Cracked Rotor System Research Articles

Experimental Investigation of Vibration Analysis of Multi-Crack Rotor Shaft

In recent years, the dynamic behaviour and diagnostic of cracked rotor have been gained momentum. In literature, several studies are available for cracked rotor systems, however very few authors have addressed the issue of multi-cracked rotor system. This paper deals with the nonlinear dynamic behaviour of multi cracked rotor system, which is analyzed experimentally and analytically with the considerations of the effects of the crack depth, crack location and the shaft's rotational speed. A new extension of Lagrangian method is used for analyzing the dynamic behaviour of a multi-cracked rotor system through Umbra Lagrangian formalism. The effects of crack depth on the shaft's stiffness and natural frequencies are analyzed experimentally. Natural frequencies have been obtained through vibration analyzer using impact hammer test under static conditions. This analysis also includes the dynamic response of rotor with breathing crack by using data acquisition system called OROS. It has been noticed that the stress concentration on the first crack has increased due to the presence of the second crack. Another interesting phenomenon is the influence of one crack over the other crack for mode shapes and for threshold speed limits. All such analysis has been carried out on experimental test rig consist of a symmetrical system, which has mild steel shaft between a pair of identical self-aligned double groove high speed bearings. The experimental results can be further validated with the analytical results.

Super-harmonic responses analysis for a cracked rotor system considering inertial excitation

In this paper, an investigation on the nonlinear vibration, especially on the super-harmonic resonances, in a cracked rotor system is carried out to provide a novel idea for the detection of crack faults in rotor systems. The motion equations of the system are formulated with the consideration of the additional excitation from an inertial environment as well as the forced excitation of the rotor unbalance. By using the harmonic balance method, the analytical solutions of the equations with four orders of harmonic exponents are obtained to analyze the nonlinear response of the system. Then through numerical calculations, the vibration responses affected by system parameters including the inertial excitation, the forced excitation, the crack and damping factors are investigated in detail. The results show that the occurrence of the super-harmonic resonances of the rotor system is due to the interaction between crack breathing and the inertial excitation. Correspondingly, the super-harmonic responses are significantly affected by the inertial excitation and the crack stiffness (or depth). The rotor unbalance, however, does not make apparent effects on the super-harmonic responses. Consequently, the super-harmonic resonances peaks can be viewed as an identification signal of the crack fault due to the application of the inertial excitation. By utilizing the inertial excitation, the super-harmonic response signals in rotor systems with early crack faults can be amplified and detected more easily.

Damping by parametric excitation in a set of reduced-order cracked rotor systems

A common tool utilized for the stability analysis of parametrically excited linear systems, such as rotors with cracked shafts, is Floquet׳s method. The disadvantage is a long calculation time needed to evaluate the monodromy matrix and instability zones. An efficient alternative is the generalized Bolotin׳s method, where the instability zones are evaluated quickly, yet the matrices that must be calculated are of large dimensions.In the present paper, the stability analysis is conducted with both Floquet׳s method and the generalized Bolotin׳s method. However, the order of the model is reduced to two modes only and stability analyses are performed for the second-order systems obtained with various combinations of the reducing modes. Then, the results of such analyses are collected in an overall stability map. The stability map obtained in this way closely reconstructs the stability map calculated with the full-order model of the rotor, yet the calculation time needed to generate the collected map as well as the dimension of the problem are considerably reduced.The approach is demonstrated with a mathematical model of the machine with the breathing crack modeled using the rigid finite element method. The rotor is not rotating, yet the stiffness of the shaft is varied periodically to simulate the parametric excitation.An interesting indication of the developing shaft crack observed in the generated stability maps is the presence of anti-resonant zones, where the rotor vibration amplitudes quickly decay. It is anticipated that this phenomenon of increased damping at specific excitation frequencies may have potential application for shaft crack detection.

