Study of the non-linear dynamic response of a rotor system with faults and uncertainties

In this paper, the quantification of uncertainty effects on the variability of the nonlinear response in rotor systems with multi-faults (such as unbalance, asymmetric shaft, bow, parallel and angular misalignments) is investigated. To take account of uncertainties in this kind of nonlinear problem, it is proposed to use the Harmonic Balance Method (HBM) with a polynomial chaos expansion (PCE). The efficiency and robustness of the proposed methodology is demonstrated by comparison with Monte Carlo simulations (MCS) for different kinds and levels of uncertainties.     

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.     

Bifurcation and chaos response of a cracked rotor with random disturbance

The Monte-Carlo method is used to investigate the bifurcation and chaos characteristics of a cracked rotor with a white noise process as its random disturbance. Special attention is paid to the influence of the stiffness change ratio and the rotating speed ratio on the bifurcation and chaos response of the system. Numerical simulations show that the affect of the random disturbance is significant as the undisturbed response of the cracked rotor system is a quasi-periodic or chaos one, and such affect is smaller as the undisturbed response is a periodic one.     

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.     

Efficient exceedance probability computation for randomly uncertain nonlinear structures with periodic loading

A method is developed to compute low-level response amplitude exceedance probabilities associated with uncertain nonlinear structures with random parameters and deterministic periodic forcing. Emphasis is focused on accurate and efficient computation in the tails of the exceedance probability distribution function associated with the largest possible response of one displacement variable for unspecified forcing frequency and normally distributed parameters. This gives a measure of system reliability when a large amplitude response exceedance of a specified threshold is designated as the mode of failure. The method exploits the First-Order Reliability Method (FORM) in which the failure surface is constructed via the Harmonic Balance Method (HBM). This combined approach is tested on a Duffing oscillator with harmonic forcing and up to three uncertain parameters, for which the frequency of multiple-solution-maximum-amplitude is found directly, and the probability computed via the Hasofer-Lind reliability index. The accuracy of the proposed HBM-FORM, in the tails of the amplitude exceedance probability, is shown for the Duffing example to be acceptably accurate, whereas the efficiency is shown to be around 1000 times faster than Direct Integration and around 200 times faster than Monte Carlo simulation.     

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.     

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 analysis of a geared rotor system considering a slant crack on the shaft

The vibration problems associated with geared systems have been the focus of research in recent years. As the torque is mainly transmitted by the geared system, a slant crack is more likely to appear on the gear shaft. Due to the slant crack and its breathing mechanism, the dynamic behavior of cracked geared system would differ distinctly with that of uncracked system. Relatively less work is reported on slant crack in the geared rotor system during the past research. Thus, the dynamic analysis of a geared rotor-bearing system with a breathing slant crack is performed in the paper. The finite element model of a geared rotor with slant crack is presented. Based on fracture mechanics, the flexibility matrix for the slant crack is derived that accounts for the additional stress intensity factors. Three methods for whirling analysis, parametric instability analysis and steady-state response analysis are introduced. Then, by taking a widely used one-stage geared rotor-bearing system as an example, the whirling frequencies of the equivalent time-invariant system, two types of instability regions and steady-state response under the excitations of unbalance forces and tooth transmission errors, are computed numerically. The effects of crack depth, position and type (transverse or slant) on the system dynamic behaviors are considered in the discussion. The comparative study with slant cracked geared rotor is carried out to explore distinctive features in their modal, parametric instability and frequency response behaviors.     

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.     

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.     

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.     

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.     

THE SWING VIBRATION, TRANSVERSE OSCILLATION OF CRACKED ROTOR AND THE INTERMITTENCE CHAOS

This paper studies the non-linear dynamic response of a cracked rotor by taking the swing vibration of disc into consideration. The results show that if a small crack appears, the frequency of transverse oscillation is synchronous with rotating speed ratio (Ω), and the frequency of swing vibration is N Ω (N=1,2,…). As the crack increases, the response becomes chaotic in some range of Ω. The deeper the crack is, the wider the chaotic range of Ω is. Routes to chaos include intermittence to chaos and quasi-period to chaos. When the crack is fairly deep, some new resonance regions develop. In these regions, the response becomes infinity rapidly. The appearance of intermittence chaos is induced by the frequent frustration of stable oscillation, which is resulted from the continuous increase of swing amplitude. Unbalance parameter U is effective in suppressing chaos. Crack angle β cannot affect the essence of response, but can influence the amplitude of synchronous response.     

DYNAMIC STIFFNESS METHOD FOR CIRCULAR STOCHASTIC TIMOSHENKO BEAMS: RESPONSE VARIABILITY AND RELIABILITY ANALYSES

The problem of characterizing response variability and assessing reliability of vibrating skeletal structures made up of randomly inhomogeneous, curved/straight Timoshenko beams is considered. The excitation is taken to be random in nature. A frequency-domain stochastic finite element method is developed in terms of dynamic stiffness coefficients of the constituent stochastic beam elements. The displacement fields are discretized by using frequency- and damping-dependent shape functions. Questions related to discretizing the inherently non-Gaussian random fields that characterize beam elastic, mass and damping properties are considered. Analytical methods, combined analytical and simulation-based methods, direct Monte Carlo simulations and simulation procedures that employ importance sampling strategies are brought to bear on analyzing dynamic response variability and assessment of reliability. Satisfactory performance of approximate solution procedures outlined in the study is demonstrated using limited Monte Carlo simulations.     

