In the field of mechanical engineering, the accurate calculation of bending strength for spur gears is fundamental to ensuring the reliability and durability of transmission systems. The basic approach involves simplifying the gear tooth as a cantilever beam and incorporating form factor and stress correction factor to account for stress concentration at the tooth root fillet. The stress correction factor is particularly critical, as it mitigates errors arising from stress concentration effects during bending stress computation. Traditional methods for determining this factor rely on experimental measurements of bending stress, such as finite element analysis, photoelastic experiments, and strain gauge measurements. However, these methods face limitations, especially for gears with small modules (less than 6 mm), where space constraints in the non-meshing region hinder direct strain measurement at the critical 30° tangent section. This study introduces a novel approach using Fiber Bragg Grating (FBG) sensors, which leverage their small cross-sectional diameter to measure bending stress in spur gears with high precision. We aim to optimize the stress correction factor for spur gear bending stress analytical calculations based on FBG measurements, thereby enhancing the reliability of gear strength design. Throughout this work, we focus on spur and pinion gears, as they are ubiquitous in power transmission, and the optimization of their stress correction factors can significantly impact overall system performance.
The determination of the stress correction factor typically involves comparing experimentally obtained bending stresses with analytically computed basic bending stress values. The accuracy of the bending stress experiment directly influences the precision of the stress correction factor, which in turn affects the fidelity of gear bending strength calculations. Conventional experimental techniques include finite element method (FEM) simulations, photoelastic experiments, and resistive strain gauge measurements. FEM is widely adopted by international standards organizations for deriving stress correction factors through extensive numerical simulations. Photoelastic methods utilize transparent materials to visualize stress distributions via polarized light interference, as employed by the American Gear Manufacturers Association. Strain gauge measurements involve attaching gauges to the tooth root; however, for gears with modules under 6 mm, practical challenges arise due to limited space in the non-meshing zone, preventing direct measurement at the hazardous section. Consequently, no prior experimental measurements exist for bending stress at the critical 30° tangent section in small-module gears under actual operating conditions. FBG sensors, with their miniature diameter (typically around 125 μm), offer a promising solution for accessing confined spaces and providing precise strain measurements. This paper presents the first step in utilizing FBG for spur gear bending stress measurement and optimizing the stress correction factor, laying groundwork for future online health monitoring of spur and pinion gears in transmission systems.
The fundamental principle of FBG strain sensing revolves around the shift in Bragg wavelength due to axial deformation. Under constant temperature and pressure, the relationship is given by:
$$\Delta \lambda_B = k_\varepsilon \varepsilon$$
where \(\Delta \lambda_B\) is the change in Bragg wavelength, \(k_\varepsilon\) is the strain sensitivity coefficient of the FBG (typically around 1.2 pm/με for silica fibers), and \(\varepsilon\) is the axial strain along the FBG. To relate this to stress, we invoke the elastic physical equations. For an isotropic material, the strain along the FBG axis (assumed as the y-direction in our coordinate system) is expressed as:
$$\varepsilon_y = \frac{1}{E} [\sigma_y – \mu (\sigma_x + \sigma_z)]$$
Here, \(E\) is the Young’s modulus, \(\mu\) is Poisson’s ratio, and \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\) are normal stresses along three mutually perpendicular directions (x: bending stress direction tangential to tooth root, y: along tooth width, z: through thickness). Combining the above equations yields:
$$\Delta \lambda_B = k_\varepsilon \frac{1}{E} [\sigma_y – \mu (\sigma_x + \sigma_z)]$$
