In mechanical transmission systems, spur gears are widely used due to their simplicity and efficiency. The mesh stiffness of spur gears is a critical parameter that influences dynamic behavior, vibration, and noise. Time-varying mesh stiffness (TVMS) arises from the alternating single and double tooth contact during gear operation. Various factors, such as tooth profile modifications, surface friction, and geometric errors, significantly affect TVMS. This study focuses on developing a comprehensive analytical model to evaluate the mesh stiffness of profile-shifted spur gears, considering nonlinear Hertzian contact stiffness, actual tooth profile curves, structural coupling effects, tooth surface friction, and geometric eccentricity. We aim to provide insights into how these factors individually and collectively alter the mesh stiffness characteristics, which is essential for accurate dynamic modeling of gear systems.
The energy method is employed for its computational efficiency compared to experimental and finite element approaches. Previous research has established foundations for calculating mesh stiffness using potential energy principles, but gaps remain in integrating multiple real-world factors. Our model incorporates revised formulas for tooth geometry due to profile shifting, frictional effects on stiffness components, and eccentricity-induced variations in pressure angle and center distance. By analyzing these aspects, we seek to enhance the understanding of spur gear dynamics under practical conditions.
Spur gears are fundamental components in many industrial applications, and their performance is heavily dependent on mesh stiffness. Profile shifting, often used to avoid undercutting or to adjust center distance, alters tooth thickness and root geometry, thereby affecting stiffness. Additionally, tooth surface friction, which changes direction at the pitch point, introduces asymmetries in stiffness calculations. Geometric eccentricity, resulting from manufacturing or assembly errors, causes periodic variations in center distance and pressure angle. Our work builds on existing energy-based models by incorporating these elements into a unified framework. We validate our approach against published data and explore parametric studies to quantify the effects of varying profile shift coefficients, friction coefficients, and eccentricity magnitudes on TVMS.
The structure of this article is as follows: First, we derive the single-tooth stiffness model including Hertzian, bending, axial, shear, and fillet foundation stiffnesses, with modifications for profile shifting and friction. Next, we present the gear pair mesh stiffness formulation for single and double tooth contact regions, accounting for structural coupling. Then, we develop the pressure angle model under geometric eccentricity. Subsequently, we analyze the impact of profile shift parameters on mesh stiffness for both angle-shifted and high-shifted spur gear pairs. Following that, we examine the individual and combined effects of tooth surface friction and geometric eccentricity. Finally, we discuss the implications of our findings for spur gear design and dynamics.
Single-Tooth Stiffness Calculation Model Considering Tooth Surface Friction
The energy method calculates mesh stiffness by considering the potential energy stored in a gear tooth under load. For a single tooth, the total potential energy comprises Hertzian energy, bending energy, axial compression energy, and shear energy, corresponding to Hertzian stiffness $k_H$, bending stiffness $k_b$, axial compression stiffness $k_a$, and shear stiffness $k_s$, respectively. The nonlinear Hertzian contact stiffness for spur gears is given by:
$$ \frac{1}{k_H} = \frac{1.275}{E^{*0.9} B^{0.8} F_m^{0.1}} $$
where $E^* = \frac{E_1 E_2}{E_1 + E_2}$ is the equivalent elastic modulus, $B$ is the face width, and $F_m$ is the meshing force at the contact point. $E_1$ and $E_2$ are the elastic moduli of the two spur gears.
