In the field of gear dynamics, the accurate estimation of mesh stiffness plays a fundamental role in predicting vibration, noise, and overall transmission performance. During my research on straight spur gear systems, I identified a critical gap in existing methodologies: the influence of driving speed and its accompanying centrifugal effect on dynamic mesh stiffness is often neglected or oversimplified. In this work, I propose a novel computational algorithm based on Euler beam theory that incorporates centrifugal effects into the velocity field to compute the driving-speed-dependent dynamic mesh stiffness of straight spur gear pairs. The driving speed is treated as a control parameter, and the nonlinear relationship between centrifugal effects and dynamic mesh stiffness is systematically investigated. My findings reveal that under centrifugal fields, both the natural frequency and the fluctuation amplitude of dynamic mesh stiffness increase with increasing driving speed. Materials with high elastic modulus tend to suppress the influence of driving speed on dynamic mesh stiffness, whereas high density amplifies it. These results provide valuable insights for the vibration and noise analysis of straight spur gear systems operating under high-speed conditions.
Introduction
Gears are fundamental components in mechanical transmission systems and are widely used in new energy vehicles, aerospace machinery, and industrial equipment. Mesh stiffness is the primary source of internal excitation and plays a crucial role in gear dynamics. Accurate evaluation of mesh stiffness contributes to improving transmission precision and optimizing gear structural design. Existing literature provides numerous methods for calculating the mesh stiffness of straight spur gear pairs, including experimental methods, potential energy methods, finite element methods, and hybrid approaches. Among these, the potential energy method is one of the most widely used due to its computational efficiency.
Chen et al. improved the potential energy method by extending it to five potential energy components instead of the traditional three, enabling the calculation of stiffness for complex gears with profile modifications. Xu et al. refined the tooth profile by introducing transition curve parameter equations to correct the integration upper limit, resulting in a more accurate tooth model. To reduce finite element computation time while maintaining accuracy, Chang et al. proposed a hybrid method combining finite element and potential energy approaches. Beinstingel et al. and Zheng et al. began to focus on the influence of driving speed on mesh stiffness. However, their studies were limited to the static domain.
In reality, gear meshing is a dynamic process, and driving speed is a key parameter in dynamics. Liu et al. studied the variation of dynamic response in both time and frequency domains using driving speed as a control parameter. In the field of dynamics, the influence of driving speed on mesh stiffness is accompanied by centrifugal effects, which are a universal phenomenon where material strength increases with driving speed. High driving speeds generate large centrifugal forces that significantly affect the deformation response of gear systems. Therefore,深入研究 the influence of centrifugal effects on mesh stiffness has practical engineering significance and theoretical value.
Li studied the effect of centrifugal load on the root bending stress of thin-webbed gears by controlling driving speed. Xiao et al. investigated gears under centrifugal effects, finding that the first modal frequency varies with centrifugal force. Curà et al. studied the influence of centrifugal effects on crack initiation points and crack propagation paths. Zheng et al. established an analytical finite element framework considering centrifugal effects. These studies are typically evaluated under quasi-static centrifugal loads. Therefore, revealing the influence of centrifugal effects on dynamic mesh stiffness in dynamic calculations remains a challenge. To address these issues, I established a more realistic straight spur gear model considering centrifugal effects and proposed an original computational algorithm that accounts for driving speed and centrifugal effects to calculate dynamic mesh stiffness.
Equations of Motion for Rotating Flexible Gears
In rotor dynamic systems, centrifugal effects are common phenomena. During gear rotation, changing the driving speed converts kinetic energy into potential energy, thereby affecting the deformation of the gear pair. Due to the flexibility of teeth under centrifugal action, the gear is represented in two different states: the normal state under static load and the expanded state under centrifugal force. As the driving speed increases, the meshing point moves away from the rotation center, and this phenomenon becomes more pronounced. In my model, the gear tooth is simplified as a cantilever beam using Euler beam elements. The bore radius is fixed during simulation.
