Finite Element Modeling Methodology for a Two-Stage Spur Gear Transmission System Under Multi-Source Time-Varying Excitation

1. Introduction

The machinery industry serves as a foundational pillar of the industrial sector, playing a critical role in the growth of the national economy and the enhancement of comprehensive national strength. Gear transmission systems, as indispensable components of mechanical power transmission, offer advantages such as high efficiency, robust power transmission capacity, and relatively low noise generation. Consequently, they are extensively deployed in aerospace, marine propulsion, mining, and numerous other industrial domains. In the contemporary era, the development trends emphasizing low vibration, low noise, and enhanced comfort have imposed increasingly stringent requirements on the vibrational performance of gear systems. Therefore, the analysis of dynamic characteristics and the prediction of vibration and noise in gear systems have become critically important for engineering practice.

This thesis focuses on a two-stage spur gear reducer. To acquire more accurate dynamic loads on gears and bearings for predictive evaluation of the transmission system’s dynamic response, a coupled dynamic model of the gear-transmission shaft-bearing system was established. This model incorporates the flexibility of the transmission shaft, the time-varying mesh stiffness of gear pairs, the time-varying stiffness of bearings, and the phase relationship between the two gear stages. The dynamic characteristics of the two-stage spur gear transmission system under multi-source time-varying excitation were obtained by solving the equations of motion using the Newmark time-domain integration method. Additionally, systematic analyses were conducted to investigate the influence of shaft stiffness, the second-stage phase relationship, and gear installation positions on the system’s dynamic response.

2. Research Status and Literature Review

Since the introduction of a mass-spring model to describe gear transmission mechanical systems by early researchers, the theoretical framework of gear system dynamics has been continuously refined. Scholars worldwide have progressively incorporated factors such as transmission error, vibration impact, and backlash. Current analytical models can be broadly classified into four categories: pure torsional models, gear coupling models, rotor coupling models, and gear-rotor-housing system models. From a methodological perspective, established approaches include the lumped mass method, the finite element method, and multi-body dynamics methods.

3. Establishment of the Dynamic Model for a Two-Stage Spur Gear Transmission System

This chapter presents a comprehensive generalized finite element method for modeling a two-stage spur gear transmission system. The system under investigation is a two-stage parallel-shaft spur gear reducer. The modeling process involves discretizing the continuous mechanical system into distinct fundamental units: shaft elements, gear meshing elements, and bearing elements. This discretization allows for a detailed representation of the system’s physical properties, including the influence of shaft geometry and material properties.

3.1 System Configuration

The physical model of the two-stage spur gear transmission system is illustrated in the related literature. It comprises an input shaft, an intermediate shaft, and an output shaft, supported by six deep groove ball bearings. The first stage consists of a pinion and gear, and the second stage consists of another pinion and gear. The primary geometric parameters for the shafts and gears are summarized in the following tables based on the DDS drivetrain diagnostics test bench parameters.

**Table 3.1: Shaft Parameters of the Two-Stage Spur Gear Transmission System**
| Parameter Name | Shaft Length (m) | Radius (mm) | Density (kg/m³) | Shear Modulus (Pa) |
|—|—|—|—|—|
| Input Shaft | 0.24 | 12.5 | 7850 | 8×10¹⁰ |
| Intermediate Shaft | 0.16 | 12.5 | 7850 | 8×10¹⁰ |
| Output Shaft | 0.18 | 12.5 | 7850 | 8×10¹⁰ |

**Table 3.2: Gear Parameters for the Two-Stage Spur Gear System**
| Parameter Name | Stage 1 Pinion | Stage 1 Gear | Stage 2 Pinion | Stage 2 Gear |
|—|—|—|—|—|
| Moment of Inertia (kg·m²) | 2×10⁻⁴ | 3.04×10⁻³ | 1×10⁻⁴ | 8.71×10⁻³ |
| Number of Teeth | 36 | 90 | 29 | 100 |
| Module (mm) | 1.5 | 1.5 | 1.5 | 1.5 |
| Pressure Angle (°) | 20 | 20 | 20 | 20 |
| Face Width (mm) | 12 | 12 | 12 | 12 |
| Mass (kg) | 0.16 | 1.3 | 0.09 | 1.6 |

3.2 Finite Element Discretization

The overall system is discretized into a series of finite elements, resulting in a total of 29 shaft elements and 95 degrees of freedom. The input shaft is divided into 12 elements, the intermediate shaft into 8 elements, and the output shaft into 9 elements. Nodes are numbered sequentially for each element. The coupling between different components, such as shaft-bearing and shaft-gear interactions, is established through shared nodes. The first stage gear pair is located on nodes 7 and 16 of the input and intermediate shafts, respectively. The second stage gear pair is located on nodes 20 (intermediate shaft) and 29 (output shaft). This assembly process forms the overall stiffness matrix for the system.

