Shock and Vibration

Volume 2015 (2015), Article ID 453216, 17 pages

http://dx.doi.org/10.1155/2015/453216

## Stability Analysis of a Breathing Cracked Rotor with Imposed Mass Eccentricity

^{1}Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, China^{2}School of Energy Power and Mechanical Engineering, China Electric Power University, Beijing 102206, China

Received 14 April 2015; Revised 8 June 2015; Accepted 15 June 2015

Academic Editor: Sundararajan Natarajan

Copyright © 2015 Huichun Peng 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

The investigation of the effects of mass eccentricity on stability of a gravity dominated cracked Jeffcott rotor would generally provide practical applicability to crack detection and instability control of the heavy loading turbo-machinery system. Based upon the numerical Floquet method, the stability and bifurcations of the periodic time-dependent rotor with a transverse breathing crack are studied with respect to the varied mass eccentricity at different rotation speed, and the stability diagrams in the parameter plane are obtained which the previous studies have not covered. The numerical response of the cracked rotor system is also analyzed by the frequency spectrum to present the vibration characters while the rotation speed approaches the critical ratio. The detailed numerical eigenvalues of the transition matrix are applied to analyze the types of the bifurcations of the cracked rotor system. Three types of bifurcations are found and responses of the cracked rotor system at these bifurcations are presented for the visualized comparisons.

#### 1. Introduction

Fatigue cracks of rotating machinery shaft might lead to catastrophic accidents. Dynamic characteristics of cracked rotor have been studied in the last three decades [1, 2]. Many factors can affect the stability of the cracked rotor system, such as crack depth ratio, crack position, the system damping ratio, mass eccentricity, and unbalance orientation angle. Although there are many researches of the influential parameters such as the crack depth ratio and damping ratio [3–8], as for heavy loading turbo-machinery system, the breathing cracked rotor coupled with mass eccentricity, which may excite abundance unstable behavior and bifurcations during the horizontal rotating motion, is in need of consideration.

As the motion equations of the cracked rotor systems are nonlinear and nonautonomous differential equations due to the open-close of the crack during the rotation of the rotor, the response is analyzed by means of the qualitative theory of nonlinear dynamics [9]. In the survey of Genesio [10], the existing methods on estimation of the region of asymptotic stability for continuous, autonomous, and nonlinear systems are mainly the perturbation methods to numerically approach to stability border lines. But owing to theoretical idealized setting, the approximation appears to be quite hard to be applied to more complicated rotor models with multivaried parameters.

Based on the stability analyses of higher linear differential equations with periodic coefficients [11], Meng and Gasch [3] and Untaroiu et al. [4] explored its applications to cracked rotor system. The transition matrix approach based on Floquet theorem, instead of perturbation methods, was employed to make the analysis of stability of a cracked rotor system. By means of evaluating the eigenvalues of the Floquet transition matrix, Guo et al. [5] investigated the effect of damping on the instability regions of a cracked rotor system in the complex plane, and Chu et al. [6–8] examined the effects of crack depth and position on the instability of multidisk cracked rotors and slant cracked rotors. The works show that the stability degree (determined by the eigenvalues of the Floquet transition matrix) can act as an efficient index to analyze the stability and bifurcation of a cracked rotor system.

On the other hand, in the field of numerical simulations of the response of a cracked rotor, scientists tried to find some more efficient ways to express the breathing motion of the rotating cracked rotor. There are primarily two types of cracks which are studied in the literature. They are called both open crack and breathing crack model [12–16]. The open crack model, which is considered to be more suitable for the small mass rotor (less than 10 kg) with a relatively high rotation speed, is proposed as the stiffness asymmetry rotors approached by Chu et al. [7, 17] and Genesio et al. [10]. As for the crack on the large quantity (larger than 20 kg) rotors, which are normally operating on a relatively low speed, usually breath from closed state in the compression zone to open state in tension zone is the case for breathing crack model [15, 16].

Bachschmid et al. [12–15] proposed the breathing cracked model as the coefficients of the Fourier series based on the finite element models to simulate the dynamic response of the actual experimental rotors. But the complicate calculation limits its application in the Jeffcott model. Mayes and Davies [16] presented a classical form for the breathing crack function to simulate the actual breathing behavior, which is assumed to be a more feasible approximation process for the cracked rotor systems.

