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 the efficiency of cutting but is also influenced by their dynamic performance. The spindle system, as a key component in direct contact with the workpiece, has a significant impact on vibration characteristics. Studying the dynamic characteristics of the spindle system is essential for improving machining accuracy and efficiency and reducing vibration. In the milling process of spiral bevel gears, the surface quality and machining efficiency are affected by the dynamic characteristics of the spindle system and process parameters. However, current research on spindle system dynamic characteristics mainly focuses on general milling or lathe machines, and there is less research on spiral bevel gear milling machine spindle systems. Additionally, the selection of process parameters often relies on experience or reference manuals, and the judgment of machining process rationality is mainly through manual observation, which may lead to inefficiencies and waste.
1.2 Research Status at Home and Abroad
- Dynamics Modeling Technology: Various methods have been used for spindle dynamics modeling, including concentrated parameter method, transfer matrix method, and finite element method. Each method has its own advantages and limitations. For example, the concentrated parameter model is simple but may not accurately simulate complex structures. The transfer matrix method is useful for analyzing slender structures but may have accuracy issues for high-order modes. The finite element method has become a widely used and powerful tool for analyzing spindle systems.
- Incentive Force Research: Studies on incentive forces have explored different ways to apply incentives during machine operation. Some use artificial excitation during machine shutdown, while others use methods like electromagnetic exciters, piezoelectric sensors, or the machine’s own vibration signals. The use of cutting force as an excitation source has also been studied, but research on the incentive forces in the spiral bevel gear milling process is relatively limited.
- Process Matching Research: Optimization of process parameters is crucial for improving machining efficiency and quality. Methods include using optimization algorithms such as artificial neural networks, simulated annealing algorithms, and genetic algorithms. Some researchers also conduct experiments or simulations to select appropriate process parameters. However, there is less research on optimizing process parameters for spiral bevel gear milling machines through system responses.
1.3 Research Contents and Ideas
This paper aims to achieve the matching of the characteristics of the spindle system of a spiral bevel gear milling machine with process parameters to improve milling efficiency and tooth surface quality. The main research contents include:
- Establishing a dynamic numerical model of the spindle system based on finite element dynamics theory and Timoshenko beam theory, and analyzing its harmonic and transient responses.
- Deriving a theoretical cutting force model for machining the pinion and analyzing the influence of process parameters on cutting force.
- Calculating cutting forces under different process parameters using simulation cutting software and analyzing their frequency components.
- Conducting experiments with selected process parameters to verify the matching effect with the spindle system.
2. Spindle System Dynamics Modeling and Analysis
2.1 Rotor Dynamics Equation Construction Principles
The dynamics model of the spindle system can be established through theoretical, experimental, or a combination of both methods. 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 the element body using the energy principle. The equation includes terms for kinetic energy, potential energy, and non-conservative forces. By assembling the element matrices, the system’s dynamics balance equation can be obtained. Different forms of mass matrices, such as consistent mass matrix and concentrated mass matrix, can be used, and the damping matrix can be determined based on system characteristics, often using Rayleigh damping.
2.2 Spindle System Dynamics Modeling Based on Beam Elements
The spindle system of the milling machine includes components such as the shaft, bearings, and milling cutter head. To simplify the model, some small-sized and less influential components are omitted, and the spindle is simplified into a stepped shaft. The Timoshenko beam model, which considers shear deformation, is used to simulate the spindle shaft system. The system is discretized into multiple elements along the axis, with each element having 6 degrees of freedom at two nodes. The unit mass matrix and stiffness matrix of the Timoshenko beam are derived, and the system’s total mass matrix and stiffness matrix are assembled by considering the coupling of bearings.
2.3 Establishment of the System Motion Differential Equation
- Unit Mass Matrix and Stiffness Matrix: The Timoshenko beam model is used to calculate the unit mass matrix and stiffness matrix. The mass matrix elements are calculated based on parameters such as density, cross-sectional area, and length, while the stiffness matrix takes into account factors like elastic modulus, shear elastic modulus, and section influence coefficient.
- System Total Mass Matrix and Stiffness Matrix: The unit matrices are assembled to obtain the system’s total mass matrix and stiffness matrix. The coupling of bearings is considered by adding the bearing stiffness matrices at the appropriate nodes after coordinate transformation.
- 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. By solving this equation, the system’s natural frequencies and characteristic vectors can be obtained.
2.4 Natural Frequencies and Modal Shapes
The natural frequencies of the spindle system are calculated using Matlab and finite element methods. The results show that the first four natural frequencies calculated by both methods are generally consistent, validating the effectiveness of the simplified model using the Timoshenko beam unit. The differences in the results may be due to model assumptions, numerical calculation precision, boundary condition settings, and program implementation. The modal shapes obtained by both methods are also compared, and they show similar characteristics, such as swinging for the first and second orders, bending deformation for the third order, and torsional deformation for the fourth order.
