Study of optimization of blade non-uniform allowance process and chatter stability domain

In addressing the phenomenon of chatter during the machining of thin-walled components, a machining strategy involving the implementation of a continuous non-uniform toolpath has been proposed as a remedy for chatter suppression. The methodology commences with a meticulous determination of machining allowances, accomplished by utilizing fitting functions. Subsequently, the stiffness coefficients and natural frequencies inherent to the workpiece-tool system are extracted through modal analysis. A stability lobes diagram is formulated, employing the principles of regenerative chatter analysis. This diagram encompasses both uniform toolpath scenarios and non-uniform continuous toolpath scenarios, with the latter established via the least squares method. The resultant stability lobes diagram effectively delineates the boundaries within which chatter remains stable, under varying depths of cut. The stability lobes diagrams substantiate the effectiveness of the non-uniform continuous toolpath constructed via the least squares method in significantly augmenting the inherent stiffness coefficient of the workpiece-tool system, thereby mitigating chatter. This approach serves as a valuable guide for the manufacturing of intricate curved thin-walled components.


Introduction
The integral impeller with complex curved surfaces is the core component of a propulsion device.Maintaining its excellent surface precision during the manufacturing process is essential.This contributes to maintaining high stability, high rigidity, and long lifespan in its operational environment.To achieve these objectives, Jiang et al. [1] proposed a method for non-uniform allocation of finishing allowances in thin-walled structural component precision machining based on the transitional state stiffness of machining features.This method effectively enhances the component's stiffness and reduces deformation.Xiong et al. [2] utilized an improved semi-discretization method to predict milling stability in helical angle-variable speed milling.They investigated the effects of cutter helix angle, radial immersion rate, and variable speed modulation coefficient on chatter suppression during the milling process.Based on the structural stiffness of thin-walled components, modal finite element analysis, and eigensensitivity analysis, Tian et al. [3] determined and optimized the distribution of allowances during semi-precision machining of thin-walled workpieces.Li et al. [4] introduced a discrete output feedback robust controller based on Linear Matrix Inequality (LMI) to suppress chatter by applying the necessary active control force by using electromagnetic actuators.Yao et al. [5] developed an intelligent spindle real-time chatter detection and suppression module.Based on the pre-obtained Stable Lobes Diagram (SLD) and a lof-based training model, this module can monitor and suppress chatter during the machining process in real time.Under different cutting conditions, it is difficult to obtain suitable parameters such as spindle speed.
In actual machining, blade vibration (chatter) is prone to occur, which affects the quality of machining.To mitigate or eliminate chatter, the study of the machine tool's chatter stability domain utilizes chatter stability lobes.These lobes effectively predict and describe the stability of cutting processes, aiding in the anticipation of machining stability.To address the issue of chatter in integral impeller machining, this study first employs finite element software to obtain the modal parameters of the manufacturing system.Subsequently, the chatter stability lobes are utilized.A comparison analysis is conducted between two models: one involving uniform allowance and the other based on the least squares method for non-uniform allowance.Finally, milling experiments are performed on the two types of blades to validate the efficacy of the least squares allowance optimization approach.The results demonstrate that this method effectively enhances the workpiece's stiffness.It ensures no sudden changes in cutting depth during tool operation, leading to the suppression of chatter during the machining process.

Allowance process design
In the milling process, the magnitude of milling forces is mainly determined by the milling depth, milling width, and feed rate.For precision machining blades, such as with uniform allowance distribution, approximately equal cutting forces are generated during cutting while maintaining a single variable.However, the stiffness varies across different positions on the blade's surface due to the uniform allowance, leading to significant cutting deformations and vibrations in the weak stiffness areas.This compromises the machining quality of the blade, limiting the quality level of the finished product.Therefore, without altering factors related to the leaf, the initial approach involves the application of the non-uniform surplus allocation method.In the vertical direction as shown in Figure 1, from the leaf base to the leaf tip were 50%, 30%, and 20% in order, and the residual amounts were 0.6 mm, 0.4 mm, and 0.16 mm, respectively.Subsequently, the inflection points are assigned numerical labels, and the coordinate information for each label is listed in Table 1.Finally, based on the collected data, equation 1 was used for fitting and the results of the fitted parameters are shown in Table 2.

