Dynamic response of cracked rotor-bearing system under time-dependent base movements

Dynamic response of cracked rotor-bearing system under time-dependent base movements is studied in this paper. Three base angular motions, including the rolling, pitching and yawing motions, are assumed to be sinusoidal perturbations superimposed upon constant terms. Both the open and breathing transverse cracks are considered in the analysis. The finite element model is established for the base excited rotor-bearing system with open or breathing cracks. Considering the time-varying base movements and transverse cracks, the second-order differential equations of the system will not only have time-periodic gyroscopic and stiffness coefficients, but also the multi-frequency external excitations. An improved harmonic balance method is introduced to obtain the steady-state response of the system under both base and unbalance excitations. The response spectra, orbits of shaft center and frequency response characteristics, are analyzed accordingly. The effects of various base angular motions, frequency and amplitude of base excitations, and crack depths on the system dynamic behaviors are considered in the discussions.     

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

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.     

Whirl and whip instabilities in rotor-bearing system considering a nonlinear force model

Linear models and synchronous response are generally adequate to describe and analyze rotors supported by hydrodynamic bearings. Hence, stiffness and damping coefficients can provide a good model for a wide range of situations. However, in some cases, this approach does not suffice to describe the dynamic behavior of the rotor-bearing system. Moreover, unstable motion occurs due to precessional orbits in the rotor-bearing system. This instability is called “oil whirl” or “oil whip”. The oil whirl phenomenon occurs when the journal bearings are lightly loaded and the shaft is whirling at a frequency close to one-half of rotor angular speed. When the angular speed of the rotor reaches approximately twice the natural frequency (first critical speed), the oil whip phenomenon occurs and remains even if the rotor angular speed increases. Its frequency and vibration mode correspond to the first critical speed. The main purpose of this paper is to validate a complete nonlinear solution to simulate the fluid-induced instability during run-up and run-down. A flexible rotor with a central disk under unbalanced excitation is modeled. A nonlinear hydrodynamic model is considered for short bearing and laminar flow. The effects of unbalance, journal-bearing parameters and rotor arrangement (vertical or horizontal) on the instability threshold are verified. The model simulations are compared with measurements at a real vertical power plant and a horizontal test rig.     

The vibration signature of chordal cracks in a rotor system including uncertainties

The aim of this paper is to investigate the effects of the presence of a transverse crack in a rotating shaft under uncertain physical parameters in order to obtain some indications that might be useful in detecting the presence of a crack in rotating system. The random dynamic response of the cracked rotor is evaluated by expanding the changing stiffness of the crack (i.e. the breathing mechanism) as a random truncated Fourier series. To avoid the use of the Monte Carlo simulations (MCS), an alternative procedure that is based on a combination of the Harmonic Balance Method and the Stochastic Finite Element Method (SFEM) using the Polynomial Chaos Expansion (PCE) is proposed. So the response of the Fourier components of the cracked rotor is expanded in the polynomial chaoses. The random dynamic response obtained by applying this procedure is compared with that evaluated through numerical integration based on the Harmonic Balance Method and the Monte Carlo simulations.     

The influence of crack breathing and imbalance orientation angle on the characteristics of the critical speed of a cracked rotor

By analyzing the limitations of weight dominance and by taking the complicated whirl of the rotor into account, general equations of motion have been developed in case of a Jeffcott rotor with a transverse crack. The angle between the crack direction and the shaft deformation direction is used to determine the closing and opening of the crack, allowing one to study the dynamic response without assuming weight dominance. Using the new equations, the dynamic response of a cracked rotor near its critical speed has been computed via a numerical method to investigate the influence of nonlinear breathing of the crack and that of the imbalance orientation angle β on the stability, critical speed and peak response of the rotor. The results show that nonlinear breathing can improve the stability of a rotor in contrast to a rotor with an open crack, and, with a reversed imbalance (70°<β<270°), that it can reduce the vibration response in contrast to an uncracked rotor. The basic characteristics of a cracked rotor near its critical speed are similar to those of an uncracked rotor. The critical speed can be determined by measuring the rotation of the center of gravity. The critical speed of a cracked rotor is located between the natural frequencies of the fully open crack and those of the fully closed crack and depends on the imbalance orientation angle. Its value is lowest at β≈90° and highest at β≈270°. The peak in the response at the critical speed is mainly determined by the imbalance orientation angle. At β≈0° and 180°, the peak corresponds to the maximum and minimum response, respectively.     