When tooth surface friction is considered, the frictional force $\mu F$ acts perpendicular to the line of action, changing direction at the pitch point. For the driving spur gear, the forces are decomposed into radial and tangential components. The radial force $F_r$ and tangential force $F_t$ are expressed as:
$$ F_r = \begin{cases}
F \sin \beta + \mu F \cos \beta & \beta < \theta_p \\
F \sin \beta & \beta = \theta_p \\
F \sin \beta – \mu F \cos \beta & \beta > \theta_p
\end{cases} $$
$$ F_t = \begin{cases}
F \cos \beta – \mu F \sin \beta & \beta < \theta_p \\
F \cos \beta & \beta = \theta_p \\
F \cos \beta + \mu F \sin \beta & \beta > \theta_p
\end{cases} $$
where $\beta$ is the angle between the force direction and the x-axis at the meshing point, and $\theta_p$ is the corresponding angle at the pitch point. The bending stiffness $k_b$, axial compression stiffness $k_a$, and shear stiffness $k_s$ are modified to account for friction and the actual tooth profile including the involute and fillet curve. The revised formulas are:
$$ \frac{1}{k_b} = \int_{\alpha}^{\frac{\pi}{2}} \frac{ \left[ (\cos \beta \mp \mu \sin \beta)(y_\beta – y_1) – x_\beta (\sin \beta \pm \mu \cos \beta) \right]^2 }{E I_{y_1}} \frac{dy_1}{d\gamma} d\gamma + \int_{\beta}^{\tau_c} \frac{ \left[ (\cos \beta \mp \mu \sin \beta)(y_\beta – y_2) – x_\beta (\sin \beta \pm \mu \cos \beta) \right]^2 }{E I_{y_2}} \frac{dy_2}{d\tau} d\tau \quad \beta \lessgtr \theta_p $$
$$ \frac{1}{k_a} = \int_{\alpha}^{\frac{\pi}{2}} \frac{ (\sin \beta \pm \mu \cos \beta)^2 }{E A_{y_1}} \frac{dy_1}{d\gamma} d\gamma + \int_{\beta}^{\tau_c} \frac{ (\sin \beta \pm \mu \cos \beta)^2 }{E A_{y_2}} \frac{dy_2}{d\tau} d\tau \quad \beta \lessgtr \theta_p $$
$$ \frac{1}{k_s} = \int_{\alpha}^{\frac{\pi}{2}} \frac{ 1.2 (\cos \beta \mp \mu \sin \beta)^2 }{G A_{y_1}} \frac{dy_1}{d\gamma} d\gamma + \int_{\beta}^{\tau_c} \frac{ 1.2 (\cos \beta \mp \mu \sin \beta)^2 }{G A_{y_2}} \frac{dy_2}{d\tau} d\tau \quad \beta \lessgtr \theta_p $$
The fillet foundation stiffness $k_f$ and the coupling-induced stiffness $k_{f21}$ and $k_{f12}$ due to adjacent teeth are also revised to include frictional effects:
$$ \frac{1}{k_f} = \frac{ (\cos \beta_1 \mp \mu \sin \beta_1) }{B E \cos \beta_1} \left[ L \left( \frac{u}{s} \right)^2 + M \frac{u}{s} + P (1 + Q \tan^2 \beta_1) \right] $$
$$ \frac{1}{k_{f21}} = \frac{ (\cos \beta_1 \mp \mu \sin \beta_1) }{B E \cos \beta_2} \left[ L_1 \left( \frac{u_1 u_2}{s^2} \right)^2 + (\tan \beta_2 M_1 + P_1) \frac{u_1}{s} + (\tan \beta_1 Q_1 + R_1) \frac{u_2}{s} + (\tan \beta_1 S_1 + T_1) \tan \beta_2 + U_1 \tan \beta_1 + V \right] $$
$$ \frac{1}{k_{f12}} = \frac{ (\cos \beta_2 \mp \mu \sin \beta_2) }{B E \cos \beta_1} \left[ L_2 \left( \frac{u_1 u_2}{s^2} \right)^2 + (\tan \beta_1 M_2 + P_2) \frac{u_2}{s} + (\tan \beta_2 Q_2 + R_2) \frac{u_1}{s} + (\tan \beta_2 S_2 + T_2) \tan \beta_1 + U_2 \tan \beta_2 + V \right] $$
For profile-shifted spur gears, the tooth geometry changes. The distance from the rack cutter tip circle center to the midline is revised as:
$$ a_1 = (h_a^* + c^*) m – r_\rho – x m $$
where $r_\rho = c^* m / (1 – \sin \alpha_0)$ is the tip radius, $x$ is the profile shift coefficient, $m$ is the module, $\alpha_0$ is the standard pressure angle, $h_a^*$ is the addendum coefficient (typically 1), and $c^*$ is the clearance coefficient (typically 0.25). The radius of the start of active profile circle is:
$$ R_c = \sqrt{ \left[ r_b \tan \alpha_0 – \frac{(h_a^* – x) m}{\sin \alpha_0} \right]^2 + r_b^2 } $$
The half-tooth angle on the base circle $\theta_b$ and the dedendum circle $\theta_f$ are modified as:
$$ \theta_b = \frac{\pi}{2Z} + \tan \alpha_0 – \alpha_0 + \frac{2x \tan \alpha_0}{Z} $$
$$ \theta_f = \frac{1}{Z} \left[ \frac{\pi}{2} + 2(h_a^* – r_c^*) \tan \alpha_0 + \frac{2 r_c^*}{\cos \alpha_0} \right] + \frac{2x \tan \alpha_0}{Z} $$
where $Z$ is the number of teeth, and $r_c^* = r_c / m$ with $r_c$ being the rack fillet radius (typically 0.38m). These modifications ensure accurate representation of profile-shifted spur gears in stiffness calculations.
