In the process of gear meshing, the driving speed plays a significant role in evaluating mesh stiffness, a factor often overlooked by many scholars, along with the accompanying centrifugal effect. In this study, we propose a novel computational algorithm based on Euler beam theory to calculate the dynamic mesh stiffness of a straight spur gear influenced by driving speed, by introducing the centrifugal effect into the velocity field. We take the driving speed as a control parameter and investigate the dynamic mesh stiffness under its influence. Our results demonstrate a nonlinear relationship between the centrifugal effect and dynamic mesh stiffness. Under the action of a centrifugal field, both the natural frequency and the fluctuation 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, while density has the opposite effect. This research provides a reference for further analysis of gear vibration and noise under centrifugal action.
Introduction
Gears are fundamental components in mechanical transmission systems, widely used in new energy vehicles and aerospace machinery. Mesh stiffness is a primary source of internal excitation and plays a vital role in gear dynamics. Therefore, accurate evaluation of mesh stiffness contributes to improving transmission accuracy and optimizing gear structure design in these fields. Existing literature provides many methods for calculating the mesh stiffness of straight spur gears, such as experimental methods, potential energy methods, finite element methods, and hybrid methods. The potential energy method has become one of the most commonly used approaches due to its high computational speed. Some scholars have improved the potential energy method by extending it to five potential energies instead of the traditional three, using a finite element approach for complex gears to address difficulties related to profile modifications. Others have refined tooth profiles by introducing transition curve parametric equations to correct integration limits, making the tooth model more accurate. To reduce finite element computation time while maintaining accuracy, researchers have proposed hybrid methods combining finite element and potential energy methods. To obtain more precise mesh stiffness, some studies have begun to focus on the calculation of mesh stiffness under driving speed. Analytical finite element methods have been developed that construct fillet-foundation stiffness as a function of driving speed. These studies investigated the effect of driving speed on mesh stiffness, but their research was limited to the static domain. In reality, gear meshing is a dynamic process, and driving speed is a key parameter in dynamics. Taking driving speed as a control parameter, researchers have studied variations in dynamic response in the time and frequency domains. In the dynamic domain, the effect of driving speed on mesh stiffness is also accompanied by the centrifugal effect. The centrifugal effect is a universal phenomenon where material strength increases with driving speed. High driving speed generates large centrifugal forces that significantly influence the deformation response of the gear system. Therefore, in-depth investigation of the centrifugal effect on gear system mesh stiffness has practical engineering significance and theoretical value.
Some works controlled driving speed so that centrifugal loads produced different effects on the tooth root bending stress of thin-webbed gears. Others studied gears under the centrifugal effect, where the first modal frequency varies with centrifugal force. Researchers have also investigated the influence of the centrifugal effect on crack initiation points and crack propagation paths. Analytical finite element frameworks considering the centrifugal effect have been established. The aforementioned studies are typically evaluated under quasi-static centrifugal loads. Therefore, revealing the influence of the centrifugal effect on dynamic mesh stiffness in dynamic calculations remains a challenge. To address these issues, we establish a more realistic straight spur gear model considering the centrifugal effect and propose an original computational algorithm that accounts for driving speed and centrifugal effect to calculate dynamic mesh stiffness. The governing equations of motion for the straight spur gear system are derived based on Hamilton’s principle, and the centrifugal force formula is derived. The gear deformation and mesh stiffness are obtained under dynamic conditions, thus providing a reference for improving the transmission performance of straight spur gears at high driving speeds and for vibration and noise reduction.
