This article focuses on the spindle system of a spiral bevel gear milling machine, an essential component in the manufacturing of spiral bevel gears. The dynamic characteristics of the spindle system and machining process parameters significantly impact the gear’s tooth surface quality and machining efficiency. Current research on spindle system dynamics is mainly centered on general milling or lathe machines, with limited studies on spiral bevel gear milling machines. Additionally, process parameter selection often relies on experience or reference manuals, and the assessment of machining process rationality is typically through manual observation. This paper aims to address these issues by studying the matching between process parameters and the dynamic characteristics of the milling machine spindle system. Through theoretical modeling, simulation, and experimental verification, a more efficient and accurate method for parameter selection and process optimization is proposed.
1. Introduction
1.1 Research Background and Significance
In modern manufacturing, machine tools play a crucial role. The machining performance of machine tools not only affects the quality of workpieces and cutting efficiency but is also significantly influenced by their dynamic performance. The spindle system, as a key part in direct contact with the workpiece, has vibration characteristics that are vital to study. These characteristics include the analysis of the inherent properties of the spindle structure, the evaluation of dynamic responses, and the study of cutting stability. Understanding these aspects can help in optimizing process parameters to avoid resonance and improve machining quality. In the case of spiral bevel gear milling machines, the current research gap in spindle system dynamics and process parameter matching calls for in-depth investigation to enhance machining efficiency and gear quality.
1.2 Current Research Status at Home and Abroad
1.2.1 Dynamics Modeling Technology
- Concentrated Parameter Model: This model concentrates the mass of the structure at several nodes and uses equivalent elastic beams to simulate elasticity. However, it has limitations in accurately simulating complex machine tool structures and considering nonlinear and coupling effects.
- Transfer Matrix Method: It is a classic method for analyzing the dynamics of slender structures. It divides the spindle system into discrete elements and uses transfer matrices to relate the state vectors of each element. Although it has advantages in simplicity and speed, it may face issues such as reduced accuracy in calculating high-order modes and difficulty in capturing nonlinear and coupling effects.
- Finite Element Method: With the development of computer technology, the finite element method has become an important tool. It can analyze the static and dynamic characteristics of the spindle system, providing a more accurate and comprehensive understanding of the system’s behavior.
1.2.2 Research on Excitation Force
Most studies on the dynamic characteristics of machine tool spindle systems have been conducted during machine shutdown. However, there are significant differences in dynamic characteristics between shutdown and operation states, especially at high speeds. New excitation methods have emerged, such as using electromagnetic exciters or piezoelectric sensors. Some researchers have also used cutting force as an excitation source, but studies on the excitation force during the spiral bevel gear milling process are relatively scarce.
1.2.3 Research on Process Matching
Process parameters play a crucial role in machining efficiency, quality, and tool durability. Current methods for optimizing process parameters include using optimization algorithms or conducting experiments and simulations. However, few studies have focused on optimizing process parameters through the system’s response, especially for spiral bevel gear milling machines.
1.3 Research Contents and Ideas of This Topic
This paper aims to establish a finite element dynamics model of the spindle system of a spiral bevel gear milling machine based on relevant theories. By analyzing the dynamic response of the system under different machining parameters and considering the cutting force as the main excitation source, the influence of process parameters on cutting force is studied. Through simulation and experimental verification, an optimal set of process parameters is obtained, and the matching effect with the spindle system is verified.
2. Dynamics Modeling and Analysis of the Spindle System
2.1 Basic Principles of Rotor Dynamics Equation Construction
The dynamics model of the spindle system can be established through theoretical modeling, experimental modeling, or a combination of both. In this paper, the finite element method is used. The basic principle involves dividing the structure into multiple elements and obtaining the dynamics equation of each element using the energy principle. By assembling the element matrices, the overall dynamics equation of the system can be obtained. Considering damping as viscous damping, the damping matrix can be determined, and for practical structures, Rayleigh damping is often used.
2.2 Dynamics Modeling of the Spindle System Based on Beam Elements
The spindle system of a milling machine includes components such as the shaft, bearings, and milling cutter head. To simplify the calculation, the system is appropriately simplified, and the Timoshenko beam model is used to simulate the spindle shaft system. The bearings are simplified as elastic supports, and their stiffness is considered constant for ease of calculation. After discretizing the system into 12 units and 13 nodes, the finite element dynamics model of the spindle system is established.
