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.
Highlights
Time-delay systems widely exist in engineering and science, where the rate of change of state is determined by both present and past state variables, such as machining processes [1, 2], wheel dynamics [3, 4], feedback controller [5, 6], gene expression dynamics [7], and population dynamics [8, 9]
For some of above applications, the time delay in the dynamic system may lead to instability, poor performance, or other types of potential damage
It is necessary for engineers and scientists to research the dynamics of these systems to reduce or avoid such problems
Summary
Time-delay systems widely exist in engineering and science, where the rate of change of state is determined by both present and past state variables, such as machining processes [1, 2], wheel dynamics [3, 4], feedback controller [5, 6], gene expression dynamics [7], and population dynamics [8, 9]. For some of above applications, the time delay in the dynamic system may lead to instability, poor performance, or other types of potential damage It is necessary for engineers and scientists to research the dynamics of these systems to reduce or avoid such problems. Compared to the finite dimensional dynamics for systems without time delay, time-delay systems have infinitedimensional dynamics and are usually described by delay differential equations (DDEs) Their stability properties can be analyzed through obtaining the stability charts that show the stable and unstable domains. SDM is a known and widely used method to determine stability charts for general time-periodic DDEs arising in different engineering problems.
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