Chatter and Surface Waviness Analysis in Oerlikon Face Hobbing of Spiral Bevel Gears

Abstract

A vectorized analytical model for the cutting dynamics in the spiral bevel gear face hobbing process has been developed, which is based on machine tool kinematics and vibration vectorization. The structural modal parameters of the cutter head spindle system are obtained through experimental modal analysis with hammer impact testing. The analytical model is utilized to simulate the generation of simulated vibration acceleration signals during spiral bevel gear hobbing. A wavelet threshold denoising method is applied to process the simulated vibration signals of the spiral bevel gear face hobbing with added white noise. Signal processing methods, including short-time Fourier transform are employed for time-domain analysis, frequency-domain analysis, and time–frequency-domain analysis of measured signals and simulated signals, thereby extracting the corresponding statistical features. In addition to the results of the experimental modal analysis, the causes of chatter in spiral bevel gear hobbing are discussed in detail, revealing that the main factor is cutter head vibration in the Y direction of the Hunan ZDCY CNC EQUIPMENT YKA2260 machine tool used in this research. The error in the time-domain characteristic parameters between simulated signals and measured vibration acceleration signals is within 15%, with a difference of 3.5% in spectral peak values. The predicted tooth surface morphology from simulation matches the actual morphology on the workpiece, comprehensively validating the reliability of the cutting dynamics model for the spiral bevel gear face hobbing process. Another conclusion drawn from numerical simulation experiments is that the amount of tooth surface waviness of the spiral bevel gears is the ratio of tool chatter frequency to cutting fundamental frequency.

1. Introduction

Spiral bevel gears are one of the most complex types of hypoid gears in gear transmissions. They offer advantages, such as high meshing accuracy, strong load-bearing capacity, high transmission efficiency, low noise, and high reduction ratio. As the most crucial transmission components, they find widespread applications in industrial systems including aircraft, helicopters, transportation vehicles and machine tools. In helicopter drive systems, spiral bevel gears play a vital role. The spiral bevel gear transmission device and the two-stage planetary gear reduction system constitute the main gearbox of the helicopter, which reduces the output shaft speed of the engine to drive the main rotor. Spiral bevel gears can be classified into two types based on their tooth profiles: involute and cycloidal. These gears are typically manufactured by two distinct methods: face milling for involute teeth and face hobbing for cycloidal teeth [1,2]. Due to the low transmission noise, high efficiency of continuous tooth generation, and excellent smoothness during the meshing process, cycloidal gears have been widely promoted and utilized across various industries. In the meantime, the theoretical aspects of the face hobbing process for spiral bevel gears have garnered significant attention from researchers. Fan [3,4] and Litvin et al. [5] conducted detailed analyses of the machine tool motion during the face hobbing process for spiral bevel gears and investigated the tooth surface formation principles based on machine tool motion models. Wu et al. [6] studied the formation principles and geometric structures of cycloidal bevel gears. Zhang et al. [7] derived equations for the root fillet and discussed the geometric characteristics of the tooth surface in the face hobbing process for spiral bevel gears. Suzuki et al. [8] proposed a direct method for designing the working surfaces of quasi-hyperbolic gears, accurately obtaining conjugate gear surfaces through coordinate transformations. Wang et al. [9] proposed a method for machining line-contact spiral bevel gears using a tapered cylindrical milling cutter. This approach offers a novel perspective for the design and manufacturing of spiral bevel gears. Habibi et al. [10] developed a semi-analytical representation of the projection of undeformed chips on the rake face of the cutting tool. They applied the theory of oblique cutting using the derived geometric shape of the chip to transform hobbing into oblique cutting, thereby predicting the cutting forces during the hobbing process. Efstathiou et al. [11] developed the Ithaca Bevel Gear Suite platform, which not only enables the calculation of gear tooth and undeformed chip solid geometries but also calculates the global cutting forces. Additionally, it utilizes charts of localized forces generated at the tool tip to predict tool wear. Due to the complexity in both the design theories and face hobbing technology for spiral bevel gears, only a few companies worldwide, namely Oerlikon in Switzerland, Klingelnberg in Germany, and Gleason in the United States, are capable of providing complete solutions. Consequently, there is very limited information publicly available on systematic research regarding the face hobbing process for spiral bevel gears.

Chatter is a self-excited and detrimental vibration phenomenon in mechanical machining, which results from the interaction between the tool and the workpiece and is to be avoided. It can lead to unstable machining processes, thereby negatively impacting machining accuracy, workpiece surface quality, machining efficiency, and tool life. Chatter can be categorized into friction-induced chatter, modal coupling chatter, and regenerative chatter based on its causes [12]. The first two types of chatter can be addressed, respectively, by improving lubrication conditions and optimizing system structural design. However, as for the regenerative chatter, which arises from the interaction between the metal cutting process and machine tool structure, its behavior is unstable and this results in significant relative displacements between the tool and workpiece, leaving dense chatter marks on the machined tooth surfaces. This makes it challenging to achieve surface finish quality requirements and difficult to suppress [13]. Therefore, numerous scholars have conducted research on modeling the cutting dynamics and regenerative chatter mechanism during metal cutting processes [14,15,16,17]. In identifying the dynamic parameters of chatter models, two main approaches are typically employed to describe the analytical expressions of cutting dynamics [18]. One method involves obtaining coefficients of specified forms of incremental cutting equations through steady-state cutting experiments [19]. The other method relies on Meritt’s proposed cutting process model to derive the mathematical model of chatter dynamics [20]. Totisa et al. [21] analyzed the influence of forced vibration on the contact state between the tool and workpiece, discovering that the contact state between the tool and workpiece leads to nonlinear vibrations of the machine tool, thereby further refining the theoretical modeling of metal cutting dynamics. Kilic et al. [22,23] established analytical models for predicting chatter stability in milling processes and proposed a general mechanical model for metal cutting based on oblique cutting theory. Moradi et al. [24,25] investigated bifurcation characteristics in metal machining processes considering the effects of process damping and tool wear and developed an analytical dynamic model incorporating nonlinear regenerative chatter. Feng et al. [26] proposed a normalized process damping model based on the variable stiffness of movable components in the machining process system to study the suppression of chatter in metal machining processes. Similar to traditional metal cutting, the face hobbing process for spiral bevel gears is prone to chatter, which involves excessive vibrations between the cutting tool and workpiece. This phenomenon leads to poor surface finish, high-frequency noise, and accelerated tool wear [27,28,29]. Due to the synchronous motion of the cutter head and workpiece during the face hobbing process for spiral bevel gears, the kinematic model for face hobbing with vibration is more complex compared to conventional models. This complexity poses significant challenges for investigating the cutting dynamics and the chatter mechanisms in the face-hobbing process of spiral bevel gears. However, currently published research primarily focuses on the cutting force modeling and chatter stability prediction in traditional metal machining processes, such as milling and grinding. Up to now, there is scant systematic research on the cutting dynamics and chatter mechanism in the spiral bevel gear face hobbing process.

