In this study, we investigate the meshing dynamic characteristics of a cylindrical straight spur gear transmission system under tooth profile modification. A comprehensive analytical model for modified straight spur gears is established, incorporating Hertzian contact stiffness, bending stiffness, shear stiffness, axial compression stiffness, and fillet foundation stiffness. Two distinct geometric designs for tooth profile modification are presented. Using numerical simulation, we systematically examine the influence of profile modification on time-varying meshing stiffness (TVMS) and the resulting dynamic behavior. Our findings reveal that for a single tooth profile modification, positive modification increases gear tooth thickness but reduces tooth stiffness and contact ratio, thereby intensifying system vibration. Conversely, negative modification enhances tooth stiffness and contact ratio, leading to smoother transmission. For compound modifications, negative compound S-gear transmission increases TVMS and significantly reduces vibration, while s0 gear modification has a minor effect on TVMS but a substantial impact on dynamic responses. This work provides a theoretical foundation for optimizing straight spur gear performance in engineering applications.
The straight spur gear is widely used in power transmission due to its simple structure, high efficiency, and low cost. However, its inherent time-varying meshing stiffness (TVMS) is a primary internal excitation source that induces vibration and noise. Tooth profile modification is a common technique to improve gear performance, but its effect on TVMS and dynamic behavior is complex and not fully understood. This study addresses this gap by developing a detailed analytical model that accurately captures the geometric and mechanical characteristics of modified straight spur gears. Unlike previous works that focused on standard gears, we specifically consider the influence of profile shift coefficients on the stiffness components and the overall system dynamics.
Time-Varying Meshing Stiffness Model
To compute the TVMS of a straight spur gear pair, we adopt the potential energy method. The gear tooth is idealized as a variable-section cantilever beam, and the total potential energy comprises five contributions: Hertzian contact energy, bending energy, shear energy, axial compression energy, and fillet foundation energy. The stiffness components are defined as follows:
$$U_h = \frac{F^2}{2k_h}, \quad U_b = \frac{F^2}{2k_b}, \quad U_s = \frac{F^2}{2k_s}, \quad U_a = \frac{F^2}{2k_a}, \quad U_f = \frac{F^2}{2k_f}$$
where \(F\) is the total meshing force. For a single tooth pair, the total stored energy can be expressed as:
$$U = \frac{F^2}{2k} = U_h + U_{b1} + U_{s1} + U_{a1} + U_{f1} + U_{b2} + U_{s2} + U_{a2} + U_{f2}$$
Thus, the overall mesh stiffness \(k\) for a single tooth pair is:
$$k = \frac{1}{\frac{1}{k_h} + \frac{1}{k_{b1}} + \frac{1}{k_{s1}} + \frac{1}{k_{a1}} + \frac{1}{k_{f1}} + \frac{1}{k_{b2}} + \frac{1}{k_{s2}} + \frac{1}{k_{a2}} + \frac{1}{k_{f2}}}$$
For a gear pair undergoing double tooth engagement, the total TVMS is the sum of the stiffnesses of the two simultaneously engaged tooth pairs:
$$k_{\text{total}} = \sum_{i=1}^{2} \frac{1}{\frac{1}{k_{h,i}} + \frac{1}{k_{b1,i}} + \frac{1}{k_{s1,i}} + \frac{1}{k_{a1,i}} + \frac{1}{k_{f1,i}} + \frac{1}{k_{b2,i}} + \frac{1}{k_{s2,i}} + \frac{1}{k_{a2,i}} + \frac{1}{k_{f2,i}}}$$
The gear tooth geometry depends on the relationship between the root circle and the base circle. Two cases are considered:
Case 1: Root circle smaller than base circle
The tooth profile consists of an involute curve from the addendum circle to the base circle, and a transition curve from the base circle to the root circle. The geometric parameters are determined using the following equations:
Tooth height \(h_{a,i}\) for pinion (p) or gear (g):
$$h_{a,i} = (h_a^* + \rho_i – \Delta q) m_n, \quad i = p, g$$
where \(\rho_i\) is the profile shift coefficient, \(h_a^*\) the addendum coefficient, \(\Delta q\) the addendum modification coefficient, and \(m_n\) the module.
