This article focuses on the spindle system of a spiral bevel gear milling machine, an essential component in gear manufacturing. The dynamic characteristics of the spindle system and the selection of processing parameters significantly impact the gear’s tooth surface quality and machining efficiency. Current research on spindle system dynamics is often limited to general milling or lathe machines, with less attention on spiral bevel gear milling machines. Moreover, the selection of processing parameters usually relies on experience or reference manuals, which may lead to suboptimal performance and potential issues. This study aims to address these gaps by analyzing the dynamic response of the spindle system under different processing parameters and establishing a matching method between the process parameters and the spindle system’s dynamic characteristics. Through theoretical modeling, simulation, and experimental verification, a more efficient and accurate way to select processing parameters is achieved, which can improve the machining quality and performance of the spiral bevel gear milling machine.
1. Introduction
1.1 Research Background and Significance
In modern manufacturing, machine tools play a crucial role in determining the quality and efficiency of workpiece production. The spindle system, as a key part in direct contact with the workpiece, has a significant impact on the machining process. Its vibration characteristics, including inherent properties, dynamic responses, and cutting stability, are essential aspects to consider. Inaccurate spindle system dynamics can lead to reduced machining accuracy, increased tool wear, and even potential safety hazards in CNC machining. For spiral bevel gear milling machines, the interaction between the spindle system dynamics and processing parameters is complex and requires in-depth study to optimize the machining process.
1.2 Current Research Status
- Dynamics Modeling Technology: Various methods such as concentrated parameter method, transfer matrix method, and finite element method have been developed for spindle system dynamics modeling. However, each method has its limitations, and the application to spiral bevel gear milling machine spindle systems needs further exploration.
- Incentive Force Research: While studies on excitation forces during machine tool operation have been conducted, research on the specific excitation forces in spiral bevel gear milling processes is relatively scarce.
- Process Matching Research: Although many methods for optimizing processing parameters exist, research on the matching of processing parameters with the spindle system dynamics of spiral bevel gear milling machines is insufficient.
1.3 Research Objectives and Methods
The main objective of this research is to establish a matching method between the processing parameters and the dynamic characteristics of the spindle system of a spiral bevel gear milling machine. This involves building a dynamic numerical model of the spindle system, analyzing the cutting force as the main excitation source, and conducting experimental verification. The research methods include theoretical analysis based on finite element dynamics and Timoshenko beam theory, simulation using software like AdvantEdge FEM, and experimental measurements with tools such as a rotating dynamometer.
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 study, the finite element method is used. The basic principle involves dividing the structure into multiple elements and obtaining the unit body’s dynamics equation using the energy principle. The equation includes terms for kinetic energy, potential energy, and non-conservative forces. After obtaining the unit mass, damping, and stiffness matrices, the system’s overall matrices are assembled, and the dynamics balance equation is derived.
2.2 Spindle System Dynamics Modeling Based on Beam Elements
The spindle system of a milling machine consists of components like the shaft, bearings, and milling cutter head. To simplify the model, certain assumptions are made, such as treating the tapered roller bearings as elastic supports with constant stiffness and considering the shaft sleeve and gears as additional distributed masses. The system is discretized into 12 units and 13 nodes along the axis, and a finite element dynamics model is established using Timoshenko beam elements, which consider shear deformation.
2.3 System Motion Differential Equation Establishment
- Unit Mass and Stiffness Matrices: The Timoshenko beam model is used to calculate the unit mass and stiffness matrices. Each node of the unit has 6 degrees of freedom, and the matrices are derived based on the element’s geometric and material properties.
- System Total Assembly Mass and Stiffness Matrices: The unit matrices are assembled to obtain the system’s total mass and stiffness matrices. The bearing stiffness matrices at nodes 5 and 9 are also considered and incorporated into the system matrices through coordinate rotation.
- Spindle System Motion Differential Equation: The final motion differential equation of the spindle system is obtained, which includes the system’s total mass, damping, and stiffness matrices, as well as the external force vector and the total displacement vector.
2.4 Natural Frequencies and Modal Shapes
By solving the motion differential equation for the system’s free vibration, the natural frequencies and corresponding modal shapes are calculated. The results obtained from Matlab and finite element calculations are compared, showing a good agreement overall, with some minor differences due to model simplifications, numerical calculation precision, boundary condition settings, and program implementation.
