1 Introduction

With the growing global concern regarding environmental degradation and conservation, strategies for enhancing energy and fuel efficiency through the reduction of materials in mechanical systems have drawn increasing interest; these strategies primarily focus on weight reduction. Numerous studies have aimed to address this challenge (Bian et al., 2015; Kroll et al., 2011; Solomon et al., 2020).

Furthermore, the improvement in gear efficiency in power transmission systems is an area of interest in the automotive and aviation sectors, where weight reduction is crucial for improving the fuel economy (Anderson & Loewenthal, 1982; Petry-Johnson et al., 2008; Simon, 1981; Xu et al., 2007). By reducing the weight of the gear—a critical component responsible for transmitting power—the fuel efficiency of the gearbox transmission can be enhanced, thereby yielding a greater weight reduction and energy efficiency (Joshi et al., 2014).

One of the most commonly applied methods for reducing gear weight involves optimizing the stress experienced by the gear teeth and the stiffness of the gear blanks through the reduction of the gear rim and web thickness (Opalic et al., 2011).

However, reducing the rim and web thicknesses can increase the likelihood of gear breakage owing to the application of a greater stress to the root of the gear than the existing stress. Therefore, the changes in stress at the root of the gear must be considered when reducing the rim and web thicknesses.

To account for the increase in the root stress resulting from a decrease in the rim thickness, the international standard ISO 6336 (ISO, 2019) employs the rim thickness factor \({Y}_{B},\) from AGMA 2101 (American Gear Manufacturers Association, 1995) rim thickness factor \({K}_{B}\), without modification. However, these standards yield highly conservative values that do not significantly contribute to gear weight reduction.

Moreover, for gears with below a backup ratio of 0.5 (obtained by dividing the gear rim thickness by the height of the gear teeth), the rim thickness factor as suggested by these standards is not feasible. Additionally, no international standards are currently available that account for the increase in tooth root stress resulting from web thickness reduction.

Several studies have focused on experimental methods (gear-tooth bending test) and analytical methods (finite element analysis; FEA) to validate the changes in gear tooth root stress according to the rim thickness. Oda et al. (Oda et al., 1981) measured the stress of the tooth root by attaching strain gauges to the tooth root of spur gears with various rim thicknesses and verified the 2D finite-element analysis model (FEM) modeled as a triangular element using the test results. To confirm the validity of the single-tooth model, the stress analysis results of the tooth root of the single-gear and adjacent-gear tooth models were compared. However, the 2D FEM model involved a limitation: the gear tooth root stress resulting from the web change was not considered.

Arai et al. (1981) conducted the static bending and bending fatigue tests to confirm the fatigue limits of solid and thin rim gears with holes. These tests revealed that the bending fatigue limit significantly decreased for values where the ratio of rim thickness to modulus was 2.5.

In contrast, Bibel et al. (1994) performed an FE analysis using a five-tooth gear model. The gear rim thickness was divided into eight equal parts to create the FE model, and a thin-rim thickness model was created by gradually removing the rim thickness from the analytical model.

With the model generated using this method, they validated the inflection point by generating a graph of the maximum tension and compression according to the ratio of tooth height to rim thickness. The maximum compressive and alternating stresses increase when the backup ratio is 1.3 or less.

Drago and Pizzigati (1980), Drago and Lutthans () measured the tooth root stress using three test gears with different rim thicknesses using the photoelastic method. Because the data were obtained with a limited number of test gears, the rim thickness factor, \({K}_{B}\), for backup ratios from 0.5 to 2.0 was presented by extrapolating the measured test data.

However, the study conducted by Drago et al. was limited to plastic gears and did not involve metal gears. Furthermore, the research focused only on the rim thickness without considering the thickness of the gear web.

The maximum point of tooth root stress generation calculated using the aforementioned international standards varies depending on the application theory of each standard and differs from the actual maximum point of tooth root stress generation that occurs in real gears.