The axial compression, bending, and shear stiffness integrals are derived in terms of the angular parameter \(\varphi\). For example, bending compliance is:
$$\frac{1}{k_b} = \int_{\varphi_2}^{\varphi_3} \frac{3 a_x (R_b – R_f \cos \varphi_3 \cos \varphi_1 – a_x \varphi \cos \varphi_1 – b_x \cos \varphi_1)^2}{E L \left[ R_b \sin \varphi_2 + r_f – \sqrt{r_f^2 – (x – d_1)^2} \right]^3} d\varphi + \int_{-\varphi_1}^{\varphi_2} \frac{3 \left[ 1 + \cos \varphi_1 ( (\varphi_2 – \varphi) \sin \varphi – \cos \varphi ) \right]^2 (\varphi_2 – \varphi) \cos \varphi}{2 E L \left[ \sin \varphi_2 + (\varphi_2 – \varphi) \cos \varphi \right]^3} d\varphi$$
Shear compliance:
$$\frac{1}{k_s} = \int_{\varphi_2}^{\varphi_3} \frac{1.2 a_x (1+\nu) \cos^2 \varphi_1}{E L \left[ R_b \sin \varphi_2 + r_f – \sqrt{r_f^2 – (x – d_1)^2} \right]} d\varphi + \int_{-\varphi_1}^{\varphi_2} \frac{1.2 (1+\nu) (\varphi_2 – \varphi) \cos \varphi \cos^2 \varphi_1}{E L \left[ \sin \varphi_2 + (\varphi_2 – \varphi) \cos \varphi \right]} d\varphi$$
Axial compression compliance:
$$\frac{1}{k_a} = \int_{\varphi_2}^{\varphi_3} \frac{a_x \sin^2 \varphi_1}{2 E L \left[ R_b \sin \varphi_2 + r_f – \sqrt{r_f^2 – (x – d_1)^2} \right]} d\varphi + \int_{-\varphi_1}^{\varphi_2} \frac{(\varphi_2 – \varphi) \cos \varphi \sin^2 \varphi_1}{E L \left[ \sin \varphi_2 + (\varphi_2 – \varphi) \cos \varphi \right]} d\varphi$$
Hertzian contact stiffness:
$$k_h = \frac{\pi E L}{4 (1 – \nu^2)}$$
Fillet foundation stiffness:
$$\frac{1}{k_f} = \frac{\cos^2 \beta}{E L} \left[ L^* \left( \frac{u_f}{S_f} \right)^2 + M^* \left( \frac{u_f}{S_f} \right) + P^* (1 + Q^* \tan^2 \varphi_1) \right]$$
where \(L^*, M^*, P^*, Q^*\) are empirical coefficients from the literature.
Case 2: Root circle larger than base circle
The tooth profile is entirely involute from addendum to root. The compliance integrals become simpler. For example, bending compliance:
$$\frac{1}{k_b} = \int_{-\varphi_1}^{\varphi_5} \frac{3 \left[ 1 + \cos \varphi_1 ( (\varphi_2 – \varphi) \sin \varphi – \cos \varphi ) \right]^2 (\varphi_2 – \varphi) \cos \varphi}{2 E L \left[ \sin \varphi_2 + (\varphi_2 – \varphi) \cos \varphi \right]^3} d\varphi$$
Shear and axial compliance are analogous.
Dynamic Model of the Gear System
A six-degree-of-freedom lumped-parameter dynamic model is developed for the straight spur gear pair. The equations of motion for the pinion and gear are:
$$m_p \ddot{x}_p + c_b \dot{x}_p + k_b x_p = -F_m$$
$$m_p \ddot{y}_p + c_b \dot{y}_p + k_b y_p = -F_f$$
$$I_p \ddot{\beta}_p = -F_m R_{b,p} – T_p$$
$$m_g \ddot{x}_g + c_b \dot{x}_g + k_b x_g = F_m$$
$$m_g \ddot{y}_g + c_b \dot{y}_g + k_b y_g = F_f$$
$$I_g \ddot{\beta}_g = -F_m R_{b,g} – T_g$$
where \(m_p, m_g\) are masses; \(I_p, I_g\) are moments of inertia; \(c_b, k_b\) are bearing damping and stiffness; \(T_p, T_g\) are input and load torques; \(F_m\) and \(F_f\) are meshing and friction forces. The meshing force is:
$$F_m = k(t) [x_p – x_g + R_{b,p} \beta_p + R_{b,g} \beta_g – e(t)] + c_m [\dot{x}_p – \dot{x}_g + R_{b,p} \dot{\beta}_p + R_{b,g} \dot{\beta}_g – \dot{e}(t)]$$
where \(e(t)\) is static transmission error, and \(c_m\) is the meshing damping coefficient given by:
$$c_m = 2 \zeta \sqrt{\bar{k}_m m_{eq}}$$
with \(\zeta\) the damping ratio, \(\bar{k}_m\) the mean mesh stiffness, and \(m_{eq} = m_p m_g / (m_p + m_g)\). The dynamic transmission error (DTE) is defined as:
$$\text{DTE} = x_p – x_g + R_{b,p} \beta_p + R_{b,g} \beta_g – e(t)$$
Case Study: Influence of Single Profile Modification
Positive Modification
We investigate the effect of positive profile shift on the pinion tooth stiffness and TVMS. Six shift coefficients (0, 0.1, 0.2, 0.3, 0.4, 0.5) are considered. The single tooth stiffness increases with positive shift, but the rate of change varies along the tooth profile. The TVMS decreases as the shift coefficient increases, and the contact ratio is reduced. The mean and standard deviation of TVMS exhibit nonlinear trends: mean decreases gradually, while standard deviation increases sharply for small shifts and then slows. Statistical indicators of DTE (RMS, SRA, PPV, KV) show an overall increasing trend with positive shift, indicating intensified vibration.