2.5 Spindle System Harmonic Response Analysis
- Harmonic Response Analysis Theory: Harmonic response analysis studies the system’s response to periodic harmonic excitation, which is relevant for the spindle system affected by cutting forces. The key is to solve the dynamics equation under harmonic excitation and obtain the displacement response data at a specific frequency.
- Spindle System Harmonic Response Analysis Based on Matlab: The modal superposition method is chosen for the harmonic response analysis of the spindle system in this study. The results show that the system’s displacements in different directions have significant peaks around 78Hz, indicating a potential resonance phenomenon. The analysis also reveals the directional and multi-modal characteristics of the system’s response to external forces and torques.
2.6 Transient Dynamics Analysis
Transient dynamics analysis considers the system’s response to non-steady and sudden external loads during the machining process. In this study, the spindle system’s transient response to impact loads during milling is analyzed using the Newmark method. The results show that different processing parameters affect the impact loads on the system, and the bearing positions have a buffering effect on the impact transmission. The transient response analysis results obtained from Matlab and finite element software are compared, validating the accuracy of the program.
2.7 Chapter Summary
This chapter establishes a dynamics model of the spindle system based on the Timoshenko beam theory, analyzes its natural frequencies, modal shapes, harmonic responses, and transient responses. The results obtained from different methods show good consistency, providing a theoretical basis for further research on process parameter matching.
3. Milling Cutting Incentive Force Analysis
3.1 Spiral Bevel Gear Milling Dynamic Force Analysis
In the milling process, the spindle system is affected by various incentive forces, with the cutting force being the most significant. The proper selection of processing parameters is crucial for balancing the tooth surface quality and machining efficiency while considering the impact of these forces on the spindle system.
3.2 Theoretical Milling Cutting Force Model
- Spiral Bevel Gear Milling Analysis: The cutting force model focuses on the machining of the small wheel using the generating method. The milling cutter head has two motion modes, and the relationship between the cutter head and the gear blank’s rotation speeds is defined by the roll ratio.
- Oblique Cutting Model: The oblique cutting model is used to describe the cutting mechanics as the cutting edges of the cutter head satisfy this model. The forces in the shear plane and the rake face are analyzed, and a coordinate system is established to describe the motion and force relationships.
- Generating Method for Machining Small Wheel’s Cutter Head Forming Surface Equation: The equation for the cutter head’s forming surface during the machining of the small wheel is derived, and the unit normal vector at any point on the cutting edge is calculated.
- Cutter Head Cutting Force Component Calculation: Based on the theoretical cutting force model, the cutting force components in different directions for the inner and outer cutters are calculated by discretizing the cutting edge and considering the oblique cutting process.
3.3 Influence Analysis of Processing Parameters Based on the Theoretical Cutting Force Model
- Influence of Feed Rate on Cutting Force: The theoretical cutting force calculations show that as the feed rate increases from 0.1mm to 0.7mm, the cutting force amplitudes in all three directions increase, with the Y direction having the largest cutting force amplitude and being identified as the main cutting force direction.
- Influence of Spindle Speed on Feed Rate: As the spindle speed increases, the cutting force amplitudes in all three directions show a linear increasing trend, with the Y direction still having the largest cutting force.
3.4 Finite Element Simulation of Spiral Bevel Gear Milling Cutting Force Based on AdvantEdge FEM
- Simulation Process: The AdvantEdge FEM software is used for cutting force simulation. The simulation process includes selecting the appropriate cutting simulation process type, setting simulation parameters, performing the simulation calculation, and post-processing the results to obtain key data such as cutting force, cutting temperature, and stress.
- Finite Element Simulation Model Establishment: The cutter head and gear blank models are established and simplified in 3D software before being imported into AdvantEdge FEM for simulation.
- Milling Cutting Force Simulation Results: The simulation results show that as the feed rate increases, the cutting force amplitudes increase, and the cutting temperature also rises. The influence of spindle speed on the cutting force is also analyzed, with the Y direction having the largest cutting force amplitude and the cutting time decreasing as the spindle speed increases.
