Abstract This article focuses on the performance analysis and test verification of spiral bevel gear transmission systems. It begins with an introduction to the importance and challenges of such systems, followed by a detailed discussion on the construction of mechanical models for key components, static and dynamic characteristics analysis methods, and experimental validation. The research aims to provide accurate and efficient analysis techniques for spiral bevel gear transmission systems, which are widely used in various industries.
1. Introduction
1.1 Background and Significance
Spiral bevel gears play a crucial role in many mechanical transmission systems due to their high transmission efficiency, strong load-bearing capacity, and compact structure. They are widely applied in aerospace, automotive, and marine industries. However, the complex geometry and meshing characteristics of spiral bevel gears pose significant challenges to the analysis of their transmission performance. Any deviation in the meshing state caused by load deformation can affect the overall performance of the transmission system. Therefore, it is essential to conduct in-depth research on the performance analysis of spiral bevel gear transmission systems to ensure their reliable operation and optimize their design.
1.2 Research Objectives and Contributions
The main objectives of this research are to develop accurate and efficient methods for analyzing the static and dynamic characteristics of spiral bevel gear transmission systems and to verify these methods through experimental tests. The contributions of this study include:
- Proposing a method based on the solid finite element for analyzing the meshing misalignment of the transmission system, which improves the calculation accuracy and efficiency.
- Constructing a dynamic model of the spiral bevel gear transmission system considering the meshing stiffness and obtaining its inherent characteristics and dynamic responses.
- Conducting experimental tests on a spiral bevel gear pair using a CNC rolling inspection machine and validating the theoretical analysis methods.
2. Mechanical Model Construction for Key Components of Spiral Bevel Gear Transmission System
2.1 Transmission Shaft Model
2.1.1 Solid Finite Element Model
The solid finite element model of the transmission shaft is constructed by appropriately simplifying its position. When meshing the shaft, a hexahedral element is used considering the installation positions of bearings and gears. This model can reflect the actual macroscopic stiffness characteristics and improve the calculation efficiency while ensuring a certain level of accuracy.
2.1.2 Beam Element Model
For the beam element model, considering that the transmission shaft is mostly a short and thick beam with a complex force situation, the Timoshenko beam element is selected. The stiffness matrix of the beam element is derived based on its section area, shear modulus, inertia moment, and polar inertia moment. Multi-node beam unit models can be obtained by deriving from two-node beam units.
2.2 Gear Pair Model
2.2.1 Equivalent Meshing Force
To simplify the tooth surface contact analysis, the gear contact is equivalent to a meshing force. The meshing node is considered as the midpoint of the tooth width. The meshing force is decomposed into axial, radial, and tangential forces according to the torque or power. The translation of the meshing force to the axis is calculated considering whether there is an offset distance or not.
2.2.2 Equivalent Rigid Body Model
The meshing stiffness of the gear contact is equivalent to a spring model, and the energy dissipation in the gear transmission process is equivalent to a damper. Thus, the gear meshing contact can be equivalent to a mechanical model containing a spring and a damper. For the construction of the gear transmission system dynamic model, considering the rotation characteristics of the gear, it is more efficient to treat the gear as a rigid disk with a certain moment of inertia.
2.3 Bearing Model
2.3.1 Structure Characteristics and Force Analysis
The cone roller bearing model consists of an inner ring, an outer ring, rolling elements, and a cage. The bearing is simplified into two parts for force analysis. The contact load and deformation of the rolling elements are calculated, and the comprehensive stiffness coefficient is obtained considering the direction perpendicular to the rolling element axis.
2.3.2 Stiffness Matrix Calculation
The stiffness of the cone roller bearing is an important performance index. Due to the nonlinear relationship between the contact load and deformation and the coupling phenomenon between the stiffness in different directions, a bearing unit considering the coupling and nonlinearity of the bearing stiffness is established. The stiffness matrix of the bearing is derived by analyzing the relationship between the load and displacement of the bearing under the action of axial force, radial force, and moment.
3. Static Characteristics Analysis of Spiral Bevel Gear Transmission System
3.1 Meshing Misalignment Calculation
The meshing misalignment of the gear pair is defined as the relative displacement of the initial installation position of the gear pair caused by the elastic or plastic deformation of the transmission shaft and bearings under load. The calculation method of the meshing misalignment amount is derived based on the offset vector of the axis intersection point after deformation.
3.2 Analysis Model Based on Solid Finite Element
3.2.1 Transmission Shaft Coupling Analysis Model
The solid finite element transmission shaft coupling analysis model is constructed to study the load deformation of the system. The model considers the installation positions of bearings and gears for meshing division. Different meshing densities are set for different regions to ensure the accuracy of displacement deformation calculation. The static analysis steps include defining materials, analysis steps, constraints, result outputs, and loads.
3.2.2 Gear Tooth Loading Contact Analysis Model Considering Meshing Misalignment
Based on the transmission shaft coupling analysis model, the meshing misalignment amount is obtained, and the theoretical spiral bevel gear model is constructed using the tooth surface equation. The meshing position and attitude of the gear in the multi-tooth meshing model are adjusted according to the meshing misalignment amount. The contact analysis of the model is carried out by setting appropriate meshing densities, coupling points, and analysis steps.
3.3 Example Analysis and Result Comparison
3.3.1 Construction of Comparison Analysis Models
To verify the proposed analysis model, a transmission shaft coupling analysis model based on beam elements and a full finite element model of the spiral bevel gear transmission system are constructed. The beam element model ensures calculation efficiency, and the full finite element model has high calculation accuracy.
