In my research, I focus on the dynamic behavior of spiral bevel gear milling machines during gear cutting, because the machining accuracy of spiral bevel gears directly governs transmission performance, noise generation, and service life. The dynamic characteristics of the milling machine are among the most important factors influencing gear cutting quality. Many existing studies examine static stiffness, single-spindle dynamics, or local tool-workpiece cutting dynamics separately, but the coupled effect among the tool, workpiece, spindle, and housing during gear cutting is still not fully unified. Therefore, I build a coupled dynamic model that integrates cutting dynamics with spindle characteristics, and I validate it through experiments. My goal is to reveal the vibration mechanism in gear cutting, propose process parameter optimization strategies, and achieve more efficient and stable gear cutting.
The gear cutting process for spiral bevel gears is complex because the tooth surface is generated by a combination of tool rotation, workpiece rotation, and relative feed motion. In my work, I treat gear cutting as a dynamic process in which the cutting force excitation, the dynamic cutting stiffness, and the machine structure interact continuously. I use the generation principle and machine kinematics to establish an analytical framework for the tool-workpiece motion trajectory during gear cutting. By analyzing the relative motion and cutting geometry, I quantify the cutting load excitation and compute the dynamic cutting stiffness. Then I incorporate this stiffness into the dynamic model of the spindle-tool-workpiece system.
I begin with the generation motion of spiral bevel gear cutting. The machine tool has a cutter spindle system and a workpiece spindle system. The cutter spindle can move linearly in the Z and Y directions and rotate, while the workpiece spindle can move in the X direction and rotate around the Z and X axes. This multi-axis motion enables the required gear cutting motion. The generating method simulates the meshing between a virtual gear and the workpiece, so the tool cutting surface and the workpiece tooth surface remain conjugate at the contact point. The roll ratio between the workpiece and the cutter is defined as
$$ R_{aq} = \frac{\omega_c}{\omega_s} $$
where $\omega_c$ is the cutter angular velocity and $\omega_s$ is the workpiece angular velocity. This ratio is a fundamental parameter in gear cutting because it determines the relative motion and the generated tooth surface.
For the instantaneous cutting speed distribution, I define the cutting speed $v_e$ as
$$ v_e = \pi R_c N $$
where $R_c$ is the nominal cutter radius and $N$ is the cutter spindle speed. The normal rake angle and the edge inclination angle are obtained from the tool geometry and the velocity vectors. These angles affect the cutting force coefficients and the dynamic stiffness in gear cutting.
The basic coordinate system for spiral bevel gear cutting includes the static coordinate system, the machine coordinate system, the cutter coordinate system, and the workpiece coordinate system. The relative position vector between the tool and the workpiece is expressed through transformation matrices that depend on the machine settings, such as the horizontal wheel position, the床位, the vertical wheel position, the radial cutter position, the root cone angle, and the swivel angle. In my model, I express the workpiece axis vector and the cutter angular position vector as
$$ \mathbf{V}_p = [\cos\Gamma, 0, \sin\Gamma] $$
$$ \mathbf{V}_{qt} = [\cos q_t, -\sin q_t, 0] $$
where $\Gamma$ is the machine root cone angle and $q_t$ is the instantaneous swivel angle. The cutter axis vector is obtained through rotation matrices. The motion trajectory of the workpiece spindle is then written as a function of the roll ratio, the swivel angle, and the feed parameters. These kinematic equations provide the position of every cutting edge during gear cutting and form the basis for calculating the instantaneous cutting area and dynamic stiffness.
The dynamic cutting stiffness is a key factor in gear cutting dynamics because it links the cutting parameters to the machine structural response. I model the contact between the tool flank face and the chip as an indentation contact. For a square cross-section indenter, the contact stiffness $K_S$ is related to the contact area $S_b$ and the equivalent elastic modulus $E_r$ by
$$ K_S = \frac{2}{\sqrt{\pi}} \beta S_b E_r $$
where $\beta$ is a constant that depends on the indenter shape. The equivalent elastic modulus is
$$ E_r = \frac{1-\nu_1^2}{E_1} + \frac{1-\nu_2^2}{E_2} $$
with $E_1$, $E_2$ being the elastic moduli and $\nu_1$, $\nu_2$ the Poisson ratios of the tool and workpiece materials. The contact area is approximated by
$$ S_b = l_s \frac{a_w}{\cos\lambda_s} $$
where $l_s$ is the contact length, $a_w$ is the cutting width, and $\lambda_s$ is the edge inclination angle. The contact length evolves with the instantaneous cutting thickness during gear cutting.