3.5 Frequency Component Analysis of Simulated Cutting Force
- Influence of Feed Rate on Cutting Force Frequency Components: As the feed rate increases, the cutting force amplitude shows an increasing trend and a tendency to shift to higher frequencies. However, the frequency distribution in all three directions is mainly concentrated in the low-frequency region, with the base frequency around 15Hz.
- Influence of Spindle Speed on Cutting Force Frequency Components: As the spindle speed increases, the first-order frequency amplitude in the X direction first decreases and then increases, while in the Y direction, it first decreases and then increases more significantly. The frequency distribution in all three directions remains mainly in the low-frequency region and away from the system’s first-order natural frequency.
3.6 Chapter Summary
This chapter analyzes the main incentive force (cutting force) in the milling process, establishes a theoretical cutting force model, and studies the influence of different processing parameters on the cutting force and its frequency components through simulation and analysis.
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 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 Milling Cutting Experiment Verification
- Introduction to the Rotating Dynamometer: The Kistler rotating dynamometer is used to measure the cutting force during the milling experiment. It can measure milling and drilling cutting forces and provides real-time data on the torque acting on the tool.
- Milling Cutting Force Experiment: The milling experiment is carried out on a YKH2235 CNC spiral bevel gear milling machine. Two sets of processing parameters are selected for comparison, and the cutting forces in three directions are measured. The experimental results show that the actual cutting force amplitudes are generally consistent with the theoretical and simulation results, with the Y direction having 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 as the feed rate increases, the spindle end vibration amplitude 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
- Frequency Analysis of Milling Cutting Force: The frequency components of the cutting force for the two sets of processing parameters are analyzed. The results show that the cutting force frequencies for the first set of parameters (spindle speed 120rpm, feed rate 0.3mm) are mainly concentrated within 50Hz and away from the system’s first-order natural frequency, indicating a good match with the spindle system’s dynamic characteristics.
- 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 system is subjected to the cutting force under the selected process parameters, it is in a stable milling state. The sensitivity of the system to the cutting force in different directions is also analyzed, with the Y direction being the most sensitive.
- Process Optimization Based on Milling Experiments: Based on the experimental results, the feed speed is optimized to keep the cutting force constant and reduce the impact load on the spindle system and the tooth surface quality. The influence of bearing stiffness on the system’s response is also studied, and it is found that adjusting the bearing stiffness can change the system’s sensitivity to the cutting force.
4.4 Chapter Summary
This chapter verifies the influence of different process parameters on the spindle system’s dynamic characteristics through theoretical analysis, experiments, and simulations. The optimal process parameters are determined, and the matching effect between the process parameters and the spindle system is verified.
5. Conclusion and Outlook
5.1 Summary of the Research
- Spindle System Dynamics Modeling: A dynamics model of the spindle system of a spiral bevel gear milling machine is established based on finite element dynamics and Timoshenko beam theory. The natural frequencies, modal shapes, harmonic responses, and transient responses of the system are analyzed, and the results are verified by comparing with finite element software calculations.
- Cutting Force Analysis: 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 different processing parameters on the cutting force and its frequency components is studied through simulation and analysis.
- Process Parameter Matching: The optimal process parameters are determined through theoretical analysis and experimental verification. The matching effect between the process parameters and the spindle system’s dynamic characteristics is analyzed, and methods for optimizing the process parameters are proposed.
5.2 Future Research Directions
- Inclusion of More Incentive Sources: Future research could consider other incentive sources such as the inertia force of moving parts and the excitation force of internal vibration sources in the milling process to make the mathematical model more accurate.
- Consideration of More Processing Parameters: More processing parameters such as tool rake angle and manufacturing errors could be considered to improve the milling process and make it more in line with practical situations.
- Analysis of More System Components: The influence of other components such as the spindle box and column on the spindle system’s dynamic characteristics could be studied to more accurately reveal the system’s response laws and provide a more reliable theoretical basis for optimizing process parameters.
In conclusion, this research provides a comprehensive analysis of the dynamic characteristics of the spindle system of a spiral bevel gear milling machine and a matching method for processing parameters. The findings and methods have practical significance for improving the machining quality and efficiency of spiral bevel gear milling machines. Future research directions offer potential for further enhancing the understanding and optimization of the milling process.
