Spectrum Cascade Research Articles

Physical consistency and invariance in machine learning of turbulent signals

This paper concerns an investigation of the invariance and consistency of deep learning of turbulent pressure fluctuations. The long-short-memory model is employed to predict wall pressure fluctuations across physical regimes featuring turbulence, shock–boundary layer interaction, and separation. The model's sensitivity to the data inputs is examined using different input data sets. Training the deep learning model based on the raw signals from different flow regions leads to large inaccuracies. It is shown that the data must be appropriately pre-processed before training for the deep learning model predictions to become consistent. Removing the mean and using the normalized fluctuating component of the signal, the deep learning predictions not only greatly improved in accuracy but, most importantly, converged and became consistent, provided that the signal sparsity remains within the inertial sub-range of the turbulence energy spectrum cascade. The power spectra of the surface pressure fluctuations reveal that the model provides high accuracy up to a certain frequency for the fully turbulent flow. The deep learning model's consistency is evidenced by being transferable across the various probe positions on the wall despite the significant differences in the turbulent flow properties in the training data set, i.e., signals obtained before, after, and inside the shock–boundary layer interaction regions. The model's prediction consistency and invariance to the turbulent signal training location(s) are promising for applying deep learning models to various turbulent flows.

Vibration characteristic investigation of an eccentric rotor system with rubbing

Assembly errors can lead to the eccentricity of rotors, further resulting in angular misalignment of rotors and inducing some secondary misalignments, such as the angular misalignment of gears and tilt misalignment of the inner ring of bearings (TMIRB). Ultimately, these misalignments can lead to rubbing faults. However, these misalignments have not raised significant concern, this article investigates their influence on the vibration responses of a dual-rotor-gear-bearing system with rubbing. First, a finite element model (FEM) of the system is built by using an in-house beam element code, and the nonlinear force model of the bearing is established based on the Hertz contact theory. Then, a new model to simulate angular misalignment of gears is proposed by changing the time-varying meshing stiffness (TVMS). Furthermore, the nonlinear dynamic characteristics of the system are investigated in detail under different misalignment angles and rotating speeds, and some interesting conclusions are summarized as follows. Specifically, analyses of the spectrum cascade of radial and axial responses reveal the presence of rich multi-time frequencies and combined frequencies. Moreover, as the misalignment angle increases, the system experiences partial rubbing, the rotating speed threshold (RST) corresponding to initial rubbing decreases, and the mesh stiffness of gears decreases. In summary, the proposed model and the results of the analysis offer valuable insights into the vibration characteristics of such systems and these findings can be instrumental in optimizing their design and operation.

Coaxial jets with disparate viscosity: mixing and laminarization characteristics

Mixing of fluids in a coaxial jet is studied under four distinct viscosity ratios, $m=1$ , $10$ , $20$ and $40$ , using highly resolved large-eddy simulations (LES), particle image velocimetry and planar laser-induced fluorescence. The accuracy of predictions is tested against data obtained by the simultaneous experimental measurements of velocity and concentration fields. For the highest and lowest viscosity ratios, standard RANS models with unclosed terms pertaining to viscosity variations are employed. We show that the standard Reynolds-averaged Navier–Stokes (RANS) approach with no explicit modelling for variable-viscosity terms is not applicable whereas dynamic LES models provide high-quality agreement with the measurements. To identify the underlying mixing physics and sources of discrepancy in RANS predictions, two distinct mixing modes are defined based on the viscosity ratio. Then, for each mode, the evolution of mixing structures, momentum budget analysis with emphasis on variable-viscosity terms, analysis of the turbulent activity and decay of turbulence are investigated using highly resolved LES data. The mixing dynamics is found to be quite distinct in each mixing mode. Variable viscosity manifests multiple effects that are working against each other. Viscosity gradients induce additional instabilities while increasing overall viscosity decreases the effective Reynolds number leading to laminarization of the turbulent jet, explaining the lack of dispersion and turbulent diffusion. Momentum budget analysis reveals that variable-viscosity terms are significant to be neglected. The scaling of the energy spectrum cascade suggests that in the TLL mode the unsteady laminar shedding is responsible for the eddies observed.

Nonlinear Responses of a Rotor-Bearing-Seal System with Pedestal Looseness

In this paper, a pedestal looseness fault model of a rotor-bearing-seal system is established. Under two working conditions of the same direction eccentricity (Working Condition 1) and reverse eccentricity (Working Condition 2), rotor orbits, vibration waveforms, spectrum cascade, and Poincaré maps are used to study the dynamic characteristics of the system when the sliding bearing support is loosened. The influence of speed, the unbalance of two discs, the looseness clearance, and the mass of bearing support on system characteristics are analyzed. The study found that Working Condition 2 is more likely to cause looseness of the bearing support. Moreover, for the rotor system in this paper, the pedestal looseness fault is more likely to occur near the second-order critical speed. Through analysis of the spectrum, it is found that the spectrum of the looseness fault will show multiple frequencies or continuous spectra, and the rotor orbits will appear “cylindrical.”