2.3 Establishment of the System Motion Differential Equation
2.3.1 Unit Mass Matrix and Stiffness Matrix
The Timoshenko beam model is chosen as it considers shear deformation, which is more suitable for actual mechanical systems. For each unit, the mass matrix and stiffness matrix are derived based on the displacements and rotations of the nodes.
2.3.2 System Assembly Mass Matrix and Stiffness Matrix
The unit mass matrices and stiffness matrices are assembled to obtain the total mass matrix and stiffness matrix of the system. The coupling of bearing stiffness matrices at specific nodes is also considered to account for the influence of bearings on the system’s dynamic characteristics.
2.3.3 Spindle System Motion Differential Equation
The motion differential equation of the spindle system is expressed as , where , , and are the total mass, damping, and stiffness matrices, respectively, and is the external force vector. The equation for free vibration can be obtained by setting , and solving this equation yields the inherent frequencies and characteristic vectors of the system.
2.4 Inherent Frequencies and Modal Shapes
The inherent frequencies of the spindle system are calculated using Matlab and finite element methods. The results show that the first four orders of inherent frequencies calculated by both methods are generally consistent, validating the effectiveness of the simplified model using the Timoshenko beam unit. The modal shapes obtained by both methods also show similar characteristics, further verifying the correctness of the theoretical model and related programs.
2.5 Harmonic Response Analysis of the Spindle System
2.5.1 Basic Theory of Harmonic Response Analysis
Harmonic response analysis is used to study the steady-state response of a vibration system under periodic excitation. For the spindle system of a milling machine, the cutting force is the main excitation source, and the harmonic response analysis helps to determine the vibration characteristics of the system at specific frequencies, which is beneficial for optimizing the system’s structure and machining process parameters.
2.5.2 Harmonic Response Analysis of the Spindle System Based on Matlab
In this paper, the modal superposition method is used for harmonic response analysis. The results show that the displacements of the spindle system in each direction have obvious peaks at around 78Hz, indicating that resonance may occur when the excitation frequency approaches 78Hz. The analysis also shows that the system’s response to external forces and torques has directional and multi-modal characteristics.
2.6 Transient Dynamics Analysis
Transient dynamics analysis considers the non-steady and sudden external loads on the spindle system during machining. These loads can cause problems such as vibration, resonance, and deformation, affecting machining accuracy and tool life. By analyzing the transient response of the spindle system under different process parameters, more accurate selection and adjustment of process parameters can be achieved to minimize the impact of impact loads on the system. The results of transient response analysis using the Newmark method show that different process parameters have a significant impact on the transient response of the spindle system, and the results obtained by Matlab and finite element software are consistent, validating the accuracy of the program.
2.7 Summary of This Chapter
This chapter establishes a dynamics model of the spindle system using the Timoshenko beam theory. The inherent frequencies and modal shapes of the system are obtained, and the harmonic response and transient response analyses are carried out. The results of these analyses provide a theoretical basis for selecting process parameters to match the dynamic characteristics of the spindle system.
3. Analysis of Milling Cutting Excitation Force
3.1 Dynamic Force Analysis of Spiral Bevel Gear Milling
During milling, the spindle system is affected by various excitation forces, and the cutting force is the most significant one. The proper selection of process parameters is crucial for balancing the improvement of tooth surface quality and machining efficiency.
3.2 Theoretical Milling Cutting Force Model
3.2.1 Analysis of Spiral Bevel Gear Milling
This paper focuses on the cutting force during the generation method of machining the pinion. The milling cutter head has two motion modes: feed motion along its axis and rotation around its axis, while the gear blank only rotates around its axis. The ratio of the rotational speeds of the cutter head and the gear blank is called the roll ratio.
3.2.2 Oblique Cutting Model
Since the micro-end cutting edges of the inner and outer blades of the milling cutter head satisfy the oblique cutting model, this model is used for mechanical modeling. By establishing a coordinate system and making certain assumptions, the cutting force can be calculated.
