New breathing functions for the transverse breathing crack of the cracked rotor system: Approach for critical and subcritical harmonic analysis

https://doi.org/10.1016/j.jsv.2010.08.022Get rights and content

Abstract

The actual breathing mechanism of the transverse breathing crack in the cracked rotor system that appears due to the shaft weight is addressed here. As a result, the correct time-varying area moments of inertia for the cracked element cross-section during shaft rotation are also determined. Hence, two new breathing functions are identified to represent the actual breathing effect on the cracked element stiffness matrix. The new breathing functions are used in formulating the time-varying finite element stiffness matrix of the cracked element. The finite element equations of motion are then formulated for the cracked rotor system and solved via harmonic balance method for response, whirl orbits and the shift in the critical and subcritical speeds. The analytical results of this approach are compared with some previously published results obtained using approximate formulas for the breathing mechanism. The comparison shows that the previously used breathing function is a weak model for the breathing mechanism in the cracked rotor even for small crack depths. The new breathing functions give more accurate results for the dynamic behavior of the cracked rotor system for a wide range of the crack depths. The current approach is found to be efficient for crack detection since the critical and subcritical shaft speeds, the unique vibration signature in the neighborhood of the subcritical speeds and the sensitivity to the unbalance force direction all together can be utilized to detect the breathing crack before further damage occurs.

Introduction

Rotordynamic systems have had many applications for many decades. Gas turbines and compressors are examples of heavy rotating machines that are driven by rotating shafts which are intensively used in power generation field and aircrafts. In addition, most of the heavy industries have a basic use of rotating machines. The extensive use of these rotordynamic systems with continuous heavy loading may yield an unpredicted failure and damage that leads to a loss in life and equipments. These damages almost always occur due to propagating fatigue cracks that lead to sudden and destructive vibration scenarios. The breathing fatigue crack has a great deal of attention in literature as one of the main causes of these dangerous damages in rotor systems. The breathing mechanism of the crack that appears in rotating machinery is mainly due to the shaft weight. Several studies have focused on two models of fatigue cracks that are affected by the static deflection of the rotor, namely the switching and breathing crack models. Finding an efficient model of the breathing crack in rotor systems may help in identifying a unique vibration signature of the cracked rotor that assists in the early detection of the crack before damage occurs due to further crack propagation.

Different techniques have been used in the literature for modeling the transverse crack in rotating shafts. The flexibility matrix method has been utilized for modeling the stiffness of the cracked rotor with breathing crack [1], [2], [3], [4], [5], [6]. The coupling of the longitudinal and bending vibration in a cracked shaft was studied for the system with an open transverse crack in Ref. [1] and two transverse breathing cracks in Ref. [2] in which the breathing mechanism was found to depend on the direction of the excitation load. It was found that there are variations in the critical frequencies as the crack depth increases for the open crack case. The analytical and the experimental results have verified the effect of the coupling on both transverse vibration directions for the breathing crack case. The finite element method (FEM) was used in modeling the equations of motion of the cracked rotor in Refs. [3], [4], [5], [6] where the flexibility matrix was also used in modeling the stiffness matrix of the cracked element.

The finite element stiffness matrix of a rod in space found in Ref. [7] was used to represent the cracked element stiffness matrix in Refs. [8], [9], [10], [11], [12] where the time-varying element stiffness matrix of the cracked element was considered. The classical breathing function proposed in Ref. [13] was used to express the time change in the stiffness of the cracked element during rotation. This results in a time-varying element stiffness matrix due to the breathing mechanism of the crack. The finite element equations of motion were solved using the harmonic balance (HB) method. The shapes of the orbits in the neighborhood of subcritical speeds and the emerged resonance peaks at these speeds can be used for crack detection in rotor systems. In addition, the shift in the critical and subcritical speeds as a function of the crack size was verified in Ref. [12] via waterfall plots.

The behavior of the cracked rotor in the neighborhood of the subcritical speeds was also studied in Refs. [14], [15], [16], [17], [18], [19], [20], [21], [22]. The transfer matrix method was employed in studying the behavior of the cracked rotor system where the second harmonic characteristics are used in detecting the crack in the system [14]. In addition, the transfer matrix method was utilized to find the cracked rotor response of a simple rotor model in Ref. [15]. It was noticed that there is a temporary whirl reversal and phase shift near to the critical and subcritical speeds due to instability in the neighborhood of these speeds.