Governing Equation of Rotating Flexible Gear
In rotor dynamic systems, the centrifugal effect is a common phenomenon. 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 (dashed lines) and the expanded state under centrifugal force (solid lines). As the driving speed increases, the meshing point in the gear moves away from the rotation center, and this phenomenon becomes more pronounced. In the gear model, we simplify the tooth as a cantilever beam using Euler beam elements. During simulation, the gear bore radius is fixed. The total displacement vector of a given point on the gear is expressed as:
$$ \mathbf{P}_T = \begin{bmatrix} x – u_r + u \cos\theta – v \sin\theta \\ z_x + v_r + u \sin\theta + v \cos\theta \end{bmatrix} $$
The velocity vector of the gear point after centrifugal expansion relative to the gear rotation center is:
$$ \dot{\mathbf{P}}_T = \begin{bmatrix} -[(u+x)\dot{\theta} – (z_x\dot{\theta}-\dot{u})]\cos\theta -[(x+u)\dot{\theta}+\dot{v}]\sin\theta \\ -[(v+z_x)\dot{\theta}+(z_x\dot{\theta}-\dot{u})]\sin\theta +[(x+u)\dot{\theta}+\dot{v}]\cos\theta \end{bmatrix} $$
where the dot denotes derivative with respect to time. The kinetic energy stored in the gear is:
$$ T_i = \frac{1}{2}\rho \int_V \dot{\mathbf{P}}_T \dot{\mathbf{P}}_T^T dV $$
Here we only consider the axial component of the strain tensor for the flexible gear. The nonlinear axial strain is:
$$ \varepsilon_{xx} = u’ + \frac{1}{2}\left[(u’)^2 + (v’)^2\right] $$
where the prime denotes derivative with respect to \(x\). Considering that the cross-section of a single tooth is symmetric about the neutral layer and using the axial strain definition, the potential energy of the gear can be expressed using only transverse displacement. The potential energy is:
$$ U_i = \frac{1}{2} \int_V E\varepsilon_{xx}^2 dV $$
According to Hamilton’s principle:
$$ \delta \int_{t_1}^{t_2} (U – T) dt = 0 $$
Using the strain energy equation and kinetic energy equation, the equation of motion for the flexible gear is derived as:
$$ (\mathbf{M}_t + \mathbf{M}_r)\ddot{\mathbf{X}} + (\mathbf{C}_r + \mathbf{C}_p)\dot{\mathbf{X}} + (\mathbf{K}_e + \mathbf{K}_v)\mathbf{X} = \mathbf{F} + \mathbf{F}_v $$
Here, \(\mathbf{M}_t\) and \(\mathbf{M}_r\) are the translational mass matrix and rotational inertia mass matrix of the gear, respectively; \(\mathbf{C}_r\) and \(\mathbf{C}_p\) are the Rayleigh damping coefficient matrix and gyroscopic damping matrix, respectively; \(\mathbf{K}_e\) and \(\mathbf{K}_v\) are the structural stiffness matrix and centrifugal stiffness matrix, respectively; \(\mathbf{F}_v\) and \(\mathbf{F}\) are the centrifugal force and meshing force, respectively. The basic matrices \(\mathbf{M}_t\), \(\mathbf{M}_r\), and \(\mathbf{K}_e\) are symmetric and depend only on material properties and gear geometry. However, the centrifugal stiffness matrix \(\mathbf{K}_v\) is symmetric and proportional to the square of the driving speed.