3.2.3 Equation of the Cutter Head Generating Surface for Machining the Pinion
The equation of the cutter head generating surface for machining the pinion is derived, and the unit normal vector at any point on the cutting edge is calculated.
3.2.4 Calculation of Cutter Head Cutting Force Components
Based on the theoretical cutting force model, the cutting force components in three directions for machining the pinion can be calculated by discretizing the cutting edge into micro-segments and considering each segment as an oblique cutting process.
3.3 Analysis of the Influence of Process Parameters on the Theoretical Cutting Force Model
3.3.1 Influence of Feed Rate on Cutting Force
According to the theoretical cutting force model, as the feed rate increases from 0.1mm to 0.7mm, the amplitudes of the theoretical cutting forces in three directions also increase, and the cutting force in the Y direction is significantly larger than those in the other two directions. When the feed rate exceeds 0.3mm, there is a significant mutation in the cutting force amplitudes.
3.3.2 Influence of Spindle Speed on Feed Rate
As the spindle speed increases, the cutting forces in three directions generally show a linear increasing trend, and the cutting force in the Y direction is still larger than those in the other two directions.
3.4 Finite Element Simulation of Spiral Bevel Gear Milling Cutting Force Based on AdvantEdge FEM
3.4.1 Simulation Process of AdvantEdge FEM Software
AdvantEdge FEM is a specialized software platform for metal cutting simulation. The simulation process includes selecting the simulation process type, setting simulation parameters, performing simulation calculations, and post-processing the results.
3.4.2 Establishment of the Finite Element Simulation Model
In the three-dimensional software, the models of the cutter head and the gear blank are established and simplified. Only the parts involved in cutting are retained to reduce the simulation time.
3.4.3 Simulation Results of Milling Cutting Force
Based on the orthogonal test design, the cutting forces in different directions under different spindle speeds and feed rates are simulated. The results show that as the feed rate increases, the cutting force amplitudes generally increase, and the cutting force in the Y direction is significantly larger than those in the other two directions. As the spindle speed increases, the cutting force amplitudes in the X direction change slightly, while those in the Y direction decrease with the increase of cutting time, and those in the Z direction remain relatively stable.
3.5 Frequency Component Analysis of the Simulation Cutting Force
3.5.1 Influence of Feed Rate on the Frequency Component of Cutting Force
As the feed rate increases to 0.4mm, the cutting force amplitude increases and shows a trend of shifting to high frequencies. Overall, changing the feed rate has little effect on the frequency distribution of the cutting force, and the frequencies are mainly concentrated in the low-frequency region.
3.5.2 Influence of Spindle Speed on the Frequency Component of Cutting Force
Changing the spindle speed has little effect on the frequency distribution of the cutting force. The frequencies are mainly concentrated in the low-frequency region and are far from the first-order inherent frequency of the system.
3.6 Summary of This Chapter
This chapter analyzes the excitation forces during the spiral bevel gear milling process and establishes a theoretical cutting force model. The influence of process parameters on the cutting force is studied through theoretical analysis and finite element simulation. The frequency components of the cutting force are also analyzed, providing a basis for further studying the influence of excitation forces on the dynamic characteristics of the spindle system.
4. Verification of Dynamic Characteristics Analysis and Process Matching Effect
4.1 Transient Response Analysis Based on Theoretical Cutting Force
Based on the calculation results of the theoretical cutting force, the transient response of the system is analyzed. Four sets of cutting force amplitudes are selected and substituted into the finite element model to study the system’s response. The results show that when the feed rate is 0.3mm and the spindle speed is 120rpm, the system has the smallest vibration amplitude and the shortest recovery time.
4.2 Verification of Milling Cutting Experiments
4.2.1 Introduction to the Rotary Dynamometer
The Kistler rotary dynamometer is used to measure the cutting force during the milling process. It integrates a piezoelectric dynamometer and a torque sensor and can measure cutting forces in milling and drilling processes.
4.2.2 Milling Cutting Force Experiments
Milling experiments are carried out using a YKH2235 CNC spiral bevel gear milling machine. Two sets of process parameters are selected for comparison. The experimental results show that the actual cutting force amplitudes in three directions are basically consistent with the theoretical calculation results and simulation results, and the cutting force in the Y direction is significantly larger than those in the other two directions.