6. Introduction
6.1 Research Background and Significance
In modern manufacturing, machine tools are the core equipment that directly affects the quality and efficiency of workpiece production. The spindle system, as a crucial component in direct contact with the workpiece, plays a vital role in determining the machining performance. Its dynamic characteristics, including inherent properties, dynamic responses, and cutting stability, have a significant impact on the surface quality of the workpiece and the lifespan of the tool.
For spiral bevel gear milling machines, the interaction between the spindle system dynamics and processing parameters is complex. Inaccurate understanding and control of these factors can lead to reduced machining accuracy, increased tool wear, and even potential safety hazards. Therefore, a detailed study of the spindle system’s dynamic characteristics and the matching of processing parameters is essential for optimizing the machining process and improving the quality of spiral bevel gears.
6.2 Current Research Status
- Dynamics Modeling Technology:
- Concentrated Parameter Method: This method simplifies the structure by concentrating the mass at several nodes and using equivalent elastic beams to simulate the elastic properties. However, it has limitations in accurately modeling 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 system into discrete elements and uses transfer matrices to relate the state vectors of each unit. Although it has advantages in program simplicity and speed, it may have reduced accuracy in calculating high-order modes and may not fully capture the system’s dynamic behavior in the presence of nonlinear and coupling effects.
- Finite Element Method: With the development of computer technology, the finite element method has become an important numerical analysis tool. It can analyze the static and dynamic characteristics of the spindle system, providing a more accurate and comprehensive understanding of the system’s performance. However, its application to spiral bevel gear milling machine spindle systems requires further exploration.
- Incentive Force Research:
- Most studies on excitation forces during machine tool operation have focused on artificial excitation during machine shutdown or on using the vibration signal of the machine itself as an excitation source. However, research on the specific excitation forces in spiral bevel gear milling processes, considering the complex cutting forces and process parameters, is relatively scarce.
- Some studies have used cutting force as an excitation source, but the methods are often limited to simple identification of natural frequencies or require complex measurement techniques.
- Process Matching Research:
- Many methods for optimizing processing parameters exist, such as using optimization algorithms or conducting experiments and simulations. However, research on the matching of processing parameters with the spindle system dynamics of spiral bevel gear milling machines is insufficient.
- Most studies have focused on using optimization algorithms to optimize process parameters, with less attention on optimizing based on the system’s response.
6.3 Research Objectives and Methods
The main objective of this research is to establish a matching method between the processing parameters and the dynamic characteristics of the spindle system of a spiral bevel gear milling machine. This involves:
- Building a dynamic numerical model of the spindle system based on finite element dynamics and Timoshenko beam theory.
- Analyzing the cutting force as the main excitation source and its influence on the spindle system’s dynamic response.
- Conducting experimental verification to validate the proposed matching method.
The research methods include:
- Theoretical analysis to derive the dynamics equations of the spindle system and calculate its natural frequencies, modal shapes, harmonic responses, and transient responses.
- Simulation using software like AdvantEdge FEM to calculate the cutting force under different process parameters and analyze its frequency components.
- Experimental measurements with tools such as a rotating dynamometer to collect cutting force and vibration signals during the milling process and verify the theoretical and simulation results.
7. Spindle System Dynamics Modeling and Analysis
7.1 Rotor Dynamics Equation Construction Principles
The dynamics model of the spindle system can be established through different methods:
- Theoretical Modeling: This method simplifies the structure and uses theoretical principles to derive the dynamics equations. However, it may have limitations due to the lack of a unified and standardized theoretical calculation method for the dynamic characteristics of joints.
- Experimental Modeling: It can accurately reflect the actual dynamics performance of the machine tool but requires a prototype and has high design costs.
- Combined Modeling: This combines the advantages of theoretical and experimental modeling but also inherits some of the limitations of experimental modeling.
In this study, the finite element method is used for theoretical modeling. The basic principle involves dividing the structure into multiple elements and applying the energy principle to obtain the unit body’s dynamics equation. The equation includes terms for kinetic energy (), potential energy (), and non-conservative forces (). The expressions for these terms are as follows:
where is the unit density, is the derivative of displacement with respect to time, is the volume of the unit body, is the deformation potential energy of the unit body, is the boundary area of the unit, is the body force, is the surface force, and is the viscous damping coefficient.
