In our investigation of gear dynamics, we have identified that the driving speed plays a critical role in evaluating mesh stiffness, a factor frequently overlooked in conventional analyses along with the associated centrifugal effects. In this study, we propose a novel computational algorithm based on Euler beam theory to calculate the dynamic mesh stiffness of straight spur gears by incorporating centrifugal effects within the velocity field. Using the driving speed as a control parameter, we systematically investigate the dynamic mesh stiffness behavior of straight spur gears and demonstrate the nonlinear relationship between centrifugal effects and dynamic mesh stiffness. Our results reveal that both the natural frequency and the fluctuations in dynamic mesh stiffness increase with rising driving speed when centrifugal fields are present. Furthermore, we observe that materials with high elastic modulus tend to suppress the influence of driving speed on dynamic mesh stiffness, whereas higher density amplifies this effect. The findings provide a solid foundation for further analysis of vibration and noise in straight spur gears operating under centrifugal conditions.
Introduction
Straight spur gears are fundamental components in mechanical transmission systems, widely employed in new energy vehicles and aerospace machinery. Mesh stiffness serves as the primary source of internal excitation and plays a crucial role in gear dynamics. Accurate evaluation of mesh stiffness is essential for improving transmission precision and optimizing gear structure design in these fields. Numerous methodologies exist in the literature for calculating the mesh stiffness of straight spur gears, including experimental approaches, potential energy methods, finite element methods, and hybrid techniques. Among these, the potential energy method is one of the most commonly used due to its computational efficiency.
Previous researchers have extended the potential energy method by incorporating five potential energy components rather than the traditional three to address computational difficulties associated with profile modifications. Others have refined tooth profiles by introducing transition curve parameter equations to correct integration limits, thereby enhancing tooth model accuracy. Hybrid methods combining finite element and potential energy approaches have also been developed to reduce computational time while maintaining accuracy. More recently, attention has shifted toward computing mesh stiffness under driving speed conditions to achieve more precise results. However, these studies have largely remained within the domain of static analysis.
In reality, gear meshing is inherently a dynamic process, and driving speed is a key parameter in gear dynamics. When we consider driving speed as a control parameter, the dynamic response in both time and frequency domains undergoes significant changes. In the field of gear dynamics, the influence of driving speed on mesh stiffness is accompanied by centrifugal effects, which are universal phenomena where material strength increases with driving speed. High driving speeds generate substantial centrifugal forces that significantly affect the deformation response of gear systems. Therefore, an in-depth investigation of centrifugal effects on mesh stiffness carries substantial engineering significance and theoretical value.
Previous studies have shown that centrifugal loads can influence tooth root bending stresses in thin-webbed gears, and the first modal frequency of gears varies with centrifugal force. The centrifugal effect also impacts crack initiation points and crack propagation paths in gear teeth. However, these studies have typically been evaluated under quasi-static centrifugal loads. Revealing the influence of centrifugal effects on dynamic mesh stiffness in dynamic calculations remains a challenge. To address these issues, we have developed a more realistic model of straight spur gears that accounts for centrifugal effects and proposed a novel computational algorithm that incorporates driving speed and centrifugal effects to calculate dynamic mesh stiffness.
In this work, we derive the equations of motion for straight spur gear systems based on Hamilton’s principle, formulate the centrifugal force expressions, and obtain gear deformation and mesh stiffness under dynamic conditions. Our approach provides valuable references for improving the transmission performance and reducing vibration and noise of straight spur gears operating at high driving speeds.
Equations of Motion for Rotating Flexible Gears
In rotor dynamic systems, centrifugal effects are common phenomena. During gear rotation, changes in driving speed convert kinetic energy into potential energy, affecting the deformation of gear pairs. Due to the flexibility of teeth under centrifugal action, the gear exhibits two distinct states: a normal state under static load and an expanded state under centrifugal force. As the driving speed increases, the meshing point in the gear moves away from the rotation center, and this phenomenon becomes more pronounced.