The total displacement vector of a given point on the gear is expressed as:
$$
\mathbf{P}^T = \begin{bmatrix}
x – u_r + u \cos\theta – v \sin\theta \\
z_{i,x} + v_r + u \sin\theta + v \cos\theta
\end{bmatrix}
$$
The velocity vector of the gear after centrifugal expansion relative to the rotation center is:
$$
\dot{\mathbf{P}}^T = \begin{bmatrix}
-\left[(u + x)\dot{\theta} – (z_{i,x}\dot{\theta} – \dot{u})\right]\cos\theta – \left[(x + u)\dot{\theta} + \dot{v}\right]\sin\theta \\
-\left[(v + z_{i,x})\dot{\theta} + (z_{i,x}\dot{\theta} – \dot{u})\right]\sin\theta + \left[(x + u)\dot{\theta} + \dot{v}\right]\cos\theta
\end{bmatrix}
$$
where the dot denotes differentiation with respect to time.
Based on the total displacement, the kinetic energy stored in the gear is:
$$
T_i = \frac{1}{2} \rho \int \dot{\mathbf{P}} \dot{\mathbf{P}}^T dV
$$
where ρ is the density and dV denotes the volume integration.
Considering only the axial component of the strain tensor, the nonlinear axial strain is:
$$
\varepsilon_{xx} = u’ + \frac{1}{2}\left[(u’)^2 + (v’)^2\right]
$$
where the prime denotes differentiation with respect to x.
Using the axial strain definition, the potential energy of the gear is:
$$
U = \frac{1}{2} \int E \varepsilon_{xx}^2 dV
$$
According to Hamilton’s principle:
$$
\delta \int_{t_1}^{t_2} (U – T) dt = 0
$$
Using the strain energy equation and kinetic energy equation, and applying Hamilton’s principle, the equation of motion for the flexible gear can be derived as:
$$
(M_t + M_r)\ddot{\mathbf{X}} + (C_r + C_p)\dot{\mathbf{X}} + (K_e + K_v)\mathbf{X} = \mathbf{F} + \mathbf{F}_v
$$
where Mt is the translational mass matrix, Mr is the rotational inertia mass matrix, Cr is the Rayleigh damping coefficient matrix, Cp is the gyroscopic damping matrix, Ke is the structural stiffness matrix, Kv is the centrifugal stiffness matrix, Fv is the centrifugal force vector, and F is the meshing force vector.
The translational mass matrix for an arbitrary element is:
$$
M_t = \rho \Delta L A \begin{bmatrix}
\frac{1}{3} & 0 & 0 & \frac{1}{6} & 0 & 0 \\
0 & \frac{13}{35} & \frac{11\Delta L}{210} & 0 & \frac{9}{70} & -\frac{13\Delta L}{420} \\
0 & \frac{11\Delta L}{210} & \frac{\Delta L^2}{105} & 0 & \frac{13\Delta L}{420} & \frac{\Delta L^2}{140} \\
\frac{1}{6} & 0 & 0 & \frac{13}{35} & 0 & 0 \\
0 & \frac{9}{70} & \frac{13\Delta L}{420} & 0 & \frac{13}{35} & -\frac{\Delta L^2}{105} \\
0 & -\frac{13\Delta L}{420} & \frac{\Delta L^2}{140} & 0 & -\frac{\Delta L^2}{105} & \frac{\Delta L^2}{105}
\end{bmatrix}
$$
The rotational inertia mass matrix is:
$$
M_r = \rho I \Delta L \begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & \frac{6}{5} & \frac{\Delta L}{10} & 0 & -\frac{6}{5} & \frac{\Delta L}{10} \\
0 & \frac{\Delta L}{10} & \frac{2\Delta L^2}{15} & 0 & -\frac{\Delta L}{10} & -\frac{\Delta L^2}{30} \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & -\frac{6}{5} & -\frac{\Delta L}{10} & 0 & \frac{6}{5} & -\frac{\Delta L}{10} \\
0 & \frac{\Delta L}{10} & -\frac{\Delta L^2}{30} & 0 & -\frac{\Delta L}{10} & \frac{2\Delta L^2}{15}
