Open Access
 Issue Int. J. Simul. Multidisci. Des. Optim. Volume 11, 2020 23 8 https://doi.org/10.1051/smdo/2020017 22 October 2020 This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## 1 Introduction

Machines produce work by transmitting power from one component to another. Gear drive is the most efficient drive to transmit the power. Gears can be classified based on the type of load and direction of transmitting torque. Spur gear is the most widely used gear due to its simplicity in manufacturing. Spur gears are subjected to bending and contact stresses. The bending stress is given by Lewis and AGMA equations. The assumptions while using these equations are as follows.

• Effect of radial component which produces compressive stress is neglected.

• Tangential component is uniformly distributed over the face width of gear.

• Stress concentration effect is negligible.

• At any time only one pair of gear is in contact and carries total load.

AGMA bending stress equation is given as: − Bending stress where

b − Face width

Yj − Spur gear geometry factor

Kv − Velocity factor or dynamic factor

In the above equation all the factors have constant values except Ft and b. Yj is calculated with calculation of Lewis form factor ‘Y’ and stress concentration factor based on a fillet radius of tooth of 0.35/P. It also depends on the number of teeth in the mating gear. This is the only factor which depends on the geometry and changes as the number of teeth changes. All other factors do not have much dependency on the geometry of the spur gear. Kv indicates the severity of impact when the pairs of teeth engage. Kv depends on pitch line velocity and manufacturing accuracy both. It is calculated by inverse of Barth's equation. Km accounts for non-uniform spread of the load across the face width. It depends on the accuracy of mounting, bearings, shaft deflection and accuracy of gears. This factor depends upon the face width but keep constant for face width 1 mm to 50 mm. From AGMA bending stress equation it can be concluded that the stress is inversely proportional to the face width.

## 2 Literature survey

This paper dear with a parametric study is conducted by varying the face width to study their effect on the bending stress of spur gear using ANSYS for modelling and stress analysis respectively.

## 3 Methodology

### 3.1 Modelling of spur gear

Here six numbers of spur gears have been modelled using AutoCAD-2017. All the design parameters have been kept same for the gears except the face width. Face width has been changed from 20 mm to 30 mm with an interval of 2 mm each time. The design parameters and relations are given in the Table 1 below.

Using the above design parameters 2D drawing has been created with the help of AutoCADand then converted into a 3D model. After modelling it was exported or saves in IGES file format for further analysis.

Table 1

Design parameter and the calculation.

### 3.2 Agma bending stress calculations (i)

where (ii) (iii) (iv) (v) (vi) (vii) For F t :

Power (P) = 5 kW = 5000 W Speed (N) = 1500 rpm (viii)  (ix)

From (ii), (iii), (iv), (v), (vi), (vii) and (ix) the equation (i) becomes: Bending stress for other gears has been computed using AGMA bending stress equation with the variation in face width (22, 24, 26, 28 and 30 mm). Also to study the effect on the material of spur gear the bending stress was calculated for different loads i.e. 2000, 2500 and 3000 N.

## 4 Material property

Structural steel has been chosen as the spur gear material. The various mechanical properties of the material are given in the Table 2.

Table 2

Properties of material (structural steel).

Table 3

Number of nodes and elements in ANSYS during meshing for various face width of spur gear.

### 4.1 Static structural analysis

Static structure analysis of the spur gear modelled in AutoCAD carried out using the Finite element analysis in ANSYS 14.5 work bench. Spur gear model was imported in Ansys and material properties were assigned. In the present study the material used is structural steel which is the most suitable material for the spur gear due to its strength. The whole gear has been divided into small elements to analyse the bending stress in each element for more accurate value of stress at critical section. In this model very fine quality triangular mesh has been generated as shown in Table 3. Relevance is set to 10 and to refine the model mesh further the refinement has been applied up to a factor of 3.

Boundary condition has been applied at the centre of the spur gear where the shaft is attached to the gear. That centre kept fixed shown in the Figure 1a with blue colour. The load applied on the face of the tooth tangentially shown in the red colour of the Figure 1b, the arrow shows the direction of force.

In the present study Von-mises stress has been computed using ANSYS for different face width to analyse the effect of the face width on bending stress at spur gear tooth. Figures 24 show the result obtained in the Ansys software. Due to the loading and boundary condition the spur gear tooth acts as the cantilever. The bending effect of load has been observed at the root fillet of the tooth of the spur gear. This region is the critical region for the tooth during meshing with the other gear.

