National Standards on gear technology (ref. EuroTrans).

Gear Calculation Method

Calculation of load capacity of spur and helical gears

Equivalent National Standards

Year of Publication

Status of Equivalence

ISO 6336-1:2006

Basic principles, introduction and general influence factors

DIN 3990-1

ANSI/AGMA 2101-D04

1987

2004

P/S

P

ISO 6336-2:2006

Calculation of surface durability (pitting)

DIN 3990-3

ANSI/AGMA 2101-D04

1987

2004

P/S

P

ISO 6336-3:2006

Calculation of tooth bending strength

DIN 3990-2

ANSI/AGMA 2101-D04

1987

2004

P/S

P

ISO 6336-5:2006

Strength and quality of materials

-

-

-

ISO 6336-6:2006

Calculation of service life under variable load

DIN 3990-6

ANSI/AGMA 6032-A94

1984

1994

P/S

NO

ISO 9083

Application to marine gears

DIN 3990-31

ANSI/AGMA 6032-A94

1990

2000

P/S

NO

Status of equivalence:

S : Strictly equivalent

P : Partially equivalent

Gear life rating is, in general terms, quantification of possible gear failure modes. Both ANSI/AGMA 2101-D04 and ISO 6336 give guidelines for both pitting resistance and bending strength calculations. However, DNV CN 41.2 considers calculations for scuffing load capacity of gears as well.

Gear rating according to ANSI/AGMA 2101-D04 uses Miner’s Rule (linear damage accumulation method) to calculate gear life based on load spectrum. This rating method is based on a reliability of 99% and the lower curves for stress cycle factors, for both pitting and tooth bending resistance. Scuffing resistance is in accordance with AGMA 925–A03.

Gear rating according to ISO 6336 also uses Miner’s Rule (see ISO 6336-6) to calculate safety factors using a load spectrum. This rating method is only available for pitting and bending fatigue lives calculation. In both cases, safety factor calculations are based on a reliability of 99%. ISO 6336:1996 does not provide a rating method for scuffing.

The gear rating procedures given in DNV CN 41.2 are mainly based on the ISO6336 Part 1 to 5 for cylindrical gears, and partly on ISO 10300 Part 1 to 3 for bevel gears and ISO Technical Reports on Scuffing and Fatigue Damage Accumulation, but especially applied for marine purposes, such as marine propulsion and important auxiliaries onboard ships and mobile offshore units. The calculation procedures cover gear rating as limited by contact stresses (pitting, spalling or case crushing), tooth root stresses (fatigue breakage or overload breakage), and scuffing resistance.

Gear life calculations

Tooth breakage and fatigue failure of gears can lead to the catastrophic failure of equipments. Because of this, effective procedures and information to evaluate the load capacity and useful life of gears are needed. In this sense, ISO 6336 has introduced useful information to consider the bending and fatigue load capacity of gears. In this section, the procedure and formulas to estimate a value of gear life expectancy is given. The procedure takes into account the pitting resistance (surface fatigue failure) and bending strength capacity (volumetric fatigue failure) of helical gears.

The distinguishing characteristic of materials associate with the lost of resistance under the action of repeated or fluctuating stresses is called fatigue failure. The study of fatigue failure is not an exact and absolute science, of which precise results can be obtained. The prediction of fatigue fracture is very often approximate and relative, with many components of the statistical calculation, and there are a great many factors to be considered, even for very simple load cases. In this sense the determination of the fatigue limit for materials with industrial purposes demands a great variety of test to define the magnitude of fatigue limit reported at a specific number of cycles.

In practice, gears are mostly operated under variable loads. Even in a continuous process the load acting on gear teeth is fluctuating due to the tooth contact process and operational conditions under which the gears shall perform. Under these variable loads a tooth breakage, which most often results in a total gear failure, must be take into account during the stages of gear design or load capacity calculation. This fact has demanded that new fatigue tests for gear materials be carried out and the fatigue resistance behavior with a high number of load cycles be analyzed.

As discussed earlier in this chapter ISO 6336 only considers two groups of gear failure modes; these are pitting and tooth breakage. The next two sections discuss useful lifetime estimation of gears based on these two failure modes.

Geometrical definitions

In the calculation of surface durability, is the common face width on the pitch diameter. In tooth strength calculations, ,  are the face widths at the respective tooth roots of the pinion and the gear. In any case, and  are not to be taken as greater than by more than one module () on either side of the contact region between mating teeth. For internal gears, , , ,  ,  ,  and  are to be taken negative. From the geometry of a gear we have the following relations.

