Investigation of the Fatigue Strength Behaviour of a Fine 2 mm Module Gear

Miniaturization is a trend is in a steep rise in every engineering field, so as the use of small mechanical components. In fact, the use of small gears that have a module m_n equal or less than 2 is always more present. Gears have a lot of different applications i.e., power transmission, automation, and manufacturing processes, only to mention some of the most relevant. It was also studied that if small sized gears are used it is possible to reduce the pollutant constituents emitted in the atmosphere [1]. This brings to the necessity to properly study and analyze gears constituents, in particular their mechanical behavior. The objective of this framework was to give importance to the phases that are used for calculating constituent’s characteristic, precisely, the admissible root tooth bending stress σ_Flim, and to evaluate results for the tested steel (39NiCrMo3). Commonly, the procedure for the design phase of gears is governed by the following standards, ANSI/AGMA 2001-D04 [2], the EU ISO 6336 [3] and the DIN 3990 [4]. Nevertheless, these approaches are founded on studies performed on gears with a module m_n  equal to 5. In fact, big sized gears (m_n≥5) were already investigated by the study [5-9] and moreover, these researches have brought out that the actual standards developed for gears have the trend to overestimate the fatigue strength of the component. More deeply, it was highlighted that the solution is hugely influenced by the size of the gear. In regard to the tooth root bending capacity, in a study carried out by Steutzger [10], it was reported that if gears having a nominal module m_n≥5 are used, it is possible to observe an incorrect computation of gears parameters. This fact was also confirmed by other researchers [11-13]. Big module nitrited and carburized gears and their tooth root bending behavior was the objective of past researches [14-16].

The comportment of gears during operation is affected by numerous aspects i.e., the surface obtained during machining operation, the heat treatment applied on the final component, the effective geometry of the gear and the temperature in which it is working, just to mention the most important. In the past years some researches on gears with normal module m_n<5 have been carried out by the study [17, 18]. Nevertheless, in order to have more reliable data for designing gears, more efforts should be put in the study of small sized gears. In this work the ISO 6336 method B was followed. This standard is based on the evaluation of two key parameters, specifically two stresses are compared. The first is called permissible stress σ_FP, that is computed according to Eq. (1), and the second one is called effective stress σ_F that is the stress that is acting on the tooth computed as Eq. (2).

$\sigma_{F 0}=\frac{F_t}{b \cdot m} \cdot Y_F \cdot Y_S \cdot Y_\beta \cdot Y_B \cdot Y_{D T}$     (1)

$\sigma_F=\sigma_{F 0} \cdot K_A \cdot K_V \cdot K_{F \beta} \cdot K_{F \alpha}$     (2)

In Eq. (3) it is possible to see the simplified formulation for the effective stress σ_F. According to the use and the geometry of the inspected gear to the correction coefficients Y_β, Y_DT and Y_B were assigned a unitary value.

$\sigma_F=\frac{F_t}{b \cdot m} \cdot Y_F \cdot Y_S \cdot K_A \cdot K_V \cdot K_{F \beta} \cdot K_{F \alpha}$      (3)

In Eqns. (4) and (5) are described respectively the form factor Y_F (Figure 1) and the stress correction factor Y_S.

$Y_F=\frac{\frac{6 h_{F e}}{m} \cdot \cos \alpha_{F e n}}{\left(\frac{s_{F n}}{m_n}\right)^2 \cdot \cos \alpha_n}$      (4)

1.png

Figure 1. Schematic representation of data needed for computing the form factor Y_F

$Y_s=(1.2+0.13 \cdot L) \cdot q_s^{\frac{1}{1.21+\frac{2.3}{L}}}$      (5)

With L described through Eq. (6) and the notch sensitivity q_s defined following Eq. (7):

$L=\frac{s_{F n}}{h_{f e}}$      (6)

$q_s=\frac{s_{F n}}{2 \cdot \rho_F}$     (7)

The application factor K_A is strongly dependent on the application in which the gear is used because it takes into account effects caused by external loads. The dynamic factor K_V has a high dependency on internal dynamics loads. Moving to K_Fα and K_Fβ it is possible to say that they are necessary to manage uneven and transverse loads that act on gears while contact occurs. The deflection of the component and defects arising from machining operation can determine the above-mentioned phenomena.

Following the International standard [3], it is possible to compute and connect σ_FP with σ_F. In other words, following the method B of the standard it is possible to relate the allowable stress with the effective one, this relation is written as Eq. (8):

$\sigma_{F P}=\sigma_{F l i m} \cdot Y_{S T} \cdot Y_{N T} \cdot Y_{\delta r e l T} \cdot Y_{\text {RrelT }} \cdot Y_X$      (8)

By comparing Eq. (8) and Eq. (3) it is possible to compute σ_Flim with Eq. (9).