3.3 Element Dynamics Models

3.3.1 Shaft Element Model

To effectively account for shaft flexibility, a spatial beam element model was adopted. Given the relatively small diameter-to-length ratio of the shaft elements, shear deformation cannot be neglected. Therefore, Timoshenko beam theory was employed over Euler-Bernoulli theory. For a typical shaft element with two nodes, each having lateral translational and torsional degrees of freedom, the nodal displacement vector is defined as:
\[
\{ \delta_e \} = [ v_i, \omega_i, \theta_i, v_j, \omega_j, \theta_j ]^T
\]
The mass and stiffness matrices for the shaft element are formulated, and Rayleigh damping is introduced:
\[
[C_s] = \alpha [M_s] + \beta [K_s]
\]
where \([M_s]\) and \([K_s]\) are the mass and stiffness matrices of the shaft element, and \(\alpha\), \(\beta\) are the Rayleigh damping coefficients.

3.3.2 Gear Meshing Element Model

For the spur gear pair, the meshing action is modeled as a spring with time-varying stiffness \(k_m(t)\). The model considers transverse vibration in two perpendicular directions and torsional vibration. The gear mesh element, with four degrees of freedom for the pinion and gear, is depicted by the generalized displacement vector:
\[
\{ X \} = [ v_p, \omega_p, \theta_p, v_g, \omega_g, \theta_g ]^T
\]
The time-varying mesh stiffness \(k_m(t)\) was calculated using the potential energy method or finite element analysis, capturing the periodic variation as teeth enter and exit mesh. A representative curve for the time-varying mesh stiffness is shown in the related figure. The elastic meshing force can be expressed as:
\[
F_k = k_m(t) ( v_p \sin\alpha + \omega_p \cos\alpha – \theta_p r_p – v_g \sin\alpha – \omega_g \cos\alpha – \theta_g r_g + e(t) )
\]
where \(r_p\) and \(r_g\) are the base circle radii of the pinion and gear, respectively, and \(e(t)\) is the comprehensive meshing error, often represented as \(e(t) = e_1 \sin(2\pi f_m t)\) with \(e_1\) being the error amplitude and \(f_m\) the meshing frequency. The corresponding element stiffness matrix \([K_e]\) is a 6×6 matrix dependent on the instantaneous mesh stiffness and the geometric parameters of the gear pair.

3.3.3 Bearing Element Model

The bearing element, representing a deep groove ball bearing, is modeled with nonlinear (or in this case, time-varying) stiffness characteristics. The bearing stiffness depends on the load distribution among the rolling elements. The stiffness is calculated based on the bearing’s geometry and load. The stiffness is time-varying and depends on the balls’ passing frequency, as rolling elements enter and leave the loaded zone. The time-varying bearing stiffness is described by:
\[
k_b(t) = k_{bs} + k_a \sin(2\pi f_b t + \beta_b)
\]
where \(k_{bs}\) is the static stiffness, \(k_a\) is the fluctuation amplitude, \(f_b\) is the bearing passing frequency, and \(\beta_b\) is the initial phase angle.

3.4 Assembly and System Equations

The global matrices for the system are assembled by combining the element matrices based on the connectivity of the nodes. The system’s global equations of motion are:
\[
[M]\{\ddot{x}(t)\} + [C]\{\dot{x}(t)\} + [K(t)]\{x(t)\} = \{F(t)\}
\]
where \([M]\), \([C]\), and \([K(t)]\) are the global mass, damping, and time-varying stiffness matrices, respectively. \(\{F(t)\}\) is the external excitation vector.

3.5 Static and Dynamic Analysis

3.5.1 Static Analysis of Shaft Deformation

The static deformation of the transmission shaft under load was analyzed. At a load torque of 10 N·m and an input speed of 500 r/min, the maximum deformation for the input shaft occurs at node 7 (gear mesh location) with a value of 0.55 μm. For the intermediate shaft, the maximum deformation is 1.98 μm at node 20, and for the output shaft, it is 1.70 μm at node 29. The deformation profiles are consistent with theoretical expectations, where maximum deformation occurs at the point of load application.