Although the breathing mechanism in rotating shafts can be accurately reproduced by 3D nonlinear finite element models, Bachschmid et al. [13, 14] showed that a simple approximate model can also accurately simulate the breathing behavior of the rotating cracked rotor. Han [18] and Patel and Darpe [19] showed that the method based on fracture mechanics, using stress intensity factors and the energy release rate to calculate the stiffness, combined with a breathing function, can increase the efficiency of the dynamic analysis. Bifurcation diagram, orbit plot, and Poincare map were employed to investigate the numerical simulation of transient response of the cracked rotor system to find the nonlinear dynamic characteristic of the cracked rotor [20–24] based on the two basic model of open-close cracked rotor, both switching and breathing models. These studies exhibited the splendid visualized comparisons and provide the nonlinear signals directly for monitoring and diagnosing fault of rotating machines. The nonlinear signals of the harmonically varying stiffness breathing models were showed to be in high compliance with the actual experimental rotors.

In view of the accurate actual experimental crack model, the previous researches of stability analyses of multidegree-of-freedom rotor system are mainly limited to investigate the influence of the crack depth ratio and the system damping ratio on the stability of cracked rotor system [3–8, 17, 25]. In fact, there are many other factors influencing the stability degree and stable region of the vibration dynamic of a rotating cracked rotor. Mass eccentricity is another important parameter that has significant influence on the stability of gravity dominated cracked rotor dynamic system. In practical experience, the mass eccentricity can be detected and modified by attaching an additional massive on the rotating machinery shaft. The research on the influence of varied mass eccentricity on the stability and bifurcations at the critical value of stability degree of cracked rotor is necessary to improve the understanding of the nonlinear dynamic characters of the cracked rotor system and consequently may help in the identification of cracks in rotating machinery. The study is presented as follows.

Firstly, we facilitate the evaluation of the transition matrix by calculating the stability degree of periodic vibration based on transient response. Secondly, in order to generate the transient response of the cracked rotor system in high compliance with the actual experimental rotors, the crack model is simulated by using stress intensity factors and the energy release rate to calculate the harmonically varying stiffness combined with a breathing function to describe the breathing motion during the rotating of the cracked rotor with weight dominance. Consequently, we apply the improved numerical transition matrix approach method to examine the dynamic response of a rotating rotor with a transverse breathing crack. We exhibit the influence of the varied mass eccentricity on the stability of the breathing cracked rotor system in different rotation speed in a stability diagram in the parameter plane. The numerical response of the cracked rotor system is also analyzed by the frequency spectrum to present the vibration characters, as the rotation speed approach the critical rotation speed, to view the differences between the dynamics of cracked and uncracked shaft which may cause the change of stabilities. Finally, the detailed numerical eigenvalues of the transition matrix are applied to analyze the types of the bifurcations in the response of the cracked rotor system. Three types of bifurcations are found and responses of the cracked rotor system at these bifurcations are presented to for the visualized comparisons.

#### 2. Transition Matrix Approach Method Based on the Floquet Theory

##### 2.1. Stability Degree of Periodic System

Given a nonlinear nonautonomous dynamic system described by the equation,

, is the state vector, and is a periodic function of with the rotation period . Suppose that is the periodic solution of this nonlinear system with period . The linear approximate disturbing equation is obtained by putting a perturbation to where : where . Suppose that is the basic solution matrix of (2); by the Floquet theory [10, 11], we have where is the Floquet matrix. Then all solutions of (2) can be expressed as Then the perturbation to is let ; from (5), we have . Matrix has same eigenvalues with the Floquet matrix , which are called Floquet multipliers. By [9–11], we know the fact that the system is Lyapunov stable if the modulus of all multipliers are less than or equal to 1 (the algebraic multipliers of all the eigenvalues with modulus 1 equal to their geometric multipliers) and unstable if there is at least one of the Floquet multipliers that has modulus greater than one.

The stability degree of the periodic system (1) is defined as [11] where are Floquet multipliers, which are eigenvalues of the discrete state transition matrix . It is obvious that the stability degree acts as an index to point out the stability of the periodic motion. The system is stable as and unstable as . denote a critical state of the system. We can use a simplified numerical method to determine the characteristic multipliers in practical computation as follows.

##### 2.2. A Numerical Method of Calculation of State Transition Matrix of Periodic Solution

To express by will be more convenient in the measurement in engineering. Considering the periodic property of , we have the following equations: Obviously, for . Considering (5), we have And the eigenvalues of matrix are also the Floquet multipliers. The information of can be obtained by numerical integration of the motion equations of the rotor system or be measured in practical engineering. So it will be a convenient method to evaluate the Floquet multipliers based on transient response. The procedures in estimation of stability of a nonlinear dynamic system based on the transient response are shown as the following strategies in Figure 1.