$\sigma_{F l i m}=\frac{F t}{b \; \cdot m} \;\; \cdot \frac{Y_F \;\; \cdot Y_S}{Y_{S T} \;\; \cdot Y_{N T} \;\; \cdot Y_{\delta r e l T} \;\; \cdot Y_{\text {RrelT }} \;\; \cdot Y_X}$      (9)

Y_ST and Y_NT are respectively the stress and the life factors. Y_δretT takes into account the notch while Y_RrelT the surface of the component. Another important factor that needs to be under review is the size factor Y_X. This coefficient, besides considering the effective dimension of the gear, evaluates how stress gradients, weak points, presence of defects etc. [3] are influenced by the size of the gear. The standard ISO 6336 method B states that the coefficient Y_X is unity if a gear with module m_n<5 is considered. This assumption may lead to an overestimation of the size of the gear. In this regard another approach for evaluating the size factor Y_X has been proposed by Dobler et al. [19], this method is suited for gears that have a fine module. The Dobler equation representing the size factor is shown in Eq. (10):

$Y_{X D o b l}=1-0.45 \cdot \log \left(\frac{m_n}{5}\right) \pm 0.075$      (10)

According to literatures [20, 21], an additional correction factor must be used for translating results of the STBF tests into results that can be compared with the one obtained with classic meshing gear test. This factor takes into account the different load history between the two types of tests. As main objective this work wants to calculate the fatigue limit (STBF) of the material and to evaluate its comportment when subjected to cyclic loads. Above all, the International standard ISO 6336 method B in combination with the Dobler et al. approach for computing the size factor were used. For the computation of the fatigue limit two similar statistic approaches were used. Both of them relies on statistics, the main difference is the number of samples needed for the correct computation of the fatigue limit. For the first approach, the classic one, a big amount of specimen is required. This method can lead to more reasonable results at the expense of the time and cost required to carry out the tests. The short stair case approach developed for small samples [22] permits to calculate the fatigue limit by means of less than six tests. The use of the last-mentioned method permits to accelerate the testing procedure. Both methods are commonly used for the computation of the fatigue limit for different materials [23, 24]. The secondary purpose of the work was to evaluate and compare the results achieved with two different statistic methods.

Results of the experimental STBF tests with the short staircase and classic approaches are shown in Table 2 and Table 3 respectively. For computing all the correction coefficient, dimensions were taken from the related CAD model. The standard deviation of the distribution is approximatively σ=64N for the shorter method, thus lower than the increment applied.

The first step to calculate the tangential force F_t is to multiply the used forces by the pressure angle α_Fen (see Figure 1). The factor k was set equal to k=-0.296. Thus, thanks to Eq. (9) the value of force with a 50% failure likelihood was set as $F_{F P_{S T B F_{\;50 \%}}} \;\; =3071.6 \mathrm{~N}$. The results of the short staircase method are shown in Figure 4.

Table 2. Results of the short staircase approach

Table 3. Results of the classic staircase approach

4.png

Figure 4. Short stair-case results

5.png

Figure 5. Classic staircase results

By applying the standard staircase, 16 specimens were tested at different load levels. Results are presented in Figure 5. In this case the standard deviation was found at σ=154N that is above the increment of 100N applied. Tests were performed on new sample gears.

By following the equations in the previous paragraphs, it was possible to compute the related stresses acting on the root tooth flank σ_F2mm=658.47 MPa for the short case and $\sigma_{F 2 m m}^*=676.25 \mathrm{MPa}$ for the classic approach. According to literature [20, 21] this value needs to be modified since tests were not done on real meshing gears, therefore another correction factor equal to 0.9 was used. At this point the effective root tooth bending stress was calculated at σ_F2mm=592.63 MPa and $\sigma_{F 2 m m}^*=608.62 M P a$ respectively for the short and classic methods. For the computation of all the other correction coefficients, needed for the calculation of σ_FP, the following considerations were taken. In this specific case the stress factor of the tested gear Y_ST was set Y_ST=2 and the life factor Y_NT equal to 1. A roughness R_a=6.3μm was measured experimentally. This value can now be translated into R_z thanks to the DIN 4778 [28]. Following the standard ISO 6336 the value of Y_RrelT related to this roughness value R_z =38.3μm is Y_RrelT=0.9458. Considering Eq. (13) it was also possible to calculate the notch sensitivity factor Y_δrelT, obtaining a value of Y_δrelT =0.9475. X^* is the relative stress gradient and ρ' the slip-layer thickness (the method for calculating those values is described in [3, 29] respectively).

$Y_{\delta r e l T}=\frac{1+\sqrt{\rho^{\prime} \cdot X^*}}{1+\sqrt{\rho^{\prime} \cdot X_T^*}}$     (13)

6.png

Figure 6. Short staircase S-N curve

7.png

Figure 7. Classic staircase S-N curve

As mentioned in the previous paragraphs, a different approach for the computation of the size factor Y_X was used. This particular correction coefficient was calculated implementing a new method developed by Dobler et al. [19] (Eq. (8)). Thanks to this new approach the resultant value of the size factor was set to Y_X=1.18, which is slightly different if considering the value suggested from the standard (=1). The last step was the one in which the admissible bending stress related to the tooth root was calculated. According to Eq. (9) values were found out at σ_Flim=280.44MPa and $\sigma_{\text {Flim }}^*=288 \mathrm{MPa}$. Figure 6 and Figure 7 display the final outputs of the experimental tests. In particular, the two S-N curves (50% failure probability) drown according to tests results are presented.