The efficient computation of the nonlinear dynamic response of a foil–air bearing rotor system

The foil–air bearing (FAB) enables the emergence of oil-free turbomachinery. However, its potential to introduce undesirable nonlinear effects necessitates a reliable means for calculating the dynamic response. The computational burden has hitherto been alleviated by simplifications that compromised the true nature of the dynamic interaction between the rotor, air film and foil structure, introducing the potential for significant error. The overall novel contribution of this research is the development of efficient algorithms for the simultaneous solution of the state equations. The equations are extracted using two alternative transformations: (i) Finite Difference (FD); and (ii) a novel arbitrary-order Galerkin Reduction (GR) which does not use a grid, considerably reducing the number of state variables. A vectorized formulation facilitates the solution in two alternative ways: (i) in the time domain for arbitrary response via implicit integration using readily available routines; and (ii) in the frequency domain for the direct computation of self-excited periodic response via a novel Harmonic Balance (HB) method. GR and FD are cross-verified by time domain simulations which confirm that GR significantly reduces the computation time. Simulations also cross-verify the time and frequency domain solutions applied to the reference FD model and demonstrate the unique ability of HB to correctly accommodate structural damping.     

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.     

Solving quantum rotor model with different Monte Carlo techniques

We systematically test the performance of several Monte Carlo update schemes for the (2+1)d XY phase transition of quantum rotor model. By comparing the local Metropolis (LM), LM plus over-relaxation (OR), Wolff-cluster (WC), hybrid Monte Carlo (HM), hybrid Monte Carlo with Fourier acceleration (FA) schemes, it is clear that among the five different update schemes, at the quantum critical point, the WC and FA schemes acquire the smallest autocorrelation time and cost the least amount of CPU hours in achieving the same level of relative error, and FA enjoys a further advantage of easily implementable for more complicated interactions such as the long-range ones. These results bestow one with the necessary knowledge of extending the quantum rotor model, which plays the role of ferromagnetic/antiferromagnetic critical bosons or Z₂ topological order, to more realistic and yet challenging models such as Fermi surface Yukawa-coupled to quantum rotor models.     

Monte Carlo simulations on corneal and total aberrations before and after LASIK

Previous Monte Carlo simulations which manipulate each Zernike coefficient of total aberrations of human eyes indicate that interactions among wave-front aberrations can provide better visual quality for both pre-LASIK eyes and post-LASIK eyes. In this paper, we go a step further for Monte Carlo simulations which are not only on total aberrations but also on corneal aberrations, before and after LASIK, for a set of eyes. The corneal aberrations after LASIK are acquired through a new reliable method. Then a series of Monte Carlo simulations (including sign simulation, value simulation and meridional simulation) are performed by manipulating each Zernike coefficient (second through sixth-order) of total aberrations as well as corneal aberrations. The results are evaluated by modulation transfer function (MTF) ratio. Total aberrations for post-LASIK eyes still show MTF advantage over randomized aberrations, with slightly change as compared to that for pre-LASIK eyes. However, true corneal aberrations before and after LASIK have no MTF advantage over random aberrations. From this research, we draw conclusions: there is apparent advantage for the complete eye's true aberrations over random aberrations, whether pre-LASIK or post-LASIK, which does not exist for any biological optical surfaces in isolation, and the ability of adaptive mechanism of human eyes, increases after LASIK.     

A new treatment for predicting the self-excited vibrations of nonlinear systems with frictional interfaces: The Constrained Harmonic Balance Method,with application to disc brake squeal

Brake squeal noise is still an issue since it generates high warranty costs for the automotive industry and irritation for customers. Key parameters must be known in order to reduce it. Stability analysis is a common method of studying nonlinear phenomena and has been widely used by the scientific and the engineering communities for solving disc brake squeal problems. This type of analysis provides areas of stability versus instability for driven parameters, thereby making it possible to define design criteria. Nevertheless, this technique does not permit obtaining the vibrating state of the brake system and nonlinear methods have to be employed. Temporal integration is a well-known method for computing the dynamic solution but as it is time consuming, nonlinear methods such as the Harmonic Balance Method (HBM) are preferred. This paper presents a novel nonlinear method called the Constrained Harmonic Balance Method (CHBM) that works for nonlinear systems subject to flutter instability. An additional constraint-based condition is proposed that omits the static equilibrium point (i.e. the trivial static solution of the nonlinear problem that would be obtained by applying the classical HBM) and therefore focuses on predicting both the Fourier coefficients and the fundamental frequency of the stationary nonlinear system.The effectiveness of the proposed nonlinear approach is illustrated by an analysis of disc brake squeal. The brake system under consideration is a reduced finite element model of a pad and a disc. Both stability and nonlinear analyses are performed and the results are compared with a classical variable order solver integration algorithm.Therefore, the objectives of the following paper are to present not only an extension of the HBM (CHBM) but also to demonstrate an application to the specific problem of disc brake squeal with extensively parametric studies that investigate the effects of the friction coefficient, piston pressure, nonlinear stiffness and structural damping.     

Absence of an Almeida-Thouless line in three-dimensional spin glasses

We present results of Monte Carlo simulations of the three-dimensional Edwards-Anderson Ising spin glass in the presence of a (random) field. A finite-size scaling analysis of the correlation length shows no indication of a transition, in contrast with the zero-field case. This suggests that there is no Almeida-Thouless line for short-range Ising spin glasses.