Fatigue verification in bend

Back to mathematical model mentioned at Chapter 5.4, concept of beam bounded to an extremity is developed and applied to a fatigue case, in order to study behavior of teeth loaded by cyclic bending stresses. That kind of verification is strictly referred to standard “U.N.I. – UNI Sperimentale 8862-2:1987”, Ref.[11] and it’s performed by the following main relation:

Eq. 5.8-1

 is “equivalent stress in root section” of the tooth.

 is “pulsating bending fatigue limit” and it’s also known as “allowable stress” but it must be distinguished by allowed stress of Chapter 5.4.

Both terms of Eq. 5.8-1 can be considered as stress, for this reason are measured in [MPa].

Equivalent stress in root section of the tooth is obtained by a well structured equation deeply described as follows.

Eq. 5.8-2

(

)

 is “face width” which represents the parameter that needs to be checked measured in [mm].

 is “module” . In this calculus can be considered a fixed parameter. It’s measured in [mm].

Plot 5.8-1: factor displayed in function of quantity and factor, Ref.[11].

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 is “mean tangential gear force”. It’s measured in [N] and, differently from tangential force calculated by Eq. 5.4-2, it’s described as follows:

Eq. 5.8-3

 is “mean torque on wheel 1”. That datum it’ difficult to own in the design preliminary phase. It needs to be esteemed or extracted by data of a similar vehicle. Risk is that, an over-esteemed value can lead to an over-sized gear. It’s measured in [Nm].

 is “pitch diameter of wheel 1”, drive wheel of the reduction stage, It’s measured in [mm].

Other factors of Eq. 5.8-2 have to be chosen on the basis of plots supplied by Ref.[11] standards.

 is “tooth form factor”. It relates the geometry of the tooth to the nominal bending stress.

It can be extracted by Plot 5.8-1. Horizontal axis displays “number of teeth of drive wheel”, indicated with . Defined value needs to be crossed with curves of “gear profile shift”, indicated with . Detailed information about that dimensionless parameter can be found at Chapter 5.12. Point identified by number of teeth and gear profile shift needs to be shifted horizontally toward vertical axis on which value of tooth form factor can be read. It’s suitable remember that this factor is a dimensionless parameter.

Plot 5.8-2: factor displayed in function of quantity and factor, Ref.[11].

 is “stress correction factor”. It’s necessary for the conversion of the nominal stress applied to tooth root. It’s determined by the application of load at the outer point of single pair of tooth in contact. Analytic calculus of this factor is quite complex and, in first approximation, it’s more useful chose the value with the help of Plot 5.8-2. As explained in the case of , Horizontal axis displays “number of teeth of drive wheel”, indicated with . Defined value

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needs to be crossed with curves of “gear profile shift”, indicated with . Point identified by number of teeth and gear profile needs shift to be shifted horizontally toward vertical axis on which value of stress correction factor can be read. It’s suitable remember that this factor is a dimensionless parameter.

 is “overlap ratio factor” which is a dimensionless parameter. Let image that meshing is extended over a single couple of teeth, condition of maximum stress appears it the root of teeth. This factor takes into account that, in the reality, force due to gear meshing isn’t applied to the top of the tooth, but in another point of the profile. To calculate the factor, standard provides this relation:

Eq. 5.8-4

{

 is “helix angle factor” which takes into account the effect of helix profile on the distribution of the stress. Its value can be extracted by Plot 5.8-3. Horizontal axis displays “helix angle”, indicated with . Defined value needs to be crossed with broken lines which represent the

“overlap ratio”, indicated with . Point identified by number of teeth and gear shift profile needs to be shifted horizontally toward vertical axis on which value of helix angle factor can be read. It’s suitable remember that this factor is a dimensionless parameter.

In the particular case of a spur gear with straight profile teeth, value of the factor is 1.

Plot 5.8-3: factor displayed in function of quantity and factor, Ref.[11].

 is “load application factor”. It takes into account the presence and the entity of overloads in the operational life of the gear. These overloads can be due to the operation of the drive machine, motors for example . Operational condition of the drive machine can be split in three cases: uniform (“uniforme”), light overloads (“sovraccarichi leggeri”), heavy overloads (“sovraccarichi pesanti”). On the other hand, operation of the driven machine, in which gear is housed, can be affected by overloads. In the same way, operational condition of the driven machine can be split in three cases: uniform (“uniforme”), light overloads (“sovraccarichi leggeri”), heavy overloads (“sovraccarichi pesanti”).