An experimental study on the nonlinear vibration phenomenon of a rotor system subjected to barrel roll flight and coupled rub-impact faults

This study focuses on the nonlinear vibration phenomenon of a rotor system subjected to aircraft barrel roll flight and coupled rub-impact faults. An experimental device is designed to imitate the rotor installed on an aircraft, which is rare because of the complex installation conditions of a rotor system. The transient nonlinear dynamic behavior of a rub-impact fault rotor and a coupled rub-impact fault rotor with misalignment are investigated for an aircraft in barrel roll flight. By combining a spectrum cascade, reassigned wavelet scalogram, and spectrogram, some novel nonlinear features of two types of rub-impact fault rotors are obtained, the most critical being the distinguishable transient harmonic components due to the time-varying influence of the aircraft barrel roll flight. This analysis contributes to a deep understanding of the complex nonlinear vibration phenomenon of coupled rub-impact fault rotors during flight maneuvers.

Numerical analysis of the dynamic behavior of a rotor-bearing-brush seal system with bristle interference

To minimize leakage and maintain the efficiency of turbomachinery, brush seal can be installed with zero clearance or interference as the flexibility. This leads to contact between the rotor and bristle pack, and may cause self-excited vibration, even instability. In this study, to establishment a mathematical model of the rotor system, a seal force model with bristle interference is proposed and the nonlinear oil-film force is applied in view of short bearing assumption. The influences of main factors containing rotor rotational speed, installing interval, disc mass, and disk eccentricity on the nonlinear characteristics of the rotor-bearing-brush seal system are conducted by adopting bifurcation diagram, spectrum cascade, axis orbit, and Poincare map. The results indicate that the vibration amplitude of the system with bristle interference is a bit larger than that without interference. The system stability is enhanced with smaller disk mass at a higher rotational speed. The proposed model could provide valuable reference for the design of a rotor-bearing-seal system.

Nonlinear Characteristics Analysis of a Rotor-Bearing-Brush Seal System

The vibration response and nonlinear dynamic behavior of a rotor-bearing-brush seal system were investigated with a new seal force model of the brush seal. The nonlinear oil–film force model was adopted based on a short bearing assumption. The dimensionless equation of motion was solved using the fourth order Runge–Kutta method. The effects of key parameters including rotor speed, installation spacing of the brush seal, disk eccentricity, disk mass, and journal mass on the nonlinear dynamic characteristics of rotor-bearing-brush seal system were determined and compared under different operating conditions with a bifurcation diagram, time history, axis orbit, poincaré map, frequency spectrum, and spectrum cascade. The results showed that the system response contained various nonlinear phenomena, such as periodic motion, multi-periodic motion, and quasi-periodic motion. The interaction of the rotor speed, installation spacing of the brush seal, disk eccentricity, disk mass, and journal mass could seriously affect the stability and working condition of the system. This study provides a theoretical support for the selection of key design parameters and further understanding of the nonlinear characteristics of rotor-bearing-seal systems with a brush seal.

Investigations on bending-torsional vibrations of rotor during rotor-stator rub using Lagrange multiplier method

The ever-increasing need of highly efficient rotating machinery causes reduction in the clearance between rotating and non-rotating parts and increase in the chances of interaction between these parts. The rotor-stator contact, known as rub, has always been recognized as one of the potential causes of rotor system malfunctions and a source of secondary failures. It is one of few causes that influence both lateral and torsional vibrations. In this paper, the rotor stator interaction phenomenon is investigated in the finite element framework using Lagrange multiplier based contact mechanics approach. The stator is modelled as a beam that can respond to axial penetration and lateral friction force during the contact with the rotor. It ensures dynamic stator contact boundary and more realistic contact conditions in contrast to most of the earlier approaches. The rotor bending-torsional mode coupling during contact is considered and the vibration response in bending and torsion are analysed. The effect of parameters such as clearance, friction coefficient and stator stiffness are studied at various operating speeds and it has been found that certain parameter values generate peculiar rub related features. Presence of sub-harmonics in the lateral vibration frequency spectra are prominently observed when the rotor operates near the integer multiple of its lateral critical speed. The spectrum cascade of torsional vibration shows the presence of bending critical speed along with the larger amplitudes of frequencies close to torsional natural frequency of the rotor. When m×1nX frequency component of rotational frequency comes closer to the torsional natural frequency, stronger torsional vibration amplitude is noticed in the spectrum cascade. The combined information from the stator vibration and rotor lateral-torsional vibration spectral features is proposed for robust rub identification.