Blade allowance modal analysis
Continuous non-uniform allowance blade and uniform allowance blade models have been established by using NX software.These models have been completed and exported in STEP format for subsequent importation into finite element analysis software.The material selected for simulation is specified as AL6061, with the following properties: a density of 2.79 g/cm³, Young's modulus of 103 GPa, and a Poisson's ratio of 0.33.We set the plane at the blade root as a fixed boundary.Considering the significant influence of the first two modal shapes on vibration, we perform the analysis through the first two modes.The modal analysis results are depicted in Figures 2 and 3.   From the modal analysis results, intrinsic frequencies and masses of the blade can be obtained, as presented in Table 3.
Table 3 shows that the natural frequencies of the blade based on the least squares method for nonuniform allowance are significantly higher than those of the uniform allowance blade.Using the modal parameters obtained from Table 3, the modal stiffness of the machine-tool-workpiece system can be calculated from Equation 2: where n  is the natural frequency of the system, k is the stiffness coefficient, and m is the mass.
For the first mode, the modal stiffness of the uniform allowance blade is calculated as: 3.12 10 For the first mode, the modal stiffness of the least squares method for a non-uniform allowance blade is calculated as:

    
For the second mode, the modal stiffness of the uniform allowance blade is calculated as: For the second mode, the modal stiffness of the least squares method for a non-uniform allowance blade is calculated as:  Based on the findings in Table 4, a notable observation is that the sub-system associated with blades employing the least squares method for non-uniform allowances demonstrates higher modal stiffness in comparison to the sub-system with uniformly allowed blades.This substantiates that the utilization of the least squares method for non-uniform allowances effectively elevates the modal stiffness within the machine-tool blade processing sub-system.

Dynamic milling force model
Prior to delving into the analysis of the milling chatter stability lobes for the blade, it is essential to establish a dynamic model for the semi-finish/finish machining stages of the impeller blade.This involves extending the two-dimensional regenerative chatter stability theory to a three-dimensional domain.Drawing insights from Altintas's work [6] on three-dimensional modeling of ball-end milling processes, the dynamic cutting force of the system can be derived from Equation 3: where N represents the number of teeth on the milling cutter; G(iω) denotes the transfer function between the tool and the machining region of the integral impeller; a p signifies the axial depth of the cut; ω c stands for the inherent frequency of the milling system; α is the matrix of directional cutting coefficients; T denotes the cutting period; K t represents the tangential milling force coefficient.The characteristic equation of the cutting force can be expressed as Equation 4.
The eigenvalues are shown in Equation 5.
The Fourier transform of is performed and substituted into the above equation to derive the critical depth of cut of the impeller based on the theory of three-dimensional chatter stability analysis, as in Equation 6:

Establishment of blade chatter stability lobes
By conducting finite element analysis on the two types of blades as previously mentioned, modal parameters for the machine-tool workpiece systems were acquired separately.Substituting the inherent frequencies and stiffness coefficients from these models into the mathematical model for blade chatter stability, we conduct an analysis to generate the chatter stability lobes for both types of blades, as depicted in Figure 5.
The solid red dotted line represents the chatter stability lobe for the least squares method with nonuniform allowance blades, while the blue line represents the chatter stability lobe for uniform allowance blades in Figure 4 and Figure 5. Comparing Figure 5 with Table 5, it becomes evident that when machining blades use uniform allowances, the critical cutting depth is significantly smaller than that of blades with non-uniform allowances.As the inherent frequency increases, the chatter stability lobes shift towards the right, yet the critical axial cutting depth remains constant.The relatively low stiffness of uniform allowance blades, due to the absence of added thickness, results in significant deformation during milling, hence leading to a smaller critical axial cutting depth.Conversely, the least squares method applied to non-uniform allowances significantly enhances stiffness through fitting techniques, leading to an enlargement of the critical axial cutting depth between the blade and the tool.

Conclusion
This study introduced a novel approach utilizing the least squares method for optimizing non-uniform allowances in impeller manufacturing.In comparison to uniform allowances, the proposed optimization technique employing the least squares method exhibited a marked enhancement in the stiffness of the tool-workpiece system.Through meticulous modal analysis of the two distinct blade configurations, it was ascertained that the inherent frequencies of the optimized blades were approximately 30% of those of the pre-optimized counterparts.By leveraging the insights from the chatter stability lobes, it was established that the optimized blades showcased an approximately 0.55~1 fold increase in the critical cutting depth, in contrast to their pre-optimized counterparts.
The outcomes of practical experimentation conclusively demonstrated that this technique, when compared to the uniform allowance process, effectively augments the machining precision and surface quality of the blades.This approach bears significant implications for guiding real-world manufacturing endeavors.

Figure 2 .
Figure 2. Comparison of first-order modal analysis of blades: (a) Uniform allowance blade; (b) Continuous non-uniform allowance blade.

Figure 3 .
Figure 3.Comparison of the second-order modal analysis of blades: (a) Uniform allowance blade; (b) Continuous non-uniform allowance blade.

Table 4 .
Two types of blade modal stiffness table.

Table 5 .
Cutting depth limited by blade.