The symmetric translational mass matrix for an arbitrary element of the gear is:
$$ \mathbf{M}_t = \rho \Delta L A \begin{bmatrix} \frac{1}{3} & 0 & 0 & \frac{1}{6} & 0 & 0\\ 0 & \frac{13}{35} & \frac{11\Delta L}{210} & 0 & \frac{9}{70} & -\frac{13\Delta L}{420}\\ 0 & \frac{11\Delta L}{210} & \frac{\Delta L^2}{105} & 0 & \frac{13\Delta L}{420} & \frac{\Delta L^2}{140}\\ \frac{1}{6} & 0 & 0 & \frac{1}{3} & 0 & 0\\ 0 & \frac{9}{70} & \frac{13\Delta L}{420} & 0 & \frac{13}{35} & -\frac{\Delta L^2}{105}\\ 0 & -\frac{13\Delta L}{420} & \frac{\Delta L^2}{140} & 0 & -\frac{\Delta L^2}{105} & \frac{\Delta L^2}{105} \end{bmatrix} $$
The rotational inertia mass matrix for an arbitrary element is:
$$ \mathbf{M}_r = \frac{\rho I}{\Delta L} \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0\\ 0 & \frac{6}{5} & \frac{\Delta L}{10} & 0 & -\frac{6}{5} & \frac{\Delta L}{10}\\ 0 & \frac{\Delta L}{10} & \frac{2\Delta L^2}{15} & 0 & -\frac{\Delta L}{10} & -\frac{\Delta L^2}{30}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\frac{6}{5} & -\frac{\Delta L}{10} & 0 & \frac{6}{5} & -\frac{\Delta L}{10}\\ 0 & \frac{\Delta L}{10} & -\frac{\Delta L^2}{30} & 0 & -\frac{\Delta L}{10} & \frac{2\Delta L^2}{15} \end{bmatrix} $$
The gyroscopic damping matrix for an arbitrary element is:
$$ \mathbf{C}_p = \rho \Delta L A \dot{\theta} \begin{bmatrix} 0 & -\frac{7}{10} & -\frac{\Delta L}{10} & 0 & -\frac{3}{10} & \frac{\Delta L}{15}\\ \frac{7}{10} & 0 & 0 & \frac{3}{10} & 0 & 0\\ \frac{\Delta L}{10} & 0 & 0 & \frac{\Delta L}{15} & 0 & 0\\ 0 & -\frac{3}{10} & -\frac{\Delta L}{15} & 0 & -\frac{7}{10} & \frac{\Delta L}{10}\\ \frac{3}{10} & 0 & 0 & \frac{7}{10} & 0 & 0\\ -\frac{\Delta L}{15} & 0 & 0 & \frac{\Delta L}{10} & 0 & 0 \end{bmatrix} $$
The symmetric elastic stiffness matrix for an arbitrary element is:
$$ \mathbf{K}_e = \frac{E}{\Delta L} \begin{bmatrix} A & 0 & 0 & -A & 0 & 0\\ 0 & \frac{12I}{\Delta L^2} & \frac{6I}{\Delta L} & 0 & -\frac{12I}{\Delta L^2} & \frac{6I}{\Delta L}\\ 0 & \frac{6I}{\Delta L} & 4I & 0 & -\frac{6I}{\Delta L} & 2I\\ -A & 0 & 0 & A & 0 & 0\\ 0 & -\frac{12I}{\Delta L^2} & -\frac{6I}{\Delta L} & 0 & \frac{12I}{\Delta L^2} & -\frac{6I}{\Delta L}\\ 0 & \frac{6I}{\Delta L} & 2I & 0 & -\frac{6I}{\Delta L} & 4I \end{bmatrix} $$
The centrifugal stiffness matrix for an arbitrary element is:
$$ \mathbf{K}_v = \frac{\rho A \Delta L \dot{\theta}^2}{210} \begin{bmatrix} -70 & 0 & 0 & -35 & 0 & 0\\ 0 & K_{22}^v & K_{23}^v & 0 & K_{25}^v & K_{26}^v\\ 0 & K_{23}^v & K_{33}^v & 0 & -K_{26}^v & K_{36}^v\\ -35 & 0 & 0 & -70 & 0 & 0\\ 0 & K_{25}^v & -K_{26}^v & 0 & K_{22}^v & -K_{23}^v\\ 0 & K_{26}^v & K_{36}^v & 0 & -K_{23}^v & K_{36}^v \end{bmatrix} $$
where
$$ K_{22}^v = -156 + \frac{504I}{A\Delta L^2},\quad K_{23}^v = -22\Delta L + \frac{42I}{A\Delta L},\quad K_{25}^v = -54\Delta L – \frac{504I}{A\Delta L^2},\quad K_{26}^v = 13\Delta L + \frac{42I}{A\Delta L},\quad K_{33}^v = -4\Delta L^2 + \frac{56I}{A},\quad K_{36}^v = \frac{3A\Delta L^2 – 14I}{A} $$
The Rayleigh damping matrix is given by:
$$ \mathbf{C}_r = \alpha_M (\mathbf{M}_t + \mathbf{M}_r) + \beta_K (\mathbf{K}_e + \mathbf{K}_v) $$
The centrifugal force acting along the gear axis is obtained by integrating the centrifugal force on a differential element \(dx\) of the gear, expressed as:
$$ \mathbf{F}_v = -\frac{\rho A \Delta L \dot{\theta}^2}{2} \left( L + \frac{\Delta L}{3}, \, 0, \, 0, \, L + \frac{2\Delta L}{3}, \, 0, \, 0 \right)^T $$
Calculation Method of Dynamic Mesh Stiffness under Centrifugal Effect