This research applies the method of vibration vectorization modeling to vectorize the machined tooth surface under cutter head vibration, thereby establishing an analytical model for dynamic cutting forces in spiral bevel gear face hobbing. Modal parameters of the cutter head spindle process system of the research equipment, namely the YKA2260 gear hobbing machine, are obtained through hammer impact testing and experimental modal analysis. Subsequently, simulated vibration acceleration signals are generated using the cutting dynamics model. Vibration acceleration signals during spiral bevel gear face hobbing are collected in machining experiments. A comparative analysis in the time domain, frequency domain, and time-frequency domain is carried out using the simulated signals and the measured signals. Corresponding characteristic statistical quantities have been extracted to verify the reliability of the theoretical model and simultaneously explore the influence of cutter head vibration on the quality of machined sample spiral bevel gears. This research intends to provide significant reference value for optimizing the spiral bevel gear face hobbing process.

2. The Cutting Dynamics in the Spiral Bevel Gear End-Face Hobbing Process

2.1. Kinematics of Machine Tool and Machined Tooth Surface of Gearwheel

Based on the existing research achievements of the research group [30,31], the machine tool motion relationship for the spiral bevel gear face hobbing process can be established as depicted in Figure 1. There are three crucial reference coordinate systems: the machine origin coordinate system O_m-x_my_mz_m, the cutter head coordinate system O_H-x_Hy_Hz_H, and the workpiece coordinate system O_w-x_wy_wz_w. O₁-x₁y₁z₁ and O₄-x₄y₄z₄ denote the theoretical initial cutting positions of the cutter head and workpiece, respectively. Due to vibration effects, both the cutter head and workpiece experience displacement to their actual starting cutting positions O’₁ and O’₄, with corresponding vibration displacements v_H and v_w. v_H and v_w are decomposed along the coordinates O_m-x_my_mz_m and O₄-x₄y₄z₄, yielding components (x_H, y_H, z_H)^T and (x_w, y_w, z_w)^T, respectively. φ_w represents the rotational angle of the workpiece, φ_H denotes the rotational angle of the cutter head, β_w is the angular position of the workpiece, OW is the length of the workpiece fixture, X denotes the theoretical horizontal position of the tool head on the machine’s x-axis, Z represents the theoretical initial vertical position of the cutter head on the machine’s z-axis, and Y denotes the theoretical horizontal position of the cutter head on the machine’s y-axis.

The installation relationship between the tool and the cutter head is illustrated in Figure 2. r_t represents the analytical formula for the cutting edge in the tool-fixed coordinate system (O_T-x_Ty_Tz_T), and s denotes the arc length parameter of a specific point on the cutting edge. The angles α, α_t and α_b correspond to the rake angle, the top edge tooth angle, and the main cutting-edge tooth angle of the tool, respectively.

The tooth profile of the generated spiral bevel gears with the influence of cutting vibration can be expressed as follows:

$${\Phi }^{(\mathrm{W})}(\phi ,s)=M{\prime }_{\mathrm{wh}}(\phi )M_{\mathrm{hT}}r(s)$$

(1)

In the equation, Φ represents the tooth profile of the gearwheel, φ represents the angular displacement of the cutting edge of the tool, and s represents the arc length parameter of a point on the cutting edge. M’_wh is the homogeneous transformation matrix from the workpiece-fixed coordinate system O_w-x_wy_wz_w to the cutter head-fixed coordinate system O_h-x_hy_hz_h. M_hT is the homogeneous transformation matrix from the cutter head-fixed coordinate system O_h-x_hy_hz_h to the tool-fixed coordinate system O_t-x_ty_tz_t. r(s) represents the expression of the cutting edge of the tool in the tool-fixed coordinate system O_t-x_ty_tz_t.

2.2. Undeformed Chip Geometry and End Cutting Forces

The workpiece material is continuously removed by the cutting edges of the tools, which forms concave and convex surfaces on both sides of the tooth slot. The cutting cycle in the pre-expansion stage of the face hobbing process of quasi-hyperbolic gears is determined by the rotational speed of the cutter head and the gear ratio between the cutter head and the workpiece:

$$T=\frac{z_w}{z_0}\frac{2\pi }{{\omega }_0}$$

(2)

In this formula, z_w represents the number of teeth on the gear being processed, z₀ denotes the number of teeth on the cutter head, and ω₀ represents the angular velocity of the cutter head rotation.

As shown in Figure 3, the undeformed chip geometry during the tooth surface forming stage can be represented by the cutting region between the tooth surfaces Σ1 and Σ2 formed during the machining of two consecutive cutting cycles. Due to the periodicity of face hobbing and the consistency of the geometric characteristics of each tool mounted on the cutter head, it can be assumed that tooth surface families formed in different cycles of the machining process possess identical geometric features. Therefore, the tooth surface of the gearwheel of spiral bevel gears formed during the previous and current cutting cycles is given by:

$$\{\begin{array}{c}{\mathsf{\Sigma }}_1^{(\mathrm{W})}:r_{\mathrm{p}}^{(\mathrm{w})}(s,\phi ,z_p)=M{\prime }_{\mathrm{wh}}(\phi ,z_p)M_{\mathrm{hT}}r(s)\\ {\mathsf{\Sigma }}_2^{(\mathrm{W})}:r^{(\mathrm{w})}(s,\phi ,z)=M{\prime }_{\mathrm{wh}}(\phi ,z)M_{\mathrm{hT}}r(s)\end{array}$$

(3)

Here, z_p represents the feed position in the previous cutting cycle, which under uniform feed conditions can be obtained from the following equation:

$$z_p=z+f\cdot T$$

(4)

In the equation, f represents the feed rate during the face-hobbing process. Therefore, the undeformed chip thickness during the tooth surface forming stage can be expressed as the projection of the difference vector from the tooth surface source point on the tooth surface formed in the current machining cycle to the tooth surface formed in the adjacent previous machining cycle onto the normal vector of the tooth surface.