In addition, the cited studies (Arai et al., 1981; Bibel et al., 1994; Drago & Pizzigati, 1980; Drago & Lutthans, 1983a, 1983b; Oda et al., 1981) did not verify the FE analysis results through tests; this resulted in a limitation—these studies could not be analyzed with a model showing the same root stress characteristics as the actual gear.

To address these research limitations, the current study accurately predicted the point where the maximum tooth root stress occurs in gears; a high-fidelity 3D FEM was developed and validated through root stress testing of gear teeth to analyze the tooth root stress of gears with thin rims and webs.

In this study, the effects of changes in the rim and web thicknesses on the root stress of gear teeth in spur gears were analyzed experimentally and analytically.

Based on a root-stress analysis model for gear teeth validated through experiments, this study presents a stress ratio, defined as the ratio of the maximum stress of gears with thickness changes in the rim and web to the maximum stress of solid gears according to the backup ratio. Four types of gears were manufactured: solid gears, gears with thin rims, gears with thin webs, and gears with intermediate rims and webs.

A load was applied to the highest point of the single-tooth contact (HPSTC) of each gear, and strain gauges were attached to the gear tooth root to measure the actual gear tooth root stress.

In contrast to previous studies, 3D FEMs were developed for each gear, and their high fidelity was confirmed through comparison with the experimental results.

Based on the developed model, a parametric study was conducted to analyze the stress ratio based on the backup ratio, considering the effects of changes in the web position, pressure angle, and module.

Moreover, the stress ratios obtained from the developed model were analyzed for different backup ratios by conducting a parametric study on the web position changes, pressure angle variations, and module changes. In addition, the stress-ratio map obtained in this study was compared with the rim thickness factor presented in the ISO 6336 standard to obtain stress ratios for gears with backup ratios of 0.3, which are not covered in the ISO 6336 standard.

2 Experimental Study

2.1 Manufacturing Test Gears and Bending Test Rig

To experimentally analyze the effect of rim and web thicknesses on gear tooth root stress, designing and manufacturing test gears with different rim and web thicknesses, as illustrated in Fig. 1 and described in Table 1, the blank shapes of the test gears are presented in Fig. 2. Four types of test gears were designed and manufactured for this study: (a) a solid gear with no rim or web, (b) a reference gear with a web thickness ratio of 0.5 and a backup ratio of 1, (c) a gear with the same web thickness ratio as the reference gear but with a backup ratio of 0.5, and (d) a gear with a web thickness ratio of 0.25 and the same rim thickness as the reference gear. The web thickness and backup ratios are defined in Eqs. (1) and (2), respectively.

Fig. 1
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Test gears used in this study

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Table 1 Geometry and specification of test gears
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Fig. 2
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Blank shapes of test gears

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Based on the method presented in ISO 6336-3 Tooth Root Stress Method B (ISO 6336-3, 2019), a gear tooth bending test device was fabricated and utilized for conducting gear-root stress measurements. To apply a line load to the HPSTC, the location where the gear root stress is maximized, a symmetric test jig was fabricated and employed, as presented in Figs. 3 and 4. In gear-bending tests, the number of gear teeth that can be subjected to a load varies depending on the size, characteristics, and gear specifications. Based on the 2D computer-aided design (CAD) model shown in Fig. 3, which was used to validate the specifications of the test gear used in this study, it was determined that in order to apply a line load to the desired location of the HPSTC, seven gear teeth must be symmetrically fixed. Therefore, a test apparatus was designed, as visualized in Fig. 4, to accommodate the seven gear teeth, and a gear tooth bending test was conducted, as depicted in Fig. 5. The gear-bending test rig was designed and manufactured in accordance with specifications reported in the literature (Meneghetti et al., 2018).