Gear Pair Mesh Stiffness Formulation
The mesh stiffness of a spur gear pair varies with the meshing position. For spur gears with a contact ratio between 1 and 2, the mesh stiffness alternates between single and double tooth contact regions. The total mesh stiffness $k_m$ for a pair is the series combination of tooth stiffnesses, fillet foundation stiffnesses, and Hertzian contact stiffness. For a single tooth pair, the stiffness $k_t$ is:
$$ k_t = \frac{1}{ \frac{1}{k_H} + \frac{1}{k_{b,p}} + \frac{1}{k_{s,p}} + \frac{1}{k_{a,p}} + \frac{1}{k_{b,g}} + \frac{1}{k_{s,g}} + \frac{1}{k_{a,g}} + \frac{1}{k_{f,p}} + \frac{1}{k_{f,g}} } $$
where subscripts $p$ and $g$ denote the pinion (driving spur gear) and gear (driven spur gear), respectively. In the single tooth contact region, $k_m = k_t$. In the double tooth contact region, considering structural coupling, the mesh stiffness becomes:
$$ k_m = \frac{1}{ \frac{1}{k_{t,1} + \frac{F_2}{F_1} \left( \frac{1}{k_{f12,p}} + \frac{1}{k_{f12,g}} \right) } + \frac{1}{ \frac{1}{k_{t,2} + \frac{F_1}{F_2} \left( \frac{1}{k_{f21,p}} + \frac{1}{k_{f21,g}} \right) } } $$
where $F_1$ and $F_2$ are the meshing forces on the first and second tooth pairs, and $k_{t,1}$ and $k_{t,2}$ are their respective tooth stiffnesses. Profile-shifted spur gears are categorized into angle-shifted (non-zero sum of shift coefficients) and high-shifted (zero sum) pairs. The geometric parameters such as center distance, working pressure angle, and tooth dimensions are revised accordingly.
To illustrate the effect of profile shifting, we vary the shift coefficient from -0.6 to 0.6 in seven steps for a spur gear with parameters from Table 1. The tooth profile and half-tooth thickness changes are summarized in Figure 2, showing how positive shift increases tooth thickness and negative shift decreases it.
| Parameter | Driving Gear | Driven Gear |
|---|---|---|
| Number of Teeth, Z | 55 | 75 |
| Module, m (mm) | 2 | 2 |
| Face Width, B (mm) | 20 | 20 |
| Bore Radius | 0.4 R_bp | 0.4 R_bg |
| Pressure Angle (°) | 20 | 20 |
| Poisson’s Ratio | 0.3 | |
| Elastic Modulus (GPa) | 206 | |
| Input Torque (N·m) | 50 | |
Pressure Angle Model Under Geometric Eccentricity
The meshing point pressure angle is crucial for TVMS calculation. For ideal spur gears, the pressure angle varies with rotation, and the single and double tooth contact boundaries are determined by gear geometry. The pressure angles at the start and end of meshing are given by:
$$ \cos \alpha_a = \frac{r_{bp}}{r_{ap}} $$
$$ \tan \alpha_b(t) = \tan \alpha_d(t) + \frac{2\pi}{Z_1} $$
$$ \tan \alpha_c = \tan \alpha_a – \frac{2\pi}{Z_1} $$
$$ \tan \alpha_d(t) = \frac{ \sqrt{ L^2(t) – (r_{bp} + r_{bg})^2 } – \sqrt{ r_{ag}^2 – r_{bg}^2 } }{ r_{bp} } $$
The driven gear pressure angle $\alpha_g(t)$ relates to the driving gear pressure angle $\alpha_p(t)$ and instantaneous center distance $L(t)$:
$$ \tan \alpha_g(t) = \frac{L(t)}{r_{bg} \sin \alpha(t)} – \frac{r_{bp} + r_{bg}}{r_{bg} \tan \alpha(t)} – \frac{r_{bp} \tan \alpha_p(t)}{r_{bg}} $$
$$ \alpha(t) = \arccos \left( \frac{r_{bp} + r_{bg}}{L(t)} \right) $$
The pressure angle at any time can be updated based on the driving gear rotation:
$$ \tan \alpha_p(t + \Delta t) = \int_t^{t+\Delta t} \omega(T) dT + \tan \alpha_p(t) $$
When geometric eccentricity exists, the gear center oscillates, causing time-varying center distance. For spur gears with eccentricity, the center distance $L(t)$ becomes:
$$ L(t) = \sqrt{ \left( a_w – e_1 \cos(\omega_p t + \theta_p) – e_2 \cos(\omega_g t + \theta_g) \right)^2 + \left( e_2 \sin(\omega_g t + \theta_g) – e_1 \sin(\omega_p t + \theta_p) \right)^2 } $$
where $a_w$ is the initial center distance, $e_1$ and $e_2$ are the eccentricities of the driving and driven spur gears, $\omega_p$ and $\omega_g$ are their angular velocities, and $\theta_p$ and $\theta_g$ are initial eccentricity angles. This variation in $L(t)$ affects the pressure angle and mesh stiffness.