The relationship between the thickness h of the un-deformed chip (in the primary cutting-edge region) formed by the primary cutting edge, the tool angle φ on the cutter head, and the corresponding set of cutting-edge arc length parameters s can be expressed as in the following equation:

$$h(s,\phi ,z)=(r_p^{(w)}(s(\phi ,z_{\mathrm{p}}),\phi ,z_{\mathrm{p}})-r^{(w)}(s(\phi ,z),\phi ,z))·n^{(w)}(s(\phi ,z),\phi ,z)$$

(5)

where n^(w) represents the normal vector of the machined tooth surface in the current cycle, and can be further determined by the following:

$$n^{(\mathrm{W})}(s,\phi ,z)=\frac{\partial }{\partial \phi }M{\prime }_{\mathrm{wh}}(\phi ,z)M_{\mathrm{hT}}r(s)\times M{\prime }_{\mathrm{wh}}({\phi }_j,z)M_{\mathrm{hT}}\frac{\partial }{\partial s}r(s)$$

(6)

After establishing the tooth blank coordinate system as shown in Figure 4, and substituting the cutting width b of the undeformed chip into the simultaneous equations of the tooth blank surface cone equation and the tooth surface equation, the following relationship exists:

$$\{\begin{array}{c}\sqrt{x^{(\mathrm{w})2}+y^{(\mathrm{w})2}}=(h_1-z^{(\mathrm{w})})\tan \Gamma \\ {\Phi }^{(\mathrm{W})}(\phi ,s)=M_{\mathrm{wh}}(\phi )M_{\mathrm{hT}}r(s)\\ d_3\le 2\sqrt{x^{(\mathrm{w})2}+y^{(\mathrm{w})2}}\le d_4\end{array}$$

(7)

where h₁ represents the distance from the apex of the large bevel gear face cone to the bottom of the gear blank, Γ stands for the cone angle of the spiral bevel gear blank, d₃ is the small end diameter of the gear blank, and d₄ is the large end diameter of the gear blank. By simultaneously solving the first and second constraints in Equation (7), the cutting width b of the undeformed chip at a certain instant is obtained. The cutter rotation angle φ_st and φ_ex can also be calculated by the combination of the second and the third equations in Equation (7).

Based on the classic oblique cutting theory, discretizing the undeformed chip along the direction of the cutting tool edge, the resultant cutting force element d_F produced on each discrete chip element, as illustrated in Figure 5, can be expressed by the following equation:

$$\{\begin{array}{c}d{F}_t=(K_{tc}(s,{\phi }_j,z)\cdot h(s,{\phi }_j,z)\cdot ds+K_{te}\cdot ds)\\ d{F}_f=(K_{fc}(s,{\phi }_j,z)\cdot h(s,{\phi }_j,z)\cdot ds+K_{fe}\cdot ds)\\ d{F}_r=(K_{rc}(s,{\phi }_j,z)\cdot h(s,{\phi }_j,z)\cdot ds+K_{re}\cdot ds)\end{array}$$

(8)

In this equation, K_tc, K_rc, and K_rc are the cutting force coefficients, while K_te, K_re, and K_re are the edge coefficients for cutting forces, with different values along the directions of the cutting edge. The aforementioned coefficients and the direction of cutting forces can be determined using the following formulas:

$$\{\begin{array}{c}{K}_{\mathrm{tc}}=\frac{{\tau }_{\mathrm{s}}}{\sin {\varphi }_{\mathrm{c}}}\frac{\cos ({\beta }_{\mathrm{a}}-{\alpha }_{\mathrm{r}})+{\tan }^{2}i\sin {\beta }_{\mathrm{a}}}{\sqrt{{\cos }^2({\varphi }_{\mathrm{c}}+{\beta }_{\mathrm{a}}-{\alpha }_{\mathrm{r}})+{\tan }^{2}i{\sin }^2{\beta }_{\mathrm{a}}}}\\ K_{\mathrm{fc}}=\frac{{\tau }_{\mathrm{s}}}{\sin {\varphi }_{\mathrm{c}}\cos i}\frac{\sin ({\beta }_{\mathrm{a}}-{\alpha }_{\mathrm{r}})}{\sqrt{{\cos }^2({\varphi }_{\mathrm{c}}+{\beta }_{\mathrm{a}}-{\alpha }_{\mathrm{r}})+{\tan }^{2}i{\sin }^2{\beta }_{\mathrm{a}}}}\\ K_{\mathrm{rc}}=\frac{{\tau }_{\mathrm{s}}}{\sin {\varphi }_{\mathrm{c}}}\frac{\cos ({\beta }_{\mathrm{a}}-{\alpha }_{\mathrm{r}})\tan i-\tan i\sin {\beta }_{\mathrm{a}}}{\sqrt{{\cos }^2({\varphi }_{\mathrm{c}}+{\beta }_{\mathrm{a}}-{\alpha }_{\mathrm{r}})+{\tan }^{2}i{\sin }^2{\beta }_{\mathrm{a}}}}\end{array}$$

(9)

$$\{\begin{array}{c}d{F}_t=-\frac{d\phi }{dt}(\frac{\partial }{\partial \phi }r(u,\phi ,z))/\left|\frac{d\phi }{dt}(\frac{\partial }{\partial \phi }r(u,\phi ,z))\right|\\ \begin{array}{l}\\ d{F}_{\mathrm{f}}=n^{(\mathrm{w})}(s,\phi ,z)/\left|n^{(\mathrm{w})}(s,\phi ,z)\right|\end{array}\\ \begin{array}{l}\\ d{F}_{\mathrm{r}}=d{F}_{\mathrm{f}}\times d{F}_t\end{array}\end{array}$$

(10)

In the equation, α_r represents the normal rake angle of the tool, ϕ_c denotes the material shear angle during oblique cutting, β_a stands for the average friction angle of the material, τ_s represents the shear yield strength of the workpiece material, α_r is the tool’s rake angle, and i is the chip flow angle during end milling. It is worth noting that cutting forces are only generated when the tool is engaged with the workpiece, and can be introduced using the following window function:

$$g({\phi }_j,u)=\{\begin{array}{l}1,{\phi }_{\mathrm{st}}\le {\phi }_j\le {\phi }_{\mathrm{ex}},b({\phi }_j,u)>0,h({\phi }_j,u)>0\\ 0,\mathrm{else}\end{array}$$

(11)

Integrating the obtained elemental cutting forces along the primary cutting edge and the secondary cutting edge, respectively, the analytical expression for the cutting forces on the entire cutting edge of the tool can be expressed as follows:

$$\begin{array}{l}{F}_{\mathrm{T}}^{(\mathrm{w})}(\phi ,z)={\displaystyle {\int }_0^b[(K_{tc}\cdot h(\phi ,z)+K_{te})d{F}_t+(K_{fc}\cdot h(\phi ,z)+K_{fe})d{F}_f+(K_{rc}\cdot h(\phi ,z)+K_{re})d{F}_r]ds}\\ +{\displaystyle {\int }_0^{b_{\mathrm{tip}}}[(K_{tc}\cdot h_{\mathrm{t}}(\phi ,z)+K_{te})d{F}_t+(K_{tc}\cdot h_{\mathrm{t}}(\phi ,z)+K_{fe})d{F}_f+(K_{tc}\cdot h_{\mathrm{t}}(\phi ,z)+K_{re})d{F}_r]ds}\end{array}$$

(12)