$${\text{Web }}\;{\text{thickness }}\;{\text{ratio}} = \frac{{{\text{Gear }}\;{\text{web }}\;{\text{thickness}}}}{{{\text{Gear}}\;{\text{ face}}\;{\text{ width}}}}$$
(1)
$${\text{Backup}}\;{\text{ ratio}} = \frac{{{\text{Gear }}\;{\text{rim }}\;{\text{thickness}}}}{{{\text{Gear }}\;{\text{tooth }}\;{\text{height}}}}.$$
(2)
Fig. 3
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Loading point for gear tooth bending test

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Fig. 4
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Schematics of gear tooth bending test device for measuring gear root stress

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Fig. 5
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Gear tooth bending test device for measuring gear root stress

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2.2 Strain Gauge Attachment on Test Gears

An FEA was performed when a load was applied to the HPSTC, as illustrated in Fig. 6a, to validate the location where the maximum stress occurred at the root of the gear tooth. All four test gears illustrated in Fig. 1 exhibited similar maximum tooth root stress locations.

Fig. 6
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Mounting position of strain gauges on gear tooth root

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Based on the analysis results, six strain gauges were attached along the gear lead direction at the gear root location where the maximum tooth root stress occurred, as depicted in Fig. 6b. As shown in Fig. 6c, strain gauges were attached to two root positions with a phase difference of 180° per test gear, as visualized in Fig. 6d, and the root stress of the gear tooth was measured accordingly.

The specifications of the strain gauges used in these tests are listed in Table 2. SAE J 1619 (SAE International, 2017) and SAE J 821042 (SAE International, n.d.) were employed as the strain gauge attachment methods.

Table 2 Specification of strain gauge used in this study
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2.3 Measuring and Analyzing Gear Tooth Root Stress

MTS 810 model was used for the gear tooth bending test device, and its specifications are listed in Table 3. The data acquisition system comprised a UCAM 550-A (KYOWA) for collecting the measurement data representing the stress levels at the roots of the gear teeth, and the device specifications are listed in Table 4. Six strain gauges were attached to the gear root, as illustrated in Fig. 6b, and strains were measured in sets of two in the lead direction at the left, center, and right positions of the gear tooth width, as depicted in Fig. 6d (SAE International, 2017). In the gear tooth bending test, the load was linearly increased from 0 to 10 kN, and the gear tooth root strain was measured accordingly. Equation (3) was used to convert the strain into stress values.

$$\sigma = E \varepsilon .$$
(3)
Table 3 Specification of test device on gear tooth bending
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Table 4 Specification of data acquisition system on gear tooth bending test
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Here, σ represents the gear root stress (MPa), E represents the Young's modulus of the gear material (GPa), and ε represents the measured strain (m/m).

The bending test results of the test gears, as plotted in Fig. 7, reveal that the maximum stress occurred at the center of the tooth width for all the test gears.

Fig. 7
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Stress results of gear tooth bending test

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Upon analyzing the tooth root bending test results of the four test gears, it was found that the stress deviation at the left and right positions of the tooth width was not considerably large. These results indicate that edge contact or contact misalignment did not occur during the gear tooth bending test.

The difference in stress at the gear root at positions 1 and 2 with a 180° phase difference, as depicted in Fig. 6c, is insignificant. The test results in Fig. 7 are reported in Table 5 as the ratio of the center stress at position 1 of the solid gear. The reference gear with a backup ratio of 1 and web thickness ratio of 0.5 showed insignificant differences at the left, right, and center positions when compared to the stress of the solid gear.

Table 5 Comparison of the ratio between the center results of solid gear position #1 and the test results
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Upon comparing the stress between the thinned-web gear with a backup ratio of 1 and web thickness ratio of 0.25 and the solid gear, an insignificant difference in stress was observed at the center position. However, the stress at the left and right positions of the thinned-web gear increased significantly compared with that of the solid gear.

For the thinned-rim gear with a backup ratio of 0.5 and a web thickness ratio of 0.5, the stresses at the left, right, and center positions increased considerably compared to those for the solid gear.