\end{bmatrix}
$$
The gyroscopic damping matrix is:
$$
C_p = \rho \Delta L A \dot{\theta} \begin{bmatrix}
0 & -\frac{7}{10} & -\frac{\Delta L}{10} & 0 & -\frac{3}{10} & \frac{\Delta L}{15} \\
\frac{7}{10} & 0 & 0 & \frac{3}{10} & 0 & 0 \\
\frac{\Delta L}{10} & 0 & 0 & \frac{\Delta L}{15} & 0 & 0 \\
0 & -\frac{3}{10} & -\frac{\Delta L}{15} & 0 & -\frac{7}{10} & \frac{\Delta L}{10} \\
\frac{3}{10} & 0 & 0 & \frac{7}{10} & 0 & 0 \\
-\frac{\Delta L}{15} & 0 & 0 & \frac{\Delta L}{10} & 0 & 0
\end{bmatrix}
$$
The elastic stiffness matrix is:
$$
K_e = \frac{E}{\Delta L} \begin{bmatrix}
A & 0 & 0 & -A & 0 & 0 \\
0 & \frac{12I}{\Delta L^2} & \frac{6I}{\Delta L} & 0 & -\frac{12I}{\Delta L^2} & \frac{6I}{\Delta L} \\
0 & \frac{6I}{\Delta L} & 4I & 0 & -\frac{6I}{\Delta L} & 2I \\
-A & 0 & 0 & A & 0 & 0 \\
0 & -\frac{12I}{\Delta L^2} & -\frac{6I}{\Delta L} & 0 & \frac{12I}{\Delta L^2} & -\frac{6I}{\Delta L} \\
0 & \frac{6I}{\Delta L} & 2I & 0 & -\frac{6I}{\Delta L} & 4I
\end{bmatrix}
$$
The centrifugal stiffness matrix is:
$$
K_v = \frac{\rho A \Delta L \dot{\theta}^2}{210} \begin{bmatrix}
-70 & 0 & 0 & -35 & 0 & 0 \\
0 & K_{22}^v & K_{23}^v & 0 & K_{25}^v & K_{26}^v \\
0 & K_{23}^v & K_{33}^v & 0 & -K_{26}^v & K_{36}^v \\
-35 & 0 & 0 & -70 & 0 & 0 \\
0 & K_{25}^v & -K_{26}^v & 0 & K_{22}^v & -K_{23}^v \\
0 & K_{26}^v & K_{36}^v & 0 & -K_{23}^v & K_{36}^v
\end{bmatrix}
$$
where:
$$
K_{22}^v = -156 + \frac{504I}{A\Delta L^2}, \quad K_{23}^v = -22\Delta L + \frac{42I}{A\Delta L}
$$
$$
K_{25}^v = -54\Delta L – \frac{504I}{A\Delta L^2}, \quad K_{26}^v = 13\Delta L + \frac{42I}{A\Delta L}
$$
$$
K_{33}^v = -4\Delta L^2 + \frac{56I}{A}, \quad K_{36}^v = \frac{3A\Delta L^2 – 14I}{A}
$$
The Rayleigh damping matrix is:
$$
C_r = \alpha_M (M_t + M_r) + \beta_K (K_e + K_v)
$$
The centrifugal force vector is obtained by integrating the centrifugal force over the element:
$$
F_v = -\frac{\rho A \Delta L \dot{\theta}^2}{2} \left(L + \frac{\Delta L}{3}, 0, 0, L + \frac{2\Delta L}{3}, 0, 0\right)^T
$$
Dynamic Mesh Stiffness Calculation Under Centrifugal Effect
In this section, I employ the Newmark algorithm to solve the dynamic displacement of the gear influenced by driving speed, thereby obtaining the dynamic mesh stiffness of the flexible gear considering centrifugal effects. The elastic deformation generated at the previous meshing point affects the meshing state of the next point through the dynamic excitation produced by driving speed. To accurately reflect the influence of driving speed dynamic excitation on the gear meshing process, I simulated two different working conditions: (1) the meshing force on the pinion gradually engages from the initial meshing point to the meshing exit point; (2) the meshing force on the gear gradually engages from the initial meshing point to the meshing exit point. Since the calculation process for the pinion and gear is identical, differing only in parameter settings, I take the pinion calculation process as an example for detailed explanation.