Stability analysis and backward whirl investigation of cracked rotors with time-varying stiffness

The dynamic stability of dynamical systems with time-periodic stiffness is addressed here. Cracked rotor systems with time-periodic stiffness are well-known examples of such systems. Time-varying area moments of inertia at the cracked element cross-section of a cracked rotor have been used to formulate the time-periodic finite element stiffness matrix. The semi-infinite coefficient matrix obtained by applying the harmonic balance (HB) solution to the finite element (FE) equations of motion is employed here to study the dynamic stability of the system. Consequently, the sign of the determinant of a scaled version of a sub-matrix of this semi-infinite coefficient matrix at a finite number of harmonics in the HB solution is found to be sufficient for identifying the major unstable zones of the system in the parameter plane. Specifically, it is found that the negative determinant always corresponds to unstable zones in all of the systems considered. This approach is applied to a parametrically excited Mathieu’s equation, a two degree-of-freedom linear time-periodic dynamical system, a cracked Jeffcott rotor and a finite element model of the cracked rotor system. Compared to the corresponding results obtained by Floquet’s theory, the sign of the determinant of the scaled sub-matrix is found to be an efficient tool for identifying the major unstable zones of the linear time-periodic parametrically excited systems, especially large-scale FE systems. Moreover, it is found that the unstable zones for a FE cracked rotor with an open transverse crack model only appear at the backward whirl. The theoretical and experimental results have been found to agree well for verifying that the open crack model excites the backward whirl amplitudes at the critical backward whirling rotational speeds.

Bifurcation analysis for 2:1 and 3:1 super-harmonic resonances of an aircraft cracked rotor system due to maneuver load

This paper focuses on the local bifurcation characteristics of an aircraft cracked rotor system mainly for the 2:1 and 3:1 super-harmonic resonances induced by the maneuver load. The motion equations of the system are formulated with the consideration of the nonlinear stiffness of the Duffing type and the breathing of a transverse crack on the shaft, as well as the maneuver load induced by the climbing and diving flight of the aircraft. By using the multiple scales method, the motion equations are analytically solved to obtain the bifurcation equations for 2:1 and 3:1 super-harmonic resonances, respectively. Furthermore, the two-state variable singularity method is employed to analyze the local bifurcation characteristics of the system affected by crack coefficient and maneuver load. For each case, two curves of hysteresis set dividing \(K-G\) parameter plane into three regions are demonstrated. Accordingly, bifurcation modes for different parameter combinations from the three regions and the two curves are obtained. The approach in this paper will provide an effective and convenient way to analyze the bifurcation characteristics of dynamical systems. The results in this paper will contribute to a better understanding of the effect of the maneuver load on the response and bifurcation characteristics of aircraft cracked rotor systems.

Stability Analysis of a Breathing Cracked Rotor with Imposed Mass Eccentricity

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.

Influence of Parameters of Cracked Rotor System on its Vibration Characteristics in Viscous Medium at Finite Region

The paper summarizes a complete analysis about vibrational characteristics of a spinning simply supported cracked shaft with fluid medium at finite region. The damping effect occurs due to external fluid is integrated in the existing analysis, with the help of navier - stokes equation. The simply supported cracked shaft is analyzed by the influence coefficient strain energy method. Here we have changing the parameter of shaft i.e. damping viscosity of fluid and the length of the shaft which accountable for the alteration of the amplitude of vibration.

Constant-excitation caused response in a class of parametrically excited systems with two degrees of freedom

This paper focuses on the response in parametrically excited systems caused by constant excitation. Taking a maneuvering cracked rotor system as an example, we formulate the vibration equations with two degrees of freedom, in which the breathing of the crack constitutes parametric excitation, and the maneuver load of the maneuvering rotor is simplified as a constant excitation, and it is supposed that the rotor system is balanced without the consideration of eccentricity. By solving the equations with harmonic balance method, each order of harmonic components related with the rotating speed and the constant excitation is derived to analyze the corresponding resonance of the system. Results show that the constant excitation plays a decisive role in the parametrically excited primary and super-harmonic resonances of the system that agrees with the gravity dominance in common cracked rotor systems without maneuver load. And the stronger the constant excitation, the greater the resonances. Moreover, the orientation of the constant excitation makes a great impact on the parametrically excited primary resonance, but does not have a significant effect on the parametrically excited super-harmonic resonances. Results implies that constant excitation may increase the parametrically excited super-harmonic resonances of the cracked rotor systems, which is disadvantageous to the operating of the system. From another point of view, however, constant excitation can be used for early detection of crack faults in rotor systems.