The gear ration is the ratio between the number of teeth on the gear and on the pinion and is given by

where is the number of teeth on the gear (wheel, the larger of the two meshing gears) and is the number of teeth on the pinion. For external gears both and are positive whereas for internal gears has a negative sign.

The transverse pressure angle, the normal pressure angles and the helix angle at the reference cylinder are related according to the following relation:

The relation between the normal module , the helix angle, the reference diameter , and the number of teeth, for both pinion and gear is given by:

The transverse pressure angle is proportional to the ratio of base diameter and the reference diameter following the relation:

whereas the pressure angle at the pitch cylinder is a function of the centre distance of the pinion and the gear

with and are the base diameters for the pinion and gear, respectively.

The transverse contact ratio between the meshing pinion and gear, for external/internal gears, is given by:

where is the tip diameter of a gear.

The overlap ration and then the total contact ration are given by the following equations:

and

Force, power, and torque

In gear design calculations force, torque or power can be used to describe the loading condition on the gear tooth and thus it is important to formulate the relationship between these three variables. The nominal tangential load, on which all ISO 6336 are based, is determined in the transverse plane at the reference cylinder. It is usually derived from either the nominal torque or from the power transmitted by the gear pair.

The load capacity rating of gears, according to ISO 6336-1, is mostly based on the input torque to the driven machine. However, in some applications, the nominal torque of the prime mover is used as a basis. The nominal tangential load, , is need to be defined for each mesh under consideration. The nominal tangential load is given by:

where is the nominal torque in the pinion/gear, is the nominal power, and is the tangential speed at the reference cylinder given by the formula:

where is the angular speed of the pinion/gear.

In practice the transmitted load is never uniform in which case all intermediate loads and their number of cycles need to be describe or measured. This type of loads is classified as a duty cycle and is generally represented by a load spectrum. In such cases, the cumulative fatigue effect of the duty cycle is considered in rating and life prediction of the gears. If the duty cycle is, for example, measured in units of torque, then the equivalent tangential load must be calculated from the equivalent torque. The equivalent torque is given by:

where and are the number of cycles and corresponding torque for bin, respectively, and is the slope of the Woehler-damage line (S-N curve), see Figure 6 below. Then, the corresponding equivalent power and the equivalent tangential force can be obtained from the equivalent torque using the relations given earlier in this section.

Figure 6 The Woehler-damage line (S-N curve).

The magnitudes of the maximum tangential load and the corresponding maximum power and maximum torque are limited by appropriate safety factor. Or in other words, , , and are required to determine the safety factor to minimize pitting damage and/or sudden tooth breakage due to loading close to static stress limit. The general equation for safety factor is

where is the safety factor for pinion and gear, respectively, is the contact stress at a given loading condition, and is the pitting or tooth bending stress limit.

General influence factors

In this section formulas for major factors, which are presently known to affect the design of gears for surface fatigue and tooth bending, are derived. General gear design assumptions and corresponding factors are also discussed in detail.

Application factor

The application factor  accounts for dynamic overloads from sources external to the gearing. The additional forces from external sources are largely dependent on the characteristics of the driving/drives machines and the masses and stiffness of the system, including shafts, couplings, and bearings used in the system.

Gears used in marine applications are subjected to cyclic peak torque (torsional vibrations) and generally designed for infinite life. For these types of gears the application factor can be defined as the ratio between the peak cyclic torque and the nominal rated torque:

where is the equivalent torque and is the nominal torque. The nominal torque is the torque used in load capacity calculations and is defined by the rated power and speed of the gear.

In all other cases, the values of  are as given in Error: Reference source not found. 

Table 6 Values of Application Factor,. (according to ISO 63336 and ABS recommendations)

Type of installation

Main gears (propulsion)

Diesel engine

with hydraulic coupling

1.05

with elastic coupling

1.30

with other type of coupling

1.50

Electric motor

1.05

Auxiliary gears

Diesel engine

with hydraulic coupling

1.00

with elastic coupling

1.20

with other type of coupling

1.40

Electric motor

1.00

Internal dynamic factor

The internal dynamic factor  accounts for the additional internal dynamic loads acting on the tooth flanks and due to the vibrations of pinion and gear. In other words, the dynamic factor is used to make allowance for the effect of gear tooth accuracy grade related to the operational speed and loadings of the gear system. This means that high grade gears require less derating by  than low accuracy gears.

The internal dynamic factor, apart from operating speed and load, depends on a number of design and manufacturing parameters. Some of these are: inertia and stiffness of rotating elements in the gear system, lubricant properties, stiffness of bearings and casing structures, critical speed ranges and internally induced vibrations, pitch deviation, balance of parts, compatibility of mating gear tooth elements, bearing fit and preload, etc. This complex interdependence of the internal dynamic factor and design and manufacturing parameters can allow only approximate values for .