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Table 5.8-1: factor displayed in function of quantity and factor, Ref.[11]

Once operational cases of drive and driven machine are determined, value of load application factor can be chosen on the Table 5.8-1. This factor is a dimensionless parameter.

 is “dynamic factor” which takes into account the effect due to rotating masses of the gear.

In first approximation it’s possible to calculate dynamic factor by following Eq. 5.8-2, where it’s the “gear maximum tangential speed”, calculated on the pitch diameter.

Eq. 5.8-5 √

 is “longitudinal load distribution factor”, it’s a dimensionless factor which takes into account of non-uniformity of the load application along the tooth profile. It can be calculated in different methods displayed in U.N.I. 8862 standard. Fine determination of this parameter is outside from thesis target.

 is “transversal load distribution factor” , it’s a dimensionless factor which takes into account of non-uniformity in pitch errors and non-uniformity in load application. It can be calculated in different methods displayed by U.N.I. 8862 standard. Fine determination of this parameter is outside from thesis target.

Back to Eq. 5.8-1, it’s necessary to describe accurately the second term of the equation the

“pulsating bending fatigue limit”.

Eq. 5.8-6

Factors of the equation are going to be explained in following words.

 is “limit of stress in bending fatigue” and depends by the chosen material. Value of the factor can be easily determined reading on Table 5.8-2.

Different types of material are, substantially, cast iron and steel. Steel types are distinguished by heat treatments like carburizing, quenching, tempering and nitridizing. Moreover, performance of any different treated materials depend on surface hardness. It’s measured in [MPa].

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Table 5.8-2: and displayed in function of the chosen material, Ref.[11].

 is “stress correction factor”, it’s a dimensionless factor which takes into account the nature of the stress: uniform, pulsating, alternated, positive alternated, and so on. It’s possible find the value of this factor on the Smith-Goodman diagram displayed in Plot 5.8-4. Following plot is strictly related to the chosen material.

Plot 5.8-4: Smith-Goodman diagram of a generic steel, Ref.[11].

 is “endurance factor” or “Woehller factor”. This dimensionless factor can be extracted by the following Plot 5.8-5.

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Plot 5.8-5: Woehller diagram related to the bending of the teeth, Ref.[11].

First parameter to chose is the “number of cycles” which is displayed on horizontal axis in elevations of base ten. Determined value needs to be crossed with one of broken lines which depend on chosen material. Identified point needs to be shifted horizontally toward vertical axis, in order to locate the value of endurance factor.

 is the “safety coefficient” a dimensionless factor, which function have been widely explained at Chapter 5.1.

 is the “relative notch sensitivity factor”. It is the ratio of the notch sensitivity factor of the gear under design compared to that of the standard test , which specifies the amount by which the theoretical stress at fatigue breakage exceeds the fatigue stress. is function of the material and of the stress gradient concerned, and can be approximated using values between 0,95 and 1,0. It’s a dimensionless factor.

Table 5.8-3: in function of material and treatment, Ref.[11].

90 is the “relative surface condition factor”. This factor considers the influence of the surface quality, especially roughness in the tooth root, on the possible root tooth stress. It is a function of the material and may be determined from the following Table 5.8-3. It’s necessary to clarify that parameter on which is based the calculus is the roughness on the root of the tooth.

 is the “dimensional factor”. It depends on dimensions of the gearwheel. Value of this factor can be extracted by Plot 2.6-1.

Plot 5.8-6: in function of the module and of the material, Ref.[11].

First step is locate the proper value of the module on the horizontal axis. Then, value of module needs to be vertically crossed with broken lines which depend on different materials. “a” curve refers to tempered steel, non-alloyed base steel and ductile iron. “b” curve refers to surface hardened steel. “c” curve refers to grey cast iron. “d” curve suitable for any loaded material.

Point located by module and quality of material, needs to be horizontally shifted toward vertical axis, in order to read the proper value of dimensional factor.