Above figure shows the variation of bending stress with different face width. Due to the load the tooth bend and the maximum stress is generated at the critical region. The maximum value of the stress is shown with the red colour at the root fillet of the tooth.

The results shown in the Figure 2a–f has been against a load of 1061.03 N. To optimize the design and to know whether the effect of face width on the bending stress of the spur gear teeth are true at higher loading condition, the applied load was incccresed. Fig. 1(a) Fixed support at spur gear (face width 20 mm); (b) loading condition and direction at 20 mm face width. Fig. 2(a) Von-mises stress at gear with face width 20 mm; (b) Von-mises stress at gear with face width 22 mm; (c) Von-mises stress at gear with face width 24 mm; (d) Von-mises stress at gear with face width 26 mm; (e) Von-mises stress at gear with face width 28 mm; (f) Von-mises stress at gear with face width 30 mm.

## 5 Results analysis

Bending stress is calculated for different face width at 1061.03 N load using AGMA equations. It is found from the results that there is a reduction in the bending stress with the increase in the face width of the spur gear. The same has been shown in Table 4. Also the bending stress is computed using ANSYS software under static structure analysis. The results obtained using Ansys are shown in Table 4. The analytical and computational results are compared and it has been found that error is within the limits. Also it is found that Increasing face width will increase the stress distribution area due to which the overall effect of load decreases at the root of mating tooth. These results signifies enhancement in tooth bending strength.

Also the Bending stress of teeth of spur gear has been determined with higher loading conditions using AGMA equation and ANSYS. The value of load increased up to the tensile yield strength of the material of spur gear (structural steel). Same trend is observed for the higher loading conditions. Table 4 and Table 5 shows the effect of increasing face width on value of bending stresses at 2, 2.5 and 3 kN. The error between the AGMA and ANSYS results are similar for all loads indicating the effect of face width on bending stress is significant and independent of loading condition.

The maximum error observed between the computational results and the AGMA results is 4%.Graphs have been plotted against these data given in the Table 4 and Table 5. Different graph have been plotted for different loading conditions, showing the variation of the bending stress at tooth of spur gear with respect to the increasing face width of the spur gear.

Figure 3 shows the variation of bending stress against face width at 1061.0329 N load. The range of face width is taken as 20–30 mm. In this graph there are two lines which show results on the basis of AGMA equation and ANSYS. The bending stress is observed to be 82.81 Mpa for face width 20 mm and load 1061.03 N. It has been observed from the results there is 34% reduction in bending stress when the value of face width varied from 20 mm to 30 mm and the corresponding reduced value of bending stress is 54.58 Mpa.

Figure 4 showing the variation of bending stress on spur gear tooth with respect to spur gear face width at a load of 2000 N, with the comparison of AGMA and ANSYS bending stress results. At 2000 N load the ANSYS bending stress at root of the tooth of spur gear for 20 mm face width was found to be 156.09 Mpa, and reduced to 102.89 Mpa for 30 mm face width which is 34% reduction in bending stress. AGMA bending stress at face width 20 mm was 149.85 Mpa which was reduced for the face width 30 mm up to 99.9 Mpa, a 33.33% reduction in stress. Comparing AGMA and ANSYS results the ANSYS shows the better reduction in results.

Figures 5 and 6 shows the decreasing slope as face width increases. The maximum stress is found to be 195.12 Mpa using Ansys at a load of 2500 N and face width of 20 mm. The bending stress decreases when face width is increased to 30 mm stress and reaches to 128.61 Mpa. As shown in graph-4 the maximum stress was observed to be 234.14 Mpa when the face width was 20 mm while it reduced to 154.33 Mpa when the face width was increased to 30 mm at a load of 3000 N. For both the loading conditions the percentage reduction remains 34% when analysing in ANSYS and 33.33% when analysing in AGMA. The same slope and identical line in each plot describe the linear relation of the effect of increasing face width on the bending stress irrespective of the load.

Figure 7 shows the comparison of different loads lines if only AGMA calculated bending stress plotted against the varying face width of spur gear tooth. Whereas in graph-6 effect of increases face width on bending stress of spur gear tooth is shown with the value of stresses solved under ANSYS.

Analysis of bending stress has been done with changing face width and varying load Using AGMA equations. Variation of the lines (different colour for different load) or the slope of each line is somewhat same, except some negligible differences. Results show that if the load is increased beyond 3000 N the material of spur gear will fail resulting in the failure of gear in the drive, because it reaches its yielding stress value. This could be resolved by increasing the face width of spur gear beyond 20 mm. Spur gear with increased face width is safe because at the increased value of face width, the bending stress will not exceed its yielding stress value.