The gear system is assumed to be an elementary single mass and spring system comprising the mass of both pinion and gear, the stiffness of the spring is the stiffness of mating teeth. For further simplification it is assumed that the effect of the loads due to torsional vibration of shafts is negligible. In practice, the dynamic tooth loads are normally smaller than what is calculated. This is because there are a number of sources of damping in the gear system, like, friction on tooth face, bearings, and couplings. However, it is assumed that the damping at the gear mesh has an average value only from a single hypothetical source.

In order to further simplify the calculation method of , the entire running speed range is divided into three ranges, subcritical, main resonance, and supercritical based on the resonance ration, The resonance ratio  is the ratio between the running speed and the resonance speed and is given by

where is in rpm, is the relative mass of gear pair, is mesh stiffness parameter. The gear relative mass is given by

where , see Error: Reference source not found.

In the cases where both gear and pinion are solid constructions, i.e. no rim, .

Figure 6 Definition of different diameters (ref. ISO 6336-1).

Once the resonance ratio is calculated one can define three main running speed ranges. The subcritical range, might lead to a limited resonance specially when the gear mesh frequency coincides with or . In the main resonance range, operations should be avoided especially with gears of coarse accuracy, see ISO 1328-1. Similarly, in supercritical range, resonance peaks are expected and thus operations should be avoided.

For reasons of safety and to incorporate the effects of stiffnesses which are not considered so far (like bearing and housing), the resonance ratio in the main resonance range is needed to be defined by an upper and lower limit as follows:

The lower limit for resonance ratio is a defined as follows: if and if then Once again, for the purpose of safety, one more region is added, intermediate resonance range, as shown in Figure 6 .

Figure 6 Resonance ranges (ISO 6336-1)

The formula for the internal dynamic load factor is given in theTable 6 , for each of these four resonance ranges.

Table 6 Internal dynamic load factor and resonance ranges.

Resonance range

With:

With:

With:

The factors to generally allow for correction of the dynamic factor in relation to pitch deviation, tooth profile deviation, and cyclic variation effect in mesh stiffness in different resonance ranges. The values of these factors are given in Table 6 below.

Table 6 Definition of factors to .

0.32

0.32

0.34

0.23

0.90

0.47

0.47

0.47

0.75

1.00

For gears without specified profile modification

When the pinion and gear are made of different materials then

The effect of tooth deviations and profile modifications affect the values of the dynamic factor significantly. The non-dimensional parameters, , and used to account for these effects are defined below:

and for gears of ISO quality grade 6 to 12 , otherwise

Where is single tooth stiffness and effective single pitch and profile deviations, and , are given by

The values of and follows ISO 1328-1 depending on the reference diameter, the normal module, and ISO grade accuracy of the pinion and gear. The calculation of the and is as given in Table 6 .

Table 6 Calculation method for running-in allowance

Gear Material types

Running-in allowance

St., St.(cast),V, V(cast)

GGG(perl.,bai) and GTS(perl.)

-

GG and GGG(ferr.)

-

Eh, IF, NT(nitr.), N(nitr.) and

N(nitrocar.)

-

Assumption: , if no data on relevant material running-in characteristics are available.

Note: can also be calculated in the same way as but using instead of

In order to simplify the calculations, in the absence of some or all relevant data, one can make a rough assumption that and use the values given in Table 6 for calculation of tooth and profile deviation parameters.

Table 6 Mean values of effective base pitch deviation.

 

ISO grade of accuracy (ISO 1328-1) 

3

4

5

6

7

8

9

10

11

12

2.8

5.1

9.8

19.5

35

51

69

100

134

191

The values of  specific to most propulsion gear, the case where , class societies assign different method of calculating the internal dynamic load factor. ABS [reference], for example, recommends the following method of calculation limited to steel gears of heavy rim sections as given in Table 6 Values of internal dynamic load factor, .

Table 6 Values of internal dynamic load factor,

Type of gear

KV

Limitations (ABS)

Spur gear

with:

where K1 has the values specified in in Error: Reference source not found.

Helical gear

If :

with:

,

where K1 has the values specified in in Error: Reference source not found

If :

Table 6 ISO grade of accuracy and Values of

 

ISO grade of accuracy (1)

Type of gear

3

4

5

6

7

8

Spur gear

0,022

0,030

0,043

0,062

0,092

0,125

Helical gear

0,0125

0,0165

0,0230

0,0330

0,0480

0,0700

(1) ISO grade of accuracy according to ISO 1328-1 1997. In case of mating gears with different grades of accuracy, the grade corresponding to the lower accuracy is to be used.

