Keywords

17.1 Introduction

Gears are commonly used in automotive, mechanical engineering and other industries. Nowadays, high power density gear applications are increasingly in demand, it is critical to design gears with improved load capacity. Gear tooth pitting due to excessive surface contact stress and root fillet fractures due to excessive tooth bending stress are two main fatigue failure modes for gears [1]. The accumulation of defects and a high concentration of stresses in micro defects are also dangerous in the event of shocks which occur during the operation of mechanisms. Micropitting on the tooth surface of the gear can cause it to break [2]. Tooth breakage causes a catastrophic failure, so gear bending stress analysis methods must be reliable and understanding the key influence parameters on bending stress is very important to improve bending strength. Pressure angle is one which plays an essential role in determining the bending stress.

A pressure angle 20° was adopted for standard gears according to ISO 53:1998(E) [3]. The standard pressure angle 20° is a compromise value and cannot meet all the needs of the different applications because of the limited load capacity on root fillet. So in many cases, non-standard pressure angles need to be designed in order to improve the gear performance.

Recently, there have been many efforts made to explore the application of non-standard pressure angle gears. Gupta [4] calculated and compared the maximum bending and contact stress for the low dedendum spur gears with different pressure angles using finite element method. Handschuh [5] investigated the effects of high pressure angle gears compared with typical gear designs, the analysis of contact and bending stress had been done on three gears- standard 3.18 module, 28 tooth and 20° pressure angle, 2.12 module, 42 tooth and 25° pressure angle and 1.59 module, 56 tooth and 35° pressure angle. Sankar [6] studied the effects of pressure angle and tip relief on the failure of a helical gear pairs. Dadhanlya [7] presented a study on the effect of pressure angle on bending stress and deformation of asymmetric involute spur gear using FEA. Oda [8] introduced a study on the effect of pressure angle, helix angle and whole tooth depth on the bending strength.

However, a very critical parameter– root fillet radius is defined by tool (hob) tip radius was not concerned in the research. Since the bending critical section occurs in the gear root fillet, root fillet radius has a great influence on gear bending strength. The root strength can be improved by using a circular fillet design or optimized fillet design according to [9,10,11]. And the maximum tool tip radius will vary with the pressure angle changing due to the geometrical relationship of basic rack [3]. So in this paper, the bending stress of a spur gear pair with identical dedendum and different pressure angles (14.5°, 17.5°, 20°, 22.5° and 25°) and the variation of root fillet radius are studied.

After reviewing the references, gear bending stress in spur gears can be evaluated with four methods namely, standard methods like ISO standards [12] and AGMA standards [13], 3D-TCA method, Finite Element Analysis (FEA) and experimental methods. In this paper three methods of predicting bending stress—ISO standard, 3D-TCA method and Finite Element Analysis are applied to example gear geometry and compared to make sure the results are valid.

17.2 Geometric Specifications of the Studied Gears

In this study, a spur gear pair with different pressure angles (14.5°, 17.5°, 20°, 22.5° and 25°) and the same dedendum (hfp/mn = 1.4) was investigated. A dedendum equal to 1.4 mn permits the finishing tool to work without interference, while maintaining the maximum fillet radius and is recommended for high-precision gears transmitting high torques. It is also suitable for gears with tooth flanks finished by grinding [3]. The geometry parameters of the spur gears have been summarised in Table 17.1.

Table 17.1 Gear parameters for spur gear pair with different pressure angles
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Tool tip radius ρfp/mn = 0.39 is equivalent to a full radius form for the fillet and is the maximum fillet radius when pressure angle equal to 20° (hfp/mn = 1.4) [3]. To ensure consistency of root radius, only investigate the influence of pressure angle on bending stress, the tool tip radius ρfp/mn of spur gear with pressure angle equal to 14.5°, 17.5° and 20° are defined as 0.39. When pressure angle is increased to be greater than 20°, the maximum tool tip radius ρfp/mn cannot reach 0.39 according to the geometric relationship of basic rack showed in Fig. 17.1.

Fig. 17.1
figure 1

The relationship between pressure angle and maximum tool tip radius (hfp/mn = 1.4)

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A basic rack hfp/mn = 1.4 gives a full radius form for the fillet as showed in Fig. 17.1. The centre of ρfp-max is on the centre of the rack space. According to the geometric relationship, the maximum tool tip radius can be derived from Eq. (17.1) [3].

$$\rho_{fp\hbox{max} } = \frac{{\left[ {\frac{{\left( {\pi m} \right)}}{4} - h_{fp} \tan \alpha_{p} } \right]}}{{\tan \left[ {\left( {90^{^\circ } - \alpha_{p} } \right)/2} \right]}}$$
(17.1)

where, hfp is the dedendum of basic rack.αp is the pressure angle of the basic rack.