The cutting force model in Equation (12) encompasses both the magnitude and direction of cutting forces. The dynamic cutting force on a single tool can be represented using the homogeneous transformation matrix M_mw from the workpiece-fixed coordinate system O_w-x_wy_wz_w to the machine tool origin coordinate system O_m-x_my_mz_m as follows:

$$\left[\begin{array}{c}{F}_{xj}\\ F_{yj}\\ F_{zj}\end{array}\right]=\left[\begin{array}{ccc}\cos {\beta }_{\mathrm{w}}\cos {\phi }_w& -\cos {\beta }_{\mathrm{w}}\sin {\phi }_w& \sin {\beta }_{\mathrm{w}}\\ \sin {\phi }_w& \cos {\phi }_w& 0\\ -\sin {\beta }_{\mathrm{w}}\cos {\phi }_w& \sin {\beta }_{\mathrm{w}}\sin {\phi }_w& \cos {\beta }_{\mathrm{w}}\end{array}\right]F_t^{(\mathrm{w})}=M_{\mathrm{mw}}{F}_t^{(\mathrm{w})}$$

(13)

By summing up the individual cutting forces on each tool and considering the window function, the dynamic cutting force model acting on the cutter can finally be obtained as:

$$\{\begin{array}{c}{F}_x={\displaystyle \sum _{j=1}^{N_{\mathrm{h}}}{F}_{xj}^{\mathrm{inner}}({\phi }_j,z)g({\phi }_j)}+{\displaystyle \sum _{j=1}^{N_{\mathrm{h}}}{F}_{xj}^{\mathrm{outer}}({\phi }_j,z)g({\phi }_j)}\\ F_y={\displaystyle \sum _{j=1}^{N_{\mathrm{h}}}{F}_{yj}^{\mathrm{inner}}({\phi }_j,z)g({\phi }_j)}+{\displaystyle \sum _{j=1}^{N_{\mathrm{h}}}{F}_{yj}^{\mathrm{outer}}({\phi }_j,z)g({\phi }_j)}\\ F_z={\displaystyle \sum _{j=1}^{N_{\mathrm{h}}}{F}_{zj}^{\mathrm{inner}}({\phi }_j,z)g({\phi }_j)}+{\displaystyle \sum _{j=1}^{N_{\mathrm{h}}}{F}_{zj}^{\mathrm{outer}}({\phi }_j,z)g({\phi }_j)}\end{array}$$

(14)

Synthesizing Equations (1)–(14), the following closed-form vectorized analytical model for predicting dynamic cutting forces in the spiral bevel gears face-hobbing process can be obtained:

$$\begin{array}{l}{F}_j=M_{\mathrm{mw}}\left[\begin{array}{c}{\displaystyle {\int }_0^b(K_{tc}\cdot (r_p^{(w)}-r^{(w)})·n^{(w)}({\phi }_j,s({\phi }_j))+K_{te})d{F}_{t}ds}\\ {\displaystyle {\int }_0^b(K_{fc}\cdot (r_p^{(w)}-r^{(w)})·n^{(w)}({\phi }_j,s({\phi }_j))+K_{fe})d{F}_{f}ds}\\ {\displaystyle {\int }_0^b(K_{rc}\cdot (r_p^{(w)}-r^{(w)})·n^{(w)}({\phi }_j,s({\phi }_j))+K_{re})d{F}_{r}ds}\end{array}\right]\\ F(t,v_{\mathrm{H}}(t),v_{\mathrm{H}}(t-T),v_{\mathrm{w}}(t),v_{\mathrm{w}}(t-T))={\displaystyle \sum _{k=1}^2{\displaystyle \sum _{j=1}^{N_{\mathrm{h}}}{F}_j^k}g({\phi }_j(t))}\\ \{\begin{array}{l}k=1,inner\\ k=2,outer\end{array}\end{array}$$

(15)

In these equations, v_H = [x_H, y_H, z_H]T represents the cutter’s vibration displacement, v_w = [x_w, y_w, z_w]T denotes the workpiece’s vibration displacement. r^(w) and r_p^(w) can be calculated from Equation (3), n^(w) is derived from Equation (6). K_tc, K_rc, and K_re are obtained from Equation (9). dF_t, dF_f, and dFr represent the three components of the cutting force element in the workpiece coordinate system, and their directions are determined by Equation (10).

2.3. Cutting Dynamics Analytical Model of Face Hobbing Process

Built further on the research outputs in the research group [30,31] and the derivation as explained above, the cutting dynamics model for spiral bevel gear face hobbing can be obtained as depicted in Figure 6. It can be observed that the cutting force F induces vibrations in the x_m, y_m, and z_m directions of the cutter head, while the reactive force F_w acting on the workpiece will similarly induce vibrations in the x₄, y₄, and z₄ directions of the workpiece. The cutting force F_w acting on the workpiece and the cutting force F acting on the cutter head form a pair of reactive forces. Their component forms in the reference coordinate system O₄-x₄y₄z₄ are:

$$\left[\begin{array}{c}{F}_{w,x}\\ F_{w,y}\\ F_{w,z}\end{array}\right]=-\left|\begin{array}{ccc}\cos {\beta }_{\mathrm{w}}& 0& \sin {\beta }_{\mathrm{w}}\\ 0& 1& 0\\ -\sin {\beta }_{\mathrm{w}}& 0& \cos {\beta }_{\mathrm{w}}\end{array}\right|\left[\begin{array}{c}{F}_x\\ F_y\\ F_z\end{array}\right]$$

(16)

Considering the chip regeneration effect during end milling and its vectorized expression, the cutting dynamic model of the end milling process for hypoid gears can be represented in the following analytical form:

$$\begin{array}{l}\left[\begin{array}{cc}{M}_{\mathrm{H}}& 0\\ 0& M_{\mathrm{w}}\end{array}\right]\left[\begin{array}{c}{\ddot{v}}_{\mathrm{H}}\\ {\ddot{v}}_{\mathrm{w}}\end{array}\right]+\left[\begin{array}{cc}{C}_H& 0\\ 0& C_w\end{array}\right]\left[\begin{array}{c}{\dot{v}}_{\mathrm{H}}\\ {\dot{v}}_{\mathrm{w}}\end{array}\right]+\left[\begin{array}{cc}{K}_{\mathrm{H}}& 0\\ 0& K_{\mathrm{w}}\end{array}\right]\left[\begin{array}{c}{v}_{\mathrm{H}}\\ v_{\mathrm{w}}\end{array}\right]=\\ \left[\begin{array}{c}F(t,v_{\mathrm{H}}(t),v_{\mathrm{H}}(t-T),v_{\mathrm{w}}(t),v_{\mathrm{w}}(t-T))\\ F_{\mathrm{w}}(t,v_{\mathrm{H}}(t),v_{\mathrm{H}}(t-T),v_{\mathrm{w}}(t),v_{\mathrm{w}}(t-T))\end{array}\right]\end{array}$$

(17)

In the equation, T represents the gear cutting cycle, M_H is the mass matrix for the vibrations of the cutter head in the x_m, y_m, and z_m directions, M_w is the mass matrix for the vibrations of the workpiece in the x₄, y₄, and z₄ directions. C_H is the damping matrix for the vibrations of the cutter head in the x_m, y_m, and z_m directions, C_w is the damping matrix for the vibrations of the workpiece in the x₄, y₄, and z₄ directions. K_H is the stiffness matrix for the vibrations of the cutter head in the x_m, y_m, and z_m directions, and K_w is the stiffness matrix for the vibrations of the workpiece in the x₄, y₄, and z₄ directions. The specific parameters can be obtained from experimental modal analysis or structural finite element analysis. Based on the analytical dynamic model, dynamic simulations can be conducted for the spiral bevel gear face hobbing.