Simulations and experimental investigation on motion stability of a flexible rotor-bearing system with a transverse crack

In the classical process for stability studies on the rotor-bearing system with crack faults, the simple discrete model is adopted for research on such problems, which neglect some needful dynamical influence factor, such as the material damping, shearing effect and gyroscopic effects, etc. Therefore, it is necessary to find a precise calculation model for simulation of the rotor-bearing system with cracks faults. In this paper, instead of the traditional simple discrete model, finite element (FE) model is adopted to investigate the motion stability of a nonlinear rotor system with crack fault. According to finite element theory, the FE model of the cracked rotor system is established firstly. It should be pointed out that the element where the crack occurs is modeled by a particular crack element and the supports at both ends are simulated by two nonlinear loads. Then, based on dimensionless and dimensionality reduction, the Newmark-β method and the shooting method are employed to study the effect of eccentricity and the depth of crack on instability speed and bifurcation feature. Furthermore, the simulation results are verified by some corresponding experiments. The simulation and experimental results show that instability speed does not change monotonically, but decreases firstly and then increases when the amount of eccentricity increases. Moreover, as the type of instability changes, the instability speed jumps concomitantly. Additionally, the presence of crack fault can disturb the oil whirl, as a result, instability speed tends to increase slightly, but it does not affect the type of instability and jumping phenomenon. This research presents an effective and convenient method which uses the finite element method (FEM) to research the motion stability of the nonlinear rotor-bearing system with cracked faults and other nonlinear force, and the proposed method can provide a theoretical reference for stability analysis and vibration control in more complex relevant rotor-bearing system.

Theoretical Analysis and Experimental Model Study on Stability of a Crack Rotor System

In this paper, the single disk rotor system with a transverse open and close crack has been taken as an example; square wave function has been adopted to express opening and closing characteristics of a transverse crack. The stability on the system has been discussed by theoretical analysis and experimental model study. The conclusions have shown that the unstable vibrations are found in the regions near the rotational speed at 2/3, 2/5and 2/7 of the critical speed in the large crack and the small damping case. The influence of some factors such as the crack depth and the damping on stability of the system is qualitatively discussed.

Crack location identification of rotating rotor systems using operating deflection shape data

Crack location identification, as one key destination of structural health monitoring, is still a challenge for operating rotor systems. The operating deflection shape (ODS), which represents a visual description of the structural vibration patterns under operating conditions, has been gaining importance for structure damage detection in recent years. The ODS carries damage information of a structure, however, it is also difficult to detect weak cracks of rotor directly. The approximate waveform capacity dimension (AWCD) method was successfully applied to damage detection of plates and beam-like structures. In this paper, a strategic approach that combines ODS and weighted AWCD is proposed for crack location identification of the rotating rotor. To eliminate the false peaks of AWCD and obtain desirable results, a weight factor and ODS curvature data are introduced to the expression of the weighted AWCD. The effectiveness of the proposed method is validated by numerical simulation and experimental investigation in a cracked rotor system. The results indicate that the proposed approach not only provides good identifying performance for incipient rotor cracks, but also effectively eliminates the fault peaks introduced by the inflexion locations of ODSs. Moreover, the proposed approach proves promising in detecting crack locations of rotating rotor systems.

Application of empirical mode decomposition to a Jeffcott rotor with a breathing crack

Fatigue damage, appearing due to developing cracks, is considered to be one of the main faults in rotating machinery. The damage in rotating components can be catastrophic, posing a potential hazard and leading to significant economic loss. In the related literature, transverse breathing cracks are considered as a primary mode of damage. Accordingly, a Jeffcott rotor with a transverse breathing crack has been examined here wherein a method for identifying the early crack propagation is proposed. The breathing functions developed in a recent publication to approximate the actual breathing mechanism of the cracked shaft are employed along with the method of empirical mode decomposition (EMD) to identify the crack vibration signature. EMD combined with the wave transform spectrum is used to decompose the measured vibration time series of the cracked rotor system into nearly monochromatic components. It is shown that the variations of the averaged amplitudes of the super-harmonic components in the neighborhood of 1/2 and 1/3 of the first critical rotational speed provide clear and robust vibration signatures indicating the early presence of the breathing crack. This signature is utilized here for early crack detection in the rotor system under consideration.

