In the analysis of gear systems, finite element methods and lumped mass approaches are commonly employed. However, when dealing with cracked gears, obtaining accurate stiffness parameters for lumped mass models remains challenging. Traditional methods often simplify tooth profiles into trapezoidal or rectangular shapes, introducing errors. This study introduces an extended Weber energy method to directly compute the meshing stiffness of irregular teeth with root cracks, avoiding simplification-induced inaccuracies. We focus on a gear shaft as the research object, combining finite element analysis of the shaft deformation with Weber-based stiffness calculations for cracked teeth. The results demonstrate the feasibility of this approach in providing reliable stiffness parameters for dynamic analyses of defective gear systems.
The gear shaft, an integral component in many mechanical transmissions, requires precise stiffness evaluation to ensure operational reliability. Cracks in gear teeth significantly alter stiffness characteristics, leading to dynamic imbalances and potential failures. Existing methods for cracked gear stiffness often rely on geometric simplifications, which may not capture true behavior. Our work integrates shaft deformation effects with tooth-level crack modeling, offering a comprehensive solution for gear shaft analysis.
Finite Element Analysis of the Gear Shaft
The gear shaft is decomposed into shaft and tooth segments for analysis. The shaft portion is treated as a variable-diameter beam, discretized into finite elements to compute deformation under operational loads. Each shaft element’s motion equation is expressed as:
$$F_e = k_e \cdot \delta_e + m_e \cdot \ddot{\delta_e}$$
where $F_e$ represents the nodal force vector, $k_e$ is the stiffness matrix, $m_e$ is the mass matrix, and $\delta_e$ is the displacement vector. For a spatially rotating gear shaft, the simplified force vector accounting for gravitational, centrifugal, and torque effects is:
$$F_e = \left[ \frac{m \cdot 9.8 – m \cdot \omega^2 \cdot e \cdot \cos\phi}{2}; \frac{-m \cdot \omega^2 \cdot e \cdot \sin\phi}{2}; T; 0; 0; 0; \frac{-W_j \cos 20^\circ}{3}; \frac{-W_j \sin 20^\circ}{3}; 0; \frac{-W_j \cos 20^\circ}{3}; \frac{-W_j \sin 20^\circ}{3}; 0; \frac{-W_j \cos 20^\circ}{3}; \frac{-W_j \sin 20^\circ}{3}; 0; 0; 0; 0; \frac{m \cdot 9.8 – m \cdot \omega^2 \cdot e \cdot \cos\phi}{2}; \frac{-m \cdot \omega^2 \cdot e \cdot \sin\phi}{2}; 0 \right]$$
The stiffness and mass matrices are derived based on beam theory, considering shear and torsional effects. The reduced stiffness matrix for a shaft element with rigid supports is:
$$k_e = \begin{bmatrix}
\frac{12EI_z}{l^3(1+\phi_y)} & 0 & 0 & -\frac{12EI_z}{l^3(1+\phi_y)} & 0 & 0 \\
0 & \frac{12EI_y}{l^3(1+\phi_z)} & 0 & 0 & -\frac{12EI_y}{l^3(1+\phi_z)} & 0 \\
0 & 0 & \frac{GJ_k}{l} & 0 & 0 & -\frac{GJ_k}{l} \\
-\frac{12EI_z}{l^3(1+\phi_y)} & 0 & 0 & \frac{12EI_z}{l^3(1+\phi_y)} & 0 & 0 \\
0 & -\frac{12EI_y}{l^3(1+\phi_z)} & 0 & 0 & \frac{12EI_y}{l^3(1+\phi_z)} & 0 \\
0 & 0 & -\frac{GJ_k}{l} & 0 & 0 & \frac{GJ_k}{l}
\end{bmatrix}$$
where $\phi_y = \frac{12EI_z}{GA_yl^2}$ and $\phi_z = \frac{12EI_y}{GA_zl^2}$ are shear influence coefficients. The consistent mass matrix is:
$$m_e = \rho A l \begin{bmatrix}
\frac{13}{35} & 0 & 0 & \frac{9}{70} & 0 & 0 \\
0 & \frac{13}{35} & 0 & 0 & \frac{9}{70} & 0 \\
0 & 0 & \frac{J}{3A} & 0 & 0 & \frac{J}{6A} \\
\frac{9}{70} & 0 & 0 & \frac{13}{35} & 0 & 0 \\
0 & \frac{9}{70} & 0 & 0 & \frac{13}{35} & 0 \\
0 & 0 & \frac{J}{6A} & 0 & 0 & \frac{J}{3A}
\end{bmatrix}$$
Using MATLAB, the shaft is divided into six segments and seven nodes. Initial displacements are computed under static loads, and transient analysis captures deformation during rotation. The shaft deformation component $q”_{12}$ is derived as:
$$q”_{12} = \sqrt{v_i^2 + w_i^2}$$
where $v_i$ and $w_i$ are nodal displacements. This deformation is incorporated into the total tooth deformation for stiffness calculation.
Weber Energy Method for Cracked Tooth Stiffness
The Weber energy method models the tooth as a non-uniform cantilever beam divided into infinitesimal rectangular elements. For a cracked gear shaft tooth, the crack is assumed to initiate at point G between the root and base circles, with depth $g$ and angle $\theta$. The affected region is excluded from load-bearing calculations. The total deformation at contact point $j$ under load $W_j$ comprises tooth body deformation $q_{bj}$, contact deformation $q_{cj}$, and fillet foundation deformation $q_{fj}$.