Table 1 summarizes the percentage changes of DTE indicators relative to standard gear (ζ=0.07).
| Shift Coefficient | RMS (%) | SRA (%) | PPV (%) | KV (%) |
|---|---|---|---|---|
| 0.1 | 2.3 | 1.9 | 0.8 | -1.2 |
| 0.2 | 5.1 | 4.3 | 2.4 | -0.5 |
| 0.3 | 8.6 | 7.2 | 4.1 | 1.3 |
| 0.4 | 12.8 | 10.9 | 6.5 | 3.8 |
| 0.5 | 17.5 | 15.2 | 9.2 | 7.4 |
Negative Modification
Negative shift coefficients (-0.1 to -0.5) are examined. The single tooth stiffness decreases, but the TVMS increases due to a higher contact ratio. The mean TVMS increases linearly with decreasing shift coefficient, while standard deviation decreases. DTE indicators RMS and SRA drop monotonically, indicating vibration reduction. KV improves slowly, and PPV changes minimally. Table 2 shows the percentage changes (ζ=0.08).
| Shift Coefficient | RMS (%) | SRA (%) | PPV (%) | KV (%) |
|---|---|---|---|---|
| -0.1 | -0.8 | -0.6 | 0.2 | -1.5 |
| -0.2 | -1.4 | -1.1 | 0.1 | -2.3 |
| -0.3 | -1.9 | -1.5 | -0.1 | -2.8 |
| -0.4 | -2.3 | -1.8 | -0.3 | -3.2 |
| -0.5 | -2.6 | -2.1 | -0.5 | -3.5 |
Compound Modification
We analyze six groups with different combinations of pinion and gear shift coefficients: Group 1 (pp=0.4, pg=0.1), Group 2 (pp=0.2, pg=0.1) – positive total shift; Group 3 (pp=0.1, pg=-0.6), Group 4 (pp=0.1, pg=-0.4) – negative total shift; Group 5 (pp=0.1, pg=-0.1) – s0 gear (zero total shift); Group 6 (pp=0, pg=0) – standard gear. Figure 2 (inserted earlier) illustrates a typical straight spur gear used in our simulation.
The TVMS results show that negative total shift increases both mean stiffness and contact ratio, while positive total shift reduces them. The s0 gear yields TVMS close to the standard gear. The mean and standard deviation of TVMS for compound shifts are summarized in Table 3.
| Group | Mean TVMS (×10⁹ N/m) | Std TVMS (×10⁸ N/m) |
|---|---|---|
| 1 | 1.52 | 3.01 |
| 2 | 1.55 | 2.87 |
| 3 | 1.67 | 1.85 |
| 4 | 1.64 | 2.05 |
| 5 | 1.60 | 2.45 |
| 6 | 1.62 | 2.35 |
Dynamic response analysis reveals that positive compound shifts (Groups 1,2) produce higher vibration indicators (RMS, SRA, PPV, KV) compared to standard, while negative compound shifts (Groups 3,4) significantly suppress vibration. For instance, at ζ=0.07, Group 3 shows a reduction of about 15% in RMS relative to the standard gear. The s0 gear (Group 5) exhibits a mild increase in vibration despite having TVMS similar to standard, indicating that the distribution of shift between pinion and gear matters.
Table 4 presents the percentage changes of DTE indicators for compound modifications at ζ=0.09.
| Group | RMS (%) | SRA (%) | PPV (%) | KV (%) |
|---|---|---|---|---|
| 1 | 18.2 | 15.6 | 10.3 | 8.9 |
| 2 | 12.4 | 10.8 | 7.1 | 5.6 |
| 3 | -14.7 | -12.3 | -8.5 | -6.1 |
| 4 | -9.3 | -7.8 | -5.2 | -3.7 |
| 5 | 2.1 | 1.8 | 1.2 | 0.9 |
| 6 | 0 | 0 | 0 | 0 |
Conclusion
In this research, we have developed a comprehensive analytical model for predicting the time-varying meshing stiffness and dynamic response of cylindrical straight spur gears with tooth profile modification. The model accounts for all major stiffness components and two distinct tooth geometries. Through numerical simulations, we have systematically evaluated the effects of single and compound profile shifts. Key conclusions are:
- Positive single-profile shift reduces TVMS and contact ratio, increasing system vibration. Negative single-profile shift improves TVMS and contact ratio, leading to smoother operation.
- For compound shifts, negative total shift (S-gear) significantly increases TVMS and reduces vibration, while positive total shift has the opposite effect. The s0 gear (zero total shift) has minor influence on TVMS but notably affects dynamic indicators due to imbalanced tooth compliance.
- The proposed model and analysis framework can guide the selection of profile shift coefficients for optimizing straight spur gear performance in engineering design.
Future work will extend this approach to helical gears and consider additional nonlinear effects such as backlash and time-varying friction.