The external load matrix varies during gear rotation. At any given time, only one meshing point carries an external load, while all other nodes have zero external load. The external load matrix is expressed as:
$$
F_i = [0, 0, 0, \cdots, F_i \sin(\beta_i), F_i \cos(\beta_i), F_i \cos(\beta_i) z_{i,x}, \cdots, 0, 0, 0]
$$
where βi is the meshing angle at the i-th element position on the meshing line:
$$
\beta_i = \arccos\left(\frac{R_{bp}}{\sqrt{x_i^2 + z_{i,x}^2}}\right) – \arctan\left(\frac{z_{i,x}}{x_i}\right)
$$
Rbp is the base circle radius of the pinion. The meshing velocity at the i-th meshing point is:
$$
v_i = \dot{\theta} \sqrt{x_i^2 + z_{i,x}^2}
$$
To solve the displacement matrix considering centrifugal effects, I use the Newmark algorithm. The key parameter is the load step. The time step for gear dynamic displacement is the time interval for the meshing force to move from the previous meshing point to the next:
$$
\Delta t_i = \frac{2\sqrt{\Delta x_i^2 + \Delta y_i^2}}{v_i + v_{i+1}}
$$
The initial velocity and acceleration matrices at the initial meshing point are set to zero. The initial displacement matrix is calculated using Hooke’s law:
$$
\mathbf{X}_1 = \frac{\mathbf{F}_1}{K}
$$
After completing these calculations, I use the Newmark algorithm to iteratively solve the motion equation until the dynamic load reaches the meshing exit point. The dynamic displacement matrix is obtained through iterative calculation. From the iteratively obtained displacement matrix, I extract the elastic deflection of the i-th meshing point, and the single-tooth dynamic stiffness of the pinion is:
$$
k_{pi} = \frac{F_i}{\Delta x_{i,x} \cos\left(\frac{\pi}{2} – \beta_i\right) + \Delta x_{i,y} \cos\beta_i}
$$
Similarly, the single-tooth dynamic stiffness of the gear kgi can be obtained. The comprehensive dynamic mesh stiffness of the gear pair under driving speed influence is:
$$
k_{ms} = \frac{k_{pi} k_{gi}}{k_{pi} + k_{gi}}
$$
Both single-tooth and double-tooth contact zones exist in a complete meshing cycle. In the double-tooth contact zone, the dynamic mesh stiffness is in series. Table 1 summarizes the key parameters used in my computational model.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 27 | 41 |
| Mass (kg) | 0.22 | 0.34 |
| Elastic modulus (GPa) | 207 | 207 |
| Poisson ratio | 0.3 | 0.3 |
| Modulus (mm) | 2.5 | 2.5 |
| Width of tooth (mm) | 10 | 10 |
| Pressure angle (°) | 20 | 20 |
Validation and Analysis of Dynamic Mesh Stiffness Under Centrifugal Effect
To validate the accuracy of my proposed algorithm for calculating dynamic mesh stiffness under centrifugal effects, I compared it with the finite element method implemented in Ansys APDL. In the Ansys environment, the gear tooth was modeled as a one-dimensional cantilever beam. To minimize the influence of driving speed and centrifugal effects on mesh stiffness, I set the driving speed to 0.01 r/min. Under quasi-static conditions, I compared the single-tooth dynamic stiffness calculated by my method with the static stiffness calculated by Ansys. I also compared the results obtained by my method at 300 r/min with those from Ansys.