2.5 Harmonic Response Analysis of the Spindle System
- Basic Theory of Harmonic Response Analysis: Harmonic response analysis studies the system’s response to periodic simple harmonic incentives. By applying a series of periodic incentives and solving the dynamics equation, the displacement response data at a specific frequency can be obtained. The solution of the dynamics equation involves expressing the displacement and external force in complex forms and calculating the node reaction forces based on inertia, damping, and stiffness forces.
- Harmonic Response Analysis Based on Matlab: In this paper, the mode superposition method is used for harmonic response analysis. The results show that the displacements in each direction of the spindle system have obvious peaks around 78Hz, indicating the possibility of resonance. The response of the spindle system to external forces and torques shows directional and multi-modal characteristics. The results obtained by Matlab are compared with those of finite element simulation, and although there are some differences in amplitudes, the overall trends are consistent.
2.6 Transient Dynamics Analysis
Transient dynamics analysis considers the system’s response to non-steady and sudden external loads during the milling process. By assuming certain process parameters and using the Newmark method, the displacements in three directions at different positions of the spindle system over time are calculated. The results show that the dynamic response has a spatial distribution characteristic, and the bearings play a buffering role in reducing the impact. Different process parameters affect the impact load on the spindle system. The transient response results obtained by Matlab are compared with those of finite element simulation, and they show similar attenuation trends, although there are differences in initial amplitudes.
2.7 Chapter Summary
In this chapter, a dynamics model of the spindle system is established using the Timoshenko beam theory. The natural frequencies and modal shapes of the system are obtained, and the harmonic and transient responses are analyzed. The results validate the effectiveness of the model and provide a basis for studying the matching of process parameters with the spindle system’s dynamic characteristics.
3. Milling Cutting Incentive Force Analysis
3.1 Cone Gear Milling Dynamic Force Analysis
In the milling process of cone gears, the spindle system is affected by various incentive forces, and the cutting force is the most important one. The choice of appropriate process parameters is crucial for balancing tooth surface quality and machining efficiency.
3.2 Theoretical Milling Cutting Force Model
- Cone Gear Milling Analysis: The cutting force model for machining the pinion using the generating method is studied. The motion relationship between the cutter head and the gear blank is analyzed, and the cutter head structure is described. The single-tooth cutting method is commonly used for cutting force modeling.
- Oblique Cutting Model: The oblique cutting mechanics model is established as the micro-end cutting edges of the inner and outer blades of the cutter head satisfy the oblique cutting model. The coordinate systems for oblique cutting are defined, and the relationships between different coordinates are derived. The cutting force acting on the chip is calculated under certain assumptions.
- Generating Method for Machining Pinion’s Cutter Head Forming Surface Equation: The equation for the forming surface of the cutter head when machining the pinion using the generating method is derived, and the unit normal vector at any point on the cutting edge is calculated.
- Cutter Head Cutting Force Component Calculation: The cutting force components in different directions for the inner and outer blades are calculated based on the oblique cutting theory and the cutter head forming surface equation.
3.3 Analysis of the Influence of Process Parameters on the Theoretical Cutting Force Model
- Influence of Feed Rate on Cutting Force: Based on the theoretical cutting force model, the influence of feed rate on the cutting force amplitude is analyzed. The results show that as the feed rate increases from 0.1mm to 0.7mm, the cutting force amplitude in three directions increases, and the Y direction has the largest cutting force amplitude.
- Influence of Spindle Speed on Feed Rate: The influence of spindle speed on the cutting force amplitude is also analyzed. The results show that as the spindle speed increases, the cutting force amplitude in three directions generally increases linearly, and the Y direction still has the largest cutting force amplitude.
3.4 Finite Element Simulation of Cone Gear Milling Cutting Force Based on AdvantEdge FEM
- Simulation Process of AdvantEdge FEM Software: The AdvantEdge FEM software is used for cutting force simulation. The simulation process includes selecting the cutting simulation process type, setting simulation parameters, calculating, and post-processing the results.
- Establishment of the Finite Element Simulation Model: The models of the cutter head and gear blank are established and simplified before being imported into the software. The materials of the cutter head and gear blank are selected, and the relative positions and motions are set.
- Simulation Results of Milling Cutting Force: Orthogonal experiments are designed to study the influence of spindle speed and feed rate on cutting force. The simulation results show that as the feed rate increases, the cutting force amplitude and temperature increase. The influence of spindle speed on the cutting force amplitude is relatively small, but it affects the cutting time.
3.5 Frequency Component Analysis of Simulation Cutting Force
- Influence of Feed Rate on the Frequency Component of Cutting Force: The frequency components of the cutting force under different feed rates are analyzed. The results show that as the feed rate increases to 0.4mm, the cutting force amplitude increases and tends to shift to higher frequencies. The frequency distribution mainly concentrates in the low-frequency region, and the Y direction has the largest amplitude at the first-order frequency.