Numerical and experimental analysis of the first-and second-mode instability in a rotor-bearing system

Aiming at the oil film instability of the sliding bearing at high speeds, a rotor test rig is built to study the non-linear dynamic behaviours caused by the first- and second-mode instability. A lumped mass model (LMM) of the rotor system considering the gyroscopic effect is established, in which the graphite self-lubricating bearing and the sliding bearing are simulated by a spring–damping model and a nonlinear oil film force model based on the assumption of short bearings, respectively. Moreover, a finite element model is also established to verify the validity of the LMM. The researches focus on the effects of two loading conditions (the first- and second-mode imbalance excitation) on the onset of instability and nonlinear responses of the rotor-bearing system by using the amplitude–frequency response, spectrum cascade, vibration waveform, orbit, and Poincare map. Finally, experiments are carried out on the test rig. Simulation and experiment all show that oil film instability can excite complicated combination frequency components about the rotating frequency and the first-/second-mode whirl/whip frequency.

Nonlinear lateral-torsional coupled motion of a rotor contacting a viscoelastically suspended stator

Interaction of a rotor with a stationary part is a kind of serious malfunction that could result in a catastrophic failure if remained undetected. Past analytical and numerical simulation work on rotor–stator interactions mainly focus on the vibrations along the lateral directions. The torsional degree of freedom (dof) is usually ignored. The present work is aimed to study the influence of a rotor to stator contact on the lateral-torsional coupled vibrations. A mathematical model consisting of interacting vibratory systems of rotor and stator is presented. The contact is modeled using contact stiffness, damping and Coulomb friction. Equations derived for kinetic, potential and dissipation energies and non-conservative external forces are used in the Langrange’s equations for deriving the motion equations for the rotor–stator system. Equations revealed that the lateral-torsional motion coupling exists twofold for the rotor. The unbalance couples lateral-torsional motion of rotor through inertia and damping matrices. Coupling due to the rotor–stator friction occurs through a force vector. The nonlinear equations are solved using a Runge–Kutta fourth-order numerical integration scheme using relatively small time step. Results obtained through the proposed model are compared with the identical rotor–stator system without torsional dof and differences are identified. Effect of several parameters such as speed, relative inertia, coefficient of friction and contact damping on the bifurcation behavior of the rotor–stator motion has been investigated. Vibration motions presented in the forms of spectrum cascade of the coast-up response, and orbit and Poincare plots of the steady-state response are exhibiting rich dynamic behavior of the system.

Order reduction and nonlinear behaviors of a continuous rotor system

An isotropic flexible shaft, acted by nonlinear fluid-induced forces generated from oil-lubricated journal bearings and hydrodynamic seal, is considered in this paper. Dimension reductions of the rotor system were carried out by both the standard Galerkin method and the nonlinear Galerkin method. Numerical simulations provide bifurcation diagrams, spectrum cascade, orbits of the disk center and Poincare maps, to demonstrate the dynamical behaviors of the system. The results reveal transitions, or bifurcations, of the rotor whirl from being synchronous to non-synchronous as the unstable speed is exceeded. The non-synchronous oil/seal whirl is a quasi-periodic motion. In the regime of quasi-periodic motion, the “windows” of multi-periodic motion were found. The investigation shows that the nonlinear Galerkin method has an advantage over the standard one with the same order of truncations, because the influences of higher modes are considered by the former.

Experimental study on the whip elimination of rotor-bearing systems with electromagnetic exciters

The elimination of whip using electromagnetic exciters (EEs) is investigated in this paper. First, mathematical models of a rotor-bearing system are derived. The threshold of instability determining if the rotor-bearing system experiences fluid-induced instability is obtained. Then, an experimental rotor-bearing system is designed and constructed to justify the elimination of whip, and orbit, spectrum, and spectrum cascade plots are obtained from the experimental data. The proposed procedure applies root locus plots to simulate the threshold of instability, and the spring model is used to compute the stiffness that the system needs to address the whip problem. Moreover, design examples are illustrated to explain the calculation process that tackles the proportional (simulating stiffness) and derivative (simulating damping) parameters of the EE to improve the stiffnesses of the rotor-bearing system. The main contribution of the present experimental work is that by using the EE it is possible to increase the stiffness of the rotor-bearing system and lead to the rotor-bearing system with a higher threshold of stability and eliminate the whip instability.

Time–frequency features of two types of coupled rub-impact faults in rotor systems

Two types of rub-impact faults in rotor systems are studied in this paper. First is crack coupled with rub-impact, and second is oil-film instability coupled with bearing rub-impact. By combining spectrum cascade, reassigned wavelet scalogram, rotor trajectory, and frequency spectrum, some novel fault features have been discovered from the vibration signals measured in both the speeding up and slowing down process. The results show that for crack coupled with rub-impact fault, the system motion is from period-three motion, to period-two motion, then to period-four motion in the slight rub-impact stage. However, in the stage of the serious rub-impact, chaotic motion occurs. For oil-film instability coupled with rub-impact fault, rotational frequency, oil-film instability frequency, and the combined frequencies of them can be observed when slight bearing rub-impact occurs. When bearing rub-impact is increasingly serious, the increase of low frequency amplitude can be viewed as the most distinguishable characteristic.

