Helical gears are widely used in high-speed and heavy-duty applications due to their smooth transmission characteristics, high load capacity, and reduced noise levels. However, during the meshing process, helical gears are inevitably subjected to tooth surface friction, which significantly influences their dynamic behavior, including time-varying meshing stiffness (TVMS). Accurate calculation of TVMS is crucial for understanding gear dynamics, vibration, and fatigue life. Traditional methods often neglect friction, leading to inaccuracies in stiffness predictions. This paper presents a comprehensive approach to compute TVMS for helical gears by incorporating tooth surface friction using the slice method and energy method. We investigate the effects of both constant and time-varying friction on TVMS and analyze how parameters like surface roughness, torque, speed, and tooth width influence the stiffness under time-varying friction conditions.
The slice method involves dividing the helical gear into infinitesimal spur gear slices along the tooth width direction. Each slice is treated as a spur gear, and its stiffness components—Hertzian contact stiffness, bending stiffness, shear stiffness, and axial compression stiffness—are calculated using the energy method. The total TVMS is obtained by integrating these components across the tooth width. The energy method is based on the principle that the potential energy stored in the gear tooth under load relates to its stiffness. For a helical gear pair, the meshing stiffness varies with time due to the changing number of contact teeth and the sliding friction between tooth surfaces. The general formula for potential energy components is given by:
$$U_h = \frac{F^2}{2k_h}, \quad U_b = \frac{F^2}{2k_b}, \quad U_s = \frac{F^2}{2k_s}, \quad U_a = \frac{F^2}{2k_a}$$
where \( U_h \), \( U_b \), \( U_s \), and \( U_a \) represent the potential energies due to Hertzian contact, bending, shear, and axial compression, respectively, and \( F \) is the meshing force along the line of action. The corresponding stiffness components for a spur gear slice are derived as follows. The bending stiffness \( k_b \), shear stiffness \( k_s \), and axial compression stiffness \( k_a \) for each slice are expressed as:
$$k_b = \sum_{i=1}^N \Delta y \int_{-\alpha’_y}^{\alpha_2} \frac{3(1 + k_1)^2 (\alpha_2 – \alpha)^2}{EA^3} d\alpha$$
$$k_s = \sum_{i=1}^N \Delta y \int_{-\alpha’_y}^{\alpha_2} \frac{1.2(1 + \nu) k_2 \cos^2 \alpha’_y}{EA} d\alpha$$
$$k_a = \sum_{i=1}^N \Delta y \int_{-\alpha’_y}^{\alpha_2} \frac{k_2 \sin^2 \alpha’_y}{2EA} d\alpha$$
where \( k_1 = \cos \alpha_y [(\alpha_2 – \alpha’_y) \sin \alpha – \cos \alpha] \), \( k_2 = (\alpha_2 – \alpha’_y) \cos \alpha \), \( A = \sin \alpha + (\alpha_2 – \alpha’_y) \cos \alpha \), \( \alpha_2 = \pi / (2Z) + \text{inv}(\alpha_0) \) is half the base circle angle, \( \alpha_y \) is the angle between the meshing force and the gear centerline, \( \alpha’_y = \alpha_y + (\gamma – \alpha_y)(i/N) \) is the angle for each slice, \( \Delta y = B/N \) is the slice width, \( Z \) is the number of teeth, \( N \) is the number of slices, \( E \) is the elastic modulus, and \( \nu \) is Poisson’s ratio. The Hertzian contact stiffness \( k_h \) is constant along the line of action and given by:
$$k_h = \frac{\pi E B}{4(1 – \nu^2)}$$
The fillet foundation stiffness \( k_f \) accounts for the flexibility of the gear body and is calculated as:
$$k_f = \frac{E \Delta y}{L^* u_f^2 / s_f^2 + M^* u_f / s_f + P^* + P^* Q^* \tan^2 \alpha’_y} \cos^2 \alpha’_y$$