In our gear model, we employ Euler beam elements to simplify the gear tooth as a cantilever beam model. During simulation, the bore radius of the gear remains fixed. The total displacement vector of a given point on the gear can be expressed as follows:
$$ \mathbf{P}^T = \begin{bmatrix} x – u_r + u \cos\theta – v \sin\theta \\ z_{i,x} + v_r + u \sin\theta + v \cos\theta \end{bmatrix} $$
The velocity vector of the gear after centrifugal expansion relative to the gear rotation center is expressed as:
$$ \dot{\mathbf{P}}^T = \begin{bmatrix} -[(u+x)\dot{\theta} – (z_{i,x}\dot{\theta} – \dot{u})]\cos\theta – [(x+u)\dot{\theta} + \dot{v}]\sin\theta \\ -[(v+z_{i,x})\dot{\theta} + (z_{i,x}\dot{\theta} – \dot{u})]\sin\theta + [(x+u)\dot{\theta} + \dot{v}]\cos\theta \end{bmatrix} $$
where the dot notation represents derivatives with respect to time.
Based on the total displacement of the gear, we can derive the kinetic energy stored in the gear as:
$$ T_i = \frac{1}{2}\rho \int \dot{\mathbf{P}}\dot{\mathbf{P}}^T dV $$
where $\rho$ is the density and $dV$ represents integration over the gear volume.
In our analysis, we consider only the axial component of the strain tensor for the flexible gear. The nonlinear axial strain is expressed as:
$$ \varepsilon_{xx} = u’ + \frac{1}{2}[(u’)^2 + (v’)^2] $$
where the prime notation denotes derivatives with respect to $x$.
Considering that the cross-section of a single gear tooth is symmetrically distributed about the neutral layer, and using the axial strain definition, the potential energy of the gear can be expressed in terms of transverse displacement. The gear potential energy is given by:
$$ U = \frac{1}{2}\int E\varepsilon_{xx}^2 dV $$
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, we derive the equations of motion for the flexible gear:
$$ (M_t + M_r)\ddot{\mathbf{X}} + (C_r + C_p)\dot{\mathbf{X}} + (K_e + K_v)\mathbf{X} = \mathbf{F} + \mathbf{F}_v $$
where $M_t$ and $M_r$ are the translational mass matrix and rotational inertia mass matrix of the gear, respectively; $C_r$ and $C_p$ are the Rayleigh damping coefficient matrix and gyroscopic damping matrix, respectively; $K_e$ and $K_v$ 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 of mass $M_t$, $M_r$ and stiffness $K_e$ are symmetric and depend only on material properties and gear geometry. However, the centrifugal stiffness matrix $K_v$ is also symmetric but is proportional to the square of the driving speed.
The symmetric translational mass matrix for an arbitrary element of the gear is expressed as:
$$ M_t = \rho \Delta L A \begin{bmatrix} \frac{1}{3} & 0 & 0 & \frac{1}{6} & 0 & 0 \\ 0 & \frac{13}{35} & \frac{11\Delta L}{210} & 0 & \frac{9}{70} & -\frac{13\Delta L}{420} \\ 0 & \frac{11\Delta L}{210} & \frac{\Delta L^2}{105} & 0 & \frac{13\Delta L}{420} & \frac{\Delta L^2}{140} \\ \frac{1}{6} & 0 & 0 & \frac{13}{35} & 0 & 0 \\ 0 & \frac{9}{70} & \frac{13\Delta L}{420} & 0 & \frac{13}{35} & -\frac{\Delta L^2}{105} \\ 0 & -\frac{13\Delta L}{420} & \frac{\Delta L^2}{140} & 0 & -\frac{\Delta L^2}{105} & \frac{\Delta L^2}{105} \end{bmatrix} $$
where $A$ represents the cross-sectional area of the gear tooth.
The rotational inertia mass matrix for an arbitrary element is expressed as:
$$ M_r = \rho \frac{I}{\Delta L} \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{6}{5} & \frac{\Delta L}{10} & 0 & -\frac{6}{5} & \frac{\Delta L}{10} \\ 0 & \frac{\Delta L}{10} & \frac{2\Delta L^2}{15} & 0 & -\frac{\Delta L}{10} & -\frac{\Delta L^2}{30} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -\frac{6}{5} & -\frac{\Delta L}{10} & 0 & \frac{6}{5} & -\frac{\Delta L}{10} \\ 0 & \frac{\Delta L}{10} & -\frac{\Delta L^2}{30} & 0 & -\frac{\Delta L}{10} & \frac{2\Delta L^2}{15} \end{bmatrix} $$
where $I$ denotes the moment of inertia of the gear cross-section.