An Experimental Investigation of Condition Monitoring for Notched Rotors Through Transient Signals and Wavelet Transform

Experimental studies considering the dynamic transient signals of a notched rotor (i.e. a rotor with a transverse open crack) are investigated in order to examine the possibility of detecting the presence of open cracks in rotating machinery for low or high rotor accelerations. Firstly, the orbital patterns, spectrum cascade plots of the Power Spectral Density, evolutions of the 2× amplitudes of rotor with and without an open transverse crack are compared and discussed in the case of low accelerations of the rotor system. Secondly, the non-stationary vibration signals of the rotor with an open transverse crack at high accelerations are investigated. The timefrequency features and a tool based on the Continuous Wavelet Transform (CWT) to detect open cracks in a rotor system are discussed. It is demonstrated that both the Continuous Wavelet Transform (CWT) and changes on the 2× harmonic components are robust indicators for crack detection.

Dynamics of a Jeffcott rotor with slant crack

A simple Jeffcott rotor model of a rotor with a slant crack is presented. Flexibility matrix of the rotor with slant crack is developed and the stiffness coefficients based on the flexibility values are used in the equations of motion. Response-dependent nonlinear breathing crack model is used to evaluate unbalance response of the cracked rotor. Comparison between rotor with slant and transverse crack is made with regard to the flexibility/stiffness coefficients and the unbalance response characteristics. The flexibility matrix for slant crack case is more populated with additional cross-coupled coefficients compared to the transverse crack case. The rotor with slant crack is found to be stiffer in lateral and longitudinal directions, but more flexible in torsion, compared to the one with transverse crack. The response of the rotor with the two types of cracks (transverse and slant) is analysed at subharmonic resonances to explore possibilities of distinctive features in their response. Spectrum cascade plot for lateral and longitudinal vibrations exhibit distinctive response features for slant and transverse crack rotors.

Nonlinear excitation of convective cell by drift wave and spectrum cascade process in finite ion temperature plasma

A nonlinear drift mode equation for finite ion temperature plasma is derived. Possibilities of explosive instabilities for the drift wave are obtained. To discuss the modulational instability of drift wave, a nonlinear evolution equation for the drift mode is deduced from a third-order secularity elimination condition using the multiple time and space scales technique. The effect of low-frequency localized structure on particle transport has been investigated and the diffusion tensor is calculated using Ficks law.

Nonlinear properties of electron acoustic wave turbulence in the geomagnetic tail

The nonlinear properties of electron acoustic waves in a magnetized plasma consisting of hot electrons, hot ions, and cold electrons are investigated. Using a fluid-guiding center model for the cold electrons and Boltzmann distributions for the hot species, a set of nonlinear mode-coupling equations is derived. Monopole and dipole-vortex solutions are shown to exist for the system of nonlinear equations. Spectrum cascade by mode-coupling in the electron acoustic wave turbulence is investigated. Relevance of our investigation to broadband electrostatic noise (BEN) in the geomagnetic tail is discussed.

Spectrum cascade processes and nonlinear stability in interchange mode turbulence

An improved equation is derived for the interchange mode in the limit k‖ = 0. The possibilities are discussed of explosive instabilities and spectrum cascade processes by mode coupling in interchange mode turbulence in two limits.

Nonlinear excitation of convective cells by interchange modes and spectrum cascade processes

Convective cell formation due to interchange modes has been studied. The spectral cascade processes between such waves have been considered in the limit k∥=0 where the influence of gravity is particularly strong. Possibilities for up conversion are found.

Nonlinear behavior and turbulence spectra of drift waves and Rossby waves

Spectrum cascade in drift wave turbulence in a magnetized plasma as well as Rossby wave turbulence in an atmospheric pressure system are studied based on a three-wave decay process derivable from the model equation applicable to both cases. The decay in the three-way interaction occurs to smaller and larger values of ‖k‖. In a region of large wavenumbers this leads to the dual cascade; the energy spectrum cascades to smaller ‖k‖ and the enstrophy spectrum to larger ‖k‖, similar to the case of two-dimensional Navier–Stokes turbulence. In a small wavenumber region a resonant three-wave decay process dominates the cascade process, and an anisotropic spectrum develops. As a consequence of the cascade, zonal flows in the direction perpendicular to the direction of inhomogeneity appear which presents a potential implication for the particle confinement in a turbulent plasma.