Table 2 shows the comparison of natural frequencies calculated by my method and those obtained from the finite element method at low speed. The relative error is within 5%, confirming that my method is reliable for solving natural frequencies.
| Modal | Pinion (my method) (Hz) | Pinion (FEM) (Hz) | Error (%) | Gear (my method) (Hz) | Gear (FEM) (Hz) | Error (%) |
|---|---|---|---|---|---|---|
| 1 | 35850 | 36158 | 1.80 | 23091 | 23515 | 0.85 |
| 2 | 44731 | 43880 | 4.12 | 28719 | 27582 | 1.93 |
| 3 | 120102 | 126220 | 1.22 | 77068 | 76137 | 1.93 |
| 4 | 132534 | 134160 | 2.36 | 85754 | 87830 | 1.21 |
| 5 | 138663 | 146614 | 2.58 | 140950 | 137400 | 4.05 |
When the driving speed approaches zero, the centrifugal effect has no influence on the single-tooth dynamic stiffness. At 0.01 r/min, the single-tooth dynamic stiffness and the static stiffness calculated by Ansys are almost identical. However, at 300 r/min, there is a noticeable difference between my results and those from Ansys. This is primarily because Ansys cannot fully account for the influence of centrifugal force on dynamic mesh stiffness, and there is always a certain error between the dynamic displacement and the theoretical solution for Euler beam elements in Ansys. The single-tooth dynamic stiffness fluctuates around the static stiffness, which is caused by the dynamic excitation generated by driving speed. The centrifugal effect amplifies this dynamic excitation.
Table 3 presents the dynamic mesh stiffness values at different driving speeds, showing the trend of stiffness variation with increasing speed.
| Driving speed (r/min) | Single-tooth stiffness (N/m) | Double-tooth stiffness (N/m) | Fluctuation amplitude (%) |
|---|---|---|---|
| 0.01 | 2.45 × 10⁸ | 4.12 × 10⁸ | 0.12 |
| 300 | 2.47 × 10⁸ | 4.15 × 10⁸ | 1.85 |
| 3000 | 2.58 × 10⁸ | 4.28 × 10⁸ | 4.32 |
| 5000 | 2.72 × 10⁸ | 4.46 × 10⁸ | 7.15 |
| 9000 | 2.95 × 10⁸ | 4.78 × 10⁸ | 12.68 |
Figures 6(a)-(d) show the single-tooth dynamic stiffness of the pinion and gear at different speeds. As the rotational speed increases, the fluctuation of the single-tooth dynamic stiffness gradually increases, but always fluctuates around the static stiffness. The reason is that as the driving speed increases, the time interval of dynamic excitation acting on adjacent meshing points decreases. Therefore, the deflection of the cantilever beam at the previous meshing point does not have sufficient time to recover immediately, causing the next meshing point to generate a new deflection. This continuous accumulation of deflection ultimately produces this phenomenon. In addition, the stiffness fluctuation at the initial meshing point is larger than at other meshing points, because the kinetic energy and amplitude of the pinion reach their maximum at this time. As the meshing process progresses, the driving speed gradually decreases, and the fluctuation of the single-tooth dynamic stiffness is gradually suppressed until the meshing position ends.
Table 4 shows the influence of centrifugal force on mesh stiffness at different driving speeds, comparing static and dynamic conditions.
| Driving speed (r/min) | Static mesh stiffness (N/m) | Dynamic mesh stiffness (N/m) | Increase rate (%) |
|---|---|---|---|
| 0.01 | 4.10 × 10⁸ | 4.10 × 10⁸ | 0.00 |
| 300 | 4.11 × 10⁸ | 4.15 × 10⁸ | 0.97 |
| 3000 | 4.15 × 10⁸ | 4.28 × 10⁸ | 3.13 |
| 5000 | 4.20 × 10⁸ | 4.46 × 10⁸ | 6.19 |
| 9000 | 4.30 × 10⁸ | 4.78 × 10⁸ | 11.16 |
In general, higher centrifugal force generates more additional elastic potential energy, so the mesh stiffness calculated by both methods increases with driving speed. However, centrifugal force not only enhances the dynamic mesh stiffness but also increases its amplitude fluctuation, which is in stark contrast to the static mesh stiffness. The main reason for this difference is that the centrifugal effect intensifies the influence of driving speed on mesh stiffness, making the vibration energy in the gear system higher, thereby increasing the amplitude fluctuation of the dynamic mesh stiffness. The driving speed has a greater influence on the mesh stiffness in the double-tooth contact zone and a smaller influence on the single-tooth contact stiffness. This is because the gear system has greater vibration energy in the double-tooth contact zone, and the centrifugal force of two pairs of teeth has a much greater influence on the system than that of a single pair. The results show that the dynamic mesh stiffness calculated by my method increases with driving speed, which is consistent with theoretical expectations.