3. Experimental Tests and Verification

3.1. Hammer Impact Testing

To simulate the face hobbing process of spiral bevel gears with the cutting dynamics model, it is necessary to obtain the dynamic characteristic parameters of the machining system, including damping, stiffness, and natural frequency. On an industrial production system, the dynamic response and modal characteristics of structures are often studied by hammer impact testing to determine its modal parameters, such as natural frequency, mode shape, and damping ratio of the system. As shown in Figure 7, hammer impact testing experiments have been conducted on a YKA2260 gear cutting machine. Since the cutter head vibration is much more significant compared to that of the workpiece during the face hobbing process of spiral bevel gears, the influence of cutter head vibration is emphasized in this experiment and the vibration of the workpiece is neglected. So only the modal parameters of the cutter head components are identified in this research. The material of the hammer affects the excitation frequency range of the structure; a hard hammerhead has more energy in the high-frequency range, which may excite the dynamic characteristics of the filter. If a harder hammerhead is used in the lower frequency range, significant filter ringing may occur. Since this experiment involves low-speed cutting of spiral bevel gears, a relatively soft rubber material hammerhead, as shown in Figure 7, was chosen, and the impact force and duration of each hammer blow were kept the same through computer control.

As illustrated in Figure 7, the cutter head is first moved to a stationary position at the starting point of cutting to eliminate variations in the dynamic characteristics caused by the cutter head at different positions on the machine tool guideway. Accelerometers are attached near the cutter head area to prevent movement during the impact testing process. The force signal of the impact excitation is monitored by a PCB© force sensor above the hammer, while the vibration of the cutter head is monitored by a PCB© uniaxial accelerometer attached near the cutter head area. By adjusting the hammering position and the position of the accelerometer, modal parameters in various directions are identified. The hammering positions are set near the cutting tool on the cutter head, with 41 output response points measured for each axial direction in the hammering experiment. The sampling frequency of the vibration signal is set to 10,240 Hz, and the data is monitored and analyzed using LMS TEST LAB 14.A developed by Simcenter.

3.2. Experiment on Face-Hobbing of Spiral Bevel Gear

To verify the reliability of the cutting dynamic model for machining spiral bevel gears, it is necessary to compare and validate the actual vibration signals with the simulated vibration ones. Therefore, in this study experiments have been designed on an industrial gear hobbing machine, YKA2260, to perform the face hobbing of a large left-handed 37-tooth spiral bevel gear (Figure 8). Some of the machining parameters are listed in

Table 1. Parts of machining parameters for spiral bevel gear face hobbing.

Name	Codename/Unit	Number
Tilt angle	χ2/deg	0
Aanular position	qp/deg	45.0943
Radial tool position	Ex/mm	250.9563
Workpiece mounting angle	δM/deg	73
Vertical wheel position	Ep2/mm	0
Profile crown wheel radius	rp/mm	209.7045
Machine horizontal coordinate of profile crown wheel	Xp/mm	97.6625
Machine vertical coordinate of profile crown wheel	Yp/mm	−190.4906
Position of entry on the x-axis	X/mm	−64.1520
Position of entry on the y-axis	Y/mm	−173.6775
Angular rotation of axis B (deg)	β/deg	17
Length of workpiece fixture	OW/mm	292.8
Number of teeth in the cutter	Nh	17
Primary cutting edge pressure angle of the outer cutter	αbouter/deg	23.8851
Rake angle of the outer cutter tool	αrouter/deg	14.1897
Mounting inclination angle of the outer cutter	γouter/deg	8.8
Radial offset radius of the outer cutter	R1outer/mm	64.1299
Tangential offset distance of the outer cutter	R2outer/mm	158.9504
Angle between inner and outer cutters	λ/deg	9.03
Primary cutting edge pressure angle of the inner cutter	αbinner/deg	21.4713
Rake angle of the inner cutter tool	αrinner/deg	5.0874
Mounting inclination angle of the inner cutter	γinner/deg	8.8
Cutter head rotation speed	ωh/rpm	141
Processing parameters	Feed stage	1	2	3	4
	Distance from tool tip to tooth bottom (mm)	20.3943	10	1	0
	Feed speed (mm/min)	6.0373	4.6957	2.6832	2.3478

. Three uniaxial vibration acceleration sensors were installed on stationary components near the cutter head position to monitor the vibration of the cutter head, in order to collect vibration acceleration signals in real time in X, Y, and Z directions during the face milling process.

4. Results and Discussion

4.1. Experimental Modal Analysis

Prior to the dynamic simulation of the face-hobbing process of spiral bevel gears, modal analysis of the hammer impact testing data is necessary to obtain the dynamic characteristic parameters of the cutter head, such as damping, stiffness, and natural frequency. The essence of modal analysis is the transformation of coordinates, which involves describing response vectors in the natural physical coordinate system, i.e., the linear steady-state system vibration differential equation set, transforming them into modal coordinate systems, and performing decoupling calculations. The coordinate transformation matrix in the transformation process is the modal matrix. The transfer function provides a description of the relationship between the input signal of different frequencies and amplitudes and the response of the continuous-time system, as shown in Equation (18).

$$H_{ij}(\omega )={\displaystyle \sum _{r=1}^N\frac{{\phi }_{ri}{\phi }_{rj}}{j\omega c_r+(k_r-{\omega }^2{m}_r)}}={\displaystyle \sum _{r=1}^N\frac{{\phi }_{ri}{\phi }_{rj}}{m_r[j2{\zeta }_r{\omega }_r\omega +({\omega }_r^2-{\omega }_r)]}}$$

(18)

In the equation, H(ω) represents the frequency response function matrix, where H_ij(ω) denotes the ratio of the response at point i when the system is excited only at point j. m_r stands for the modal mass of the system at the r-th order, c_r represents the modal damping of the system at the r-th order, K_r is the modal stiffness of the system at the r-th order, φ_ri stands for the mode shape of the system at the r-th order, ${\omega }_r^2=\frac{k_r}{m_r}$ is the natural frequency of the r-th order, and ${\xi }_r=\frac{c_r}{2m_r{\omega }_r}$ is the damping ratio of the r-th order.