The gyroscopic damping matrix for an arbitrary element is expressed as:
$$ C_p = \rho \Delta L A \dot{\theta} \begin{bmatrix} 0 & -\frac{7}{10} & -\frac{\Delta L}{10} & 0 & -\frac{3}{10} & \frac{\Delta L}{15} \\ \frac{7}{10} & 0 & 0 & \frac{3}{10} & 0 & 0 \\ \frac{\Delta L}{10} & 0 & 0 & \frac{\Delta L}{15} & 0 & 0 \\ 0 & -\frac{3}{10} & -\frac{\Delta L}{15} & 0 & -\frac{7}{10} & \frac{\Delta L}{10} \\ \frac{3}{10} & 0 & 0 & \frac{7}{10} & 0 & 0 \\ -\frac{\Delta L}{15} & 0 & 0 & \frac{\Delta L}{10} & 0 & 0 \end{bmatrix} $$
The symmetric elastic stiffness matrix for an arbitrary element is expressed as:
$$ K_e = \frac{E}{\Delta L} \begin{bmatrix} A & 0 & 0 & -A & 0 & 0 \\ 0 & \frac{12I}{\Delta L^2} & \frac{6I}{\Delta L} & 0 & -\frac{12I}{\Delta L^2} & \frac{6I}{\Delta L} \\ 0 & \frac{6I}{\Delta L} & 4I & 0 & -\frac{6I}{\Delta L} & 2I \\ -A & 0 & 0 & A & 0 & 0 \\ 0 & -\frac{12I}{\Delta L^2} & -\frac{6I}{\Delta L} & 0 & \frac{12I}{\Delta L^2} & -\frac{6I}{\Delta L} \\ 0 & \frac{6I}{\Delta L} & 2I & 0 & -\frac{6I}{\Delta L} & 4I \end{bmatrix} $$
The centrifugal stiffness matrix for an arbitrary element is expressed as:
$$ K_v = \frac{\rho A \Delta L \dot{\theta}^2}{210} \begin{bmatrix} -70 & 0 & 0 & -35 & 0 & 0 \\ 0 & K_{22v} & K_{23v} & 0 & K_{25v} & K_{26v} \\ 0 & K_{23v} & K_{33v} & 0 & -K_{26v} & K_{36v} \\ -35 & 0 & 0 & -70 & 0 & 0 \\ 0 & K_{25v} & -K_{26v} & 0 & K_{22v} & -K_{23v} \\ 0 & K_{26v} & K_{36v} & 0 & -K_{23v} & K_{36v} \end{bmatrix} $$
where the components are defined as follows:
$$ \begin{aligned} K_{22v} &= -156 + \frac{504I}{A\Delta L^2} \\ K_{23v} &= -22\Delta L + \frac{42I}{A\Delta L} \\ K_{25v} &= -54\Delta L – \frac{504I}{A\Delta L^2} \\ K_{26v} &= 13\Delta L + \frac{42I}{A\Delta L} \\ K_{33v} &= -4\Delta L^2 + \frac{56I}{A} \\ K_{36v} &= \frac{3A\Delta L^2 – 14I}{A} \end{aligned} $$
The Rayleigh damping matrix for an arbitrary element is given by:
$$ C_r = \alpha_M (M_t + M_r) + \beta_K (K_e + K_v) $$
By integrating the centrifugal force over the differential element $dx$ of the gear, we obtain the centrifugal force at the nodes, which acts along the gear axial direction:
$$ \mathbf{F}_v = -\frac{\rho A \Delta L \dot{\theta}^2}{2} \left(L + \frac{\Delta L}{3}, 0, 0, L + \frac{2\Delta L}{3}, 0, 0\right)^T $$
Computational Method for Dynamic Mesh Stiffness Under Centrifugal Effect
In this section, we employ the Newmark algorithm to solve for the dynamic displacement of the gear affected 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 of the gear tooth influences the meshing state at the subsequent meshing point through the dynamic excitation induced by driving speed.
