Stability Analysis in Milling Based on the Localized Differential Quadrature Method

Abstract

Chatter stability analysis is an effective way to optimize the cutting parameters and achieve chatter-free machining. This paper proposes a milling chatter stability analysis method based on the localized differential quadrature method (LDQM), which has the advantages of simple principle, easy application, and high computational efficiency. The milling process, considering the regeneration effect, is modeled using linear periodic delay differential equations (DDE), then the state transition matrix during the adjacent cutting period is constructed based on the LDQM, and finally, the stability of the milling process is obtained based on the Floquet theory. The accuracy and computation efficiency of the proposed method are verified through two benchmark milling models. The simulation results demonstrate that the proposed method in this paper can accurately and quickly obtain the chatter stability lobe diagram (SLD) of the milling process, which will provide guidance for optimizing the process parameters.

Stability Analysis of Milling Based on the Barycentric Rational Interpolation Differential Quadrature Method

“…Experimental-based methods [2][3][4] Time-domain simulation methods [5][6][7][8] Dynamic analysis methods [9][10][11][12][13][14][15][16][18][19][20][22][23][24][25][26][27][28][29]33] From Table 1, it can be seen that the number of studies based on dynamic analysis methods is the highest, indicating that such methods have good application prospects.…”

“…As mentioned earlier, when the number of discrete nodes is large, the classical DQM produces an illconditioned differential matrix, resulting in erroneous stability prediction results. Figure 3b shows the SLD obtained by the LDQM [33], from which it can be seen that when the parameter l is too large, the LDQM may still yield unstable results. In Figure 3c,d, the SLD is obtained by the method proposed in Section 3.…”

“…To address these issues, Zong Z. et al [32] introduced the localized differential quadrature method (LDQM) and successfully applied it to the problem of two-dimensional wave equations. Mei et al [33] derived a chatter stability analysis method based on the LDQM. In their works, the derivative of the state vector in the milling dynamics equation is represented as the weighted sum of state vector values at local nodes, which can generate a sparse weight matrix and improve the stability of numerical calculations.…”