Stability Analysis on a Crack Rotor System in the Special Speeds

In this paper, the single disk rotor system with a transverse open and close crack has been taken as an example; the stability problem on the system in the special speeds has been discussed by theoretical analysis and experimental study. First, the conditions, positions and areas of the stable vibrations and the unstable vibrations on a rotor system with a transverse crack have been studied quantitatively theoretically. Not only the conclusions of other authors are verified，but also that the unstable vibrations are found in the regions near the rotational speed at 2/5and 2/7 of the critical speed in the large crack and the small damping case. Then the influence of some factors such as the crack depth parameters and the damping on stability of the system is qualitatively discussed.

Study the Stability of Bending-Torsional Coupled Vibration of a Rotor System with a Crack and Variable Perturbation Method

The differential equations of bending-torsional coupled vibration have been established with a switching crack for a Jeffcott rotor system with a transverse crack as the research object. Variable perturbation method has been used to study the stability of bending-torsional coupled vibration of a rotor system with a crack. The theoretical result shows that, when it is close to the natural frequency of 2 times, 3/2 times and 1 times, unstable vibrations exist and conditions of stable vibration. According to the above order, unstable region becomes narrow. While in 1/4 times, 1/6 times, the system is stable. The conclusions provide valuable theoretical basis for a crack rotor system in the real working conditions when corresponding measures need taking.

On the finite element modeling of the asymmetric cracked rotor

The advanced phase of the breathing crack in the heavy duty horizontal rotor system is expected to be dominated by the open crack state rather than the breathing state after a short period of operation. The reason for this scenario is the expected plastic deformation in crack location due to a large compression stress field appears during the continuous shaft rotation. Based on that, the finite element modeling of a cracked rotor system with a transverse open crack is addressed here. The cracked rotor with the open crack model behaves as an asymmetric shaft due to the presence of the transverse edge crack. Hence, the time-varying area moments of inertia of the cracked section are employed in formulating the periodic finite element stiffness matrix which yields a linear time-periodic system. The harmonic balance method (HB) is used for solving the finite element (FE) equations of motion for studying the dynamic behavior of the system. The behavior of the whirl orbits during the passage through the subcritical rotational speeds of the open crack model is compared to that for the breathing crack model. The presence of the open crack with the unbalance force was found only to excite the 1/2 and 1/3 of the backward critical whirling speed. The whirl orbits in the neighborhood of these subcritical speeds were found to have nearly similar behavior for both open and breathing crack models. While unlike the breathing crack model, the subcritical forward whirling speeds have not been observed for the open crack model in the response to the unbalance force. As a result, the behavior of the whirl orbits during the passage through the forward subcritical rotational speeds is found to be enough to distinguish the breathing crack from the open crack model. These whirl orbits with inner loops that appear in the neighborhood of the forward subcritical speeds are then a unique property for the breathing crack model.

Nonlinear Dynamic Analysis of a Cracked Rotor-Bearing System With Fractional Order Damping

Fatigue cracking of the rotor shaft is an important fault observed in the rotating machinery of key industries, which can lead to catastrophic failure. Nonlinear dynamics of a cracked rotor system with fractional order damping is investigated by using a response-dependent breathing crack model. The fourth-order Runge–Kutta method and tenth-order continued fraction expansion-Euler (CFE-Euler) method are introduced to simulate the proposed system equation of fractional order cracked rotors. The effects of the derivative order of damping, rotating speed ratio, crack depth, orientation angle of imbalance relative to the crack direction, and mass eccentricity on the system dynamics are demonstrated by using a bifurcation diagram, Poincaré map, and rotor trajectory diagram. The simulation results show that the rotor system displays chaotic, quasi-periodic, and periodic motions as the fractional order increases. It is also observed that the imbalance eccentricity level, crack depth, rotational speed, fractional damping, and crack angle all have considerable influence on the nonlinear behavior of the cracked rotor system. Finally, the experimental results verify the effectiveness of the theoretical analysis.