Face load distribution factors  and

The face load distribution factors,   for contact stress and  for tooth root bending stress, account for the effects of non-uniform distribution of load across the face width of a gear mesh. The gear tooth manufacturing accuracy, the alignment of axes of mating gear elements, elastic deflection of transmission system units (like gears, shafts, bearings, housing, and foundation) and bearing clearances; they all play a role in uneven distribution of load over the gear face width. In addition to these, the operational variables, like operating temperature, speed, tangential load, additional shaft load, and gear geometry also affect the value of load distribution factors.

The face load factor for contact stress  is a ratio between the specific transverse loading at the reference cylinder and the corresponding maximum local loading on the tooth surface. The face load factor for tooth root stress depends on the same parameters as in and also on the face width to tooth depth ratio

The face load distribution factor according to ISO 6336-1 is calculated considering the effect of the mean load intensity across tooth face , mesh stiffness factor , and effective total gear mesh misalignment The value of is calculated as follows:

If  ,

and otherwise

The effective equivalent misalignment is dependent on the combined effect of manufacturing error and elastic deformations of the pinion and pinion shaft. Assuming no or negligible helix modification

where is the amount by which the initial misalignment is reduced running-in since operation was commenced, see Table 6 Calculation method for running-in allowance , and is the initial misalignment.

The mesh misalignment factor is the maximum separation the tooth flanks of the meshing teeth of mating gears. For most iindustrial transmissions gears it is assumed that . The approximate value of the equivalent misalignment is given by

where is the external diameter of the pinion shaft, for stiffened shaft and for non-stiffened shaft assuming , is the length of the shaft, and is the distance between the pinion and the supporting bearing, see Figure 6 Shaft, bearing and gear arrangement..

Figure 6 Shaft, bearing and gear arrangement.

Table 6 Calculation method for running-in allowance

Gear Material types

Running-in allowance

St., St.(cast),V, V(cast)

GGG(perl.,bai) and GTS(perl.)

-

GG and GGG(ferr.)

-

Eh, IF, NT(nitr.), N(nitr.) and

N(nitrocar.)

-

Note: If the material of the pinion and the gear are different both and assume the average values of the pinion and the gear materials.

Once the face load factor for contact stress is calculated, the face load factor for tooth root stress can be calculated by modifying with face width to tooth depth ration as follows

where is the smaller of and  however when use

Transverse load distribution factors  and

The transverse load distribution factors  for contact stress and  for tooth root bending stress, account for the effects of pitch and profile errors on the transversal load distribution between two or more pairs of teeth in mesh. In other words, the factors  and account for the non-uniform distribution of transverse loads between contacting gear teeth. The main factors affecting these factors are the deflections due to loading, gear profile modifications, tooth manufacturing accuracy, and running-in effects. If a gear possesses optimum profile modification, high manufacturing accuracy and even load distribution over the face width, both  and approach unity.

The main geometric parameter affecting transverse load factors is the average distance between base pitches of the pinion and the gear. Under this assumption, the values of  and are given as follows:

For gears with contact ratio

and if the contact ration

where the determinant tangential load in transverse plane , is the mesh stiffness factor, is running-in allowance, and is the larger of the base pitch deviation of pinion or gear. See Table 6 for the limiting conditions for both transverse load factors.

Table 6 Limiting conditions for transverse load factors  and

If , then assume

If , then assume

If , then assume

If , then assume

Tooth and mesh stiffness parameters and

The stiffness parameters represent the required load capacity over a face width, in the direction of applied load; amounting for the deformation of of a single pair of deviation-free teeth in contact. The single tooth stiffness is defined as the maximum stiffness of single pair of spur gear teeth. For helical gears is the maximum stiffness normal to the helix of one tooth pair. Mesh stiffness factor is the mean value of the stiffnesses of all the teeth in a mesh. Some of the factors affecting tooth and mesh stiffnesses are geometric parameters of gears (like number of teeth, helix angle, contact ratio, ... ), load normal to tooth flank, shaft-hub connection, surface roughness, mesh misalignment, and gear material properties.

The tooth stiffness factor , under the assumption that helical gear is made of steel with helix angle and is subjected to specific loading , is to be determined as follows:

Under all the assumptions made in the previous paragraph and under low specific loading , the tooth mesh stiffness factor is calculated as follows:

If the pinion and gear are made of different materials the following correction should be made

where and are the modulus of elasticity of the pinion and the gear, respectively.