From the Eq. (17.1), we can see that the maximum tool tip radius ρfp is correlated to pressure angle and is reduced with pressure angle increasing. The maximum tool tip radius ρfp/mn of spur gear with pressure angle equal to 22.5° and 25° showed in Table 17.1 are determined according to Eq. (17.1) equal to 0.308 and 0.208 respectively.

17.3 Methods for Calculating Gear Bending Stress

17.3.1 ISO Standard

ISO standard is a commonly used international standard to determine gear stress. The bending stress equation in ISO 6336-3:2006 [12] based on cantilevered beam bending is given as Eq. (17.2):

$$\sigma_{F} = K_{A} K_{V} K_{F\alpha } K_{F\beta } \frac{{F_{t} }}{{bm_{n} }}Y_{F} Y_{S} Y_{\beta } Y_{B} Y_{DT}$$
(17.2)

where σF0 is the nominal tooth root stress; YF is the form factor, which is defined in Eq. (17.3):

$$Y_{F} = \frac{{\frac{{6h_{Fe} }}{{m_{n} }}cos\alpha_{Fen} }}{{\left( {\frac{{S_{Fn} }}{{m_{n} }}} \right)^{2} cos\alpha_{n} }}$$
(17.3)

YS is the stress correction factor. Equations (17.4), (17.5) and (17.6) define YS, L and qs respectively:

$$Y_{s} = \left( {1.2 + 0.13L} \right)q_{s}^{{\left[ {\frac{1}{{1.21 + \frac{2.3}{L}}}} \right]}}$$
(17.4)
$$L = \frac{{s_{Fn} }}{{h_{Fe} }}$$
(17.5)
$$q_{s} = \frac{{s_{Fn} }}{{2\rho_{F} }}$$
(17.6)

Yβ is the helix angle factor.

YB is the rim thickness factor.

YDT is the deep tooth factor.

\({\text{K}}_{{\text{A}}} {\text{K}}_{{\text{V}}} {\text{K}}_{{{\text{F}}\beta }} {\text{K}}_{{{\text{F}}\alpha}}\) are the load correction factors.

sFn is the tooth root chord at the critical section.

hFe is the bending moment arm for tooth root stress relevant to load application at the outer point of single pair tooth contact.

ρF is the tooth root fillet radius at the critical section.

The determination of sFn, hFe and ρF is showed in Fig. 17.2.

Fig. 17.2
figure 2

The determination of sFn, hFe and ρF from [12]

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17.3.2 3D-TCA Method

Analytical Tooth Contact Analysis (TCA) method is another method to predict gear tooth root bending stress. A3D-TCA software named GATES [14] (Gear Analysis for Transmission Error and Stress) which was initially developed and tested at Design Unit, Newcastle university is used to calculate gear bending stress in this study. It is an FE based analysis package, using a full 3D FEA stiffness model to estimate the gear stiffness and then using contact analysis to estimate the load distribution, stress and other functional parameters.

The bending stress determined in GATES is similar to ISO 6336 standard, calculating the bending stress at 30 degree tangent position, except the load distribution is calculated from the TCA, stiffness variation across the face width and compressive load is considered.

A critical point for bending stress determined by GATES is extended contact which occurs as the loaded tooth is restored to its original un-deflected state at the end of active profile [15]. If tip relief is not applied, the extended contact will increase the contact ratio thus lowering the actual HPSTC (for spur gear, the bending stress is calculated with load applied at the highest point of single tooth contact (HPSTC)) and then lower the actual bending stress. If too much tip relief is applied, the loading point will increases which will increase the bending stress. Therefore bending stress derived from GATES must be calculated with correct tip relief to make sure the gear loaded at exactly the same position as ISO (HPSTC). In this study, to ensure the results from GATES are valid, the correct tip relief are applied to gears.

17.3.3 Finite Element Analysis

In recent years, finite element analysis is widely used to evaluate gear stress, resulting in an abundance of published research [16, 17]. According to [16], in this study, the finite element analysis of gear bending stress is set as follows:

  1. (1)

    Three teeth are used to calculate gear stress.

  2. (2)

    Boundary conditions: fix the three free faces as showed in Fig. 17.3.

    Fig. 17.3
    figure 3

    The boundary conditions of bending stress calculation in FEA

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  3. (3)

    Mesh types: automatic mesh method.