Model Validation
To validate our TVMS model, we use spur gear parameters from literature (Pair 2 in reference [6]), as listed in Table 2. We compute TVMS for different bore radii (15 mm, 25 mm, 35 mm) and compare with finite element results. The comparison shows close agreement, with a maximum error of 6.41%, confirming the accuracy of our approach.
| Parameter | Driving Gear | Driven Gear |
|---|---|---|
| Number of Teeth, Z | 50 | 50 |
| Module, m (mm) | 3 | 3 |
| Face Width, B (mm) | 20 | 20 |
| Bore Radius (mm) | 15, 25, 35 | 15, 25, 35 |
| Pressure Angle (°) | 20 | 20 |
| Poisson’s Ratio | 0.3 | |
| Elastic Modulus (GPa) | 206.8 | |
| Input Torque (N·m) | 50 | |
Influence of Profile Shift Parameters on Mesh Stiffness
We analyze the effect of profile shift on TVMS for angle-shifted and high-shifted spur gear pairs using parameters from Table 1. The mesh stiffness is computed over one meshing cycle with 20 points, starting from the single tooth contact point.
Angle-Shifted Spur Gear Pairs
For angle-shifted pairs, we vary the shift coefficient of the driving gear ($X_1$) from -1.0 to 1.0 while keeping the driven gear shift ($X_2$) zero, and vice versa. The resulting center distances and contact ratios are summarized in Tables 3 and 4.
| Case | A1 | B1 | C1 | D1 | E1 | F1 |
|---|---|---|---|---|---|---|
| X1 | -1.0 | -0.6 | -0.2 | 0.2 | 0.6 | 1.0 |
| X2 | 0 | 0 | 0 | 0 | 0 | 0 |
| Center Distance (mm) | 128.3 | 129.2 | 130.0 | 130.8 | 131.6 | 132.3 |
| Contact Ratio | 1.92 | 1.89 | 1.83 | 1.75 | 1.67 | 1.57 |
| Case | A2 | B2 | C2 | D2 | E2 | F2 |
|---|---|---|---|---|---|---|
| X1 | 0 | 0 | 0 | 0 | 0 | 0 |
| X2 | -1.0 | -0.6 | -0.2 | 0.2 | 0.6 | 1.0 |
| Center Distance (mm) | 128.3 | 129.2 | 130.0 | 130.8 | 131.6 | 132.3 |
| Contact Ratio | 1.90 | 1.87 | 1.82 | 1.76 | 1.69 | 1.61 |
As $X_1$ increases from -1 to 1, the center distance increases from 128.3 mm to 132.3 mm, and the contact ratio decreases from 1.92 to 1.57. The TVMS amplitude increases in both single and double tooth regions, and the proportion of double tooth contact decreases. The first harmonic amplitude of TVMS rises from $1.47 \times 10^7$ N/m to $5.98 \times 10^7$ N/m, while the second harmonic initially increases then decreases. Similar trends are observed when shifting the driven gear, with TVMS amplitude increasing and contact ratio decreasing.
High-Shifted Spur Gear Pairs
For high-shifted spur gears, the sum of shift coefficients is zero, so the center distance remains constant at 130.4 mm. The shift cases are listed in Table 5.
| Case | A3 | B3 | C3 | D3 | E3 | F3 |
|---|---|---|---|---|---|---|
| X1 | -1.0 | -0.6 | -0.2 | 0.2 | 0.6 | 1.0 |
| X2 | 1.0 | 0.6 | 0.2 | -0.2 | -0.6 | -1.0 |
| Center Distance (mm) | 130.4 | 130.4 | 130.4 | 130.4 | 130.4 | 130.4 |
| Contact Ratio | 1.70 | 1.77 | 1.80 | 1.78 | 1.73 | 1.63 |
When the absolute shift coefficients are large (e.g., cases A3 and F3), the TVMS amplitude is smaller than for smaller shifts (e.g., cases C3 and D3). The first harmonic amplitude is higher for larger shifts, while the second and third harmonics are lower. This indicates that profile shifting can optimize stiffness characteristics in spur gears.
