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

Jin, Gang; Qi, Houjun; Li, Zhanjie; Han, Jianxin; Li, Hua

doi:10.1155/2017/9490142

Cited by 3 publications

(2 citation statements)

References 37 publications

(53 reference statements)

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“…M Pospíšil and F Jaroš, 21 studied the approximate solution of DDEs using the Laplace transformation. G Jin et al., 22 studied, the stability analysis method of the periodic DDEs with time-periodic and multiple delays. D Lehotzky et al., 23 extended the stability and convergence analysis of spectral-element-method for the periodic time DDEs with distributed and multiple delays.…”

Section: Introductionmentioning

confidence: 99%

Applications of Legendre spectral collocation method for solving system of time delay differential equations

Khan

Ali

2020

Advances in Mechanical Engineering

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The numerical techniques are regarded as the backbone of modern research. In literature, the exact solution of time delay differential models are hardly achievable or impossible. Therefore, numerical techniques are the only way to find their solution. In this article, a novel numerical technique known as Legendre spectral collocation method is used for the approximate solution of time delay differential system. Legendre spectral collocation method and their properties are applied to determined the general procedure for solving time delay differential system with detail error and convergence analysis. The method first convert the proposed system to a system of ordinary differential equations and then apply the Legendre polynomials to solve the resultant system efficiently. Finally, some numerical test problems are given to confirm the efficiency of the method and were compared with other available numerical schemes in the literature.

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Section: Introductionmentioning

confidence: 99%

Applications of Legendre spectral collocation method for solving system of time delay differential equations

Khan

Ali

2020

Advances in Mechanical Engineering

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“…e regenerative chatter is caused by the forces generated during the dynamic cutting process and is not dependent on the external periodic forces. e disastrous nature of chatter vibration brings numerous negative effects including poor surface finish, aggravated tool wear, excessive noise, and damages to machine tool components [4,5]. erefore, extensive research studies have been conducted to prevent the chatter occurrence by analytical stability prediction, real-time detection, chatter control technique, and so on [6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Robust Chatter Stability Prediction of the Milling Process considering Uncertain Machining Positions

Deng

Zhou

Yang

et al. 2020

Mathematical Problems in Engineering

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Milling stability is a function of the tool point frequency response functions (FRFs), which vary with the movements of the moving parts within the whole machine tool work volume. The position-dependent tool point FRFs result in uncertain prediction of the stability lobe diagram (SLD) for chatter-free machining parameter selection. Taking the variations of modal parameters to represent the variations of tool point FRFs, this paper introduces the edge theorem to predict the robust milling chatter stability. The application of the edge theorem requires the minimum and maximum modal parameters within the machining space defined by the machining position and machining allowance information. Then, radial basis function artificial neural networks (RBFANNs) are used to predict the position-dependent modal parameters in X and Y directions based on the sample information of machining positions and related modal parameters at the tool point. Moreover, sample machining spaces are determined based on the aforementioned sample positions, and the trained RBFANNs are used to obtain corresponding sample extreme modal parameters. On this basis, RBFANNs for predicting the position and machining allowance-dependent extreme modal parameters can also be trained, and they are combined with the edge theorem and zero exclusion condition to calculate robust pairs of the spindle speed (n) and limiting axial cutting depth (aplim) and then plot the robust SLD (RSLD). A case study was performed on a real three-axial vertical machining center, and the plotted RSLD considering position variations was compared with the traditional SLD. Results of the chatter tests show that the RSLD can provide more reliable (ap, n) pairs to guarantee the milling stability, validating the feasibility of the proposed robust milling chatter stability prediction method.

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Stability Analysis Method for Periodic Delay Differential Equations with Multiple Distributed and Time-Varying Delays

Jin

Zhang

et al. 2020

Mathematical Problems in Engineering

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Dynamic stability problems leading to delay differential equations (DDEs) are found in many different fields of science and engineering. In this paper, a method for stability analysis of periodic DDEs with multiple distributed and time-varying delays is proposed, based on the well-known semidiscretization method. In order to verify the correctness of the proposed method, two typical application examples, i.e., milling process with a variable helix cutter and milling process with variable spindle speed, which can be, respectively, described by DDEs with the multidistributed and time-varying delays are considered. Then, comparisons with prior methods for stability prediction are made to verify the accuracy and efficiency of the proposed approach. As far as the milling process is concerned, the proposed method supplies a generalized algorithm to analyze the stability of the single milling systems associated with variable pith cutter, variable helix cutter, or variable spindle speed; it also can be utilized to analyze the combined systems of the aforementioned cases.

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

Cited by 3 publications

References 37 publications

Applications of Legendre spectral collocation method for solving system of time delay differential equations

Applications of Legendre spectral collocation method for solving system of time delay differential equations

Robust Chatter Stability Prediction of the Milling Process considering Uncertain Machining Positions

Stability Analysis Method for Periodic Delay Differential Equations with Multiple Distributed and Time-Varying Delays

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