For gear teeth under bending, the through-thickness stress \(\sigma_z\) is negligible (\(\sigma_z \approx 0\)). Thus, measuring \(\sigma_x\) directly via FBG requires understanding the relationship between \(\sigma_y\) and \(\sigma_x\). Due to geometric constraints, FBGs can only be aligned along the tooth width direction (y-axis) in the non-meshing region, as shown in Figure 1 (refer to conceptual diagram). Therefore, we measure \(\varepsilon_y\) and must establish a correlation with \(\sigma_x\). Based on elasticity theory for plane stress problems, we hypothesize that at any point on the tooth root surface, the stress ratio \(\sigma_y / \sigma_x\) depends on the distance from that point to the gear end-face and follows an exponential decay along the tooth width. Specifically, as tooth width \(w\) approaches zero, \(\sigma_y / \sigma_x \to 0\), and as \(w \to \infty\), \(\sigma_y / \sigma_x \to \mu\). This leads to the proposed relationship:
$$\frac{\sigma_y}{\sigma_x} = \mu (1 – a^w)$$
where \(w\) is the tooth width (for a point at distance \(d\) from end-face, \(w = 2d\) for symmetric consideration), and \(a\) is a constant related to material elasticity (\(0 < a < 1\)). To validate this, we constructed a 3D finite element model of a spur gear with 50 mm face width and computed \(\sigma_y / \sigma_x\) at various points from the end-face to the center. The results, plotted in Figure 2, confirm an exponential trend. Substituting into the strain equation gives:
$$\sigma_x = -\frac{E}{a^w} \varepsilon_y$$
The negative sign indicates compressive strain in the y-direction under tensile bending stress, but magnitude is considered. Thus, by measuring \(\varepsilon_y\) with FBG, we can deduce the bending stress \(\sigma_x\). This forms the basis for our experimental methodology.
We conducted experiments on standard involute spur gears with parameters listed in Table 1. The gear pair represents common spur and pinion configurations in industrial applications.
| Parameter | Value |
|---|---|
| Module, \(m\) (mm) | 4 |
| Pressure angle, \(\alpha\) (degrees) | 20 |
| Number of teeth, \(z\) | 31 |
| Face width, \(b\) (mm) | 30 |
| Tool fillet radius, \(r_f\) (mm) | 0.38\(m\) (1.52 mm) |
| Material | Steel (E = 200 GPa, μ = 0.3) |
The loading condition assumes no load distribution irregularities or dynamic effects. The analytical bending stress according to GB/T 3480-1997 is calculated as:
$$\sigma_{F0} = \frac{F_t}{b m} Y_F Y_S$$
where \(F_t\) is the equivalent tangential force at the pitch circle, \(Y_F\) is the form factor, and \(Y_S\) is the stress correction factor. For our gear, \(Y_F = 1.3981\) and \(Y_S = 1.9387\) as per standard formulas. The tangential force relates to applied torque \(T\) via \(F_t = 2000T / d\), with \(d\) as pitch diameter. We computed bending stresses for torques from 100 to 600 N·m in increments of 100 N·m, as summarized in Table 2.
| Torque, \(T\) (N·m) | Analytical Bending Stress, \(\sigma_{F0}\) (MPa) |
|---|---|
| 100 | 36.43 |
| 200 | 72.86 |
| 300 | 109.29 |
| 400 | 145.72 |
| 500 | 182.15 |
| 600 | 218.58 |
For FBG measurements, we selected a gear with 30 mm face width, as beyond this width, the strain \(\varepsilon_y\) at the center becomes negligible (under 0.001). The FBG sensors were bonded along the tooth width direction at the 30° tangent point on the fillet curve, with the sensitive region centered 10 mm from one end-face (optimal per our hypothesis). Three adjacent teeth were instrumented to average results. The gear was statically loaded via a hydraulic press, with force calibrated to torque using \(F_t’ = 2000T / d_b\), where \(d_b\) is base diameter. Initial state served as zero strain reference, and wavelength shifts were converted to strain changes using \(k_\varepsilon = 1.2\) pm/με. The measured strains \(\varepsilon_y\) were then used to calculate bending stress \(\sigma_x\) via the derived formula, with constant \(a\) determined from FEM fitting as \(a = 0.85\) for this gear geometry. Results are in Table 3.
| Torque, \(T\) (N·m) | FBG Measured Strain, \(\varepsilon_y\) (με) | Calculated Bending Stress, \(\sigma_x\) (MPa) |
|---|---|---|
| 100 | -42.5 | 30.291 |
| 200 | -85.0 | 60.582 |
| 300 | -127.5 | 90.873 |
| 400 | -162.9 | 116.115 |
| 500 | -205.4 | 146.406 |
| 600 | -255.0 | 181.746 |
Plotting bending stress against torque (Figure 3) reveals a linear relationship, validating the measurement consistency. Comparing with analytical values, the FBG-measured stresses are consistently lower, indicating potential overestimation by the standard stress correction factor.