In this section, we use the Newmark algorithm to solve the gear dynamic displacement affected by driving speed, thereby obtaining the dynamic mesh stiffness of the flexible gear considering the centrifugal effect. The elastic deformation produced at the previous meshing point will influence the meshing state of the next meshing point due to the dynamic excitation generated by the driving speed. The meshing process of the gear pair is simulated using a single-tooth gear model. To more accurately reflect the influence of driving speed dynamic excitation on the gear meshing process, we simulate two different conditions: (a) the meshing force on the pinion gradually moves from the initial meshing point B to the meshing exit point A; (b) the meshing force on the gear gradually moves from the initial meshing point A to the meshing exit point B. Since the calculation process for the pinion and gear is identical except for parameter settings, we detail the pinion calculation process below.
The external load matrix \(\mathbf{F}_i\) changes as the gear rotates; at a given time, only one meshing point \(i\) has an external load, and all other nodes have zero external load. The external load matrix is expressed as:
$$ \mathbf{F}_i = \begin{bmatrix} 0 & 0 & 0 & \cdots & F_i \sin(\beta_i) & F_i \cos(\beta_i) & F_i \cos(\beta_i) z_{i,x} & \cdots & 0 & 0 & 0 \end{bmatrix} $$
where \(\beta_i\) is the meshing angle at the \(i\)-th element position on the meshing line, given by:
$$ \beta_i = \arccos\left(\frac{R_{bp}}{\sqrt{x_i^2 + z_{i,x}^2}}\right) – \arctan\left(\frac{z_{i,x}}{x_i}\right) $$
where \(R_{bp}\) is the base circle radius of the pinion. The meshing velocity at the \(i\)-th meshing point can be obtained from the pinion driving speed \(\dot{\theta}\) and the meshing point coordinates:
$$ v_i = \dot{\theta} \sqrt{x_i^2 + z_{i,x}^2} $$
To solve the displacement matrix \(\{\mathbf{X}_i\}\) of the flexible gear under centrifugal effect affected by driving speed, we use the Newmark algorithm to solve the equation of motion. The key parameter is the load step size. We take the average of two consecutive meshing velocities for the interval between two meshing points. The time step \(\Delta t_i\) for gear dynamic displacement is the time interval for the meshing force to move from the previous meshing point to the next meshing point, calculated as:
$$ \Delta t_i = \frac{2\sqrt{\Delta x_i^2 + \Delta y_i^2}}{v_i + v_{i+1}} $$
where \(\Delta x_i\), \(\Delta y_i\) are the elastic deflections of the meshing point in the \(x\) and \(y\) directions, respectively. At the initial meshing point, the initial velocity matrix \(\dot{\mathbf{X}}_1\) and initial acceleration matrix \(\ddot{\mathbf{X}}_1\) are set to zero. We use traditional Hooke’s law to calculate the initial displacement matrix \(\mathbf{X}_1\):
$$ \mathbf{X}_1 = \frac{\mathbf{F}_1}{K} $$
After the above parameters are computed, we use the Newmark algorithm to iteratively calculate \(\mathbf{X}_i\), \(\dot{\mathbf{X}}_i\), and \(\ddot{\mathbf{X}}_i\) until the dynamic load moves to the meshing exit point. The dynamic displacement matrix \(\mathbf{X}_i\) affected by driving speed is thus obtained through iteration. From the iterated \(\mathbf{X}_i\), we extract the elastic deflections \(\Delta x_{i,x}\) and \(\Delta x_{i,y}\) at the \(i\)-th meshing point. Then the single-tooth dynamic stiffness value \(k_{pi}\) of the pinion at that meshing point can be expressed by elastic deflection as:
$$ k_{pi} = \frac{F_i}{\Delta x_{i,x} \cos(\frac{\pi}{2} – \beta_i) + \Delta x_{i,y} \cos \beta_i} $$
Similarly, we obtain the single-tooth dynamic stiffness value \(k_{gi}\) of the gear at the \(i\)-th meshing point. The combined dynamic mesh stiffness of the gear pair affected by driving speed during single-tooth meshing can be expressed as:
$$ k_{ms} = \frac{k_{pi} k_{gi}}{k_{pi} + k_{gi}} $$
A complete meshing cycle contains both single-tooth and double-tooth meshing regions. In the double-tooth meshing region, the dynamic mesh stiffness is in series. To more accurately compute the dynamic mesh stiffness with centrifugal effect, we draw a detailed flowchart of the algorithm (the algorithm flow is described in the following steps: start, input parameters, build mass/stiffness matrices, apply centrifugal force, Newmark time integration, output displacement, compute stiffness per tooth, combine for mesh stiffness, end). The figure below illustrates the gear model used in our analysis.