To calculate the cutting force coefficients, I use an orthogonal cutting model and then extend it to oblique cutting. The shear force on the shear plane is
$$ F_s = \tau_s S_s = \tau_s a_c a_w $$
where $\tau_s$ is the shear stress on the shear plane, $a_c$ is the uncut chip thickness, and $a_w$ is the cutting width. The force components on the tool in the y and z directions are
$$ F_z = \frac{F_s \cos(\beta – \gamma_0)}{\cos(\phi + \beta – \gamma_0)} $$
$$ F_y = \frac{F_s \sin(\beta – \gamma_0)}{\cos(\phi + \beta – \gamma_0)} $$
where $\beta$ is the friction angle, $\gamma_0$ is the tool rake angle, and $\phi$ is the shear angle. For oblique cutting, the cutting force coefficients $K_{tc}$, $K_{rc}$, and $K_{ac}$ are derived by transforming the orthogonal cutting forces into the normal plane. These coefficients are essential for predicting the dynamic cutting force in gear cutting.
The instantaneous dynamic uncut chip thickness is expressed as
$$ a_c = \Delta x \sin\theta_j + \Delta y \cos\theta_j $$
where $\Delta x$ and $\Delta y$ are the relative dynamic displacements between the current cutting edge and the previous cutting edge, and $\theta_j$ is the angular position of the cutting edge. The time delay between teeth is
$$ \tau = \frac{2\pi}{Z \omega_c} $$
where $Z$ is the number of cutter teeth. The dynamic force in gear cutting is then
$$ F_{c,j} = a_x K_{tc} a_c $$
$$ F_{d,j} = a_y K_{rc} a_c $$
By resolving the forces into the x and y directions, I obtain the dynamic cutting force vector that excites the spindle-tool-workpiece system.
I calculate the dynamic cutting stiffness for different gear cutting parameters. The contact length $l_s(\theta_j)$ is obtained from a quadratic equation that depends on the instantaneous uncut chip thickness. The contact area $S_b(\theta_j)$ is then computed, and the dynamic stiffness is
$$ K_s(\theta_j) = \frac{2}{\sqrt{\pi}} \beta S_b(\theta_j) E_r $$
I use a CAD-based numerical simulation to reconstruct the gear cutting process, because the cutting area changes continuously along the tooth surface. The simulation steps include solving the tool-workpiece kinematics, building a parametric solid model of the tool and blank, and performing Boolean operations to extract the instantaneous cutting area. The dynamic stiffness is then calculated for each cutting step.
| Feed rate (mm/min) | 40 | 60 | 80 | 100 |
|---|---|---|---|---|
| 100 r/min – max stiffness (N/mm) | 221958 | 271842 | 313896 | 350946 |
| 120 r/min – max stiffness (N/mm) | 214740 | 263002 | 303689 | 339535 |
| 140 r/min – max stiffness (N/mm) | 205859 | 252124 | 291128 | 325491 |
| 160 r/min – max stiffness (N/mm) | 197199 | 241519 | 278882 | 311799 |
From these results, I observe that the dynamic stiffness increases with feed rate but decreases with cutter speed. At the same cutter speed, increasing the feed rate by 20 mm/min raises the maximum dynamic stiffness by about $3.1 \times 10^4$ N/mm. At a fixed feed rate, increasing the cutter speed reduces the stiffness because of inertia effects and the dynamic coupling of the cutting force. The fluctuation of the dynamic stiffness during a single gear cutting step is about $1.5 \times 10^5$ N/mm. This time-varying stiffness is a major source of vibration in gear cutting.
For the dynamic model, I simplify the cutting system by treating the tool-chip contact zone as a concentrated vibration source with time-varying stiffness. The dynamic differential equation of the gear cutting system in the x-y plane is
$$ m_{x1}\ddot{x} + c_{x1}\dot{x} + k_{x1}x = F_x $$
$$ m_{y1}\ddot{y} + c_{y1}\dot{y} + k_{y1}y = F_y $$
where $m_{x1}$, $c_{x1}$, $k_{x1}$ are the modal mass, damping, and stiffness in the x direction, and $m_{y1}$, $c_{y1}$, $k_{y1}$ are the corresponding quantities in the y direction. I then extend this model to include the spindle-tool and spindle-workpiece systems. The cutting point stiffness is decomposed into normal, tangential, and radial components:
$$ K_t = K_s \cos\gamma \cos\lambda + \mu K_s \sin\lambda \tan\eta $$
$$ K_r = K_s \sin\gamma \cos\lambda $$
where $\lambda$ is the edge inclination angle, $\gamma$ is the rake angle, $\mu$ is the friction coefficient, and $\eta$ is the chip flow angle. This decomposition allows me to couple the cutting stiffness with the spindle structures in the x, y, and z directions.