4.2.3 Milling Cutting Vibration Experiments
The vibration signals at the spindle end during the milling process are collected. The results show that as the feed rate increases, the vibration amplitude at the spindle end increases. When the spindle speed is 120rpm and the feed rate is 0.3mm, the spindle system has a smaller vibration amplitude, which is beneficial for ensuring the tooth surface quality.
4.3 Process Matching
4.3.1 Frequency Analysis of Milling Cutting Force
The frequency components of the cutting force collected in the experiments are analyzed. The results show that when the spindle speed is 120rpm and the feed rate is 0.3mm, the cutting force frequencies are mainly concentrated in the [0 – 100]Hz range and are far from the first-order inherent frequency of the system, indicating good matching with the spindle system’s dynamic characteristics.
4.3.2 Response Analysis Based on Milling Excitation Force
The response of the spindle system under the action of the milling excitation force is studied. The results show that when the spindle system is simultaneously subjected to cutting forces in three directions, the system is in a stable milling state. The system is more sensitive to the cutting force in the Y direction, and measures can be taken to reduce the sensitivity in the Y direction to improve the tooth surface quality.
4.3.3 Process Optimization Based on Milling Experiments
Based on the experimental results, the process parameters are optimized. By adjusting the feed speed to keep the cutting force amplitude constant, the milling time is reduced, and the impact of impact loads on the spindle system and tooth surface quality is minimized. The adjustment of bearing stiffness can also change the sensitivity direction of the spindle system to the cutting force, ensuring the stability of the main cutting force and improving the tooth surface quality.
4.4 Summary of This Chapter
This chapter verifies the influence of the excitation force amplitudes caused by different process parameters on the dynamic characteristics of the spindle system through theoretical analysis, experiments, and simulations. The optimal process parameters are obtained, and the matching effect with the spindle system is verified. The sensitivity of the spindle system to the cutting force is analyzed, and measures for process optimization are proposed.
5. Summary and Outlook
5.1 Summary of the Full Text
This paper has established a dynamics model of the spindle system of a spiral bevel gear milling machine, analyzed its dynamic characteristics, established a cutting force model, and studied the influence of process parameters on the cutting force and the dynamic characteristics of the spindle system. Through experimental verification, an optimal set of process parameters is obtained, and the matching effect with the spindle system is verified. The research results provide a theoretical basis and practical guidance for optimizing the machining process of spiral bevel gears.
5.2 Outlook
In future research, several aspects need to be further explored to enhance the understanding and optimization of the spiral bevel gear milling process.
Firstly, regarding the excitation sources, the current study mainly focuses on the cutting force. However, in actual milling operations, there are other factors that can affect the spindle system’s dynamics. For example, the inertial force of moving parts and the excitation force from internal vibration sources of the machine tool also play roles. By considering these additional excitation sources, a more comprehensive and accurate model can be developed. This will involve detailed analysis of the interactions between different forces and their impacts on the spindle system’s vibration characteristics.
Secondly, more process parameters should be investigated. In this paper, only the influence of spindle speed and feed rate on the cutting force has been studied in detail. Other parameters such as the tool rake angle and the actual manufacturing errors of the tool can also significantly affect the milling process. Understanding the combined effects of these multiple parameters on the cutting force and the spindle system’s dynamics will lead to more precise process optimization.
Finally, the influence of multiple components on the spindle system’s dynamic characteristics needs to be considered. Currently, the focus has been on the spindle – bearing system. However, components such as the spindle box and the column also have an impact on the overall dynamic behavior of the spindle system. By analyzing the contributions of these different components, a more accurate understanding of the system’s response θ§εΎ can be achieved. This will provide a more reliable theoretical basis for optimizing the machining process parameters, ultimately leading to improved machining quality and efficiency of spiral bevel gears.
In conclusion, this research has laid a solid foundation for the study of the spindle system of spiral bevel gear milling machines and the matching of machining process parameters. Future research directions identified above will further contribute to the development and optimization of this important manufacturing process.