Verification and Analysis of Dynamic Mesh Stiffness under Centrifugal Effect
To verify the accuracy of the proposed algorithm for calculating dynamic mesh stiffness under centrifugal effect, we compared our results with those obtained using the Ansys method. In the context of Ansys APDL software, the straight spur gear tooth was modeled as a one-dimensional cantilever beam. To minimize the influence of driving speed and centrifugal effect on mesh stiffness, we set the driving speed to \(\dot{\theta}_p = 0.01\) r/min. Under quasi-static conditions, we compared the single-tooth dynamic stiffness (STDS) computed by our method with the static stiffness computed by Ansys. We also present the comparison between the STDS obtained by our method at \(\dot{\theta}_p = 300\) r/min considering centrifugal effect and the single-tooth dynamic stiffness computed by Ansys. In this section, the STDS is computed in the Fortran environment; the Ansys results are obtained by simulating and extracting Euler beam elements in the APDL environment. The parameters of the straight spur gear pair are listed in Table 1.
| Parameter | Pinion/Gear |
|---|---|
| Number of teeth | 27/41 |
| Mass (kg) | 0.22/0.34 |
| Elastic modulus \(E\) (GPa) | 207 |
| Poisson’s ratio | 0.3 |
| Modulus (mm) | 2.5 |
| Width of tooth (mm) | 10 |
| Pressure angle (°) | 20 |
When the driving speed approaches zero, the centrifugal effect has no influence on the STDS. At this point, the STDS value tends to the single-tooth static stiffness. Our results show that the STDS at \(\dot{\theta}_p = 0.01\) r/min is almost identical to the single-tooth static stiffness computed by Ansys. However, when the driving speed is \(\dot{\theta}_p = 300\) r/min, there is a noticeable error between the STDS and the dynamic mesh stiffness computed by Ansys. The main reason is that Ansys cannot calculate the influence of centrifugal force on dynamic mesh stiffness; additionally, when performing dynamic calculations for Euler beam elements, the integration point in Ansys’s built-in finite element theory is at a certain distance from the tip, so the dynamic displacement always deviates from the theoretical solution. The single-tooth dynamic stiffness fluctuates around the single-tooth static stiffness, which is caused by the dynamic excitation generated by the driving speed, and the centrifugal effect amplifies this dynamic excitation. Therefore, the STDS exhibits larger fluctuation amplitudes compared to the single-tooth dynamic stiffness computed by Ansys at \(\dot{\theta}_p = 300\) r/min.
To further verify the role of the proposed algorithm in stiffness calculation, we compared the dynamic mesh stiffness and static mesh stiffness. The dimensionless time \(t/t_c\) is normalized to one meshing cycle \(t_c\). The comparison shows that the dynamic mesh stiffness fluctuates around the static mesh stiffness, and the fluctuation increases with increasing driving speed.