By substituting the expressions for , , and into the equation and using the relationships , , , and (where is the displacement of the unit node, is the shape function, is the strain transformation matrix, and is the elasticity matrix), the unit mass matrix (), damping matrix (), stiffness matrix (), and load vector () can be obtained as follows:
The system’s overall mass matrix (), damping matrix (), and stiffness matrix () are then assembled from the unit matrices, and the dynamics balance equation of the system is given by:
where , , and are the node displacement, velocity, and acceleration of the structure, respectively, and is the load vector of the structure nodes.
In practice, the damping characteristics of the actual structure are difficult to accurately measure. Commonly, the overall damping matrix is determined based on the system’s overall characteristics rather than calculating the damping matrix of each unit directly. For small damping cases, the damping matrix can be expressed as a Rayleigh damping matrix:
where and are the mass and stiffness proportional coefficients of the proportional damping, which can be calculated using the following formulas:
where and are the damping coefficients, and and are the first two natural frequencies.
7.2 Spindle System Dynamics Modeling Based on Beam Elements
The spindle system of a milling machine consists of several components, including the shaft, bearings, and milling cutter head. To simplify the model for analysis, certain assumptions are made:
- The tapered roller bearings are simplified as elastic supports with the pivot point located on the spindle axis.
- The influence of load and spindle speed on the bearing stiffness is ignored, and the bearing stiffness is considered a constant.
- The shaft sleeve, gears, etc. are equivalent to the shaft material with the same density and are regarded as additional distributed masses acting on the unit nodes.
- The influence of additional structural factors such as threaded holes and chamfers is ignored.
After these simplifications, the spindle system is discretized into 12 units along the axis, forming 13 nodes. Each unit is connected at the nodes on the axis, and the mass of the system is concentrated at each node. Thus, a finite element dynamics model of the spindle system is established using Timoshenko beam elements.
The Timoshenko beam model is chosen because it considers shear deformation, which is more suitable for analyzing mechanical systems with smaller width-to-diameter ratios of shaft segments. In contrast, the Euler – Bernoulli beam model does not consider shear deformation and is only applicable to cases where the width-to-diameter ratio of the shaft segment is large.
7.3 System Motion Differential Equation Establishment
- Unit Mass and Stiffness Matrices:
- For the Timoshenko beam model, when considering an arbitrary unit with adjacent nodes and , each node has 6 degrees of freedom. The displacement vector of the unit in the local coordinate system is given by:
- The unit mass matrix () and stiffness matrix () are derived based on the geometric and material properties of the unit. The expressions for these matrices are complex and involve parameters such as the cross-sectional area (), length (), elastic modulus (), shear elastic modulus (), polar moment of inertia (), and section moment of inertia () of the beam element, as well as a section influence coefficient () that depends on the cross-sectional shape of the element.
- System Total Assembly Mass and Stiffness Matrices:
- The unit mass and stiffness matrices are assembled to obtain the system’s total mass and stiffness matrices. The assembly process involves considering the coupling between adjacent units and incorporating the bearing stiffness matrices at nodes 5 and 9.
- The bearing stiffness matrices are first transformed to the same coordinate system as the shaft’s stiffness matrix through coordinate rotation. The rotation matrix is given by:
- After transforming the bearing stiffness matrices, they are added to the corresponding positions of the shaft’s stiffness matrix to obtain the system’s total stiffness matrix. The total mass matrix is obtained in a similar manner.
- Spindle System Motion Differential Equation:
- The final motion differential equation of the spindle system is given by:
where is the system’s total mass matrix, is the total damping matrix, is the total stiffness matrix, is the total external force vector acting on the system, and is the total displacement vector of the system. - For the system’s free vibration, the differential equation is:
Solving this equation yields the system’s natural frequencies and characteristic vectors.
- The final motion differential equation of the spindle system is given by:
7.4 Natural Frequencies and Modal Shapes
By solving the free vibration differential equation of the system, the natural frequencies and corresponding modal shapes are calculated. The results are obtained using Matlab programming and are compared with those from finite element calculations. The comparison shows that the first four natural frequencies calculated by both methods are generally consistent, validating the effectiveness and correctness of the simplified model using Timoshenko beam elements and the accuracy of the written program.