For a cracked tooth, the tooth body deformation $q_{bij}$ for element $i$ is:
$$q_{bij} = \frac{W_j}{E_e} \left\{ \cos^2 \beta_j \left[ \frac{L_i^3 + 3L_i^2 S_{ij} + 3L_i S_{ij}^2}{3I_i} \right] – \cos \beta_j \sin \beta_j \left[ \frac{L_i Y_j + 2L_i Y_j S_{ij}}{2I_i} \right] + \cos^2 \beta_j \left[ \frac{12(1+\nu)L_i}{5A_i} \right] + \sin^2 \beta_j \left( \frac{L_i}{A_i} \right) \right\}$$
Here, $E_e$ is the effective elastic modulus, $\beta_j$ is the load angle, $S_{ij}$ is the distance between elements $i$ and $j$ along the tooth profile, $Y_j$ is the half-thickness at $j$, and $L_i$, $A_i$, $I_i$ are the width, cross-sectional area, and moment of inertia of element $i$, respectively. For cracked teeth, $A_i$ and $I_i$ are modified based on the crack geometry.
The crack line equation, given crack tip coordinates $(x_2, y_2)$ and load point $(x_1, y_1)$, is:
$$y” = \frac{y_1 – y_2}{x_1 – x_2} (x” – x_1) + y_1$$
The tooth profile in the rotated coordinate system is derived from the involute equation:
$$\begin{cases}
x = r_b (\cos \varphi + \varphi \sin \varphi) \\
y = r_b (\sin \varphi – \varphi \cos \varphi)
\end{cases}$$
Transformed to the local coordinate system $(x’, y’)$:
$$\begin{cases}
x’ = \frac{\sqrt{x^2 + y^2}}{\sqrt{\tan^2(\beta – \gamma) + 1}} \\
y’ = \pm \tan(\beta – \gamma) \frac{\sqrt{x^2 + y^2}}{\sqrt{\tan^2(\beta – \gamma) + 1}}
\end{cases}$$
where $\beta$ is the rotation angle and $\gamma = \arctan(y/x)$. The modified area $A_i$ and inertia $I_i$ for each element are calculated based on the intact and cracked regions.
The contact deformation $q_{cj}$ is evaluated using:
$$q_{cj} = \frac{1.275}{E_e^{0.912} B^{0.8} W_j^{0.1}}$$
where $E_e$ is the combined modulus of the mating gears. The fillet foundation deformation $q_{fj}$ depends on the tooth width-to-thickness ratio. For narrow teeth ($B/H_p < 5$):
$$q_{fj} = \frac{W_j \cos^2 \beta_j}{BE} \left[ 5.306 \left( \frac{L_f}{H_f} \right)^2 + 2(1-\nu) \left( \frac{L_f}{H_f} \right) + \frac{1.534(1 + 0.4167 \tan^2 \beta_j)}{1+\nu} \right]$$
For wide teeth ($B/H_p > 5$):
$$q_{fj} = \frac{W_j \cos^2 \beta_j}{BE} \left[ \frac{5.306 (L_f/H_f)^2}{1-\nu^2} + \frac{2(1-\nu-2\nu^2)(L_f/H_f)}{1+\nu} + \frac{1.534(1 + 0.4167 \tan^2 \beta_j)}{1+\nu} \right]$$
with $L_f = x_j – x_M – y_j \tan \beta_j$ and $H_f = 2y_M$. The total deformation for a gear pair is:
$$q_{12} = q_{bj1} + q_{bj2} + q_{fj1} + q_{fj2} + q_{cj} + q”_{12}$$
and the meshing stiffness $K$ is:
$$K = \frac{W_j}{q_{12}}$$
Case Study: Stiffness Calculation for a Cracked Gear Shaft
A practical example involves a gear shaft transmitting 4 kW power at 247 rpm. The gear parameters are:
| Parameter | Pinion | Gear |
|---|---|---|
| Module (mm) | 4 | 4 |
| Teeth | 73 | 25 |
| Face Width (mm) | 78 | 84 |
| Material | 45 Steel | ZG310-570 |
A root crack of depth 5 mm and angle 10° is introduced in the pinion. The shaft finite element analysis yields initial displacements, and transient analysis provides deformation data. The following table summarizes computed deformations at key engagement points:
| Deformation Component | Value (mm) |
|---|---|
| Tooth Body Deformation (Pinion) | 3.2e-3 |
| Tooth Body Deformation (Gear) | 2.8e-3 |
| Contact Deformation | 1.1e-3 |
| Fillet Foundation Deformation | 0.9e-3 |
| Shaft Deformation | 0.6e-3 |
| Total Deformation | 8.6e-3 |
The meshing stiffness $K$ is calculated over the engagement phase, as shown in the table below:
| Profile Angle (rad) | Meshing Stiffness (kN/µm) |
|---|---|
| 0.30 | 27.2 |
| 0.35 | 27.8 |
| 0.40 | 28.4 |
| 0.45 | 28.6 |
| 0.50 | 28.2 |
| 0.55 | 27.6 |
Comparatively, the stiffness of an intact gear shaft tooth is higher, peaking at approximately 39 kN/µm. The crack reduces overall stiffness and shifts the peak position, highlighting the sensitivity of gear shaft dynamics to defects.
Conclusion
The extended Weber energy method effectively computes the meshing stiffness of cracked gear shafts by integrating shaft deformation analysis with tooth-level crack modeling. This approach eliminates the need for geometric simplifications, providing accurate stiffness parameters for lumped mass analyses. The case study confirms that cracks reduce stiffness and alter engagement characteristics, emphasizing the importance of precise stiffness evaluation in gear shaft design. Future work could extend this method to helical gears or complex crack patterns, further enhancing its applicability in mechanical system dynamics.