I build the spindle system model using Timoshenko beam elements. The tool spindle and workpiece spindle are discretized into multiple segments with different outer and inner diameters. The bearing stiffness is represented by a 6×6 stiffness matrix that includes radial, axial, and tilting stiffness components. The bearing stiffness is computed from the contact deformation of the tapered rollers. The contact stiffness of the bearing outer race is
$$ K_{no} = \left[ G_o + G_i \left( \frac{\sin(\alpha_o + \alpha_f)}{\cos(\alpha_i + \alpha_f)} \right)^{0.9} \right]^{-1.11} $$
where $G_o$ and $G_i$ are the compliance coefficients of the outer and inner races. The full bearing stiffness matrix is obtained by differentiating the load-displacement relations with respect to the six degrees of freedom. I then assemble the global stiffness matrix of the spindle system by summing the beam element stiffness matrices and the bearing stiffness matrices at the corresponding nodes.
| Component | Young’s modulus (GPa) | Poisson ratio | Density (kg/m³) | Yield strength (MPa) |
|---|---|---|---|---|
| Tungsten alloy cutter | 250 | 0.28 | 15000 | 1750 |
| Alloy steel spindle | 210 | 0.30 | 7860 | 890 |
The equation of motion for the spindle system is
$$ \mathbf{M}\ddot{\mathbf{X}} + \mathbf{C}\dot{\mathbf{X}} + \mathbf{K}\mathbf{X} = \mathbf{F} $$
where $\mathbf{M}$, $\mathbf{C}$, and $\mathbf{K}$ are the global mass, damping, and stiffness matrices. For free vibration analysis, I neglect damping and solve
$$ \mathbf{M}\ddot{\mathbf{X}} + \mathbf{K}\mathbf{X} = 0 $$
The natural frequencies and mode shapes are extracted using the Lanczos method. I analyze the tool spindle and workpiece spindle separately, and then I couple them through the cutting point stiffness. The coupled model shows that the first natural frequency of the tool spindle increases from 47.4 Hz to 67.3 Hz when the cutting stiffness is included. The workpiece spindle first frequency increases from 103.3 Hz to 127.3 Hz. Some higher-order frequencies remain unchanged because they are dominated by local structural modes.
| Mode | Tool spindle (Hz) | Workpiece spindle (Hz) | Coupled system (Hz) |
|---|---|---|---|
| 1 | 47.436 | 103.33 | 56.64 |
| 2 | 56.64 | 184.85 | 67.304 |
| 3 | 496.37 | 787.57 | 127.30 |
| 4 | 621.66 | 860.35 | 183.09 |
| 5 | 1070.3 | 958.72 | 499.36 |
| 6 | 1389.0 | 1326.7 | 621.66 |
The coupled modal analysis reveals that the cutting point stiffness significantly affects the low-order modes. The first coupled frequency is particularly important because it is close to the excitation frequencies that occur during gear cutting. If the cutting force frequency or the bearing frequency coincides with this coupled frequency, resonance can occur, leading to poor surface finish and accelerated tool wear.
I also analyze the spindle housing using a solid finite element model. The housing, spindle, tool, and workpiece are meshed with tetrahedral elements. The bearing connections are modeled as spring-damper elements with the computed stiffness. The tool and workpiece are connected by a spring that represents the dynamic cutting stiffness. The natural frequencies of the housing-tool system and the housing-workpiece system are extracted. The first natural frequency of the tool housing is about 687.8 Hz, which is much higher than the typical gear cutting excitation frequencies. This indicates that the tool housing has good vibration resistance. The workpiece housing has a lower natural frequency, so the workpiece spindle relies more on its own stiffness to resist vibration.