Study on the Non-Linear Dynamic Stability of a Cracked Rotor System

The non-linear dynamic responses with one degree are studied and solved by the multiple-scale method and Newton-Raphson method.The prime resonance vibrations are discussed.It is analyzed that the values of external excitation,the internal damping of the shaft and the depth of the crack have effect on the amplitude frequency response curves.

Dynamic modeling and simulation of transverse cracks in rotating shafts through dissipative coupling

The present paper deals with the dynamic modeling and simulation of a cracked rotor through dissipative coupling. The analysis is presented for a rotor with one transverse crack on the surface of the rotor. The paper initially elucidates a few basic concepts of modeling of cracked rotor systems. Later, the cracked rotor system is modeled with bond graphs as it facilitates the system modeling from the physical paradigms. The mathematical stiffness matrixes for cracked elements are computed and a detailed integrated bond graph model is created, which will portray the exact and accurate dynamics of a cracked rotor system. The simulation result presents interesting phenomena of limiting dynamics, which is also validated through experiments. The whole analysis is also validated with the simulation results.

Parametric instability of a rotor-bearing system with two breathing transverse cracks

Abstract When the rotor rotates at a constant speed, the transverse crack opens and closes alternatively, due to gravity, and thus a “breathing effect” occurs. This variance in shaft stiffness is time-periodic, and hence a parametrically excited system is expected. The parametric excitation from the time-varying stiffness causes instability and severe vibration under certain operating conditions. Current research mostly focused on the rotor with single transverse crack. There are few studies on the multi-cracked rotor system. In fact, the interaction between the multiple parametric excitations with various phasing and amplitude, which are induced by the multiple breathing transverse cracks, would make the instability behavior of the system differ distinctly from that of the single cracked rotor system. Moreover, how the instability regions change with various crack breathing mechanisms should also be investigated. Thus, the parametric instability of a rotor-bearing system with two breathing transverse cracks is studied in the paper. First, the finite element equations of motion are established for the cracked rotor system. Two types of crack breathing mechanisms, of which one is more accurate (new) and the other is empirical (old), are adopted in the finite element formulation. Then, a generalized Bolotin's method is introduced for determining the boundaries of the primary and secondary instability regions. Based upon these, instability analysis for a practical used rotor-bearing system with single and two cracks are conducted, respectively. The instability regions induced by the single transverse crack with new and old breathing mechanisms are compared with each other. For the two-cracked rotor system, the variations of the unstable boundaries with crack depths, orientation angles and positions are observed and discussed in detail. It is shown from the results that the dynamic instability of the two-cracked rotor-bearing system indeed have some unique features that differ from that of the single cracked rotor system.

Cohesive zone model for a transverse breathing crack in a rotor

AbstractThe breathing mechanism of a transversely cracked rotor and its influence on a rotor system that appears due to the shaft weight is studied. This breathing mechanism is based on experimental and simulation result for the crack shape reported in the literature. If the crack depth is small, the crack closure line is a straight line while for larger crack depths the crack closure becomes more curved. For both cases, a method is proposed for the evaluation of the stiffness losses in the cross section that contains the crack. This method is based on a cohesive zone model (CZM) instead of linear elastic fracture mechanics (LEFM) approach, because LEFM is valid only for the fully open crack and cannot be extended to other intermediate situations. As the crack is closed, the stress intensity factor (SIF) will not appear at the boundary between the closed cracked areas and the open cracked areas. The CZM is developed for mode‐I plane strain conditions and accounts explicitly for triaxiality of the stress state by using constitutive relations. The proposed model gives more realistic results than models based on LEFM for the stiffness losses of the crack rotor system for a wide range of the crack depth. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)