3.5.2 Static Transmission Error

The static transmission error (STE) for the first and second stage gear pairs is calculated as:
\[
STE_1 = (x_{18} \sin\alpha + x_{19} \cos\alpha – \theta_1 r_{p1}) – (x_{45} \sin\alpha + x_{46} \cos\alpha + \theta_1 r_{g1})
\]
\[
STE_2 = (x_{57} \sin\alpha + x_{58} \cos\alpha – \theta_2 r_{p2}) – (x_{84} \sin\alpha + x_{85} \cos\alpha + \theta_2 r_{g2})
\]
The results show that when the gear teeth enter or leave the single-double tooth meshing transition, the abrupt change in mesh stiffness causes significant fluctuations in the static transmission error.

3.5.3 Modal Analysis

The modal properties of the two-stage spur gear system were obtained by solving the eigenvalue problem. The first 20 natural frequencies are presented in the following table.

**Table 3.3: First 20 Natural Frequencies of the System**
| Order | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|—|—|—|—|—|—|—|—|—|—|—|
| Freq (Hz) | 128 | 205 | 1483 | 1595 | 1603 | 1760 | 2908 | 3231 | 4587 | 6484 |
| Order | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| Freq (Hz) | 6689 | 6691 | 7183 | 8805 | 8927 | 8948 | 12183 | 12185 | 12723 | 12842 |

The analysis reveals that the system exhibits not only torsional and translational vibrations of the gear pairs but also significant bending modes of the transmission shafts. The first mode is predominantly the torsional vibration of the first-stage spur gear pair. The third mode is associated with the second-stage spur gear pair’s torsion. Modes 2 and 4 are primarily translational vibrations of the gear teeth. Modes 6 and 9 show first-order bending and yawing vibrations of the parallel transmission shafts. Higher-order modes, such as 10 and 17, exhibit complex shaft bending forms, which are typically not captured by simpler lumped parameter models.

3.5.4 Dynamic Response Analysis

Using the Newmark time-domain integration method, the system’s dynamic response can be obtained under a given operating condition. At an input speed of 500 r/min and a load torque of 100 N·m, the dynamic load time histories and frequency spectra for both gear stages were calculated. The first-stage dynamic load has a maximum value of 690 N, while the second-stage dynamic load has an maximum value of 2950 N. The time-domain signals are relatively stable and periodic, with occasional vibration impacts. In the frequency domain, the dominant components are the gear meshing frequencies and their harmonics. The first-stage meshing frequency is 286 Hz, and the second-stage meshing frequency is 92 Hz. Notably, due to the bending-torsion coupling, the first-stage spur gear system shows a significant component at the second-stage meshing frequency, whereas the second-stage system is less affected by the first-stage meshing frequency. This indicates that the second-stage spur gear pair has a greater influence on the vibration of the first stage, while the reverse influence is smaller.

The dynamic loads on the bearings were also computed. The maximum radial loads on the bearings of the input shaft, intermediate shaft, and output shaft were found to be 120 N, 172 N, and 725 N, respectively. The imbalance in bearing loads and the flexibility of the shaft contribute to shaft bending and skewing. The frequency spectra of the bearing loads include the meshing frequencies, their harmonics, the system natural frequencies, and sideband components associated with the bearing passing frequency.

4. Influence of Shaft Stiffness and Bearing Stiffness Variations

4.1 Effect of Shaft Flexibility on Dynamic Response

The influence of shaft flexibility on the system’s dynamic performance was investigated. By comparing a rigid shaft assumption (E=2.1×10¹² Pa) with a flexible shaft (E=2.1×10¹¹ Pa), it was found that the flexible shaft reduces the amplitude of dynamic loads on the gears and attenuates the beating phenomenon. The frequency spectrum of the dynamic load becomes simpler when shaft flexibility is considered, as the flexible shaft acts as a vibration isolator, damping the transmission of vibration. The natural frequencies of the system are lower when shaft flexibility is incorporated.