  4. (4)

    Element size is defined in Eqs. (17.7) and (17.8):

    $${\text{M}}_{\text{bending}} = 0.2\rho_{\text{F}} + 0.15\,\,{\text{For }}\rho_{\text{F}} > 0. 5 {\text{mm}}$$
    (17.7)
    $${\text{M}}_{\text{bending}} = 0.366\rho_{\text{F}}^{0.614} \,\,{\text{For }}\rho_{\text{F}} \le 0. 5 {\text{mm}}$$
    (17.8)
  5. (5)

    Maximum principal stress is used to represent the gear stress.

  6. (6)

    Use smaller element sizes to verify the validity of the maximum bending stress to ensure the consistency of the results.

17.4 Results and Discussions

In this paper three independent methods of calculating bending stress—ISO standard, 3D-TCA method and Finite Element Analysis are used to make sure the results are valid. The bending stresses for spur gears with different pressure angle (14.5°, 17.5°, 20°, 22.5° and 25°) from GATES and FEA are showed in Figs. 17.4 and 17.5. The bending stresses from ISO and TCA and FEA are summarized in Table 17.2 and Fig. 17.6. In this study, only bending stresses of the pinion are calculated and compared.

Fig. 17.4
figure 4

Bending stress from GATES with tip relief for spur gear

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Fig. 17.5
figure 5

Bending stress from FEA for spur gear

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Table 17.2 Bending stresses (MPa) for gears with different pressure angle from ISO, GATES and FEA
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Fig. 17.6
figure 6

Bending stresses for spur gears with different pressure angle from ISO, GATES and FEA

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From Table 17.2, it can be seen that:

  1. (1)

    The variations of bending stresses from ISO, GATES with correct tip relief and FEA with pressure angle changing have the same trends, which can be clearly seen from Fig. 17.6.

  2. (2)

    The bending stress is minimum when pressure angle equal to 14.5°.

  3. (3)

    Bending stress is reduced when pressure angle changing from 17.5° to 20°.

  4. (4)

    While bending stress is increased when pressure angle changing from 20° to 22.5° and 25°.

There are some reasons to explain the results.

  1. (1)

    The transverse contact ratio εα is 2.048 greater than 2.0 when pressure angle equal to 14.5°, which means there are at least 2 teeth sharing load when gear meshing as shown in Fig. 17.7. This results in lower bending stresses, without considering the effects of manufacturing deviations and misalignment.

    Fig. 17.7
    figure 7

    Number of teeth sharing load condition when 2 < εα < 3 for spur gears

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  2. (2)

    When pressure angle is increased from 17.5° to 20°, the tooth root chord at the critical section sFn is increased, the bending moment arm hFe is also slightly higher, while the root radius ρF is almost same. The detailed sFn, hFe and ρF are showed in Table 17.3, which will lead to a lower form factor YF and a slightly higher stress correction factor Ys, while the decrement of form factor is greater than the increment of stress correction factor, so bending stress decreases when pressure angle changes from 17.5° to 20°.

    Table 17.3 The details of bending stress key factors in the calculation from ISO
    Full size table
  3. (3)

    When pressure angle is increased from 20° to 22.5° and 25°, the tooth root chord at the critical section sFn and bending moment arm hFe are also increased, and the root radius ρF is largely reduced due to the significant declining of tool tip radius ρfp/mn from 0.39 to 0.308 and 0.208. The calculated SFn, hFe and ρF are showed in Table 17.3. The large reduction root radius ρF leads to significant increase in notch parameter qs, and dramatic increases stress correction factor Ys which is greater than the form factor YF reduction. So bending stress increases when pressure angle changes from 20° to 22.5° and 25°.

17.5 Conclusions

From the analysis presented in this study, the following conclusions can be drawn:

  1. (1)

    The variations of bending stresses from ISO, GATES with correct tip relief and FEA with pressure angle changing have the same trends. The results in this study are valid.

  2. (2)

    For basic rack hfp/mn = 1.4, according to the geometric relationship, the maximum tool tip radius (ρfp/mn) of spur gear with pressure angle equal to 20°, 22.5° and 25° are equal to 0.39, 0.308 and 0.208 respectively. It can greatly affect the bending stress, so we must consider its influence when we investigate the effect of pressure angle on bending stress.

  3. (3)

    Smaller pressure angle such as 14.5° in this study, can result in high contact ratio spur gears (εα > 2), at least two teeth sharing the load when gears are in mesh, so it can get lower bending stress. But we must make sure this is maintained when considering manufacturing deviations, elastic deflections and misalignment of shaft and micro modifications, the gears always have transverse contact ratio greater than 2.0, otherwise, the benefit of lower bending stress will disappear.

  4. (4)

    For the studied gears with hfp/mn = 1.4, the calculated bending stress reduced when pressure angle changing from 17.5° to 20° (1 < εα < 2). And the bending stress increases when pressure angle changes from 20° to 22.5° and 25° (1 < εα < 2).