To further investigate the influence of driving speed and its accompanying centrifugal effect on mesh stiffness, we display the single-tooth dynamic stiffness of the pinion and gear at different speeds. It can be observed that as the rotation speed increases, the fluctuation of the single-tooth dynamic stiffness gradually increases, but it always fluctuates around the static stiffness. The reason is that with increasing driving speed, 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 enough time to recover immediately, causing the next meshing point to generate a new deflection. This cumulative deflection process eventually 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 moment. As the meshing process proceeds, the driving speed gradually decreases, and the fluctuation of the single-tooth dynamic stiffness is gradually suppressed until the meshing position ends. With increasing driving speed, the enhanced centrifugal effect increases the fluctuation amplitude of the dynamic single-tooth stiffness. This is because the time interval between two meshing points becomes smaller with increasing driving speed, causing slower deflection recovery.
To more comprehensively reveal the influence of the centrifugal effect on mesh stiffness, Figure [reference] shows a comparative analysis of static mesh stiffness and dynamic mesh stiffness under different centrifugal force conditions. Generally, higher centrifugal force generates more additional elastic potential energy, so the mesh stiffness of both methods increases with increasing driving speed. However, it must be pointed out that the 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 exacerbates the influence of driving speed on mesh stiffness, making the vibration energy in the gear system higher, thus increasing the amplitude fluctuation of dynamic mesh stiffness. Among these, the driving speed has a greater influence on the mesh stiffness in the double-tooth meshing region and a smaller influence on the single-tooth meshing region. This is because the gear system has greater vibration energy in the double-tooth meshing region, and the centrifugal force of the two pairs of teeth has a much greater impact on the system than that of the single pair. The results show that the dynamic mesh stiffness calculated by the new method increases with increasing driving speed, which is consistent with the theoretical qualitative analysis.
Dynamic Mesh Stiffness and Natural Frequency Analysis of Gears Made of Different Materials
Table 2 shows the natural frequencies obtained by the new algorithm and the finite element method at low speed. The relative error between them is within 5%. Therefore, the new algorithm is reliable in solving natural frequencies. Based on this, we studied the influence of centrifugal effect on the natural frequency of flexible gears under two different mass matrix conditions: (1) without considering the rotational inertia mass matrix; (2) considering both the rotational inertia mass matrix and the translational inertia mass matrix.
| Modal | Pinion (New algorithm/Hz) | Pinion (FEM/Hz) | Error (%) | Gear (New algorithm/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 |
We plotted Campbell diagrams of the pinion and gear at different driving speeds with only the translational inertia mass matrix. The results show that as the rotation speed increases, the centrifugal effect does not completely affect every natural frequency of the gear system. The first, third, and fifth natural frequencies of the pinion remain unchanged with increasing driving speed. The second and fifth natural frequencies of the gear increase significantly with driving speed. The results also indicate that as the number of gear teeth decreases, the frequency bifurcation becomes more prominent. The reason is that with increasing tooth number, the support of the gear body to the tooth increases, and the influence of the centrifugal effect on the gear natural frequency decreases.
We also plotted Campbell diagrams of the pinion and gear at different driving speeds with both translational and rotational inertia mass matrices. It is found that the rotational inertia mass matrix has a non-negligible influence on the natural frequencies of the gear system, and when coupled with the centrifugal effect, more diverse frequency characteristics appear. At 12790 r/min, a frequency veering phenomenon can be observed for the first mode, indicating the possible presence of strong coupling of higher rotation modes in the gear system. It also exhibits a special frequency due to centrifugal hardening at different driving speeds when considering rotational inertia mass, which is consistent with previous conclusions on the centrifugal effect, further verifying the influence of centrifugal hardening on the natural frequency of the gear system.