4.2 Influence of Shaft Stiffness on Natural Frequencies and Vibration Modes

The shaft stiffness significantly affects the system’s natural frequencies and mode shapes. By varying the shaft stiffness through a factor \(k\) (0.25×k, 0.5×k, 1.0×k, 2.0×k, 4.0×k, and 10.0×k), it was observed that an increase in shaft stiffness leads to a higher natural frequency for each mode. For example, the first natural frequency increases by 41% when the shaft stiffness is doubled from 0.25×k to 0.5×k. Furthermore, changes in shaft stiffness can alter the type and position of the primary vibration mode. The most sensitive modes to shaft stiffness changes are those involving the vibration of the input shaft and the second-stage spur gear.

4.3 Influence of Shaft Stiffness on Static Deformation and Bearing Loads

As the shaft stiffness increases, the static deformation of the shaft decreases. The maximum deformation decreases at every node. Moreover, the distribution of bearing loads between the two bearings supporting a shaft becomes more balanced with increasing shaft stiffness. This means that a more flexible shaft leads to a greater load difference between the two ends, increasing the potential for shaft misalignment and reducing bearing life.

4.4 Effect of Shaft Stiffness on System Resonance Characteristics

The resonance characteristics were studied by analyzing the dynamic gear load amplitude as a function of input speed for different shaft stiffness values. For the original shaft stiffness (k), resonance peaks occur at input speeds of 668 r/min, 1671 r/min, and 2244 r/min. These correspond to the excitation frequencies aligning with key natural frequencies of the system. As the shaft stiffness increases, the system’s natural frequencies shift, and more resonance regions emerge. For instance, at 4.0×k, additional resonances appear at 5252 r/min, 6302 r/min, and 6350 r/min. The increase in shaft stiffness tends to excite more torsional vibration modes, increasing the overall vibration energy and the likelihood of resonance. This emphasizes the importance of not overlooking the effect of shaft flexibility in the design and analysis of spur gear transmission systems.

4.5 Influence of Bearing Time-Varying Stiffness

The influence of the bearing’s time-varying stiffness on the system’s dynamic response was investigated. For the gear dynamic load, the influence was negligible, as the gear’s response is dominated by the meshing frequency and its harmonics. However, the bearing’s time-varying stiffness significantly affects the bearing’s own dynamic response. The analysis of the bearing’s dynamic load reveals a component at the bearing passing frequency \(f_b\) in the low-frequency range. Additionally, sidebands appear near the 17th harmonic of the meshing frequency for the second stage, indicating an amplitude modulation phenomenon. This makes the vibration signal more complex and provides critical information for fault diagnosis.

5. Influence of Assembly Relationships on System Dynamic Characteristics

5.1 Effect of the Second-Stage Phase Relationship

The angle between the shaft axes \(\beta\) plays a crucial role in defining the phase relationship between the two stages of the spur gear system. The allowable range for this angle, ensuring proper meshing without interference, was found to be from 54° to 306°. The simulation results indicate that the angle between the shaft axes has a negligible effect on the dynamic load fluctuations of the gear pairs, with variations of only 7% for the first stage and 2% for the second stage. However, the shaft angle significantly affects the static deformation of the intermediate shaft. When the mesh forces of the two stages oppose each other (at \(\beta = 277^\circ\)), the net force on the intermediate shaft is minimized, resulting in the smallest deformation of 7.548 μm. In contrast, at \(\beta = 57^\circ\), the deformation is maximized at 9.563 μm, a 26.7% increase.

5.2 Effect of Gear Installation Position on System Dynamics

The effect of the gear installation position on the system’s dynamic performance was explored by varying the location of the second-stage spur gear along the intermediate shaft. The results show that the transmission shaft deformation is minimized when the gear is positioned near the bearings, while the maximum deformation occurs when the gear is at the midpoint of the shaft. This is consistent with the theoretical analysis of a simply supported beam under a concentrated load. When the gear is located closer to the middle of the shaft, the bending moment is maximized, leading to larger deflections. Conversely, placing the gear closer to the bearings reduces the bending moment and deflection. This principle is critical for the rational layout of components on a shaft to enhance system stiffness and stability.

The gear installation position also influences the system’s natural frequencies and mode shapes. As shown in the analysis, changing the position of the second-stage gear from location 2 (closer to the left bearing) to location 4 (closer to the right bearing) can shift the system’s resonance speeds and alter the vibration modes. The system may transition from a torsional vibration mode to a bending vibration mode, which affects the dynamic load amplitude and the overall vibrational energy. By examining the system’s response across a range of input speeds, it becomes evident that a thoughtful selection of the gear installation position can be an effective strategy to avoid resonance conditions and reduce vibration levels. For example, when the gear is in a certain position, the system’s natural frequency might be shifted away from the excitation frequency, thereby mitigating the resonance response.

