Generalized Numerical Differentiation Method for Stability Calculation of Periodic Delayed Differential Equation: Applic
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International Journal of Precision Engineering and Manufacturing https://doi.org/10.1007/s12541-020-00409-6
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Generalized Numerical Differentiation Method for Stability Calculation of Periodic Delayed Differential Equation: Application for Variable Pitch Cutter in Milling Ding Chen1 · XiaoJian Zhang1 · Han Ding1 Received: 13 February 2020 / Revised: 8 July 2020 / Accepted: 24 August 2020 © Korean Society for Precision Engineering 2020
Abstract This paper proposes a generalized numerical differentiation method for stability prediction of the non-autonomous delayed differential equations (DDEs) with periodic coefficients and discrete delays. Firstly, the periodic DDE is described in statespace form and the period of a system is equally discretized. Then, the discrete first derivatives versus time are approximated by a linear combination of the state function values at multiple neighboring sampling grid points based on the finite-difference formulas. Such that, the original DDE is approximated as a series of algebraic equations and the Floquet transition matrix can be constructed on one period. At last, the system stability is determined according to the Floquet theory by checking the eigenvalues. The delayed damped Mathieu equation is regarded as a typical case to verify the effectiveness and efficiency of the presented method. The stability diagrams and rate of convergence are computed in comparison with those via the benchmark algorithm (the semi-discretization method). As an application, the presented method is used to predict the stability of milling with variable pitch cutter, and the computational result agrees well with the experimentally verified example. Keywords Numerical differentiation method · Stability · Delayed differential equation · Milling
1 Introduction Time delay always occurs in various mechanical systems, dynamic networks, and machining processes with feedback mechanisms or regenerative mechanisms, where the rates of change of the current state depend on both the past and the present states [1]. The typical examples include output feedback control of dynamical systems [2], regenerative machine tool chatter [3–6], communication delays in remote control and network [7], etc. These systems can be generally represented as delayed differential equations (DDEs) which can describe dynamical models including past effects. The finite time delay introduces an infinite-dimensional phase space, even for a simple system [8]. In manufacturing processes, machine tool chatter is mainly caused by the regenerative chatter [9], the first and * XiaoJian Zhang [email protected] 1
State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
widely used analytical method for stability prediction is the zero-order approximation (ZOA) frequency method [10], but it is confined to its application in machining. In the past decade, the discrete time domain methods catch a lot of
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