When I include the cutting stiffness in the solid model, a new low-order frequency appears at about 66.04 Hz. This frequency is generated by the tool-workpiece coupling and is sensitive to the cutting stiffness. I vary the cutting stiffness according to the calculated values for different gear cutting parameters. The first coupled frequency changes from 65.71 Hz to 66.04 Hz, which is a small variation. This means that the dynamic cutting stiffness has a limited effect on the coupled system frequency within the normal gear cutting range. However, the appearance of this new frequency is important because it can be excited by the gear cutting process.
| Spindle speed (r/min) | 40 mm/min (Hz) | 60 mm/min (Hz) | 80 mm/min (Hz) | 100 mm/min (Hz) |
|---|---|---|---|---|
| 100 | 65.78 | 65.89 | 65.98 | 66.04 |
| 120 | 65.76 | 65.87 | 65.96 | 66.02 |
| 140 | 65.73 | 65.85 | 66.93 | 66.00 |
| 160 | 65.71 | 65.82 | 65.91 | 65.97 |
I perform harmonic response analysis under cutting force excitation. The cutting force is applied at the cutting point, and the displacement and acceleration responses are extracted at the housing end cover. The results show that when the cutting stiffness decreases from $3.5 \times 10^5$ N/mm to $2.2 \times 10^5$ N/mm, the vibration amplitude increases significantly. The system enters a sensitive region when the stiffness is below $3.0 \times 10^5$ N/mm. This provides a guideline for selecting gear cutting parameters that maintain sufficient dynamic stiffness.
To validate the dynamic model, I design a series of gear cutting experiments using a computer numerical control spiral bevel gear milling machine. The cutter has eight inner blades and eight outer blades, with a total of sixteen teeth. The blade material is high-speed steel, the rake angle is 20 degrees, and the edge inclination angle is 5 degrees. The workpiece material is 20CrMnTi. I use a three-axis accelerometer and a signal acquisition system to measure vibration. The sensors are mounted on the tool spindle housing and the workpiece spindle housing near the spindle. I also use a rotary dynamometer to measure the cutting force and torque during gear cutting.
| Parameter | Value |
|---|---|
| Number of teeth | 11 |
| Module | 4.024 |
| Spiral angle | 49.48° |
| Face width | 27.64 mm |
| Pitch cone angle | 19.48° |
| Pressure angle (concave) | 17.33° |
| Pressure angle (convex) | 20.27° |
| Shaft angle | 90° |
| Outer cone distance | 88.26 mm |
| Pinion offset | 23 mm |
I set the machine adjustment parameters for the pinion rough cutting. The root cone angle is 17.57 degrees, the horizontal wheel position is 1.78 mm, the床位 is -0.97 mm, the vertical wheel position is 23.86 mm downward, the radial cutter position is 69.83 mm, and the roll ratio is 4.09179. The swivel angle ranges from 48.51 degrees to 86.05 degrees. These parameters ensure that the gear cutting process matches the design tooth surface.
I design a gradient process parameter experiment. The cutter spindle speed levels are 100, 120, 140, and 160 r/min. The feed rate levels are 0, 40, 60, 80, and 100 mm/min. The zero feed rate case serves as a reference because it represents the non-cutting state. For each parameter combination, I record the cutting force and vibration acceleration. I use fast Fourier transform to analyze the frequency content of the signals.
| Test number | Spindle speed (r/min) | Feed rate (mm/min) | Total cutting force (N) | Workpiece housing vibration (m/s²) | Tool housing vibration (m/s²) |
|---|---|---|---|---|---|
| 1 | 100 | 40 | 838.13 | 0.67868 | 1.9631 |
| 2 | 100 | 60 | 998.38 | 0.49692 | 1.8987 |
| 3 | 100 | 80 | 1321.4 | 0.5672 | 1.9023 |
| 4 | 100 | 100 | 1351.0 | 0.62525 | 1.9449 |
| 5 | 120 | 40 | 777.13 | 0.44687 | 2.3434 |
| 6 | 120 | 60 | 922.23 | 0.5103 | 2.4569 |
| 7 | 120 | 80 | 1044.4 | 0.56461 | 2.4553 |
| 8 | 120 | 100 | 1198.7 | 0.58021 | 2.5225 |
| 9 | 140 | 40 | 735.52 | 0.48392 | 2.3419 |
| 10 | 140 | 60 | 896.35 | 0.51829 | 2.4839 |
| 11 | 140 | 80 | 1028.3 | 0.58699 | 2.4689 |
| 12 | 140 | 100 | 1121.4 | 0.64185 | 2.6426 |
| 13 | 160 | 40 | 712.75 | 0.49793 | 2.3499 |
| 14 | 160 | 60 | 845.69 | 0.53636 | 2.4326 |
| 15 | 160 | 80 | 994.74 | 0.59705 | 2.4703 |
| 16 | 160 | 100 | 1112.8 | 0.66156 | 2.4940 |