The parameter is the theoretical single tooth stiffness assuming specific loading of , is adjustment factor for low specific loading, correction factor for gear geometrical changes (rims and webs), and is factor to consider basic rack profiles. For solid disc gears , , and

The theoretical single tooth stiffness is given by:

where is the minimum value for teeth pair flexibility and is given by:

If and for specific loading satisfying , one can only make an error between -8% and +5% in approximating using the series given by The number of teeth of virtual spur gears, on which the above calculation for tooth stiffness is based on, is given by

The mesh stiffness factor is to be determined following the procedure given in Error: Reference source not found.

Table 6 Calculation procedure for mesh stiffness factor .

Type of gear

Limitations

Application

Spur gears

Helical gears

-

-

Note: For spur gears with , .

Useful lifetime estimation under linear pitting

According to ISO 6336-2, such pitting that involves formation of pits and increases linearly or progressively with time under unchanged service conditions is termed as linear pitting. Calculation for time estimation when linear pitting occurs is based on the contact stress at the pitch point of the meshing gears, or at the inner point of single pair tooth contact. The surface contact stress shall be less than its permissible value for preventing failure and vice versa. For any reason, if this limit of surface durability of meshing gear tooth is exceeded, particles will break off from tooth flanks leaving pits of different size and shape. Under unchanged operational condition pitting tend to increase progressively in time.

Gears operating under different operational environment require different level of tolerance for pitting damage. However, progressive increase of area of pits is always unacceptable. On the other hand, after initial pit the rate of generation of pits could reduce (degressive pitting) or cease in some cases (arrested pitting). These kinds of pitting damages can be considered tolerable. The scope of this section is formulation of to analysis methods for linear pitting: a pitting damage that progresses linearly in time after initial pitting.

The main assumption is gear life calculation models for surface durability is that the contact stress on the gear tooth at the pitch point should be less that the permissible contact stress. The ratio of these two stress components is generally used as a safety factor against pitting damage which should be higher than the agreed minimum limit. Furthermore, the general guidelines presented in Table 6 should always be followed in calculations of surface stress.

Table 6 General guidelines for calculation of contact stress

Type of gear

Parameter range

Calculation reference point

Limitation

Spur gears

Pinion

Inner point of single pair tooth contact

-

Gear

Pitch point

-

Helical gears

Pinion

and gear

Pitch point

Linear interpolation (1)

and

No method of calculation (not in the scope of this paper)

-

Linear interpolation between two limit values: stress of spur gear with same number of teeth and stress of helical gear with overlap ratio .

For example, in case of helical gears is determined at the pitch point of a gear only in the case when . The calculation procedure for contact stress according to ISO 6336-2 is described below.

Contact stress 

The contact stress should be calculated separately for both pinion and gear elements and is to be determined as follows:

for the pinion

for the gear

Where is the nominal contact stress, the stress induced by nominal static torque in error-free gearing, and is calculated using the following formulae:

where the Z parameters are as defined in Error: Reference source not found.

Table 6 Definition of the Z parameters.

Symbol

Name

Description

Single pair tooth contact factor for pinion

Transforms stress from pitch point to inner point of tooth contact

Single pair tooth contact factor for wheel

Same as above but for wheel

Zone factor

Transform at reference cylinder to at pitch cylinde

Elasticity factor

Compensates for difference in gear material properties

Contact ratio factor

Accounts for the effective length of contact lines

Helix angle factor

Accounts for variation of load along contact line

Permissible contact stress 

The permissible contact stress  is to be determined separately for pinion and wheel using the following formula:

where is pitting stress limit and the remainder of the parameters are discussed below.

The two important stresses related to permissible contact stress are; the reference and static permissible contact stresses. The reference permissible contact stress is the value of permissible contact stress when the life factor , .

Unfinished ............???????????????????

Gear life under tooth surface pitting

Gear life under tooth bending

Gear life and lubricant cleanliness

Gear life and lubricant film thickness

Gear life and stress cycles

Service life under variable loading

Damage accumulation rule

Failure rate and reliability models

Preliminary results and conclusions

Expert judgement

Methods and techniques

Use the courseware to get this started.

Calibration and validation

Available data

Describe the available data.

Calibration and validation

Show results before and after calibration.

Results

Implementation plan

Use DLP project plan. Extend it with a good risk analysis and some other PM things.

Business strategy

Use Porter 5 forces analysis.

Don’t forget about the little customers.

Do a QFD exercise with shitty and extended system.

Consider DLP to be an extra dimension, which gives predictability.

Discussion

Conclusion

Recommendations