The dynamics of a cracked rotor with an active magnetic bearing

The dynamic characteristics of a cracked rotor with an active magnetic bearing (AMB) are theoretically analyzed in this paper. The effects of using optimal controller parameters on the dynamic characteristics of the cracked rotor and the effect of a crack on the stability of the active control system are discussed. It is shown that the dynamic characteristics of the cracked rotor with AMBs are clearly more complex than that of the traditional cracked rotor system. Adaptive control with AMBs may hide the fault characteristics of the cracked rotor, rather than helping to diagnose a crack; this will depend on the controller strategy used. It is very difficult to detect a crack in the rotor with an AMB support system when the vibration of the rotor system is fully controlled. Monitoring the super-harmonic components of 2× and 3× revolution in the sub-critical speed region can be used as an index to detect a crack in the rotor with an AMB system. If the effect of the crack is not taken into account at the design stage of the controller, then the rotor-AMB system will lose its stability in some cases when cracks appear.     

Rigid finite element model of a cracked rotor

The article introduces a new mathematical model for the cracked rotating shaft. The model is based on the rigid finite element (RFE) method, which has previously been successfully applied for the dynamic analysis of many complicated, mechanical structures. In this article, the RFE method is extended and adopted for the modeling of rotating machines. An original concept of crack modeling utilizing the RFE method is developed. The crack is presented as a set of spring–damping elements of variable stiffness connecting two sections of the shaft. An alternative approach for approximating the breathing mechanism of the crack is introduced. The approach is simple and allows one to intuitively and systematically prepare and analyze the model of a cracked rotor.The proposed method is illustrated with numerical and experimental results. The experiments conducted for the uncracked free–free rotor as well as the numerical results obtained with other software confirm the accuracy of the RFE model. The numerical analysis conducted for a set of cracked rotors has shown that, depending on the eccentricity and its angular location, the breathing behavior of the crack may take different forms. In spite of this, the frequency spectra for different cracks are almost identical.Due to its simplicity and numerous advantages, the proposed approach may be useful for rotor crack detection, especially if methods utilizing the mathematical model of the rotor are applied.     

Dynamic behavior analysis of cracked rotor

In the present study, the additional slope is used to consider the crack breathing, and is expressed explicitly in the equation of motion as one of the inputs to produce the bending moment at the crack position. Inversely, the additional slope is calculated by integrating on the crack region based on a fracture mechanics concept. The response of a cracked rotor is formulated based on the transfer matrix method. The transient behavior due to the crack breathing is considered by introducing a ‘moving’ Fourier-series expansion concept to the additional slope. The time-varying harmonic components of the additional slope are used to calculate the harmonic responses. The application considered is a general rotor model composed of multiple shafts, disks and cracks, and resilient bearings at both ends. Verification analysis is carried out for a simple rotor model similar to those found in the literature. Using the additional slope, the cracked rotor behavior is explained by the crack depth and rotation speed increase. It is shown that region on the crack front line having the dominant stress intensity factor value moves from the central area to both ends, as the crack depth increases. The result matches well with the crack propagation pattern shown in a bench mark test in the literature. Whirl orbits near the critical and sub-critical speed ranges of the rotor are discussed. It is shown that there exists some speed range near the critical speed, where the temporary whirl direction reversal and phase shift exist. When an unbalance is applied, the peculiar features, such as the whirl direction reversal and phase shift, disappear.     

Detecting cracked rotors using auxiliary harmonic excitation

Cracked rotors are not only important from a practical and economic viewpoint, they also exhibit interesting dynamics. This paper investigates the modelling and analysis of machines with breathing cracks, which open and close due to the self-weight of the rotor, producing a parametric excitation. After reviewing the modelling of cracked rotors, the paper analyses the use of auxiliary excitation of the shaft, often implemented using active magnetic bearings to detect cracks. Applying a sinusoidal excitation generates response frequencies that are combinations of the rotor spin speed and excitation frequency. Previously this system was analysed using multiple scales analysis; this paper suggests an alternative approach based on the harmonic balance method, and validates this approach using simulated and experimental results. Consideration is also given to some issues to enable this approach to become a robust condition monitoring technique for cracked shafts.     

Experimental investigations of the response of a cracked rotor to periodic axial excitation

In this paper, the coupling of lateral and longitudinal vibrations due to the presence of transverse surface crack in a rotor is explored. A crack in a rotor is known to introduce coupling between lateral and longitudinal vibrations. Steady state unbalance response of a cracked rotor with a single centrally situated crack subjected to periodic axial impulses is investigated experimentally. The cracked rotor is excited axially using an electrodynamic exciter at a frequency equal to its bending natural frequency in both non-rotating and rotating conditions. The resulting time domain and frequency domain signals of the cracked rotor are studied. Spectral response of the cracked rotor with and without axial excitation is found to be distinctively different. When excited axially, it shows prominent presence of rotor bending natural frequency. However for an uncracked rotor, the response is similar with or without axial excitation. It is thus proposed that the response of the rotor to axial impulse excitation could be used for more reliable diagnosis of rotor cracks.     

