Mathematical Problems in Engineering

Volume 2017, Article ID 9490142, 8 pages

https://doi.org/10.1155/2017/9490142

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

^{1}Tianjin Key Laboratory of High Speed Cutting and Precision Machining, Tianjin University of Technology and Education, Tianjin 300222, China^{2}National-Local Joint Engineering Laboratory of Intelligent Manufacturing Oriented Automobile Die & Mould, Tianjin University of Technology and Education, Tianjin 300222, China^{3}Tianjin 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.

#### 1. Introduction

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]. However, for some of above applications, the time delay in the dynamic system may lead to instability, poor performance, or other types of potential damage. Therefore, 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 infinite-dimensional 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. For example, a stable milling process can be realized by choosing the corresponding parameter from a stability lobe diagram (SLD), which is a function of spindle speed and depth of cut parameters. Thus, more and more attention has been paid on this issue and many analytical and numerical methods have been developed to derive the stability conditions for the system parameters.

By using the -subdivision method, Bhatt and Hsu [10] determined stability criteria for second-order scalar DDEs. Budak and Altıntaş [11, 12] and Merdol and Altintas [13] proposed a method in frequency domain called multifrequency solution. By employing a shifted Chebyshev polynomial approximation, Butcher et al. [14, 15] presented a new technique to study the stability properties of dynamic systems by obtaining an approximate monodromy matrix. Insperger and Stepan [16–18] proposed a known method called semidiscretization method (SDM), which is based on the discretization of the DDEs and approximates their infinite-dimensional phase space by a finite discrete map in time domain. Bayly et al. [19] carried out a temporal finite element analysis for solving the DDEs, which are written in the form of a state space model and discretizing the time interval of interest into a finite number of temporal elements. Based on the direct integration scheme, Ding et al. [20], Liu et al. [21], and Jin et al. [22] used a full-discretization method to gain stability chart efficiently. Recently, Khasawneh and Mann [23] and Lehotzky et al. [24] presented a numerical algorithm called spectral element method. This method has good efficiency because of its highly accurate numerical quadratures for the integral terms.

SDM is a known and widely used method to determine stability charts for general time-periodic DDEs arising in different engineering problems. In this paper, based on SDM, a generalized method for periodic DDEs with multiple time-periodic delays is proposed to obtain the stability chart of DDEs. The structure of the paper is as follows. In Section 2, the mathematical model is introduced. In Section 3, two typical examples are used to verify the effectiveness of the proposed method. In Section 4, conclusions with a brief discussion are presented.

#### 2. Mathematical Model

The general form of linear, time-periodic DDEs with multiple time-periodic delays can be expressed aswhere is the state, is the input, is a constant matrix, and , respectively, are and periodic coefficient matrices that satisfy and , , and , is an constant matrix, is the time period, and is the number of time-delays. Note that (1) can also be written in the form with .

Consider that the period is divided into number of discrete time intervals, such that each interval length . Introduce symbol to represent the th time interval, . Here, means the th time node and is equal to . Thus, considering the idea in [2], the averaged delay for the discretization interval is defined as follows:where

The number of intervals related to the delay item can be approximately obtained bywhere indicates the operation that rounds positive number towards zero.

Substituting (3) into (2) and solving it as an ordinary differential equation over the discretization period with initial condition , the following equation is derived:

Substituting into (6), then it can be equivalently expressed asIn , , , , and are defined as follows: where , , , , and . Here, it should be noted that the approximation of in (8) is the same as that for the so-called zero-order SDM in [2].

Substituting (8) into (7) leads towhereClearly, , , , and can be expressed as follows:where denotes the identity matrix. Let andthen combining (9) and (12), one can be recast into a discrete map aswhere each matrix is given bywhere . The horizontal position of the discrete input matrices and in (14) depends on the value of corresponding to and they, respectively, begin from the column of and for a single DOF system as opposed to the column of and for a two-DOF one.

Based on (13) and (14), the following mathematical expressions can be established by coupling the solutions of the successive time intervals in period : where is the Floquet transition matrix that gives the connection between and . According to the Floquet theory, the stability of the system is determined using the following criterion. If the moduli of all the eigenvalues of the transition matrix are less than unity, the system is stable. Otherwise, it is unstable.

Here, it should be noted that the matrix can be reduced because only the delayed positions show up in the governing equation of the milling process. Thus, the size of the approximation vector in (14) could be reduced by removing the delayed values of the velocities, such that the size of vector can be decreased to for a single DOF system and for a two-DOF system. This can give some additional improvement in the computational time for the proposed method.

#### 3. Verification of Method

There are several numerical and semianalytical techniques to determine the stability conditions for periodic DDEs. However, most of them were developed with the aim of constructing stability charts for milling processes, such as the analysis of the milling system with runout [25], with variable pitch/helix cutter [26–30], with variable-spindle speed [31–33], or with serrated cutter [34, 35]. In order to verify the proposed method, two typical milling operations are chosen and considered. The first is the varying spindle speed process, which can be described by a DDE with time-periodic delay in general. The other is the milling process with variable pitch cutters, which is often characterized by a DDE with multiple delays. Both methods are known means to influence and to prevent chatter vibration in milling.

##### 3.1. Milling with Varying Spindle Speed

Generally, the mathematical models for milling processes with spindle speed variation can be written as that is, (2) is degenerated into one with one-time-periodic delay. For a single DOF system in [2], the matrices in (16) have the form where is the mode mass, is the natural frequency, is* the* damping ratios, and is the specific directional factor and has the formwhere is the axial depth of cut, is the number of teeth, and are the linearized cutting coefficients in tangential and radial directions, denotes whether the th tooth is cutting, and the angular position of tooth iswhere is the spindle speed and is assumed to change in the form of a sinusoidal wave, which is periodic at a time period , with a nominal value, , and an amplitude , as shown in Figure 1. For this sinusoidal modulation, the shape function is modeled aswhere is the ratio of the speed variation amplitude to the nominal spindle speed and is the ratio of the speed variation frequency to the nominal spindle speed.