where \( L^*, M^*, P^*, Q^* \) are coefficients obtained from polynomial functions based on gear geometry, and \( u_f \) and \( s_f \) are geometric parameters. The single-tooth stiffness \( k_d \) for a pair of meshing teeth is then:
$$k_d = \frac{1}{\frac{1}{k_{f1}} + \frac{1}{k_{b1}} + \frac{1}{k_{s1}} + \frac{1}{k_{a1}} + \frac{1}{k_{f2}} + \frac{1}{k_{b2}} + \frac{1}{k_{s2}} + \frac{1}{k_{a2}} + \frac{1}{k_h}}$$
For multiple pairs of helical gears in contact, the comprehensive stiffness \( k_t \) is the sum of the stiffnesses of all engaged tooth pairs. If \( n \) pairs are engaged, the total stiffness is:
$$k_{1,i} = \frac{1}{\frac{1}{k_{f1,i}} + \frac{1}{k_{b1,i}} + \frac{1}{k_{s1,i}} + \frac{1}{k_{a1,i}}}, \quad k_{2,i} = \frac{1}{\frac{1}{k_{f2,i}} + \frac{1}{k_{b2,i}} + \frac{1}{k_{s2,i}} + \frac{1}{k_{a2,i}}}$$
$$k_t = \sum_{i=1}^n \frac{1}{k_{1,i} + k_{2,i} + \frac{1}{k_h}}$$
To incorporate tooth surface friction, we model the time-varying friction coefficient \( \mu \) using a thermomechanical model based on rough surface elastohydrodynamic lubrication. The friction coefficient is given by:
$$\mu = e^{f(SR_k, P_h K, v_0, R_a)} P_h K^{b_2} |SR_k|^{b_3} V_e K^{b_6} v_0^{b_7} R_k^{b_8}$$
where \( f = b_1 + b_4 |SR_k| P_h K \lg v_0 + b_5 e^S + b_9 e^{R_a} \) and \( S = -|SR_k| P_h K \lg v_0 \). Here, \( SR_k \) is the slide-to-roll ratio, \( P_h K \) is the contact stress, \( V_e K \) is the entrainment velocity, \( v_0 \) is the absolute viscosity, \( R_a \) is the surface roughness, and \( b_i \) (i=1 to 9) are regression coefficients. The friction force \( F_f \) is decomposed into components along the meshing direction:
$$F_f = \mu F, \quad F_{f1} = F_f \cos \alpha_y, \quad F_{f2} = F_f \sin \alpha_y$$
When the meshing point is below the pitch line (meshing-in phase), the friction force direction is away from the pitch line; above the pitch line (meshing-out phase), it is toward the pitch line; and at the pitch line, the friction force is zero due to reversal of sliding direction. The stiffness components with friction are modified as follows. For bending stiffness:
$$k_b = \begin{cases}
\sum_{i=1}^N \Delta y \int_{-\alpha’_y}^{\alpha_2} \frac{3\{1 + f(\alpha) – M_b\}^2 (\alpha_2 – \alpha)^2}{EA^3} d\alpha & \text{(meshing-in)} \\
\sum_{i=1}^N \Delta y \int_{-\alpha’_y}^{\alpha_2} \frac{3\{1 + f(\alpha) + M_b\}^2 (\alpha_2 – \alpha)^2}{EA^3} d\alpha & \text{(meshing-out)}
\end{cases}$$
For shear stiffness:
$$k_s = \begin{cases}
\sum_{i=1}^N \Delta y \int_{-\alpha’_y}^{\alpha_2} \frac{1.2(1 + \nu) k_2 (\cos \alpha’_y – M_s)^2}{EA} d\alpha & \text{(meshing-in)} \\
\sum_{i=1}^N \Delta y \int_{-\alpha’_y}^{\alpha_2} \frac{1.2(1 + \nu) k_2 (\cos \alpha’_y + M_s)^2}{EA} d\alpha & \text{(meshing-out)}
\end{cases}$$
For axial compression stiffness:
$$k_a = \begin{cases}
\sum_{i=1}^N \Delta y \int_{-\alpha’_y}^{\alpha_2} \frac{(\alpha_2 – \alpha) \cos \alpha (\sin \alpha’_y + M_a)^2}{2EA} d\alpha & \text{(meshing-in)} \\
\sum_{i=1}^N \Delta y \int_{-\alpha’_y}^{\alpha_2} \frac{(\alpha_2 – \alpha) \cos \alpha (\sin \alpha’_y – M_a)^2}{2EA} d\alpha & \text{(meshing-out)}
\end{cases}$$
where \( f(\alpha) = \cos \alpha’_y [(\alpha_2 – \alpha’_y) \sin \alpha – \cos \alpha] \), \( M_b = \mu \{\alpha’_y + \alpha_2 + \sin \alpha’_y [(\alpha_2 – \alpha) \sin \alpha – \cos \alpha]\} \), \( M_s = \mu \sin \alpha’_y \), and \( M_a = \mu \cos \alpha’_y \). The Hertzian contact stiffness and fillet foundation stiffness remain unchanged as they are not affected by friction. The overall calculation procedure for TVMS with friction involves slicing the helical gear, computing stiffness components for each slice, integrating along the tooth width, and summing over all engaged pairs.