Structural damage assessment in a cantilever beam with a breathing crack using higher order frequency response functions

Vibration measurements offer an effective, inexpensive and fast means of non-destructive testing of structures and various engineering components. There are mainly two approaches to crack detection through vibration testing; open crack model with emphasis on changes in modal parameters and secondly, the breathing crack model focusing on nonlinear response characteristics. The open crack model based on linear response characteristics can identify the crack only at an advanced stage. Researchers have shown that a structure with a breathing crack behaves more like a nonlinear system, similar to that of a bilinear oscillator and the nonlinear response characteristics can very well be investigated to identify the presence of the crack. In the present study, the bilinear restoring force is approximated by a polynomial series and a nonlinear dynamic model of the cracked structure is developed using higher order frequency response functions. The effect of crack severity on the response harmonic amplitudes are investigated and a new procedure is suggested whereby the crack severity can be estimated through measurement of the first and second harmonic amplitudes.     

DYNAMICS OF A TWO-CRACK ROTOR

The effect of the presence of the single transverse crack on the response of the rotor has been a focus of attention for many researchers. In the present work a simple Jeffcott rotor with two transverse surface cracks has been studied. The stiffness of such a rotor is derived based on the concepts of fracture mechanics. Subsequently, the effect of the interaction of the two cracks on the breathing behavior and on the unbalance response of the rotor is studied. When the angular orientation of one crack relative to the other is varied, significant changes in the dynamic response of the rotor are noticed. A special case of practical importance of a two-crack rotor is one when one of the cracks is assumed to remain open always whereas the other can breathe like a fatigue crack. This simulates a transverse crack in an asymmetric rotor. Effect of orientation of the breathing crack with respect to the open crack on the dynamic response is studied in detail. The results of the present study will be useful in diagnosing fatigue cracks in real rotors, which invariably have some asymmetry.     

DYNAMIC ANALYSIS OF A SPIRAL BEVEL-GEARED ROTOR-BEARING SYSTEM

Spiral bevel gears can transmit motion between two rotors, which are commonly perpendicular to each other. In this paper, the dynamic analysis of a spiral bevel-geared rotor-bearing system is studied. Firstly, the constraint equation describing the relationship between the generalized displacements of spiral bevel gear pairs is derived briefly. Then the modelling of coupled axial-lateral-torsional vibration of the rotor system geared by spiral bevel gears is discussed. Finally, the mechanism of coupled vibration of the spiral bevel-geared rotor system is analyzed theoretically and the dynamic behavior of the system is investigated numerically. The conclusions are characterized as follows. The influences of the critical speeds in rigid journal supports, stability threshold speed and unbalanced responses in hydrodynamic journal bearings are not remarkable in comparison with the spur bevel-geared system under the same conditions. However, the critical speeds and stability threshold speed are essentially affected by boundary conditions such as the torsional stiffness, and meanwhile the effect of the unbalanced responses is not prominent under the concerned rotating speeds except that around the resonance peaks. The steady state response due to torsional excitation is also analyzed, and the results show that it cannot be neglected either in the torsional direction or in the lateral and axial directions in the spiral bevel-geared rotor system.     

Crack detection by measurement of mechanical impedance of a rotor-bearing system

Measurements were done to determine the influence of a transverse crack on the mobility of a rotor-bearing system. The results show that the mobility changes substantially due to the presence of crack and follows definite trends with the crack depth and direction of force application. The significant changes in mobility were observed at the natural frequency and at the running frequency of the rotor system. Good agreement between the experimental results and numerical simulations has been observed. The measurement of mobility has been suggested for crack detection and condition monitoring of rotor-bearing systems.     

ANALYSIS OF THE RESPONSE OF A CRACKED JEFFCOTT ROTOR TO AXIAL EXCITATION

The coupling of lateral and longitudinal vibrations due to the presence of transverse surface crack in a rotor is explored. Steady state unbalance response of a Jeffcott rotor with a single centrally situated crack subjected to periodic axial impulses is studied. Partial opening of crack is considered and the stress intensity factor at the crack tip is used to decide the extent of crack opening. A crack in a rotor is known to introduce coupling between lateral and longitudinal vibrations. Therefore, lateral vibration response of a cracked rotor to axial impulses is studied in detail. Spectral analysis of response to periodic multiple axial impulses shows the presence of rotor bending natural frequency as well as side bands around impulse excitation frequency and its harmonics due to modulations caused by rotor running frequency. It is concluded that the above approach can prove to be a useful tool in detecting cracks in rotors.     