The reasons for the minor differences between the two calculation results may include:
- Model Simplification: The simplified model used in this study may not fully consider all the details of the actual system, such as complex geometric shapes, nonlinear characteristics, or other factors, which may affect the accuracy of the calculation results.
- Numerical Calculation Precision: The accuracy of the numerical calculation method used, including the choice of calculation parameters and the convergence of the calculation, can have an impact on the results.
- Boundary Condition Settings: Different methods are used to simulate the bearings in Matlab and finite element calculations. In Matlab, the bearing stiffness matrix is added directly, while in finite element calculations, a spring unit (Combination214) is used. These differences in boundary condition settings can lead to differences in the calculated natural frequencies and modal shapes.
- Program Implementation: Errors in the writing and implementation of the calculation program, such as the choice of algorithm and program logic, can also cause differences in the results.
Despite these minor differences, the overall consistency of the results validates the reliability and accuracy of the adopted calculation methods. The modal shapes obtained from both methods are also compared, showing similar characteristics such as swinging for the first and second orders, bending deformation for the third order, and torsional deformation for the fourth order.
7.5 Spindle System Harmonic Response Analysis
- Harmonic Response Analysis Theory:
- The harmonic response analysis studies the system’s response to external harmonic excitation. For the spindle system of a milling machine, the cutting force is a typical harmonic force. The key to harmonic response analysis is to solve the dynamics equation under harmonic excitation and obtain the displacement response data at a specific frequency.
- The dynamics equation for harmonic response analysis is given by:
where , , and are the total mass matrix, damping matrix, and stiffness matrix of the system, respectively, , , and are the node displacement, velocity, and acceleration of the system, respectively, and is the external load acting on the system. - Assuming the solution of the equation is , where is the maximum displacement of the system under the external load, is the unit complex number, is the phase angle, and is the excitation frequency (angular frequency), the equation can be further transformed and analyzed to obtain the relationships between the forces and displacements in different directions.
- Spindle System Harmonic Response Analysis Based on Matlab:
- In this study, the modal superposition method is chosen for the harmonic response analysis of the spindle system. This method has advantages in terms of simplicity and applicability to large-scale systems, especially considering that the spindle system usually has a large scale and the modal shapes of the system have already been obtained.
- The results of the harmonic response analysis using Matlab show that the displacements of the spindle system in different directions have significant peaks around 78Hz. This indicates a potential resonance phenomenon when the excitation frequency is around 78Hz, which can have a significant impact on the system’s stability and performance.
- The analysis also reveals the directional and multi-modal characteristics of the system’s response to external forces and torques. When a force or torque is applied in a particular direction, the response in that direction is more prominent, but there are also responses in other directions, indicating the complexity of the system’s dynamic behavior.
7.6 Transient Dynamics Analysis
- Motivation for Transient Dynamics Analysis:
- The spindle system of a milling machine is subjected to sudden and periodic loads during the machining process, which can cause vibration, resonance, and deformation problems, thereby affecting the machining accuracy, tooth surface quality, and tool lifespan.
- Transient dynamics analysis is necessary to understand the system’s response to non-steady and sudden external loads and to identify potential vibration problems, so as to select appropriate process parameters and develop optimization strategies.
- Transient Response Analysis Method:
- In this study, a specific milling process scenario is assumed, where the spindle reaches a rated speed of 120rpm before the cutter head contacts the gear blank, with a feed rate of 0.2mm. During the milling process, the cutter head is subjected to tangential (X-direction), radial (Y-direction), and axial (Z-direction) forces with magnitudes of 200N, 500N, and 200N, respectively, and the excitation is considered a pulse excitation with a pulse width of 0.005s.
- The Newmark method is used to calculate the displacements of the spindle system in three directions at different positions as a function of time.
- Analysis of Transient Response Results:
- The results of the transient response analysis show that the displacements of the spindle system in different directions and at different positions have different characteristics. For example, when subjected to an X-direction impact load, the displacements at the bearing positions are smaller…