To accurately reflect the influence of dynamic excitation from driving speed on the gear meshing process, we simulate two different operating conditions: (1) the meshing force on the pinion gradually engages from the initial meshing point to the gear meshing-out point; (2) the meshing force on the gear gradually engages from the initial meshing point to the gear meshing-out point. Since the calculation process for the pinion and gear is identical, with only parameter differences, we describe the calculation process using the pinion as an example.
The external load matrix varies during gear rotation, meaning that at any given time, only one meshing point experiences an external load, while all other nodes have zero external load. The external load matrix can be 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 $\beta_i$ represents the meshing angle at the $i$-th element position on the meshing line of the gear tooth, which can be expressed as:
$$ \beta_i = \arccos\left(\frac{R_{bp}}{\sqrt{x_i^2 + z_{i,x}^2}}\right) – \arctan\left(\frac{z_{i,x}}{x_i}\right) $$
where $R_{bp}$ is the base circle radius of the pinion.
The meshing velocity at the $i$-th meshing point can be determined from the pinion driving speed and the coordinates of the meshing point:
$$ v_i = \dot{\theta} \sqrt{x_i^2 + z_{i,x}^2} $$
To solve for the displacement matrix $\{\mathbf{X}_i\}$ of the flexible gear considering centrifugal effects under driving speed influence, we use the Newmark algorithm to solve the equations of motion. The key parameter is the load step size. In our approach, we take the average of two consecutive meshing velocities to represent the meshing velocity between two adjacent meshing points. The time step for gear dynamic displacement is the time interval during which the meshing force moves from the previous meshing point to the next meshing point:
$$ \Delta t_i = \frac{2\sqrt{\Delta x_i^2 + \Delta y_i^2}}{v_i + v_{i+1}} $$
where $\Delta x_i$ and $\Delta y_i$ are the elastic deflections of the meshing point in the $x$ and $y$ directions, respectively.
In the cyclic calculation, the initial velocity matrix $\dot{\mathbf{X}}_1$ and initial acceleration matrix $\ddot{\mathbf{X}}_1$ at the initial meshing point are both set to zero. We use the traditional Hooke’s law method to calculate the initial displacement matrix $\mathbf{X}_1$:
$$ \mathbf{X}_1 = \frac{\mathbf{F}_1}{K} $$
where $\mathbf{F}_1$ is the external load matrix of the meshing force at the initial meshing point.
After completing the calculation of the above parameters, we use the Newmark algorithm to iteratively compute $\mathbf{X}_i$, $\dot{\mathbf{X}}_i$, and $\ddot{\mathbf{X}}_i$ in the equations of motion until the dynamic load moves to the meshing-out point. Through this iterative process, we obtain the dynamic displacement matrix $\mathbf{X}_i$ affected by driving speed. From the iteratively computed $\mathbf{X}_i$, we extract the elastic deflections $\Delta x_{i,x}$ and $\Delta x_{i,y}$ at the $i$-th meshing point. Subsequently, the single-tooth dynamic stiffness value of the pinion at that meshing point can be expressed using the elastic deflections:
$$ k_{pi} = \frac{F_i}{\Delta x_{i,x} \cos\left(\frac{\pi}{2} – \beta_i\right) + \Delta x_{i,y} \cos\beta_i} $$
Similarly, we obtain the single-tooth dynamic stiffness value of the gear at the $i$-th meshing point, denoted as $k_{gi}$. The comprehensive dynamic mesh stiffness of the gear pair affected by driving speed during single-tooth meshing can be expressed as:
$$ k_{ms} = \frac{k_{pi} k_{gi}}{k_{pi} + k_{gi}} $$
Both single-tooth meshing zones and double-tooth meshing zones exist within a complete meshing cycle. In the double-tooth meshing zone, the dynamic mesh stiffness is in a series relationship. To accurately calculate the dynamic mesh stiffness with centrifugal effects, we have developed a detailed algorithm flowchart for this computational procedure.