6. Experimental Validation

To verify the effectiveness of the finite element modeling method, a series of experiments were conducted on a gear transmission test rig. The experimental setup consisted of a two-stage spur gear reducer, an electric motor, a brake, and associated control and data acquisition systems. Accelerometers were mounted on the bearing housings to measure the vibration acceleration in the radial direction. The experiments were performed at various input speeds and load torques.

In one test case, the input speed was 960 r/min with a load torque of 10 N·m, and the sampling frequency was 10 kHz. The experimental time-domain signal and its frequency spectrum were analyzed. The dominant frequency components in the spectrum were the first and second-stage meshing frequencies (\(f_{m1} = 576 \, \text{Hz}\) and \(f_{m2} = 186 \, \text{Hz}\)) and their harmonics. Low-frequency components corresponding to the bearing passing frequency \(f_b\) were also observed. A resonance peak was noted at the second harmonic of the first-stage meshing frequency (1152 Hz), which is close to the 5th natural frequency of the system. Another peak was observed at the 7th harmonic of the second-stage meshing frequency (1299 Hz), closely aligning with the 6th natural frequency.

The simulation results, obtained from the developed finite element model, exhibited a strong correlation with the experimental measurements. The frequency components matched well, with minor discrepancies attributed to the unmodeled dynamic characteristics of the test rig’s foundation. The acceleration amplitude as a function of input speed was also compared. Both the simulated and experimental curves showed similar trends, with resonance peaks occurring at approximately 1700 r/min and 2200 r/min. These correspond to situations where a harmonic of the meshing frequency excites a system natural frequency. The close agreement between the simulated and experimental results confirms the validity and accuracy of the finite element modeling approach for two-stage spur gear transmission systems. The minor differences between the simulation and experimental results can be attributed to the use of Rayleigh damping in the model, the neglect of air and lubricant coupling effects, and the influence of the foundation on the measured signals.

7. Conclusions

In this thesis, a comprehensive investigation into the dynamic modeling and analysis of a two-stage spur gear transmission system was conducted. The following key conclusions can be drawn:

1. A generalized finite element modeling method for multi-shaft spur gear transmission systems was successfully developed. The model considers the flexibility of transmission shafts, time-varying mesh stiffness of gear pairs, time-varying bearing stiffness, and the phase relationship between gear stages. The validity of the model was verified through experimental comparison, demonstrating its capability to accurately predict system dynamics.
2. The analysis revealed that the two-stage spur gear system exhibits complex dynamic behavior, including torsional and bending vibrations of the gear pairs and transmission shafts. The dynamic loads on the gears and bearings are periodic, and their frequency components include the meshing frequencies, their harmonics, and sidebands due to bearing and meshing stiffness variations.
3. The flexibility of the transmission shaft plays a significant role in attenuating gear dynamic loads and simplifying the frequency content, acting as a vibration isolator. An increase in shaft stiffness raises the natural frequencies of the system but can also excite more torsional vibration modes, leading to more resonance regions and higher vibration energy. Therefore, selecting an appropriate shaft stiffness is a key design consideration for avoiding torsional resonance.
4. The bearing time-varying stiffness has a minimal effect on the gear dynamic response but significantly influences the bearing’s own vibration characteristics. The presence of the bearing passing frequency component and the associated sidebands in the bearing load spectrum provides valuable diagnostic information.
5. The assembly relationships, including the shaft angle and gear installation positions, significantly influence the system’s natural frequencies, mode shapes, and static deformations. While the shaft angle has a minor effect on the gear dynamic load, it substantially affects the static deformation of the intermediate shaft. The gear installation position affects both the static deflection and the dynamic resonance behavior. A rational placement of the gears, closer to the bearings, can reduce shaft deformation and potentially mitigate resonance issues.

In summary, this research work provides a robust and efficient finite element methodology for the dynamic analysis of two-stage spur gear systems. The findings offer valuable insights into the influence of key system parameters and assembly relationships, providing a theoretical basis for the low-vibration design, fault diagnosis, and optimization of gear transmission systems.