The test results revealed that the stress of the gear increased significantly only at the left and right positions when the web thickness ratio was 0.25, whereas when the backup ratio was 0.5, the stress at the left, right, and center positions increased significantly.

3 Results and Validation of the FE Model for Gear Tooth Root Stress

3.1 FE Modeling for Analyzing Gear Tooth Root Stress

A gear FEM was developed using Abaqus/CAE (Dassault Systems, 2013) to predict the gear-tooth root stress with respect to changes in the gear rim and web geometry. The FE element used in the gear FEM, a linear brick element (C3D8) with eight nodes and six faces, was utilized in accordance with previous studies (Chen & Tsay, 2002; Li, 2002). The material properties of the gear FEM are listed in Table 6.

Table 6 Material properties of gear the FEM
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To apply a load to the gear HPSTC and determine the point of maximum gear-tooth root stress, the gear FEM generated a fine gear-tooth mesh. To investigate the stress difference owing to changes in the web thickness and web position, a fine mesh was generated for the face width of the gear. The details of the developed gear FEM are listed in Table 7.

Table 7 Specifications of the gear FEMs
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Moreover, to characterize the gear-tooth root stress, loading and boundary conditions were applied, as illustrated in Fig. 8a. The nodes on the inner surface of the gear are coupled to the master nodes, which are subsequently restrained to five degrees of freedom, except in the rotational direction. The gear loading condition, as depicted in Fig. 8b, is established as follows: to implement the same analysis conditions as the test conditions, a load was applied to the HPSTC point of the gear, and the nodes at the X-axis symmetric point in Fig. 3 were restrained in all degrees of freedom.

Fig. 8
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Loading and boundary conditions for gear FEA

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The tested gear has end reliefs measuring 3 mm on each side of the gear face, as shown in Fig. 9. To reflect the effect of the end relief applied to the gear, a linearly decreasing load was applied toward the left and right ends at the point where the end relief began. This proportional load reduction was determined through trial and error by comparing the analysis and test results of the solid gear. Blank shapes of the four tested gears were generated using the FEM, as depicted in Fig. 10.

Fig. 9
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Proportionally decreasing loading-condition pattern considering end relief

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Fig. 10
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Blank shapes for developed FE model

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3.2 Validation for Gear FE Model

Table 8 lists the relative errors between the measured stresses at positions 1 and 2, as illustrated in Fig. 6c, and stress obtained via FEA. The relative error was calculated using Eq. (4).

$${\text{Relative error}} = \frac{{{\text{Test Result }} - {\text{ FEA result }}}}{{\text{Test result}}}.$$
(4)
Table 8 Comparison between FEA results and test results
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The maximum values of the relative errors of the center and end points of the face width (face width: 0–10 mm and 30–40 mm) analyzed for the four test gears were 4.56% and 16.64%, respectively, which were well correlated.

The maximum error occurred in the thinned-rim gear, which was 50% thinner than the reference gear. As shown in Fig. 11b–e, the stress variation with respect to the face width position exhibited a higher magnitude at the end point (0–10 mm; 30–40 mm) of the face width than at the center point (10–30 mm) of the face width. In particular, the stress variation in the thinned-rim gear (Fig. 11d) was significantly larger than that in magnitude compared to the other tested gears.

Fig. 11
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FEA results of maximum tooth-root stress according to gear face width

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This result implies that a small error in the attachment position of the strain gauges during the gear root stress measurement can lead to a significant increase in the relative error between the measured and analyzed stresses. These results conclusively indicate that large relative errors occurred at the left and right positions of the thinned-rim gear.

4 Parametric Study & Results Using Gear FE Model

Based on the verified FE model, the stress ratios of the gear roots were analyzed with respect to the backup ratio by varying the web position, pressure angle, and the number of modules. The stress ratio of the gear roots is defined using Eq. (5).

$${\text{Stress ratio}} = \frac{{\text{maximum gear tooth root stress of target gear }}}{{\text{maximum gear tooth root stress of solid gear}}}.$$
(5)

The stress ratio is calculated using the maximum root stress, which is the most critical factor affecting the gear failure. As demonstrated in Fig. 12, the maximum tooth root stress point and stress values are obtained through FEA.