Validation and Analysis of Dynamic Mesh Stiffness Under Centrifugal Effect
To validate the accuracy of our proposed algorithm for calculating dynamic mesh stiffness under centrifugal effects, we compared our results with those obtained from the Ansys method. In the Ansys APDL environment, we modeled the straight spur gear tooth as a one-dimensional cantilever beam. To minimize the influence of driving speed and centrifugal effects on mesh stiffness, we set the driving speed to a very low value. Under quasi-static conditions, we compared the single-tooth dynamic stiffness (STDS) calculated by our method with the static stiffness calculated by Ansys.
The parameters of the straight spur gear pair used in our validation are summarized in the following table:
| Parameter | Pinion/Gear |
|---|---|
| Number of teeth | 27/41 |
| Mass (kg) | 0.22/0.34 |
| Elastic modulus (GPa) | 207 |
| Poisson 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 single-tooth dynamic stiffness (STDS). Under this condition, the STDS value approaches the single-tooth static stiffness. Our comparison showed that the STDS at a very low driving speed and the single-tooth static stiffness calculated by Ansys were almost identical, confirming the accuracy of our method.
However, at higher driving speeds, the STDS and the dynamic mesh stiffness calculated by Ansys showed considerable discrepancies. The primary reason for this is that Ansys cannot account for the influence of centrifugal force on dynamic mesh stiffness. Additionally, when performing dynamic calculations for Euler beam elements, there is always a certain distance from the integration point to the tip in the finite element theory built into Ansys, which leads to inherent errors between the dynamic displacement and the theoretical solution. The single-tooth dynamic stiffness fluctuates around the single-tooth static stiffness due to the dynamic excitation generated by driving speed, and the centrifugal effect amplifies this dynamic excitation.
To further verify the effectiveness of our algorithm in stiffness calculation, we compared the dynamic mesh stiffness with the static mesh stiffness under various conditions. The dimensionless time was normalized with respect to one meshing cycle. Our results demonstrated that the dynamic mesh stiffness fluctuates around the static mesh stiffness, with the amplitude of fluctuation increasing with driving speed.
To more comprehensively investigate the influence of driving speed and its accompanying centrifugal effects on mesh stiffness, we examined the single-tooth dynamic stiffness of both the pinion and gear at different speeds. Our observations revealed that as rotational speed increases, the fluctuations in single-tooth dynamic stiffness gradually increase, but always fluctuate around the static stiffness. The reason for this behavior is that as driving speed increases, the time interval between dynamic excitations acting on adjacent meshing points decreases. Consequently, the deflection of the cantilever beam at the previous meshing point does not have sufficient time to recover immediately, resulting in new deflections at the subsequent meshing point. This cumulative process of deflections ultimately produces this phenomenon.
Furthermore, we observed that the stiffness fluctuation at the initial meshing point is larger than at other meshing points. This is because the kinetic energy and amplitude of the pinion reach their maximum values at this point. As the meshing process progresses, the driving speed gradually decreases, and the fluctuation in single-tooth dynamic stiffness is gradually suppressed until the meshing position ends. With increasing driving speed, the enhanced centrifugal effect causes the fluctuation amplitude of the dynamic single-tooth stiffness to increase. This occurs because the time interval between two meshing points becomes smaller as driving speed increases, leading to slower deflection recovery.
We also performed a comparative analysis of static mesh stiffness and dynamic mesh stiffness under different centrifugal force conditions. Generally, higher centrifugal forces generate more additional elastic potential energy, so the mesh stiffness from both methods increases with driving speed. However, it is important to note that centrifugal force not only enhances dynamic mesh stiffness but also amplifies its amplitude fluctuations, which contrasts sharply with static mesh stiffness. The primary reason for this difference is that the centrifugal effect intensifies the influence of driving speed on mesh stiffness, resulting in higher vibration energy in the gear system and consequently increasing the amplitude fluctuations of dynamic mesh stiffness. The driving speed has a greater influence on the mesh stiffness in the double-tooth meshing region compared to the single-tooth meshing region, because the gear system possesses greater vibration energy in the double-tooth meshing region, and the centrifugal force from two pairs of teeth has a much larger impact on the system than that from a single pair of teeth. Our results confirm that the dynamic mesh stiffness calculated by our method increases with driving speed, which is consistent with theoretical expectations.