To optimize \(Y_S\), we first derived experimental stress correction factors by dividing measured bending stress by the basic bending stress (\(\sigma_{\text{basic}} = F_t Y_F / (b m)\)). Results are in Table 4.
| Torque, \(T\) (N·m) | Basic Bending Stress (MPa) | Experimental \(Y_S\) |
|---|---|---|
| 100 | 18.791 | 1.612 |
| 200 | 37.583 | 1.612 |
| 300 | 56.374 | 1.612 |
| 400 | 75.166 | 1.545 |
| 500 | 93.957 | 1.558 |
| 600 | 112.748 | 1.612 |
The average experimental \(Y_S\) is 1.592, which is 21.8% lower than the standard value of 1.9387. This suggests that the standard formula overestimates stress concentration for this spur and pinion gear configuration. To generalize, we investigated the influence of tool fillet radius \(r_f\) on \(Y_S\). Analytical calculations for \(r_f\) from 0.25\(m\) to 0.38\(m\) show that bending stress decreases with increasing \(r_f\) (Figure 4), primarily due to reduction in \(Y_S\) (Figure 5), while \(Y_F\) increases slightly. Using FEM, we computed stress correction factors for three \(r_f\) values (0.25\(m\), 0.30\(m\), 0.38\(m\)) by dividing FEM stress by Lewis formula stress. Comparing FEM-derived \(Y_S\) with analytical \(Y_S\) (Figure 6) reveals a linear correlation:
$$Y_{S,\text{FEM}} = c \cdot Y_{S,\text{analytical}}$$
where \(c\) is a slope constant. For \(r_f = 0.38m\), we compute the optimization constant as:
$$k_{\text{opt}} = \frac{Y_{S,\text{experimental}}}{Y_{S,\text{analytical}}} = \frac{1.592}{1.9387} = 0.821$$
Thus, the optimized stress correction factor formula becomes:
$$Y_S = 0.821 \left(1.2 + 0.13L\right) q_s^{1/(1.21 + 2.3/L)}$$
where \(L = s_{Fn} / h_{Fe}\) (tooth thickness to bending moment arm ratio) and \(q_s = s_{Fn} / (2\rho_F)\) (fillet parameter). This optimized formula applies to spur gears with pressure angle 20° and tool fillet radii around 0.38\(m\). For other \(r_f\) values, the constant may vary linearly, but further experiments are needed for validation.
Our findings have significant implications for spur and pinion gear design. First, the stress distribution along tooth width follows an exponential decay, with measurable strain occurring within 15 mm from the end-face. For FBG placement, the optimal location is 5–15 mm from the end-face, centered at 10 mm, to avoid stress relief near ends. Second, the standard stress correction factor is conservative, overestimating bending stress by approximately 21.8% for gears with \(r_f = 0.38m\). This overestimation may lead to oversized gear designs, reducing efficiency and increasing cost. By integrating FBG measurements, we can refine analytical models for more accurate strength predictions. Future work should extend this methodology to helical and bevel gears, and explore dynamic loading conditions for real-time monitoring of spur and pinion systems in operation.
In conclusion, this study demonstrates the efficacy of FBG sensors in measuring bending stress in small-module spur gears, overcoming limitations of traditional methods. The optimized stress correction factor, derived from experimental data, enhances the accuracy of gear bending strength calculations, contributing to more reliable and economical gear design. As spur and pinion gears remain fundamental components in countless mechanical transmissions, such advancements hold promise for improved performance and longevity across industries.