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Mathematical Problems in Engineering
Volume 2017, Article ID 9490142, 8 pages
https://doi.org/10.1155/2017/9490142
Research Article

A Method for Stability Analysis of Periodic Delay Differential Equations with Multiple Time-Periodic Delays

1Tianjin Key Laboratory of High Speed Cutting and Precision Machining, Tianjin University of Technology and Education, Tianjin 300222, China
2National-Local Joint Engineering Laboratory of Intelligent Manufacturing Oriented Automobile Die & Mould, Tianjin University of Technology and Education, Tianjin 300222, China
3Tianjin Jinhang Institute of Technical Physics, Tianjin, China

Correspondence should be addressed to Jianxin Han; nc.ude.ujt@nixnaijnah

Received 28 June 2017; Accepted 14 August 2017; Published 17 September 2017

Academic Editor: Renming Yang

Copyright © 2017 Gang Jin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Delay differential equations (DDEs) are widely utilized as the mathematical models in engineering fields. In this paper, a method is proposed to analyze the stability characteristics of periodic DDEs with multiple time-periodic delays. Stability charts are produced for two typical examples of time-periodic DDEs about milling chatter, including the variable-spindle speed milling system with one-time-periodic delay and variable pitch cutter milling system with multiple delays. The simulations show that the results gained by the proposed method are in close agreement with those existing in the past literature. This indicates the effectiveness of our method in terms of time-periodic DDEs with multiple time-periodic delays. Moreover, for milling processes, the proposed method further provides a generalized algorithm, which possesses a good capability to predict the stability lobes for milling operations with variable pitch cutter or variable-spindle speed.