Effects of Tooth Surface Friction and Geometric Eccentricity
Tooth Surface Friction in Profile-Shifted Spur Gears
With a friction coefficient of 0.1, we compute TVMS for the cases in Table 4. Friction causes stiffness to increase from the root to just before the pitch point and decrease from the pitch point to the tip. For positive shift coefficients ($X_2 > 0$), the first to fourth harmonic amplitudes are lower with friction, while for negative shifts ($X_2 < 0$), they are higher. This is because negative shifts position the friction direction change point in the double tooth region near the driving gear tip, increasing stiffness during meshing.
Profile shifting alters the position of the pitch point within the meshing interval. For angle-shifted spur gears, as the shift coefficient changes from -1 to 1, the pitch point moves from the double tooth region to the single tooth region and back to the double tooth region. In high-shifted pairs, the pitch point remains fixed, but its relative position shifts. This movement affects how friction influences TVMS.
For cases A2 ($X_2 = -1$) and F2 ($X_2 = 1$), we compute single tooth stiffness and TVMS with friction coefficients of 0, 0.05, and 0.1. In case A2 (pitch point near root), friction reduces TVMS in both single and double tooth regions as the coefficient increases. In case F2 (pitch point near tip), friction increases TVMS. Thus, the combined effect of profile shift and friction significantly alters spur gear mesh stiffness.
Geometric Eccentricity in Spur Gears
We analyze geometric eccentricity for non-shifted spur gears ($X_1 = X_2 = 0$) under four conditions: no eccentricity, driving gear eccentricity $e_1 = 0.1$ mm, driven gear eccentricity $e_2 = 0.1$ mm, and both gears eccentric $e_1 = e_2 = 0.1$ mm. Eccentricity increases the peak-to-peak variation of TVMS. When both spur gears have eccentricity, the fluctuation is most pronounced.
For profile-shifted cases A3 to F3, we compute TVMS peak-to-peak values under different eccentricities. Eccentricity consistently increases peak-to-peak values, with larger eccentricities causing greater increases, regardless of profile shift.
Frequency domain analysis reveals that eccentricity introduces sidebands around the meshing frequency and its harmonics. For single gear eccentricity, sidebands appear at intervals equal to the rotational frequency of the eccentric spur gear. When both spur gears are eccentric, additional sidebands emerge at intervals equal to the difference in rotational frequencies ($\Delta f_{pg}$). This is due to the modulated center distance affecting pressure angle and stiffness.
Combined Effects of Multiple Factors
We evaluate the combined impact of profile shift, friction, and eccentricity using case F3 ($X_1 = 1$, $X_2 = -1$), with a friction coefficient of 0.1 and eccentricities $e_1 = e_2 = 0.1$ mm. The TVMS shows a periodic fluctuation due to eccentricity, and the friction-induced stiffness change occurs at a specific rotation angle (0.05 rad). However, under combined shift and eccentricity, the entire meshing interval may occur after the pitch point, so friction direction does not change, simplifying the stiffness variation. This highlights the complex interactions in spur gears under practical conditions.
Conclusion
This study develops an advanced analytical model for calculating time-varying mesh stiffness of profile-shifted spur gears, incorporating tooth surface friction, geometric eccentricity, and actual tooth geometry. Key findings include:
- Profile shifting significantly alters mesh stiffness, center distance, and contact ratio. For angle-shifted spur gears, positive shifts increase stiffness and reduce contact ratio. For high-shifted pairs, larger shift magnitudes decrease stiffness amplitudes but increase first harmonic content.
- Tooth surface friction affects stiffness differently depending on the pitch point position relative to the meshing interval. Profile shifting can move the pitch point into double tooth regions, changing how friction influences stiffness.
- Geometric eccentricity increases TVMS peak-to-peak variations and introduces characteristic sidebands in the frequency domain. Combined eccentricities of both spur gears produce additional sidebands at the difference frequency.
- When multiple factors are combined, their interactions can lead to unique stiffness behaviors, such as eliminated friction direction changes or amplified fluctuations.
These insights provide a foundation for accurate dynamic modeling of spur gear systems, enabling better design and optimization for reduced vibration and noise. Future work could extend this model to helical gears or include thermal effects for broader applicability.