Fig. 12
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Maximum stress point of target gears

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A parametric study was conducted for variations in the web position, pressure angle, and module, and the results are presented in the form of a stress-ratio map in Table 9.

Table 9 Parametric study cases
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In each model, the web thickness ratios (Eq. 1) was varied at four types (0.2, 0.4, 0.6, and 0.8), and the backup ratio (Eq. 2) was varied at eleven types (0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.2, 1.5, and 2.0). Therefore, 308 FEA simulations were conducted in this parametric study.

The x-axis of the graphs (parametric study results) presented in Figs. 14, 15 and Fig. 16 represents the backup ratio, whereas the y-axis represents the stress ratio for each parametric study case. In each figure, the numbers following the model name (e.g., A-1, A-2, A-3, and A-4) represent the web thickness. A smaller value indicates a smaller web thickness factor.

4.1 Parametric Study #1: Web Position Variation

In this study, the module was fixed at 4, and the pressure angle was fixed at 20°. As shown in Fig. 13, parametric studies were performed for three different web positions.

Fig. 13
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Definition of gear web position

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As shown in Fig. 14, the stress ratio increased as the backup ratio decreased when the backup ratio was less than 1.0, and the stress ratio increased rapidly when the backup ratio was less than 0.5.

Fig. 14
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Stress-ratio map according to web position variation

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Considering the web thickness and backup ratio simultaneously, the stress ratio increased by more than 20% when the backup ratio was 0.5 or less and the web thickness ratio was 0.4 or less.

As shown in Fig. 12, as the web position was changed to 0.5 and 1 to obtain an asymmetric gear blank, the maximum stress of the gear occurred at the point wherein the web and rim met and not at the midpoint of the tooth width. Additionally, the maximum stress increased by 1.3% compared to that for gears with symmetrical blanks featuring webs in the middle.

4.2 Parametric Study #2: Pressure Angle Variation

In this study, the module was fixed at 4 and the web position at 0. Parametric studies were performed at three different pressure angles (16°, 18°, and 20°).

As plotted in Fig. 15, the difference in the maximum tooth-root stress according to the pressure-angle change in the range of 16°–20° was approximately 1%, which was insignificant.

Fig. 15
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Stress-ratio map according to pressure angle variation

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The variation in the pressure angle during changes in the rim and web thickness implies that it does not have a significant impact on the variation in the maximum root stress. Similar to the results of the parametric study based on the web-position change performed previously, the stress ratio increased rapidly when the backup ratio was less than 0.5.

Considering the backup ratio and web thickness simultaneously, the stress ratio increased rapidly when the backup ratio was less than 0.5 and the web thickness ratio was less than 0.4.

4.3 Parametric Study #3: Module Variation

The pressure angle was fixed at 20°, and the web position was set to 0. Parametric studies were performed using three different modules (2, 3, and 4).

As indicated in Fig. 16, no variation occurred in the maximum root-stress ratio despite the change in the module; only a difference in the absolute stress values was manifested.

Fig. 16
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Stress-ratio map according to module variation

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Similar to the results of the previous parametric study based on web-position variation and pressure-angle variation, the stress ratio increased rapidly when the backup ratio was less than 0.5.

Considering both the rim thickness and web thickness simultaneously, the stress ratio increased rapidly when the backup ratio was less than 0.5 and the web thickness ratio was less than 0.4.

5 Comparison Between ISO 6336 Standard and Stress-Ratio Map

Figure 17 presents a comparison between the stress-ratio map of the web position variation, pressure angle variation, and module variation, and the \({Y}_{B}\) factor graph, which indicates the rim thickness factor of the external gears presented in ISO 6336-3.