From the experimental data, I identify the natural frequencies of the tool spindle and workpiece spindle. The measured first natural frequency of the tool spindle is 48.0 Hz, which is within 1.1 percent of the model prediction of 47.436 Hz. The workpiece spindle first frequency is 104.5 Hz, which is within 1.2 percent of the model prediction. The bearing frequency and the cutting frequency are also identified. The theoretical cutting frequency is
$$ f_c = \frac{N Z}{60} $$
For a cutter speed of 100 r/min and 16 teeth, the cutting frequency is 26.67 Hz. The second harmonic is 53.33 Hz, and the third harmonic is 80.0 Hz. The theoretical bearing frequency is approximately
$$ H_q = Z_q \frac{N}{60} \left( 1 + \frac{d_q}{D_n} \cos\theta \right) $$
For the cutter spindle bearing, the first bearing frequency is about 59.67 Hz at 100 r/min. These frequencies are used to exclude non-natural frequencies from the vibration spectrum. After excluding the cutting and bearing frequencies, I identify a new frequency at about 66.5 Hz in the gear cutting state. This frequency corresponds to the coupled tool-workpiece mode predicted by the model at 66.04 Hz. The error is 0.69 percent, which confirms the accuracy of the coupled dynamic model.
I observe that the coupled frequency becomes less prominent as the cutter speed increases. This is because the cutting contact time decreases, and the influence of the cutting stiffness on the overall system response is reduced. However, the coupled frequency still has significance for gear cutting because it can cause resonance if the cutting force harmonics coincide with it. Therefore, I use this frequency as a constraint in process parameter optimization.
I analyze the effect of cutting stiffness on the spindle dynamic response. The cutting force measurements show that the main excitation frequency in the X and Y directions is the spindle rotation frequency, while the main excitation in the Z direction is the cutting frequency. The theoretical cutting frequency and the measured cutting frequency agree within 0.1 percent. The vibration acceleration spectra show that the low-frequency range below 400 Hz is most affected by the gear cutting process. In the high-frequency range above 900 Hz, the vibration amplitude does not change significantly with feed rate, but the peak positions shift slightly. This shift is caused by the change in the coupled natural frequency due to the dynamic cutting stiffness. The experimental results confirm that the dynamic cutting stiffness has a small but measurable effect on the coupled system frequency.
I build a multi-objective optimization model that considers the machine structural characteristics. The objectives are to minimize the total cutting force, the vibration acceleration, and the cutting time, while avoiding resonance with the coupled frequency, the cutting frequency, and the bearing frequency. The weighted objective function is
$$ \min \left( w_1 \frac{a}{a_{\max}} + w_2 \frac{F_c}{F_{\max}} + w_3 \frac{f_n}{f_{\max}} – w_4 \frac{T}{T_{\max}} \right) $$
where $a$ is the vibration acceleration, $F_c$ is the cutting force, $f_n$ is the coupled natural frequency, and $T$ is the cutting time. The weights are chosen as $w_1 = 0.2$, $w_2 = 0.6$, $w_3 = 0.1$, and $w_4 = 0.1$, because the cutting force has the strongest influence on tool life and machining stability. The constraints are
$$ |f_n – f_{cut}| \ge \Delta f $$
$$ |f_n – f_z| \ge \Delta f $$
$$ N_{\min} \le N \le N_{\max} $$
$$ v_{\min} \le v \le v_{\max} $$
$$ F_c \le F_{cut} $$
where $f_{cut}$ is the cutting frequency, $f_z$ is the bearing frequency, and $\Delta f$ is the frequency avoidance margin. I fit the objective functions using the experimental data. The vibration acceleration in the tool spindle Y direction is
$$ a = -0.097696 – 0.030218 N + 0.000304 N^2 – 0.000285 v^2 – 0.000076 Nv + 2.762125 $$
The cutting force in the Y direction is
$$ F_c = -898.3 + 10.15 N + 14.52 v – 0.0469 N^2 – 0.07362 Nv – 0.002152 v^2 $$
The coupled natural frequency is
$$ f_n = 65.6277 + 0.0131 N – 7.3438 \times 10^{-4} v – 3.4375 \times 10^{-6} N^2 + 4 \times 10^{-5} Nv $$
The cutting time is approximated by
$$ T \approx \frac{39.978 \times 60}{v} $$
I use the NSGA-II algorithm to solve the multi-objective optimization problem. The Pareto front provides a set of non-dominated solutions. After twenty independent runs, the average optimal parameters are a spindle speed of 140.99 r/min and a feed rate of 41.71 mm/min. The predicted cutting time is 57.58 s, the cutting force is 569.22 N, the maximum vibration acceleration is 1.96 m/s², and the coupled natural frequency is 66.78 Hz.