We analyze the effects of constant and time-varying friction on TVMS using a helical gear pair with parameters listed in the table below. The gear pair has a helix angle of 15 degrees, module of 3.5 mm, and tooth width of 35 mm, among other specifications. The total contact ratio for helical gears is between 2 and 3, meaning two to three tooth pairs are engaged simultaneously, and the friction coefficient varies periodically for each pair.
| Parameter | Symbol | Value |
|---|---|---|
| Number of teeth (pinion) | \( Z_1 \) | 40 |
| Number of teeth (gear) | \( Z_2 \) | 40 |
| Module (mm) | \( m_n \) | 3.5 |
| Tooth width (mm) | \( B \) | 35 |
| Pressure angle (degrees) | \( \alpha \) | 20 |
| Helix angle (degrees) | \( \beta \) | 15 |
| Torque (N·m) | \( T \) | 1000 |
| Speed (r/min) | \( n \) | 1000 |
| Poisson’s ratio | \( \nu \) | 0.3 |
| Elastic modulus (N/m²) | \( E \) | 2 × 10⁸ |
| Surface roughness (μm) | \( R_a \) | 1.6 |
Under constant friction, the TVMS decreases compared to the frictionless case. The single-tooth stiffness reduces during the meshing-in phase and increases during the meshing-out phase due to friction direction changes. At the pitch line, stiffness exhibits a sudden change as friction force reverses. Higher constant friction coefficients lead to greater reductions in TVMS. For instance, with a constant friction coefficient of 0.05, TVMS drops by approximately 5%, while with 0.1, it decreases by about 10%. This is because friction introduces additional energy dissipation and alters the load distribution along the tooth profile.
Under time-varying friction, the TVMS is influenced by operational parameters. We vary surface roughness, torque, speed, and tooth width individually to study their effects. The time-varying friction coefficient \( \mu \) changes during meshing: it decreases initially, reaches zero at the pitch line, and then increases. The friction coefficient increases with surface roughness and torque but decreases with speed. Tooth width has a complex effect due to increased contact length. The following table summarizes the impact of these parameters on TVMS and friction coefficient.
| Parameter | Range | Effect on Friction Coefficient | Effect on TVMS |
|---|---|---|---|
| Surface roughness \( R_a \) (μm) | 0.4, 0.8, 1.2, 1.6 | Increases with roughness | Decreases with roughness |
| Torque \( T \) (N·m) | 500, 1000, 1500, 2000 | Increases with torque | Increases with torque |
| Speed \( n \) (r/min) | 500, 1000, 1500, 2000 | Decreases with speed | Increases with speed |
| Tooth width \( B \) (mm) | 25, 30, 35, 40 | Increases with width | Increases with width |
Specifically, as surface roughness increases from 0.4 μm to 1.6 μm, the average friction coefficient rises by about 20%, causing a reduction in TVMS due to increased energy loss and altered contact conditions. For torque, higher input torque (e.g., from 500 N·m to 2000 N·m) increases the friction coefficient slightly but also stiffens the gear pair, leading to a net increase in TVMS. Speed has an inverse relationship; increasing speed from 500 r/min to 2000 r/min reduces friction due to better lubrication, resulting in higher TVMS. Tooth width enlargement from 25 mm to 40 mm increases the contact area, raising both friction and stiffness, with TVMS showing a significant boost.
The load-sharing transmission error (LSTE) is another critical parameter affected by TVMS. It is defined as the deviation from ideal meshing and calculated as \( \text{LSTE} = F / K \), where \( F \) is the meshing force and \( K \) is the TVMS. Under time-varying friction, LSTE increases with surface roughness due to reduced stiffness, but decreases with torque, speed, and tooth width. For example, with surface roughness of 1.6 μm, LSTE is about 15% higher than at 0.4 μm, whereas with tooth width of 40 mm, LSTE is 10% lower than at 25 mm. This inverse relationship between TVMS and LSTE highlights the importance of friction in gear design for minimizing vibrations.
In conclusion, we have developed a method to calculate the time-varying meshing stiffness of helical gears considering tooth surface friction. By combining the slice method and energy method with a time-varying friction model, we accurately predict stiffness variations. Our results show that friction significantly alters TVMS: constant friction reduces stiffness, with higher coefficients leading to greater reductions, while time-varying friction causes stiffness to decrease with surface roughness but increase with torque, speed, and tooth width. These findings emphasize the need to incorporate friction in gear dynamics models to improve performance and reliability in practical applications involving helical gears. Future work could explore thermal effects and material nonlinearities on friction and stiffness.