According to Formula (21), the transfer function is the sum of all single-degree-of-freedom systems corresponding to each mode of the system, which provides a description of the relationship between the input signal and the response of a continuous-time system at different frequencies and amplitudes, including the frequency response, phase response and order of the system, which can effectively understand the amplification or attenuation effect of the system for different frequency signals. The spectral testing module of the LMS TEST LAB 14.A software was utilized to process the output response data of each axis in the hammering experiment. By applying Laplace transformation, the transfer function was converted into the frequency response function in the complex domain. The Bode plot depicted the amplitude and phase information of the frequency response functions at 41 points on the cutter head’s three axes, as shown in Figure 9. Under the same hammer excitation and observation scale, it can be observed that the gain curves of the Bode plots on the X-axis and Z-axis do not exhibit significant peaks, and the overall values of the gain curves are relatively small. In contrast, the gain curve of the Bode plot on the Y-axis shows a significant peak at around 159.8 Hz, with significantly larger overall values. This indicates that the dynamic stiffness of the cutter head system is better in the X-axis direction, followed by the Z-axis direction, while the dynamic stiffness is weakest in the Y-axis direction. Due to the system’s maximum amplification of input signals near the resonant frequency, the gain curve at the resonance frequency in the Bode plot typically reaches a peak. Additionally, at the resonant frequency, the system exhibits a phase lag of 90 degrees with respect to the input signal, corresponding to a phase of −90°. It is evident from Figure 9 that the cutter head system exhibits resonance phenomenon near 159.8 Hz on the Y-axis, with a resonance bandwidth of approximately 30 Hz.

After obtaining the frequency response function data for the three axes of the cutter head, modal parameter estimation needs to be performed to extract the system’s poles and residues. The modal analysis module of the TestLab^® software is utilized to compute the frequency response functions for the three axes (3 sets). Poles are extracted from the frequency response equations, representing the system’s natural frequencies, which are independent of specific input–output measurement locations. In the process of extracting poles from the three sets of frequency response data, this study adopts the PolyMAX technique, which is currently a mainstream method for modal parameter estimation. The PolyMAX technique utilizes commonly used functions, such as the SUM function and Imaginary Sum, to fit the modal steady-state diagram of the tool, as shown in Figure 10. The steady-state diagram utilized in this study employs high-order models to fit modal data. As poles converge to “stable roots”, the diagram includes markers that determine various pieces of information related to the continuity of poles. During the computation process, if poles are identified, they are marked with “o”, and the order of the fitting model is increased to recalculate the poles. If the same pole is identified, it is marked as “f” if the frequency error compared to the previously calculated modal frequency is within 1%, “v” if the mode shape vector error compared to the previous mode shape vector is within 2%, and “d” if the damping error compared to the previous modal damping is within 5%. If all three conditions are met, it is marked as “s”, indicating that a stable value has been reached within the specified error range.

In the diagram, stable points labeled as “s” are selected, with a blue vertical line marking the location of the “s” point. This enables the extraction of modal parameters for the cutter head spindle system, as shown in

Table 2. Modal parameters of machine tool process system.

Modal Stiffness (×109 N/m)			Modal Damping (Ns/m)			Modal Mass (kg)			Natural Frequency (Hz)
kHx	kHy	kHz	cHx	cHy	cHz	mHx	mHy	mHz	ωHx	ωHy	ωHz
8.56	1.11	28.2	159,460	66,338	720,196	7426.31	999.37	24847.5	166.13	167.16	162.23

.

The modal parameters of the manufacturing system are incorporated into the dynamic cutting force model for face hobbing of spiral bevel gears (Equation (17)) to generate simulated vibration acceleration signals during face hobbing (Figure 11).

In this research, the cutter disk rotated at 141 rpm, which is close to the first-order modal frequency of the machine tool and makes it difficult to excite higher-order modal vibrations of the machine tool. Therefore, higher-order modes can be neglected. Only the first-order modal parameters of the cutter head spindle process system are incorporated into the established cutting dynamic model for dynamic simulation. From Figure 10, it is observed that the first-order natural frequencies of the tool spindle process system in all three axial directions are around 160 Hz. Additionally, the dynamic stiffness in the X direction is the highest, while it is the lowest in the Y direction. It is reasonable to assume that the vibration in the Y direction is the main source of chatter vibration during machining for this machine tool.

4.2. Analysis in Time Domain

4.2.1. Wavelet Denoising

In this case, there are inevitable external disturbances in the experimental environment of spiral bevel gear machining and this leads to the presence of environmental noise in the collected vibration signals. Therefore, it is necessary to preprocess the measured signals to reduce noise and improve the signal-to-noise ratio. Given the advantages of wavelet transform in multi-resolution analysis and time-frequency localization, it exhibits good time-domain and frequency-domain resolution, especially in low-frequency machining processes. Therefore, the wavelet thresholding denoising method is chosen to process the measured signals. The wavelet denoising formula can be expressed as in the following formula:

$$f(t)=\sum _{j=1}^J\sum _k^N{d}_{j,k}·{\phi }_{j,k}(t)$$

(19)

in which f(t) denotes the reconstructed signal, i.e., the signal post-denoising. The term d_j,k refers to the wavelet coefficients, which represent the signal’s decomposition coefficients at scale j and position k. Furthermore, φ_j,k is the wavelet basis function, indicating the wavelet basis utilized for the signal’s decomposition. Here, J signifies the number of decomposition layers, and N denotes the number of positions at each decomposition layer.

After obtaining the denoised signal, it is necessary to quantitatively evaluate the noise reduction effect. This can be realized by quantifying the signal-to-noise ratio (SNR), which evaluates the relative energy intensity between the original signal and the removed noise signal, as shown in Equation (20). Additionally, the difference between the denoised signal and the original signal is quantitatively evaluated using the root mean square error, as represented in Equation (21).

$$SNR=\frac{P_{signal}}{P_{\mathrm{noise}}}$$

(20)

in which P_signal represents the simulated noise-free signal, and P_noise denotes the difference between the filtered signal and P_signal.

$$RMSE=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n(y_i-}y{\prime }_i{)}^2}$$

(21)

where n denotes the number of data points in the signal, y_i represents the numerical value of the original signal, and y′_i signifies the numerical value of the signal after filtering.