| Parameter | Average value | Standard deviation |
|---|---|---|
| Spindle speed | 140.99 r/min | 4.66 r/min |
| Feed rate | 41.71 mm/min | 2.12 mm/min |
| Cutting time | 57.58 s | 2.70 s |
| Cutting force | 569.22 N | 8.30 N |
| Maximum vibration acceleration | 1.96 m/s² | 0.0103 m/s² |
| Coupled natural frequency | 66.78 Hz | 0.016 Hz |
I validate the optimized parameters through gear cutting experiments. The measured maximum cutting force in the Y direction is 570.38 N, which is within 0.24 percent of the predicted value. The measured maximum vibration acceleration is 1.943 m/s², which is within 0.87 percent of the predicted value. The measured coupled frequency is 67 Hz, which is within 0.33 percent of the predicted value. The cutting time is 57.87 s, which is within 0.5 percent of the predicted value. These results confirm the reliability of the multi-objective optimization model.
| Metric | Experimental value | Pareto prediction | Error |
|---|---|---|---|
| Maximum cutting force (Y) | 570.38 N | 569.22 N | 0.24% |
| Maximum vibration acceleration | 1.943 m/s² | 1.96 m/s² | 0.87% |
| Coupled frequency | 67 Hz | 66.78 Hz | 0.33% |
| Cutting time | 57.87 s | 57.58 s | 0.50% |
I compare the optimized parameters with the original conservative parameters and the orthogonal optimization result. The original parameters are 100 r/min and 40 mm/min. The orthogonal optimization gives 160 r/min and 40 mm/min. The Pareto optimization gives 140.99 r/min and 41.71 mm/min. The total cutting force decreases from 838.13 N to 727.82 N, a reduction of 13.2 percent. The maximum workpiece housing acceleration decreases from 0.67868 m/s² to 0.48106 m/s², a reduction of 29.1 percent. The cutting time is reduced by 3.52 percent compared with the orthogonal optimization. The comprehensive score of the Pareto optimization is 0.7145, which is higher than the orthogonal optimization score of 0.7041. This shows that the Pareto optimization provides a better balance among cutting force, vibration, and efficiency.
| Parameter set | Total cutting force (N) | Workpiece housing acceleration (m/s²) | Cutting time (s) | Score |
|---|---|---|---|---|
| Original | 838.13 | 0.67868 | 59.98 | 0.6521 |
| Orthogonal optimization | 712.75 | 0.49793 | 59.98 | 0.7041 |
| Pareto optimization | 727.82 | 0.48106 | 57.87 | 0.7145 |
The error sources in the optimization include the fitting error of the objective functions, the limited number of experimental data points, and the nonlinear behavior of the gear cutting process. The fitting functions have a goodness of fit above 90 percent, but the remaining error propagates through the optimization. In future work, I plan to include the effect of cutting temperature on the dynamic stiffness, because temperature changes can alter the contact stiffness and the thermal deformation of the spindle. I also plan to extend the model to include other machine tool components, such as the column, the base, and the motors, to capture the electromechanical coupling effects. Finally, I intend to improve the beam element model to better represent the dynamic characteristics of the tool-workpiece system and to make the modeling approach more general for different machine tool configurations.
In summary, I have developed a coupled dynamic model for gear cutting on spiral bevel gear milling machines. The model includes the generation motion, the dynamic cutting stiffness, the spindle-bearing system, and the tool-workpiece coupling. I validated the model through modal analysis and gear cutting experiments. The results show that the cutting coupling introduces a new low-order frequency that can affect machining stability. The dynamic cutting stiffness has a limited effect on the coupled frequency within the normal parameter range, but it still influences the vibration amplitude. Using the coupled model, I optimized the gear cutting parameters with the NSGA-II algorithm and achieved a better balance among cutting force, vibration, and efficiency. The optimized parameters reduced the cutting force by 13.2 percent and the vibration acceleration by 29.1 percent compared with the original conservative parameters. These findings provide a scientific basis for improving the stability and efficiency of gear cutting in spiral bevel gear milling machines.