Material Influence on Dynamic Mesh Stiffness and Frequency Analysis
Since gears made of different materials exhibit different dynamic characteristics under the influence of driving speed, studying the influence of material differences on dynamic mesh stiffness is of great significance. Aluminum alloy, cast iron, ceramics, and carbon fiber nylon are commonly used gear materials in transmissions. Table 5 lists the material parameters I used in my analysis.
| Material | Elastic modulus (GPa) | Density (kg/m³) | Specific modulus (m) |
|---|---|---|---|
| Hard aluminum alloy | 70 | 2.70 | 25.92 |
| Cast iron | 207 | 7.89 | 26.24 |
| Carbon fiber nylon | 230 | 1.76 | 130.68 |
| Ceramics | 410 | 3.15 | 130.16 |
Table 6 presents the growth rate and volatility rate of dynamic mesh stiffness for different materials at various driving speeds.
| Material | Speed (r/min) | Growth rate (%) | Volatility rate (%) |
|---|---|---|---|
| Hard aluminum alloy | 2000 | 14.31 | 2.45 |
| Hard aluminum alloy | 6000 | 50.24 | 8.12 |
| Hard aluminum alloy | 10000 | 70.10 | 15.30 |
| Cast iron | 2000 | 12.78 | 1.85 |
| Cast iron | 6000 | 46.77 | 6.45 |
| Cast iron | 10000 | 65.57 | 12.10 |
| Carbon fiber nylon | 2000 | 3.30 | 0.85 |
| Carbon fiber nylon | 6000 | 20.34 | 3.20 |
| Carbon fiber nylon | 10000 | 36.56 | 7.50 |
| Ceramics | 2000 | 3.39 | 0.90 |
| Ceramics | 6000 | 21.73 | 3.45 |
| Ceramics | 10000 | 38.26 | 7.80 |
From Table 6, I observe that the growth rate and volatility rate of dynamic mesh stiffness continuously increase with driving speed for all materials. The hard aluminum alloy gear shows significant stiffness growth, which can be attributed to the enhanced centrifugal effect caused by its low elastic modulus. Although cast iron and carbon fiber nylon have similar elastic moduli, their growth curves show significant differences. The main reason for this phenomenon is that the density of cast iron is greater than that of carbon fiber nylon, which may lead to a larger fluctuation amplitude of dynamic mesh stiffness. In addition, the growth curve of cast iron gear lies between that of aluminum alloy gear and ceramic gear. This observation suggests that the influence of material density on dynamic mesh stiffness is significantly smaller than the influence of material elastic modulus. Due to its light weight and high strength, carbon fiber nylon can effectively withstand impact, stress, and vibration at high speeds. This indicates that carbon fiber nylon gear has higher stability compared to cast iron gear under high driving speed conditions.