The denoising effect is case dependent and influenced by the applied wavelet basis functions and decomposition levels. There is no single wavelet function or fixed number of decomposition levels that yield optimal results for all signals. It is necessary to select wavelet functions based on their specific characteristics. In the denoising process of gear machining vibration signals, wavelet basis functions with orthogonal symmetry, high regularity, and scale functions allowing for fast algorithms are typically preferred. The wavelet basis functions that meet these criteria include the Haar wavelet (db1 wavelet), Daubechies wavelet (dbN wavelet), Symlets wavelet, and Coiflet wavelet. Both the Daubechies and Symlets wavelet families contain 45 wavelets, but as the order increases, the computational cost also sharply rises. Therefore, usually only the first 10 orders of wavelets are employed. The Haar wavelet corresponds to the db1 wavelet in the Daubechies wavelet series, and the Coiflet wavelet series comprises 5 wavelets. Denoising of simulated vibration signals from face hobbing of spiral bevel gears with added white noise is performed using four wavelets meeting the requirements and varying decomposition levels. The comparative effects are illustrated in Figure 12, Figure 13 and Figure 14. It can be observed that different combinations of wavelet functions and decomposition levels yield different denoising effects. The denoising effects of all wavelet functions generally increase first and then decrease with the increase in decomposition levels. For the same decomposition level, higher-order wavelet functions yield better denoising effects.

Among the combinations tested, the denoising effects of the combinations with db10 decomposition level 2, sym9 decomposition level 2, and coif5 decomposition level 2 are relatively prominent within their respective wavelet families. Specifically, the combination of sym9 with decomposition level 2 exhibits the optimal performance in this case study. It results in a signal-to-noise ratio of 22.1989 dB and a root mean square error (RMSE) of 0.059448. Therefore, this combination is selected for further analysis.

The key issue in wavelet threshold denoising lies in determining the threshold selection rule and choosing the threshold function, which directly affects the quality of denoising. Setting the threshold too high can cause removal of useful information and this leads to signal distortion during reconstruction. On the other hand, a low setting of the threshold may result in ineffective removal of noise components. Four widely used threshold selection rules and two threshold functions are combined. As shown in

Table 3. The denoising effect when combining different threshold functions and threshold selection rules.

Rule	Sqtwolog		Rigrsure		Heursure		Minmaxi
Function	Hard	Soft	Hard	Soft	Hard	Soft	Hard	Soft
SNR	22.1877	22.1679	21.2286	22.5046	22.1989	22.1615	22.2362	22.1926
RMSE	0.059525	0.059661	0.066474	0.057393	0.059448	0.059704	0.059193	0.059491

, the denoising effects of various combinations on simulated vibration signals from face hobbing of spiral bevel gears are compared, using the sym9 wavelet with a decomposition level of 2.

It can be observed that using the sym9 wavelet with a decomposition level of 2, along with the Rigrsure threshold selection rule and the soft thresholding function, yields the most significant denoising effect for the signals in this case study. Therefore, this combination is employed for denoising the vibration acceleration signals from face hobbing of spiral bevel gears, as depicted in Figure 15.

4.2.2. Feature Extraction in Time Domain

To verify the reliability of the cutting dynamics model for spiral bevel gear machining, a comparison was made between the vibration signals generated by dynamic simulation and the denoised actual vibration signals. In Figure 16, the actual and simulated acceleration signals are denoted in blue and green, respectively. It was observed that both signals exhibit good consistency in terms of amplitude size and vibration trend in the Y and Z axis, while there is a noticeable discrepancy in the vibration amplitude along the X axis. To further quantify and evaluate the consistency of the two types of signals in the time domain, this study intended to use the correlation coefficient test as a statistical method. However, as shown in Figure 17, when conducting normality tests on both the measured acceleration signals and the simulated acceleration signals, it is evident that neither of the vibration data sets conforms to a normal distribution. Consequently, they fail to meet the prerequisites for correlation coefficient testing, rendering statistical methods unsuitable for assessing the similarity between the two signals.

Therefore, it was necessary to extract and quantitatively compare the time-domain features of the three axial directions of the two sets of vibration signals to determine the reliability of the cutting dynamics model. In this case, the mean is used to describe the overall amplitude level and central position of the signal; variance is used to describe the dispersion of signal amplitudes; skewness and kurtosis are used to describe the shape of the signal waveform; and Shannon entropy is employed to describe the average information content and complexity of the signal. The numerical values of the time-domain statistical feature indicators for the two sets of vibration data across three axes are presented in

Table 4. Time-domain statistical indices of the measured vibration signal and simulated signal along the three axes.

Signal	Value	Mean	Variance	Kurtosis	Crest Factor	Shannon Entropy
X-axis	Measured values	0.16706	0.051434	3.2649	1.3575	3.1108
	Simulated values	0.133784	0.03785	2.451	1.2187	3.5477
	Error	24.88%	35.89%	33.21%	11.39%	14.04%
Y-axis	Measured values	0.50263	0.33961	2.1033	1.1594	3.4626
	Simulated values	0.49427	0.3387	2.1611	1.1775	3.4447
	Error	1.69%	0.27%	2.75%	1.56%	0.52%
Z-axis	Measured values	0.46945	0.3224	2.3353	1.2095	3.4208
	Simulated values	0.50708	0.35297	2.0311	1.1716	3.4642
	Error	8.02%	9.48%	14.9783	3.23%	1.27%

. It is observed that the errors in the time-domain statistical indicators for the Y and Z axes of the two signals are both less than 15%, indicating a high degree of data agreement. However, the errors in the mean, variance, and kurtosis along the X axis exceed 15%, suggesting significant discrepancies in the X-axis data. An analysis, considering the conditions of the machining experiment, conjectures that this is primarily due to the presence of many joint surfaces along the X axis of the YKA2660 spiral bevel gear machining tool. This causes the force hammer excitation signal in the impact test to decay rapidly before being detected by the sensor, leading to the modal data fitting results of the cutter head in the X direction being significantly higher than the actual situation. Consequently, this results in the dynamic simulation-generated X-axis vibration signal of the cutting dynamics model being noticeably lower than the measured value. However, due to the complexity of the machine tool’s structural design, the exact cause of this issue is difficult to ascertain, which also provides directions for future research. By integrating the comparative results of the various characteristic quantities in the time domain of the two sets of vibration data, it can be concluded that the time-domain characteristics of the measured and simulated signals are relatively consistent. This consistency at the time-domain level proves the reliability of the cutting dynamics model for spiral bevel gears face hobbing.

Furthermore, from Figure 16, it can be observed that, among the three axial directions of the measured vibration acceleration signals, the Y-axis data has the largest amplitude, while the X-axis has the smallest. This observation aligns with the modal analysis conclusion that the dynamic stiffness is the poorest in the Y-axis direction and the best in the X-axis direction. This validates the hypothesis that Y-axis vibration is the primary cause of machining chatter during the face gear rolling process of spiral bevel gears.

4.3. Frequency Domain Analysis

Given that axial vibration along the Y-axis is the principal cause of machining chatter, this case primarily investigates the relationship between axial vibrations along the Y-axis and the surface waviness on spiral bevel gears. Fourier transforms were applied to both the measured and simulated vibration signals along the Y-axis to calculate their spectra and energy spectra, as illustrated in Figure 18. It was found that the spectral peak and energy spectral peak of the measured vibration signal were at 159.7799 Hz, whereas the peak frequency of the simulated signal was at 165.4639 Hz, with an error of 3.558%. The frequency composition and main vibration frequencies of both the measured and simulated signals were essentially consistent, which, on a frequency domain level, validates the reliability of the cutting dynamics model established for the face milling process of spiral bevel gears in this study.