The nonlinear behavior of the cracked rotor was studied in Ref. [16] where new peaks of vibration have appeared at half and one third of the critical speeds. A theoretical cracked beam model was used for detecting cracks in power plant rotating machines [17]. The vibration amplitudes in the neighborhood of the first subcritical speed (1/2 first critical speed) were used in detecting the crack while a good match was found between the numerical and experimental results. The nonlinear dynamic behavior of the cracked Jeffcott rotor with switching and breathing crack models was also studied in Ref. [18]. Chaos and bifurcation were observed only in the case of a switching crack. An experimental analysis of a cracked rotor in the neighborhood of the subcritical speeds was performed in Ref. [19]. The effects of the crack depth and the additional eccentricity were verified experimentally via the shapes of the orbits, response and waterfall plots for the shaft with an open crack. The cracked rotor response during the passage through subcritical speeds was discussed in Refs. [20], [21] where the two loops orbit appears in the neighborhood of the 1/2 the critical speed. This behavior of the orbit before and after the critical speed can be utilized as an indication of a propagating crack in the rotor system.

A review of the strain energy release rate approach (SERR) for different modeling techniques of open, switching and breathing cracks and their corresponding methods of solution was introduced in Ref. [22]. Some of these modeling techniques have already been overviewed in this introduction.

Most of the above techniques have considered some assumptions in modeling the breathing crack. Hence, the breathing mechanism of the breathing crack was almost an approximation of the actual breathing of the crack. In this study, the actual breathing mechanism is presented and new breathing functions of the breathing crack are introduced. The correct time-varying stiffness matrix is formulated and incorporated to the global stiffness matrix in the finite element model of the cracked rotor with breathing crack. The harmonic balance method is employed for finding the response, orbits and critical and subcritical speeds of a cracked rotor system. The analytical results of this approach are compared with some published results in which other techniques or forms for breathing mechanism were used. It is found that some of the previous studies have used an approximate formula of the breathing function [8], [9], [10], [11], [12], [13], [23]. It is shown that the new breathing functions introduced in this study are considerably more accurate than the previously used functions in the literature. It is found that for small breathing crack depths, high vibration amplitudes with unique whirl orbits appear during the passage through the subcritical speeds. These amplitudes of vibration that appear for these small crack depths were barely observable when the old breathing function was used. The unique whirl orbits that appear for small breathing crack depths in the neighborhoods of subcritical speeds can be used as an early indication of breathing crack propagation.

Section snippets

Actual breathing mechanism of the breathing crack model

An approach for calculating the accurate breathing mechanism of the crack in a cracked rotor was introduced in Ref. [24]. In this approach a linear stress/strain distribution was assumed in the crack location to approximate the actual breathing of the crack found via three-dimensional nonlinear finite element calculations. An excellent agreement has been found between the simplified linear model and the nonlinear finite element model for finding the accurate breathing mechanism of the crack.

Comparison with previous model for breathing mechanism

The classical form of the breathing crack function proposed in Ref. [13] and used in Refs. [8], [9], [10], [11], [12], [23] to describe the breathing mechanism of the crack in a cracked rotor system is given byf(t)=12(1±cos(Ωt))The plus sign of the cosine term in this function is used when the crack is fully open and symmetric with the negative Y-axis at t=0 while the negative sign is used when the crack is fully closed and symmetric with the positive Y-axis at t=0. The sign change of the

Theoretical results and analysis

The same finite element model used in Ref. [12] is used here as shown in Fig. 10, Fig. 11. The undamped rotor-bearing-disk system is divided into 18 elements where the unbalance mass me is attached either to the right or left disk as shown in Fig. 11 at distance d from the shaft centerline. The values of the physical parameters are given in Table 3.

It is found that 4–6 harmonics are sufficient for the HB solution and give nearly the same shapes of orbits for a wide range of the rotor speeds in

Experimental results

The similarity between the theoretical whirl orbits in the neighborhood of the subcritical whirl speeds in this study has an excellent agreement with those experimental orbits in Ref. [20]. However, the Spectra-Quest MFS-RDS rotordynamic simulator, shown in Fig. 19, was used here for finding the experimental whirl orbits for node 2 of the finite element model of the rotor system in the neighborhood of 1/2 of the first pair of the critical speeds. The physical parameters of the MFS-RDS

Conclusions

An efficient model for the correct breathing mechanism of the transverse breathing crack in a cracked rotor system is introduced in this study. The correct time-varying stiffness matrix of the cracked rotor depends on the time-varying area moments of inertia of the cracked element. Hence, two new breathing functions are introduced here and used in formulating the correct time-varying stiffness matrix of the cracked element. These new functions are considerably more accurate than the previously

Acknowledgement

Financial support of NASA under Grant no. GR0002488 is gratefully acknowledged.

References (26)

Cited by (160)

  • Rotor crack breathing under unbalanced disturbance

    2024, Journal of Sound and Vibration
View all citing articles on Scopus
View full text