## 4 Conclusions  
This paper presents an improved method to gain the SLDs for milling process with multiple and distributed time delays. After the milling dynamic of the VHCVSS process, taking into account regeneration effect, is modeled by linear DDEs with multiple and distributed time delays, the presented method is applied to deal with the problems about the stability prediction for VHCVSS process for the first time. By comparing with the existing research and time-domain simulations, the presented method has been validated. The effectiveness and feasibility of chatter suppression by the combination of VHC and VSS milling are explored and investigated for associated 1- and 2-DOF systems. Moreover, the following conclusion can be drawn:  
1. The presented method is effective for obtaining the SLDs for milling processes with multiple and distributed time delays. Thus, the theory in [15; 32] is developed and its application scope is widened.
2. The presented method is applied to dealing with the stability prediction problem with regard to the new process, i.e., VHCVSS for the first time. Obviously, the method is not only applicable to the cases associated with VPC, VHC, or VSS, but also to their combination cases, e.g., VPCVSS and VHCVSS.
3. Compared with Processes I, II, and III, application of Process IV can result in nearly 2-fold as high as the minimum depth of cut for Processes I and II and are about 1.3-fold for Process III for some special domain, and can respectively lead to the average increases of stable area by 30.4%, 23.5%, and 1.5% at different RVF values. This indicates that one can improve the chatter suppression performance of the cutting system through VHCVSS in practice. The proposed method can also provide a good help for the subsequent process optimization to achieve the chatter-free maximum removal rate.  
Figure 10: Contribution factor diagram of Processes II and III to the combination effect under different \(\beta\).  
4. For Process IV, the increasing of the helix angle increment \(\Delta\beta\) plays a positive role in enhancing the stability area for VHCVSS. Thus, bigger values of \(\Delta\beta\) should be suggested in practice.
5. The contribution of Process III to the cutting stability of Process IV is more than 85% and 61% for the associated 1- and 2-DOF systems, respectively. Thus, essentially, VHCVSS is a process in which VHC plays an absolutely leading role for milling stability but VSS plays an auxiliary role.  
Conceptualization, G.J. and W.L.; Methodology, J.H.; Validation, G.H.; Formal analysis, Z.L. and W.L.; Investigation, G.S.; Data curation, G.H.; Writing--original draft preparation, G.J.; Funding acquisition, G.J. and J.H. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (grant numbers 51405343) and Tianjin Science and Technology Planning Project (grant numbers 18 JCZDJJC10050, 20 JCYBJC00490, 18 JCQNJC73100, 20 JQQNJC01070, 20 JQCQNJC01060).  
Not applicable.  
The MATLAB data corresponding to the figures in this paper have been deposited and can be obtained through the following website ([[[https://pan.baidu.com/s/1](https://pan.baidu.com/s/1) Be2]([https://pan.baidu.com/s/1](https://pan.baidu.com/s/1) Be2) QKh5 NLsAKhv3 cRzQZwg, Extraction code: en95]).  
The authors declare