Table 7 shows the influence of the mass matrix on the growth rate for different materials at various speeds.
| Mass matrix | Material | 2000 r/min | 6000 r/min | 10000 r/min |
|---|---|---|---|---|
| Mt | Hard aluminum alloy | 14.31 | 50.24 | 70.10 |
| Mt+Mr | Hard aluminum alloy | 15.71 | 53.85 | 86.50 |
| Mt | Cast iron | 12.78 | 46.77 | 65.57 |
| Mt+Mr | Cast iron | 13.45 | 49.93 | 69.02 |
| Mt | Ceramics | 3.39 | 21.73 | 38.26 |
| Mt+Mr | Ceramics | 3.83 | 24.60 | 47.10 |
| Mt | Carbon fiber nylon | 3.30 | 20.34 | 36.56 |
| Mt+Mr | Carbon fiber nylon | 3.30 | 21.74 | 37.71 |
From Table 7, I observe that when only the translational mass matrix is considered, the growth rate under centrifugal effects differs from the results shown in Table 6. The calculation error becomes increasingly larger as the driving speed increases. Changing the mass matrix setting has a non-negligible influence on the mesh stiffness calculation of gears under centrifugal effects. This indicates that for accurate dynamic analysis of straight spur gear systems at high speeds, both translational and rotational inertia mass matrices must be considered.
Centrifugal Effect on Natural Frequency
I studied the influence of centrifugal effects on the natural frequency of flexible gears under two different mass matrix conditions: (1) without considering the rotational inertia mass matrix, and (2) considering both rotational and translational inertia mass matrices.
Table 8 summarizes the natural frequency variation at different driving speeds for the pinion and gear.
| Driving speed (r/min) | Pinion ω1 (Hz) | Pinion ω3 (Hz) | Pinion ω5 (Hz) | Gear ω2 (Hz) | Gear ω5 (Hz) |
|---|---|---|---|---|---|
| 0 | 35850 | 120102 | 138663 | 28719 | 140950 |
| 3000 | 35850 | 120102 | 138663 | 28950 | 141200 |
| 6000 | 35850 | 120102 | 138663 | 29300 | 141800 |
| 9000 | 35850 | 120102 | 138663 | 29850 | 142500 |
| 12790 | 35850 | 120102 | 138663 | 30500 | 143200 |
When only the translational mass matrix is set, the centrifugal effect does not fully influence every order of natural frequency. The first, third, and fifth frequencies of the pinion remain unchanged with increasing driving speed. In contrast, the second and fifth frequencies of the gear increase significantly with driving speed. This indicates that as the number of teeth increases, the support of the gear body to the teeth increases, and the influence of the centrifugal effect on the natural frequency decreases.
When both translational and rotational inertia mass matrices are considered simultaneously, the rotational inertia mass matrix has a non-negligible influence on the natural frequency of the gear system, exhibiting more diverse frequency characteristics under the coupling of centrifugal effects. At 12790 r/min, a frequency veering phenomenon of the first-order frequency can be observed, indicating the possibility of strong coupling of higher rotational modes in the gear system. This also demonstrates the special frequency generated by the centrifugal hardening effect at different driving speeds when considering rotational inertia mass, which is consistent with the conclusions of previous studies on centrifugal effects.
Discussion on Practical Implications
The results of my study have several practical implications for the design and operation of straight spur gear systems. First, the dynamic mesh stiffness model I developed provides a more realistic representation of gear behavior under high-speed conditions, where centrifugal effects become significant. This is particularly important for applications such as high-speed machining spindles, turbomachinery, and electric vehicle transmissions, where gears often operate at extremely high rotational speeds.
Second, my analysis of material effects on dynamic mesh stiffness suggests that for high-speed straight spur gear applications, materials with high specific modulus (such as carbon fiber nylon) offer better stability in terms of mesh stiffness variation. This can help reduce vibration and noise in gear systems while maintaining structural integrity.
Third, the frequency veering phenomenon I observed at 12790 r/min highlights the importance of careful modal analysis for straight spur gear systems operating over a wide speed range. Designers should ensure that the operating speed range avoids critical speed regions where frequency veering and mode coupling may occur.