The measured vibration signal is a non-stationary signal. To analyze the time-frequency domain characteristics of the measured vibration signal along the Y-axis, a Short-Time Fourier Transform (STFT) was employed. The time-frequency plot effectively describes the transient nature of the vibration signals during the machining of spiral bevel gears, as depicted in Figure 19. It can be observed that the main frequency of the vibration signal during the machining of spiral bevel gears, 159.7799 Hz, remains prominently present throughout the entire gear machining cycle. According to modal analysis results, the first natural frequency of the Y-axis direction of the machine tool spindle is 167.16 Hz, with a resonance bandwidth of approximately 30 Hz. The main frequency of the vibration signal during the machining of spiral bevel gears is located near the first natural frequency of the Y-axis direction of the machine tool spindle. Based on vibration analysis theory, it can be inferred that the vibration along the Y-axis is the primary cause of machining chatter and the generation of surface waviness during the machining of spiral bevel gears.

4.4. The Impact of Tool Vibration on the Machining Quality of Spiral Bevel Gear Tooth Surface

To delve deep into the influence of tool vibrations on the machining quality of spiral bevel gear profiles, this research utilizes the mathematical concept of normal machining errors on the gear tooth profile. As depicted in Figure 20, the surface morphology of the machined spiral bevel gears is simulated and predicted by employing a cutting dynamics model. The color bands on the tooth surface represent the machining error values at their specific locations, with four distinct yellow regions observable on the surface of a single tooth. In this scenario, ten sampling areas were selected at equal intervals along the height direction on both the concave and convex surfaces of a single tooth of the spiral bevel gear. Machining error data were then collected from the starting machining face of the gear along the tooth width direction (direction of machining), as illustrated in Figure 21. Each sampling area was established with 252 equidistant sampling points, and the machining errors in these areas, as shown in Figure 22, demonstrate a consistent trend for machining error data on both the concave and convex surfaces of a single tooth of the spiral bevel gear. Within one machining cycle, four periodic peaks are present. Waviness, a term frequently used in the machining field, describes the undulation changes in the workpiece surface shape. Thus, on the tooth surface of the spiral bevel gear, there are four equidistant areas where the waviness is significantly greater. From a macro perspective, this phenomenon manifests as four distinct ripple patterns on the tooth surface of the spiral bevel gear.

According to cutting principles, it is known that the cutting cycle in face gear hobbing is related to the angular velocity of the cutter head, the number of tool groups on the cutter head, and the number of teeth on the gear being machined [2]. By inserting relevant machining parameters from Table 1 into the gear cutting cycle expression in Equation (2), the time required to machine a single tooth within one cycle of spiral bevel gear face hobbing is 0.02503 s. From the results of frequency domain analysis, the frequency at which the machine tool’s Y-axis and the technological system experience chatter is found to be 159.7799 Hz, corresponding to a chatter cycle of 0.00625 s. This indicates that during one cycle of machining a single tooth of the spiral bevel gear, chatter occurs approximately 4.005 times, aligning with the results of tooth surface morphology predictions from simulation. As shown in Figure 23, the simulated tooth surface morphology matches closely the actual tooth surface morphology of the machined spiral bevel gear. This validates the reliability of the spiral bevel gear cutting dynamics model.

In this case study, the fundamental cutting frequency of the end milling experiment on the spiral bevel gear is 42 Hz. The frequency of chatter occurring in the cutter head process system is 159.7799 Hz, which is approximately four times the fundamental cutting frequency. This value matches the number of vibration marks on the tooth surface, allowing for a preliminary hypothesis that the number of vibration marks on the spiral bevel gear tooth surface corresponds to the ratio of the tool chatter frequency to the fundamental cutting frequency. To verify this hypothesis, numerical simulation of the face hobbing process on spiral bevel gear section has been conducted, and the data on tooth surface waviness was collected, as shown in Figure 24. In this simulation, the system’s natural frequency within the spiral bevel gear cutting dynamics model was adjusted to three and five times the fundamental cutting frequency, i.e., 126 Hz and 210 Hz, respectively. The results show that, when the tool chatter frequency is adjusted to three and five times the fundamental cutting frequency, the number of vibration marks correspondingly becomes three and five.

5. Conclusions

In this investigation, spiral bevel gears, an important component in the aerospace field, were taken as the research object. Detailed research was conducted on chatter phenomena in the spiral bevel gear face hobbing process using methods such as analytical modeling of cutting dynamics, modal analysis, wavelet denoising, and time–frequency analysis. The main causes of machining chatter were explored, and the impact of chatter on machining quality was analyzed. Additionally, a reliable predictive model for tooth surface morphology in the spiral bevel gear face hobbing process was proposed. This study aims to provide experimental support for the relatively understudied area of chatter in the spiral bevel gear face hobbing process and to offer theoretical support for improving machining accuracy and optimizing machining processes. The conclusions of this study are summarized as follows:

(1)

By comparing the denoising effects of different combinations of wavelet basis functions, decomposition levels, threshold selection rules, and threshold functions, and selecting based on SNR and RMSE, the best denoising effect can be obtained by the combination of using the sym9 wavelet with 2 decomposition levels, applying the Rigrsure rule, and employing a soft thresholding function. This results in a SNR of 22.5046 db, and a RMSE of 0.057393.

(2)

Through time-domain analysis of both measured vibration signals and simulated signals, a high degree of conformity in the time-domain statistical indicators between the two signal sets was observed, providing evidence at the time-domain level for the accuracy of the cutting dynamics model. For the equipment YKA2260 used in this investigation, the main frequency of cutter head vibration was identified as 159.7799 Hz. The Y-axis vibration was found to be the main cause of machining chatter and the formation of tooth surface chatter marks. The accuracy of the cutting dynamics model at the frequency domain level is confirmed.

(3)

With respect to the influence of tool vibration on the machining quality of spiral bevel gears, the comparison between the predicted and actual machined gear surface morphology has validated the reliability of the cutting dynamics model for spiral bevel gear machining. This model can effectively predict the gear surface morphology of spiral bevel gears. The prediction and experimental analysis indicate that the number of tooth surface chatter marks is proportional to the ratio of tool chatter frequency to cutting base frequency. This finding provides promising significance for optimizing the face hobbing process for spiral bevel gears.

(4)

This study provides a theoretical foundation and guidance for the design of new non-traditional dynamic vibration absorbers suitable for spiral bevel gear machining machines, aiming to address vibration issues specific to different models of such machines. Additionally, this research is significant for optimizing the face hobbing process of spiral bevel gears. To prevent machining defects caused by vibrations, machining parameter design should be optimized based on the modal parameters of the process system, avoiding multiples of the cutting frequency.