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 common gear materials used in transmissions. Their parameters are listed in Table 3.
| Material | Elastic modulus (GPa) | Density (kg/m³) | Specific modulus (m) |
|---|---|---|---|
| Hard aluminum alloy | 70 | 2.7 | 25.92 |
| Cast iron | 207 | 7.89 | 26.24 |
| Carbon fiber nylon | 230 | 1.76 | 130.68 |
| Ceramics | 410 | 3.15 | 130.16 |
We compared the growth rate and volatility rate of different materials. It can be seen that the growth rate and volatility rate of dynamic mesh stiffness continue to rise with increasing driving speed. The dynamic mesh stiffness of aluminum alloy gear shows a significant stiffness growth, which is attributed to the enhanced centrifugal effect caused by low elastic modulus. Although the elastic modulus of cast iron and carbon fiber nylon are similar, their growth curves show obvious differences. The main reason is that the density of cast iron is greater than that of carbon fiber nylon, which may lead to larger fluctuation amplitudes in dynamic mesh stiffness. In addition, the growth curve of cast iron gear lies between those of aluminum alloy gear and ceramic gear. This observation indicates that the influence of material density on dynamic mesh stiffness is significantly smaller than that 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 suggests that under high driving speed conditions, carbon fiber nylon material gears have higher stability compared to cast iron gears.
We further studied the influence of two different mass matrix conditions on the stiffness growth caused by centrifugal effect for four materials: (1) without considering the rotational inertia mass matrix; (2) considering both mass matrices. The numerical simulation results are shown in Table 4.
| Mass matrix | Material | 2000 r/min | 6000 r/min | 10000 r/min |
|---|---|---|---|---|
| \(\mathbf{M}_t\) | Hard aluminum alloy | 14.31 | 50.24 | 70.10 |
| \(\mathbf{M}_t + \mathbf{M}_r\) | Hard aluminum alloy | 15.71 | 53.85 | 86.50 |
| \(\mathbf{M}_t\) | Cast iron | 12.78 | 46.77 | 65.57 |
| \(\mathbf{M}_t + \mathbf{M}_r\) | Cast iron | 13.45 | 49.93 | 69.02 |
| \(\mathbf{M}_t\) | Ceramics | 3.39 | 21.73 | 38.26 |
| \(\mathbf{M}_t + \mathbf{M}_r\) | Ceramics | 3.83 | 24.60 | 47.10 |
| \(\mathbf{M}_t\) | Carbon fiber nylon | 3.30 | 20.34 | 36.56 |
| \(\mathbf{M}_t + \mathbf{M}_r\) | Carbon fiber nylon | 3.30 | 21.74 | 37.71 |
Table 4 shows that the growth rates obtained with only the translational inertia mass matrix are different from the results in Figure [reference], and the calculation error becomes larger as the driving speed increases. Therefore, the influence of changing the mass matrix setting on the calculation of mesh stiffness under centrifugal effect cannot be ignored.
Conclusion
In this study, the governing equations of motion for a straight spur gear system were derived based on Hamilton’s principle. The model combines the centrifugal effect with meshing deformation, extending the gear dynamic equations. We proposed an original computational algorithm based on a finite element analysis framework to calculate the influence of the centrifugal effect on dynamic mesh stiffness. The numerical analysis results show that the driving speed, elastic modulus, centrifugal effect, and density jointly affect the dynamic mesh stiffness of the gear system. The main conclusions are drawn as follows:
(1) The dynamic mesh stiffness always fluctuates around the static mesh stiffness; as the driving speed increases, the dynamic mesh stiffness exhibits obvious centrifugal hardening and fluctuation phenomena. Moreover, the dynamic excitation generated by the driving speed has a greater influence on double-tooth meshing and a smaller influence on single-tooth meshing.
(2) The dynamic mesh stiffness calculated by the proposed model is more realistic than that of traditional models, especially under high-speed driving conditions or for flexible material gears. Under the influence of the centrifugal effect, frequency veering of the gear natural frequency occurs at 12,790 r/min.
(3) Under the centrifugal effect, the influence of rotational mass on vibration characteristics becomes more significant as the rotation speed increases. Different gear systems exhibit significant differences at 6,000 r/min, so using two mass matrices yields higher accuracy in calculations under high driving speeds. Flexible gears should select an appropriate driving speed based on specific operating conditions, which is beneficial for improving the transmission performance of the gear system.