General harmonic balance solution of a cracked rotor-bearing-disk system for harmonic and sub-harmonic analysis: Analytical and experimental approach

Abstract

The effect of crack depth of a rotor-bearing-disk system on vibration amplitudes and whirl orbit shapes is investigated through a general harmonic balance technique and experimental verification. Two models of the crack, which are the breathing and the open crack models, are considered. Finite element models and general harmonic balance solutions are derived for breathing and open cracks which are valid for damped and undamped rotor systems. It is found via waterfall plots of the system with a breathing crack that there are large vibration amplitudes at critical values of crack depth and rotor speed for a slight unbalance in the system. The high vibration amplitudes at the backward whirl appear at earlier crack depths than those of the forward whirl for both crack models. Resonance peaks at the second, third and fourth subcritical speeds emerge as the crack depth increases. It is shown that the unique signature of orbits for the breathing crack model which have been verified experimentally can be used as an indication of a breathing crack in the shaft. In addition, the veering in the critical frequencies has been noticed in the open crack case.

Introduction

Damage detection in rotor dynamic systems has had a great deal of attention in the past few decades. Destructive vibration amplitudes may appear in rotating shafts that are used in different industrial applications due to propagating cracks. These high amplitudes of vibration, which appear due to cracks, yield an unpredicted failure and a possible damage of machine components. A breathing crack in the transverse direction of the shaft is one of these dangerous damage scenarios in rotor-dynamic systems. The breathing mechanism of this type of crack in heavy-duty rotating machineries is mainly due to the gravity force that caused by the shaft weight and leads to crack propagation with time. This crack model has been used extensively in modeling damage in beams and shafts. Some studies have focused on the open crack model while several studies have focused on the switching and breathing crack models in rotating shafts.

The coupling of longitudinal and bending vibration in a cracked shaft with an open transverse crack model has been studied by deriving the local flexibility matrix of the cracked shaft [1]. As a result, the frequency equation was derived and solved for the natural frequencies of the system. It has been noticed that an instability region exists and that there are variations in the critical frequencies as the crack depth increases. The same issue has been studied again in [2] but with a stationary shaft that has two breathing cracks. The cracks were modeled with a compliance matrix where the breathing mechanism was found to depend on the excitation load direction. The analytical and experimental results have verified the effect of the coupling on both vertical and horizontal vibrations. The nonlinear dynamics of the flexible cracked Jeffcott rotor on a simple rigid support was studied in [3] with both switching crack and breathing crack models. Chaos and bifurcation were observed only in the case of a switching crack. However, the breathing crack model can represent the physical crack mechanism in the rotor better than the switching crack and often describes crack propagation in real life applications better than other models.

In [4], [5], [6] the characteristics of the sub- and super-harmonics of the cracked rotor with breathing cracks were used for crack detection in rotor systems. The harmonic balance method was employed in solving the cracked rotor system with a breathing crack model. It was found that with an increase in the crack size, new resonance peaks emerged at the second, third and fourth super-harmonic frequency components which can be used as an indication of crack propagation. The nonlinear behavior of the cracked rotor was also studied in [7] on a well known simple rotor. The results showed that the peaks appear at half and one third of the critical frequencies which are caused by the second and third super-harmonics. The flexibility matrices of the cracked element of the shaft were utilized for modeling the breathing crack [8], [9], [10]. In addition, the finite element method (FEM) was employed in solving the cracked rotor system. In [10] it was shown that the transverse breathing crack can be detected through the characteristics of the second and third harmonic components while the slant rotor crack can be detected by observing the sub- and super-harmonic components. It is also observed that the transverse breathing crack is highly sensitive to the mechanical impedance when compared with a slant crack.

Some researchers employed the transfer matrix method in the cracked rotor analysis. The global and local asymmetry transverse crack models have been employed to predict the rotor system response characteristics via the transfer matrix method where the second harmonic characteristics are used in monitoring the crack in the system [11]. In addition, the transfer matrix method was utilized to find the cracked rotor response of a simple rotor model where the temporary whirl reversal and phase shift were observed to occur near the critical and subcritical speeds since there is an unstable range at some neighborhood of these critical speeds [12]. An experimental analysis of a cracked rotor was performed in [13] where the effect of the crack depth and the additional eccentricity was verified experimentally via the orbits, time histories and waterfall plots of the shaft with an open crack. Most of the above techniques have considered discrete depths of the crack at different locations along the shaft. In [14] the sign change of the stress intensity functions (SIFs) was used in a breathing crack modeling for a FEM of a rotor-disk system with a fatigue transverse crack. The flexibility matrix was calculated and the FEM equations of motion were solved using the Newmark method of direct numerical integration. The effect of coupled torsional and lateral vibration has been investigated. In addition, wavelet transforms (WTs) was also employed for investigating the transient features of bending vibration at resonance. A theoretical cracked beam model is used for detecting cracks in power plant rotating machines in [15]. The study included theoretical, numerical and experimental analysis of the model. The bending vibration amplitudes in the neighborhood of the first subcritical speed have allowed the detection of a crack in which a good match was found between the numerical and experimental results. A review of 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 [16].

In this study the harmonic balance technique is employed for finding the critical and subcritical vibration speeds of a rotor-disk-bearing system for harmonic and sub-harmonic analysis. The solution is employed in studying the behavior of the shaft with either open or breathing cracks. The theoretical results are experimentally verified using Spectra-Quest rotor-dynamic simulator system with the same parameters used in the theoretical model. The development of methods to track more severe crack depths and their corresponding orbit shapes is addressed in this research in addition to the veering phenomenon in the critical frequencies of the cracked rotor where an exchange of modes takes place [17], [18], [19]. It is found that there are slight breathing crack depths at which high vibration amplitudes with a unique whirl orbit shapes appear. These amplitudes of vibration, which appear at these low crack depths, are larger at the backward whirling frequency than those at the forward whirling frequency. The unique whirl orbit shapes that appear at low breathing crack depths and are verified experimentally can be used as an early indication of the breathing crack propagation. Thus, the appearance of a breathing crack in a rotor system may explain a sudden and destructive damage in rotor-dynamic systems.