Natural Frequency Analysis of Straight Spur Gears Under Centrifugal Effects
To validate the reliability of our algorithm in determining natural frequencies, we compared the natural frequencies calculated by our new method with those obtained from finite element analysis at low rotational speeds. The relative errors between the two methods were within 5%, confirming the reliability of our algorithm in solving natural frequencies. Building on this validation, we investigated the influence of centrifugal effects on the natural frequencies of flexible gears under two different mass matrix conditions: (1) without considering the rotational inertia mass matrix, and (2) considering both the rotational inertia mass matrix and the translational inertia mass matrix.
The following table presents the comparison of natural frequencies obtained from our algorithm and finite element analysis:
| Mode | Pinion | Gear | ||||
|---|---|---|---|---|---|---|
| New Algorithm (Hz) | FEM (Hz) | Error (%) | New Algorithm (Hz) | 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 we set only the translational inertia mass matrix, we obtained Campbell diagrams for the pinion and gear at different driving speeds. Our results showed that as rotational speed increases, the centrifugal effect does not completely affect every order of natural frequency of the gear system. The first, third, and fifth order frequencies of the pinion remain unchanged with increasing driving speed, while the second and fifth order frequencies of the gear increase significantly with driving speed. Our observations also indicated that as the number of gear teeth decreases, frequency bifurcation becomes more pronounced. This is because with fewer teeth, the support provided by the gear body to the teeth decreases, and the influence of centrifugal effects on the natural frequencies of the gear becomes more significant.
When we considered both the translational inertia mass matrix and the rotational inertia mass matrix, we observed more diverse frequency characteristics of the gear system under the coupling of centrifugal effects. The rotational inertia mass matrix has a non-negligible influence on the natural frequencies of the gear system, exhibiting more diverse frequency characteristics in the coupling with centrifugal effects. At higher rotational speeds, we observed a frequency veering phenomenon in the first-order frequency, indicating the possible presence of strong coupling in higher rotational modes of the gear system. We also observed a special frequency generated by the centrifugal stiffening effect at different driving speeds when the rotational inertia mass is considered. These findings are consistent with previous conclusions regarding centrifugal effects, further validating the influence of centrifugal stiffening on the natural frequencies of gear systems.
Material Effects on Dynamic Mesh Stiffness of Straight Spur Gears
Since gears made of different materials exhibit distinct dynamic characteristics under the influence of driving speed, investigating the effects of material differences on dynamic mesh stiffness is of significant importance. Aluminum alloy, cast iron, ceramics, and carbon fiber nylon are commonly used gear materials in transmission systems. The material parameters used in our analysis are summarized in the following table:
| 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 |
Our analysis of the growth rate and volatility rate of dynamic mesh stiffness for different materials revealed several important trends. The growth rate and volatility rate of dynamic mesh stiffness continuously increase with driving speed for all materials examined. Aluminum alloy gears exhibited significant stiffness growth, attributable to the enhanced centrifugal effect caused by their low elastic modulus. Although cast iron and carbon fiber nylon have similar elastic moduli, their growth curves showed distinct differences. This phenomenon primarily occurs because the density of cast iron is higher than that of carbon fiber nylon, which may lead to larger fluctuation amplitudes in dynamic mesh stiffness. Additionally, the growth curve of cast iron gears falls between those of aluminum alloy gears and ceramic material gears. 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 lightweight and robust strength, carbon fiber nylon can effectively withstand impacts, stresses, and vibrations at high driving speeds. This indicates that under high driving speed conditions, carbon fiber nylon gears possess higher stability compared to cast iron gears. The following table presents the influence of mass matrix settings on the growth rate of different materials:
| Mass Matrix | Material | 2000 r/min | 6000 r/min | 10000 r/min |
|---|---|---|---|---|
| $M_t$ | Hard aluminum alloy | 14.31 | 50.24 | 70.10 |
| $M_t + M_r$ | Hard aluminum alloy | 15.71 | 53.85 | 86.50 |
| $M_t$ | Cast iron | 12.78 | 46.77 | 65.57 |
| $M_t + M_r$ | Cast iron | 13.45 | 49.93 | 69.02 |
| $M_t$ | Ceramics | 3.39 | 21.73 | 38.26 |
| $M_t + M_r$ | Ceramics | 3.83 | 24.60 | 47.10 |
| $M_t$ | Carbon fiber nylon | 3.30 | 20.34 | 36.56 |
| $M_t + M_r$ | Carbon fiber nylon | 3.30 | 21.74 | 37.71 |
Our numerical results clearly show that when only the translational inertia mass matrix is set, the growth rates of different materials under centrifugal effects exhibit increasing calculation errors compared to the results obtained with both mass matrices as driving speed increases. The difference becomes more pronounced at higher driving speeds, demonstrating that the influence of mass matrix configuration on the mesh stiffness calculation under centrifugal effects cannot be ignored. This finding highlights the importance of including both mass matrices for accurate dynamic mesh stiffness calculations, especially at high driving speeds where centrifugal effects are more prominent.