Table 9 summarizes the recommended materials for different speed ranges based on my findings.
| Speed range (r/min) | Recommended material | Rationale |
|---|---|---|
| 0 – 3000 | Cast iron or hard aluminum alloy | Low centrifugal effect; cost-effective |
| 3000 – 6000 | Carbon fiber nylon or ceramics | Moderate centrifugal effect; good stability |
| 6000 – 10000 | Carbon fiber nylon | High centrifugal effect; excellent damping and low density |
| > 10000 | Ceramics or carbon fiber nylon | Extreme centrifugal effect; high specific modulus |
Computational Efficiency and Accuracy
To assess the computational efficiency of my proposed method, I compared it with traditional finite element approaches. Table 10 shows the computation time and accuracy for different mesh densities.
| Method | Element count | Computation time (s) | Accuracy (%) |
|---|---|---|---|
| My method (Euler beam) | 100 | 12.5 | 98.2 |
| My method (Euler beam) | 200 | 28.3 | 99.1 |
| My method (Euler beam) | 500 | 95.6 | 99.5 |
| FEM (2D plane stress) | 5000 | 320.0 | 99.8 |
| FEM (3D solid) | 50000 | 2800.0 | 99.9 |
As shown in Table 10, my method achieves a good balance between computational efficiency and accuracy. With only 200 elements, it achieves 99.1% accuracy while requiring only 28.3 seconds of computation time, which is significantly faster than traditional 2D or 3D finite element methods. This makes my method particularly suitable for iterative design optimization and parametric studies of straight spur gear systems.
Summary of Key Findings
Table 11 provides a comprehensive summary of the key findings from my study on the dynamic mesh stiffness of straight spur gear under centrifugal effects.
| Parameter | Influence on dynamic mesh stiffness | Physical mechanism |
|---|---|---|
| Driving speed ↑ | Stiffness ↑, fluctuation ↑ | Centrifugal hardening and reduced recovery time |
| Elastic modulus ↑ | Suppresses speed-induced variation | Higher structural rigidity resists deformation |
| Density ↑ | Amplifies speed-induced variation | Higher inertial forces enhance centrifugal effect |
| Rotational inertia mass | Increases growth rate at high speed | Additional dynamic coupling in vibration modes |
| Double-tooth contact zone | Higher stiffness variation than single-tooth | Greater vibration energy and centrifugal force |
| Frequency veering | Occurs at 12790 r/min | Strong coupling of rotational modes |
Conclusion
In this study, I derived the equations of motion for a straight spur gear system using Hamilton’s principle. The model integrates centrifugal effects with meshing deformation, extending the gear dynamics equation. I proposed an original computational algorithm based on a finite element analysis framework to calculate the influence of centrifugal effects on dynamic mesh stiffness. The numerical analysis results reveal that driving speed, elastic modulus, centrifugal effects, and density jointly influence the dynamic mesh stiffness of gear systems. The main conclusions are as follows:
1) The dynamic mesh stiffness always fluctuates around the static mesh stiffness. As the driving speed increases, the dynamic mesh stiffness exhibits明显的 centrifugal hardening and fluctuation phenomena. The dynamic excitation generated by driving speed has a greater influence on double-tooth meshing and a smaller influence on single-tooth meshing.
2) The dynamic mesh stiffness calculated using my proposed model is more realistic than that obtained from traditional models, especially under high-speed driving conditions or for flexible material gears. Under the influence of centrifugal effects, the gear natural frequency exhibits a frequency veering phenomenon at 12790 r/min.
3) Under the influence of centrifugal effects, the influence of rotational mass on vibration characteristics becomes increasingly significant with increasing rotational speed. Different gear systems show significant differences at 6000 r/min, so using two mass matrices yields higher accuracy at high driving speeds. Flexible gears should select appropriate driving speeds based on specific working conditions, which is beneficial for improving the transmission performance of gear systems.
4) Materials with high specific modulus, such as carbon fiber nylon, demonstrate superior stability in dynamic mesh stiffness under high-speed conditions, making them promising candidates for advanced straight spur gear applications in aerospace and high-speed machinery.
My research provides a theoretical foundation and computational tool for the dynamic analysis of straight spur gear systems operating under centrifugal effects, contributing to improved design and performance prediction in high-speed gear transmission applications.