Figures 7 and 8 shows the comparison of bending stress at different loads which helps in estimating the variation of the stress at different face width and loads. Here in Figure 7, the AGMA stresses variation for different loads are similar. It does not change considerably and has a similar decreasing curve for every load. Whereas Figure 8 shows that the slope of the stress line curve is sharper for 3000 N load, as compared to the other loads. The overall reduction for all loads are same but the reduction when face width reduced from 20 mm to 24 mm is more for higher loads than in the lower loads as observed in the Figure 8.

In ANSYS software finite numbers of very small elements were analysed due to which it shows the maximum value of stress that could develop in spur gear material. Due to this the ANSY results are greater than that of AGMA results but with a very small error value.

Table 4

Value of bending stresses on the teeth of spur gear using AGMA equation and ANSYS software applied load of 1061.03 N and 2000 N with varying face width.

Table 5

Value of bending stresses on the teeth of spur gear using AGMA equation and ANSYS software applied load of 2500 N and 3000 N with varying face width. Fig. 3Comparison of the effect of increases in face width on bending stress with AGMA and ANSYS at 1061.03 N of load. Fig. 4Comparison of the effect of increases in face width on bending stress with AGMA and ANSYS at 2000 N load. Fig. 5Comparison of the effect of increases face width on bending stress with AGMA and ANSYS at 2500 N of load. Fig. 6Comparison of the effect of increases face width on bending stress with AGMA and ANSYS at 3000 N of load. Fig. 7Comparison of the effect of increases face width on bending stress with analytical value (AGMA) at varying loads. Fig. 8Comparison of the effect of increases face width on the Von-mises stress with software values (ANSYS) at varying load.

## 6 Conclusion

The Results showed reduction in bending stress with an increase in the face with negligible error. Bending stress values obtained using both the methods do not exceed the yield stress and hence safe and within the acceptance imit. The analysis showed that the gear will not fail if face width is more than 20 mm, as the bending stress at greater face width will not cross the yield stress value of the material. An increase in the face width of the spur gear evidently increases the load carrying capacity of gear. Increase in bending strength has been achieved with increase in face width of spur gear. The value of load does not affect the variation of bending stress with respect to face width. This implies that the reduction in bending stress at the tooth of the spur gear is same at any value of load. Increasing load beyond 3000 N will fail the spur gear having face width 20 mm, as it crosses the yielding limit of the material of gear.

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Cite this article as: Hardial Singh, Deepak Kumar, Effect of face width of spur gear on bending stress using AGMA and ANSYS, Int. J. Simul. Multidisci. Des. Optim. 11, 23 (2020)

## All Tables

Table 1

Design parameter and the calculation.

Table 2

Properties of material (structural steel).

Table 3

Number of nodes and elements in ANSYS during meshing for various face width of spur gear.

Table 4

Value of bending stresses on the teeth of spur gear using AGMA equation and ANSYS software applied load of 1061.03 N and 2000 N with varying face width.

Table 5

Value of bending stresses on the teeth of spur gear using AGMA equation and ANSYS software applied load of 2500 N and 3000 N with varying face width.

## All Figures Fig. 1(a) Fixed support at spur gear (face width 20 mm); (b) loading condition and direction at 20 mm face width. In the text Fig. 2(a) Von-mises stress at gear with face width 20 mm; (b) Von-mises stress at gear with face width 22 mm; (c) Von-mises stress at gear with face width 24 mm; (d) Von-mises stress at gear with face width 26 mm; (e) Von-mises stress at gear with face width 28 mm; (f) Von-mises stress at gear with face width 30 mm. In the text Fig. 3Comparison of the effect of increases in face width on bending stress with AGMA and ANSYS at 1061.03 N of load. In the text Fig. 4Comparison of the effect of increases in face width on bending stress with AGMA and ANSYS at 2000 N load. In the text Fig. 5Comparison of the effect of increases face width on bending stress with AGMA and ANSYS at 2500 N of load. In the text Fig. 6Comparison of the effect of increases face width on bending stress with AGMA and ANSYS at 3000 N of load. In the text Fig. 7Comparison of the effect of increases face width on bending stress with analytical value (AGMA) at varying loads. In the text Fig. 8Comparison of the effect of increases face width on the Von-mises stress with software values (ANSYS) at varying load. In the text

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