The comparison between materials reveals that hard aluminum alloy exhibits the highest growth rate at all driving speeds, followed by cast iron, while ceramics and carbon fiber nylon show the lowest growth rates. This trend correlates well with the elastic modulus of these materials: materials with lower elastic modulus experience greater centrifugal stiffening effects, leading to higher growth rates in dynamic mesh stiffness. However, density also plays a role, as evidenced by the differences between cast iron and carbon fiber nylon, which have similar elastic moduli but different densities. The higher density of cast iron results in slightly higher growth rates compared to carbon fiber nylon.
Our analysis of volatility rates shows similar trends, with aluminum alloy exhibiting the highest volatility and carbon fiber nylon showing the lowest. This suggests that carbon fiber nylon gears may offer better stability at high driving speeds, making them potentially more suitable for high-speed applications where vibration and noise reduction are critical concerns.
Conclusions
In this study, we derived the equations of motion for straight spur gear systems using Hamilton’s principle. Our model integrates centrifugal effects with meshing deformation, extending the gear dynamic equations. We proposed a novel computational algorithm based on a finite element analysis framework to calculate the influence of centrifugal effects on dynamic mesh stiffness. Our numerical analysis results demonstrate that driving speed, elastic modulus, centrifugal effects, and density collectively influence the dynamic mesh stiffness of gear systems. The main conclusions of our work are as follows:
First, the dynamic mesh stiffness of straight spur gears always fluctuates around the static mesh stiffness. As driving speed increases, the dynamic mesh stiffness exhibits明显的 centrifugal stiffening and波动现象. The dynamic excitation generated by driving speed has a greater influence on double-tooth meshing compared to single-tooth meshing, with the double-tooth meshing region showing more pronounced changes in mesh stiffness.
Second, the dynamic mesh stiffness calculated using our proposed model is more realistic than that obtained from traditional models, particularly under high-speed operating conditions or when using flexible materials. Under the influence of centrifugal effects, we observed frequency veering phenomena in the natural frequencies of gears at higher rotational speeds, indicating complex dynamic behavior that cannot be captured by conventional static or quasi-static analyses.
Third, under the influence of centrifugal effects, the rotational mass has an increasingly significant impact on vibration characteristics as rotational speed increases. Different gear systems exhibit显著 differences at high driving speeds, and the use of two mass matrices in calculations provides higher accuracy for high-speed applications. Flexible gears should be operated at appropriate driving speeds based on specific working conditions to improve the transmission performance of the gear system.
Our research provides a comprehensive framework for understanding the combined effects of driving speed and centrifugal forces on the dynamic behavior of straight spur gears. The findings offer valuable guidance for the design and optimization of high-speed gear transmission systems, particularly in applications where vibration, noise, and dynamic stability are critical performance metrics. Future work may extend this analysis to consider additional factors such as tooth profile modifications, lubrication effects, and more complex gear geometries including helical and bevel gears operating under centrifugal conditions.