Non-linear dynamics of flexible multi-bearing rotors

A general analysis has been developed to computer simulate steady state and transient vibration phenomena of complex rotor-bearing-support systems. A central feature of this analysis is a proper handling of various highly non-linear effects (most notably journal bearings) which dominate the dynamic phenoména encountered during large amplitude rotor-bearing vibrations. There are a number of potential causes of large amplitude rotor vibration, such as high rotor imbalance (e.g., loss of turbine blades at running speed), critical speed operation, journal bearing dynamic instability (oil whip), earthquakes, and shock. Failure mode analysis requires the evaluation and understanding of such potentially large dynamic forces and displacements. The paper presents development of the analysis, comparison with experiment and examples of its use in industrial applications.     

Vibration of cracked shafts in bending

Cracked rotating shafts exhibit a certain particular dynamic response due to the local flexibility of the cracked section. In this response, most of the features of the response of a shaft with dissimilar moments of inertia can be identified. Moreover, the non-linear behavior of the closing crack introduces the characteristics of non-linear systems. For many practical applications, the system can be considered bi-linear and analytical methods can be applied. A de Laval rotor with an open crack is investigated by way of application of the theory of shafts with dissimilar moments of inertia. Furthermore, analytical solutions are obtained for the closing crack under the assumption of large static deflections, a situation common in turbomachinery. Finally, a solution is developed for the case in which the local flexibility function is found experimentally.     

Dynamic stability of rotor-bearing systems subjected to random axial forces

This paper investigates the lateral vibration of a spinning disk-shaft system supported by a pair of ball bearings and subjected to a pair of random axial forces at both ends. The axial forces are assumed as the sum of a static force and a random process with a zero mean. Due to the random axial forces, the rotor-bearing system may experience parametric random instability under certain situations. In this work, the finite element method is applied to yield a set of discretized system equations first. The set of discretized system equations is partially uncoupled by the modal analysis procedure suitable for gyroscopic systems. The stochastic averaging method is then adopted to obtain Ito's equations for the response amplitudes of the system. Finally the first- and second-moment stability criteria are utilized to determine the stability boundaries of the system. Numerical results show that the rotor-bearing system is always stable in the sense of the first-moment stability, and the effects of the average axial compressive force and the disk mass, which will lower all frequencies of the system, tend to destabilize the second-moment stability of the system.     

Nonlinear parametric instability of wind turbine wings

Nonlinear rotor dynamic is characterized by parametric excitation of both linear and nonlinear terms caused by centrifugal and Coriolis forces when formulated in a moving frame of reference. Assuming harmonically varying support point motions from the tower, the nonlinear parametric instability of a wind turbine wing has been analysed based on a two-degrees-of-freedom model with one modal coordinate representing the vibrations in the blade direction and the other vibrations in edgewise direction. The functional basis for the eigenmode expansion has been taken as the linear undamped fixed-base eigenmodes. It turns out that the system becomes unstable at certain excitation amplitudes and frequencies. If the ratio between the support point motion and the rotational frequency of the rotor is rational, the response becomes periodic, and Floquet theory may be used to determine instability. In reality the indicated frequency ratio may be irrational in which case the response is shown to be quasi-periodic, rendering the Floquet theory useless. Moreover, as the excitation frequency exceeds the eigenfrequency in the edgewise direction, the response may become chaotic. For this reason stability of the system has in all cases been evaluated based on a Lyapunov exponent approach. Stability boundaries are determined as a function of the amplitude and frequency of the support point motion, the rotational speed, damping ratios and eigenfrequencies in the blade and edgewise directions.     

IDENTIFICATION OF SPEED-DEPENDENT BEARING PARAMETERS

Bearing dynamic characteristics have been a major unknown in the modelling and analysis of large turbo-generators. An identification algorithm for bearing dynamic characterization by using unbalance response measurements is developed for multi-degree-of-freedom (m.d.o.f.) flexible rotor-bearing systems. The algorithm identifies the bearing dynamic parameters, consisting of four effective stiffness and four damping coefficients for each bearing, utilizing frequency domain synchronous unbalance response measurements from the accelerometers attached to the bearing housings in the horizontal and vertical directions, for a minimum two different unbalance configurations. The procedure of identifying bearing dynamic coefficients by using the proposed algorithm is presented and demonstrated through a numerical example. Adding noise to the simulated signal checks the robustness of the algorithm against measurement noise. Combinations of regularization and the generalized singular value decomposition (SVD) are used to tackle an ill-posed problem due to the nearly circular orbit of the rotor at the bearings, as a special case for nearly isotropic bearings. It is demonstrated that by measuring noisy bearing responses with the direction of rotation of the rotor both in the clockwise and anticlockwise directions, the bearing estimation problem for circular orbit becomes well-conditioned. The regularization algorithm is tested for an experimental rotor-bearing rig. The response reproduction capabilities are excellent even in the presence of measurement noise.