- Influence of Spindle Speed on the Frequency Component of Cutting Force: The frequency components of the cutting 力 under different spindle speeds are analyzed. The results show that as the spindle speed increases, the amplitude of the first-order frequency in the X direction first decreases and then increases, and the amplitude of the second-order frequency first increases and then decreases. The frequency distribution mainly concentrates in the low-frequency region, and the Y direction has the largest amplitude at the first-order frequency.
3.6 Chapter Summary
In this chapter, the cutting force is identified as the main incentive force in the milling process. A theoretical cutting force model is established, and the influence of process parameters on cutting force is analyzed. The cutting force is simulated using AdvantEdge FEM software, and the frequency components are analyzed. The results provide a basis for further studying the influence of incentive forces on the dynamic characteristics of the spindle system.
4. Dynamic Characteristics Analysis and Verification of Process Matching Effect
4.1 Transient Response Analysis Based on Theoretical Cutting Force
Based on the calculated results of theoretical cutting force, the transient response of the spindle system is analyzed. Four sets of cutting force amplitudes are selected and substituted into the finite element model to study the transient response. The results show that when the feed rate is 0.3mm and the spindle speed is 120rpm, the spindle system has the smallest vibration amplitude and the shortest recovery time.
4.2 Milling Cutting Experiment Verification
- Introduction to the Rotary Dynamometer: The Kistler rotary dynamometer is used to measure the cutting force in the milling process. It integrates a piezoelectric dynamometer and a torque sensor and can measure cutting forces in milling and drilling processes.
- Milling Cutting Force Experiment: Milling experiments are conducted 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 peaks in three directions are generally consistent with the theoretical and simulation results, and the Y direction has the largest cutting force amplitude.
- Milling Cutting Vibration Experiment: The vibration signals at the spindle end during the milling process are collected. The results show that 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 tooth surface quality.
4.3 Process Matching
- Frequency Analysis of Milling Cutting Force: The frequency components of the cutting force collected in the experiment are analyzed. The results show that when the spindle speed is 120rpm and the feed rate is 0.3mm, the cutting force frequency components are mainly concentrated in the low-frequency region and are far from the first-order natural frequency of the spindle system, indicating good matching.
- Response Analysis Based on Milling Incentive Force: The response of the spindle system to the cutting force in three directions is analyzed. The results show that when the spindle system is subjected to cutting forces in three directions, it is in a stable milling state. The sensitivity of the spindle system to cutting forces in different directions is calculated, and it is found that the Y direction is the most sensitive.
- Process Optimization Based on Milling Experiments: The process parameters are optimized based on the experimental results. By adjusting the feed speed to keep the cutting force constant, the milling time is reduced, and the impact load on the spindle system and tooth surface quality is reduced. The influence of bearing stiffness on the sensitivity of the spindle system to cutting forces is also studied.
4.4 Chapter Summary
Tables and Figures
Table 1: Comparison of Natural Frequencies Calculated by Matlab and Finite Element Method
| Order | Matlab Calculation Results (Hz) | Finite Element Calculation Results (Hz) |
|---|---|---|
| 1st | 68.542 | 57.648 |
| 2nd | 77.150 | 73.639 |
| 3rd | 1447.0 | 1446.9 |
| 4th | 2231.9 | 2249.8 |
Table 2: Orthogonal Experiment Results for Milling Cutting Force Simulation
| Number | Spindle Speed (rpm) | Feed Rate (mm) | Fx (N) | Fy (N) | Fz (N) |
|---|---|---|---|---|---|
| 1 | 100 | 0.1 | 168 | 480 | 195 |
| 2 | 100 | 0.2 | 246 | 654 | 223 |
| 3 | 100 | 0.3 | 388 | 714 | 200 |
| 4 | 100 | 0.4 | 899 | 1494 | 594 |
| 5 | 120 | 0.1 | 165 | 556 | 156 |
| 6 | 120 | 0.2 | 202 | 680 | 200 |
| 7 | 120 | 0.3 | 480 | 768 | 212 |
| 8 | 120 | 0.4 | 996 | 1578 | 689 |
| 9 | 140 | 0.1 | 175 | 454 | 199 |
| 10 | 140 | 0.2 | 225 | 684 | 207 |
| 11 | 140 | 0.3 | 446 | 856 | 207 |
| 12 | 140 | 0.4 | 1079 | 1685 | 656 |
| 13 | 160 | 0.1 | 146 | 474 | 182 |
| 14 | 160 | 0.2 | 263 | 650 | 246 |
| 15 | 160 | 0.3 | 368 | 787 | 223 |
| 16 | 160 | 0.4 | 1100 | 1979 | 701 |