3.3.2 Meshing Misalignment Calculation Results
The displacement deformation diagrams of the transmission shaft of the three models are obtained through finite element static analysis. The deformation results of the three models are compared, and it is found that the solid finite element analysis model is closer to the full finite element model. The meshing misalignment amount is calculated using the displacement deformation amounts of the solid finite element model and the full finite element model, and the calculation results show high consistency.
3.3.3 Loading Imprint and Transmission Error
The loading imprint and transmission error of the gear are obtained by substituting the meshing misalignment amount into the corresponding models and analyzing them. The loading imprints of the two models are compared, and it is found that they have a high degree of fit. The transmission errors of the two models are calculated, and the results show that the curves of the two models are almost identical, and the amplitude deviation is within a certain range.
4. Dynamic Characteristics Analysis of Spiral Bevel Gear Transmission System
4.1 Analysis Model Considering Meshing Stiffness
Based on the multi-tooth meshing model considering meshing misalignment, the multi-tooth meshing stiffness is calculated. A dynamic model of the spiral bevel gear transmission system considering meshing stiffness is constructed, which includes transmission shaft dynamics models, gear pair meshing dynamics models, and bearing dynamics models.
4.2 Modal Analysis of the Model
The modal analysis of the spiral bevel gear transmission system model considering meshing stiffness is carried out. The material, analysis steps, and constraints are defined. The first 20 orders of inherent frequencies and vibration modes of the transmission system are obtained. The vibration modes in different orders show different deformation characteristics.
4.3 Harmonic Response Analysis of the Model
The harmonic response analysis of the model is carried out to study the dynamic response of the transmission system under the excitation of the meshing force. The multi-tooth meshing force is obtained through static analysis, and its Fourier transform is carried out to obtain the contact force spectrum. The amplitude of the contact force spectrum is applied to the meshing node of the rigid disk of the gear pair, and the dynamic characteristics analysis of the transmission system is carried out based on the modal superposition method. The vibration acceleration amplitude-frequency characteristic curves of the transmission shaft are obtained, and the frequencies corresponding to the peaks of the curves are compared with the inherent frequencies of the transmission system.
5. Test Experiments
5.1 Experimental Preparation
5.1.1 Experimental Objects
The theoretical working surface imprint and transmission error are preset. Based on the meshing misalignment amount calculation results, the contact area between the convex surface of the large wheel and the concave surface of the small wheel is taken as the research area. The gear pair samples are obtained by adjusting the cutting and grinding parameters of the gears.
5.1.2 Experimental Scheme
The gear pair is installed and adjusted according to the meshing misalignment. The gear pair is rotated slowly to observe its meshing state. Red lead powder is applied to the large wheel, and a constant torque is applied to the large wheel while gradually increasing the speed of the small wheel. Acceleration sensors and microphones are arranged at appropriate positions to collect vibration and noise signals.
5.1.2 Experimental Equipment
A CNC rolling inspection machine is used for the experiment, and acceleration sensors, microphones, data acquisition cards, and sound calibrators are selected as experimental equipment. The acceleration sensor has high sensitivity and stability, and the microphone has high sensitivity and a wide frequency response range. The data acquisition card has high-speed and high-precision data processing capabilities, and the sound calibrator can calibrate the microphone to ensure the accuracy of the measured data.
5.2 Experimental Results
5.2.1 Static Characteristics Experimental Results
The contact imprint of the convex surface of the large wheel, the transmission error, and the light-load transmission error order amplitude of the gear pair are obtained under actual light-load test conditions. The experimental results are compared with the theoretical results and the results obtained by the solid finite element step-by-step spiral bevel gear transmission system analysis model, and it is found that they are in good agreement.
5.2.2 Dynamic Characteristics Experimental Results
The vibration acceleration amplitude-frequency characteristic curves of the two spindles in three directions and the meshing noise amplitude-frequency characteristic curve are obtained by extracting the real-time signals of the acceleration sensors and microphones and processing them. The experimental results are compared with the dynamic analysis calculation results of the spiral bevel gear system dynamics model considering meshing stiffness, and it is found that the frequencies corresponding to the amplitudes are basically the same, and the corresponding errors are within a certain range.
5.3 Scheme Optimization
Based on the experimental results, the optimization scheme is to make the load distribution on the entire tooth surface more uniform and reduce the amplitude during the meshing process. The contact trace diagonal is increased, and the tooth profile curvature radius of the small wheel is appropriately increased to increase the contact path. The experimental results after optimization show that the contact imprint and transmission error of the gear pair are improved, and the vibration and noise levels are reduced.
6. Conclusion and Outlook
6.1 Research Summary
In this study, a static analysis model based on the solid finite element and a dynamic analysis model considering the meshing stiffness of the spiral bevel gear transmission system are constructed. The performance analysis of the spiral bevel gear transmission system is carried out using these two models, and the test experiments are completed using a CNC rolling inspection machine. The research results show that the proposed methods have high accuracy and efficiency in analyzing the meshing misalignment, transmission error, and loading imprint of the transmission system. The dynamic analysis method can accurately obtain the inherent characteristics and dynamic responses of the transmission system. The experimental results verify the scientific and correct nature of the static and dynamic characteristics analysis methods of the spiral bevel gear transmission system.
6.2 Future Research Directions
Although certain research results have been achieved in this study, there are still some limitations. Future research can be carried out in the following directions:
- Increasing the meshing force of multiple nodes on the contact trace to improve the calculation accuracy of the transmission system when analyzing the meshing misalignment.
- Considering the influence of time-varying stiffness on the acceleration amplitude curve characteristics of the transmission system to improve the accuracy of the analysis results when constructing the spiral bevel gear transmission system model considering the meshing stiffness.
- Considering other dynamic indicators in the dynamic characteristics analysis of the spiral bevel gear transmission system model to improve the performance of the transmission system.