Fig. 17
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Comparison between parametric stress-ratio map and ISO 6336 results

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In the rim thickness factor presented in ISO 6336-3, the stress ratio started to increase from a back-up ratio of 1.2. Moreover, the stress-ratio map based on web position variation, pressure angle variation, and module variation performed in this study revealed that the back-up ratio increased from 1.0. Furthermore, the \({Y}_{B}\) factor presented in ISO 6336-3 was extremely conservative when compared with the stress-ratio map of the parametric study conducted in this study.

6 Conclusion

In this study, the effects of the rim and web thicknesses of a spur gear on the tooth-root stress were analyzed using a FEM that was verified through testing.

For the FEM validation test, a gear tooth bending test device was implemented based on ISO 6336-3 Tooth root stress Method B (ISO, 2019) and the test device of Meneghetti et al. (2018). Four test gears (solid, reference, thinned-rim, and thinned-web gears) were designed and manufactured to measure the tooth root stress for FEM validation.

After detecting the point of maximum tooth-root stress predicted through FEA, the stress at the tooth root was measured by attaching a strain gauge to the point.

Comparing the measured gear tooth-root stress data and the analyzed values of the developed FEM, revealing that the FEM reflected the test results well, with an error range of the gear center-tooth root stress of 0.98–4.56%.

Using the developed 3D FEM, a parametric study was performed on the change in the gear tooth-root stress for varying thicknesses of the spur gear rim and web according to the variations in the web position, pressure angle, and module.

The results obtained from the stress-ratio map of the parametric study for web position variation, pressure angle variation, and module variation are as follows:

  1. (1)

    When the web position of the gear was moved from the center to the left or right to create an asymmetric blank, the point where the maximum tooth root stress occurred changed from the center to the position where the web was moved. The maximum tooth-root stress of gears whose web position was moved to the left or right instead of the center increased by 1.3% compared with those of gears whose webs were moved to the center. When designing gears with asymmetrical gear-blank shifting of the web position, the increase in the maximum stress point along with the change in the maximum tooth-root stress point must be considered.

  2. (2)

    The stress ratio tendency was analyzed according to the variation in pressure angle(16°, 18°, and 20°), revealing that the maximum stress difference between the back-up ratio of 0.3 and web thickness ratio of 0.2, where the largest stress occurred in the parametric study case, was approximately 1%. Therefore, a pressure angle change from 16° to 20° had an insignificant effect on the change in the tooth root stress between the gears.

  3. (3)

    Upon analyzing the stress ratio against the backup ratio according to the module change (3, 4, 5), no change was observed in the stress ratio, except for the difference in the absolute value of stress, due to the application of the same load.

  4. (4)

    A common result obtained from the three parametric studies was that the stress ratio started increasing at a backup ratio of 1.0. From the point where the web thickness ratio was less than 0.4 and the backup ratio was less than 0.5, the stress ratio increased sharply.

  5. (5)

    Upon comparing the stress-ratio maps obtained for different web positions, pressure angles, and modules with the rim thickness factor (\({Y}_{B}\)) presented in ISO 6336-3, the \({Y}_{B}\) factor in ISO 6336-3 was found to be more conservative than the results obtained in this study. While ISO 6336-3 provides a \({Y}_{B}\) factor that can predict stress ratios for gears with backup ratios greater than 0.5, this study reports stress ratios for gears with backup ratios as low as 0.3.

  6. (6)

    This study demonstrates that to reduce the weight of the spur gear, mutual consideration of the rim and web thicknesses is necessary when reducing the thickness of the rim and web. The results of this study can serve as fundamental data for establishing lightweight design standards for spur gears. However, the study was conducted under the assumption of a uniform load acting on the gear, and gear-mesh misalignment or gear-edge contact phenomena were not considered. Further studies are planned to analyze the effect of gear tooth-root stress on the rim and web thickness under the conditions of gear-mesh misalignment or gear-edge contact.