An analytical study of rotor dynamics coupled with thermal effect for a continuous rotor shaft

This paper presents an analytical analysis of a continuous rotor shaft subjected to universal temperature gradients. To this end, an analytical model is derived to investigate the generic thermal vibrations of rotor structures. The analytical solutions are obtained in a rotating frame and include parameters related with both the thermal environment and the rotor dynamic structures. This provides an insight into the mechanisms for the rotor thermal vibration. Furthermore, numerical results based on the analytical solutions are given. An index denoting the temperature gradients is proposed for the occasions with nonlinear cross-sectional temperature distributions. Finally, the factors influencing the thermal vibrations are analyzed. The results show that the thermal vibration is affected by many factors including the shaft size, rotational speeds, heating locations, critical speed, etc. Moreover, it is investigated how the convection coefficient and the heat conductivity influence the thermal vibrations in order to provide an insight into the management of thermal vibrations from the perspective of thermal aspects.     

Robust estimation of excitations in mechanical systems using M-estimators—Experimental applications

This second part of the study presents some experimental applications to mechanical systems in which the results of excitation estimation, obtained using traditional least squares and M-estimate, are compared.The first case presented is a single input–multiple outputs system: a simple test-rig for the study of the vibrations of a two-degrees of freedom system is employed to identify the constraint displacement that causes the measured mass vibrations in presence of heavy noise.The second case is a multiple inputs–multiple outputs system: a rotor test-rig is used to identify the positions, the amplitudes and the phases of two unbalances using the vibrations measured in the bearings. In this case, also an additional theoretical part is introduced about the basics of model-based identification in the frequency domain applied to rotor dynamics.The last case is again a single input–multiple outputs system, but in an industrial application: experimental vibrations of a 320 MW steam turbo-generator are used to identify position and amount of a known balancing mass in an on-field real case.Moreover, whilst in the numerical examples presented in the first part the knowledge of the system was perfect, in these cases some uncertainties are present also in the system model.Finally, the paper introduces the use of the M-estimate technique to evaluate the adequacy the model of the system, by means of the analysis of the weights attributed to the measures as a function of the frequency of the excitation.     

IDENTIFICATION OF MULTIPLE FAULTS IN ROTOR SYSTEMS

Many papers are available in the literature about identification of faults in rotor systems. However, they generally deal only with a single fault, usually an unbalance. Instead, in real machines, the case of multiple faults is quite common: the simultaneous presence of a bow (due to several different causes) and an unbalance or a coupling misalignment occurs often in rotor systems. In this paper, a model-based identification method for multiple faults is presented. The method requires the definition of the models of the elements that compose the system, i.e., the rotor, the bearings and the foundation, as well as the models of the faults, which can be represented by harmonic components of equivalent force or moment systems. The identification of multiple faults is made by a least-squares fitting approach in the frequency domain, by means of the minimization of a multi-dimensional residual between the vibrations in some measuring planes on the machine and the calculated vibrations due to the acting faults. Some numerical applications are reported for two simultaneous faults and some experimental results obtained on a test-rig are used to validate the identification procedure. The accuracy and limits of the proposed procedure have been evaluated.     

Identification of unbalance in a rotor bearing system

Model based methods for fault identification in rotating systems are gaining importance for the last three decades due to their ability to identify both location and severity of the fault. Model based methods are of different types. Among them, equivalent loads minimization method is one method. In this method, fault is identified in a rotor bearing system by minimizing difference between equivalent loads estimated in the system due to the fault and theoretical fault model loads. This method has a limitation that the error in identified fault parameters increases with decrease in number of measured vibrations. Thus a comprehensive methodology for fault identification with minimum error even in case of fewer measured vibrations is attempted in the present work. Two different approaches: equivalent loads minimization and vibration minimization method are applied for the identification of unbalance fault in a rotor system. Unbalance fault is identified using proposed methods by measuring transverse vibrations at only one location.     

Transverse vibration of a rotor system driven by a Cardan joint

The transverse vibration of a rotor system driven by a Cardan joint is analyzed and the effect of the transmitted torque on the dynamic stability of the system evaluated. As a result of the analysis, the following facts are proved: when the driving shaft and driven shaft (rotor shaft) are included, both parametric and self-excited vibrations arise due to transmitted torque; asymmetrical stiffness of the rotor supports has the effect of stabilizing this self-excited vibration.     

QUASI-PERIODIC VIBRATION OF CRACKED ROTOR ON FLEXIBLE BEARINGS

A cracked rotor on flexible bearings is studied in this paper. The vibration of such a system has many complexities because of the crack and bearing flexibility. However, if the properties of the bearings are known, the system can be simplified by supposing that, the vibration due to weight is dominant. Equations of motion are derived, and a linear system in which the crack has been considered as an external disturbance described by a series of trigonometric functions is obtained. Consequently, the quasi-periodic vibrations of the rotor and bearings are established by harmonic balance method and approximate values of the vibration determined by truncating the higher order terms. It is believed that the simulated results will be useful for crack detection in the case of weight-dominant rotors.     

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.     

非共振条件下高斯光束斜入射法布里-珀罗干涉仪的透射特性

基于多光束干涉原理,研究了当高斯光束的载波频率与法布里—珀罗(F P)干涉仪的谐振频率存在偏差(即非共振)时,高斯光束斜入射F P干涉仪的透射特性。结果表明,与共振情况相比,非共振条件下透射光束的峰值强度及其所对应的位置、以及光斑的大小随入射角的变化都发生了显著的改变。     

Intra and inter-ionic multiphonon transitions in rare earth doped crystals

A theory has been developed for non-radiative multiphonon transitions in ionic crystals doped by RE-impurities. The degeneration of final electronic states has been taken into account in the framework of the first cumulant approximation. An analytical expression has been obtained for the rate of non-resonant energy transfer caused by electrostatic inter-ionic interaction and numerical estimations for Yb → Tm and Yb → Ho transitions in YAlO₃ and YLiF₄ crystals are listed. The intra-center multiphonon non-radiative relaxation has been considered in the case where acoustical modes of lattice vibrations are the promoting ones.     

Dynamic behaviours of a full floating ring bearing supported turbocharger rotor with engine excitation

The rotor dynamic behaviour of turbochargers (TC) has been paid significant attention because of its importance in their healthy operation. Commonly, the TC is firmly mounted on engines and they will definitely suffer from the vibrations originated from engines in operation. However, only a limited number of papers have been published with consideration of this phenomenon. In this paper, a finite element model of a TC rotor supported by nonlinear floating ring bearings has been established. The nonlinear bearing forces have been calculated by a newly proposed analytical method. An efficient numerical integration approach has been employed to conduct the investigation including the traditional unbalance and the considered engine excitation effects in question. The results show that the unbalance will place considerable influence on the rotor response at a low working speed. At high speeds, the effect will be prevented by the dominant sub-synchronous vibrations, which also prohibit the appearance of a chaotic state. The novel investigation with the proposed model considering engine excitation reveals that the engine induced vibration will greatly affect the TC rotor response at relatively lower rotor speeds as well. At higher speed range, the dominant effect of sub-synchronous vibrations is still capable of keeping the same orbit shapes as that without engine excitation from a relative viewpoint.     

On the bifurcations of non-resonant solutions in autonomous systems

The behaviour of a non-linear system represented by a set of autonomous ordinary differential equations containing two real parameters is considered. At a certain critical value of these parameters it is assumed that the linearized system has two pairs of imaginary eigenvalues which are not even nearly commensurable. This non-resonant case is analyzed via a perturbation technique, and asymptotic formulate type results are obtained.     

RESPONSE OF A PLATE TO DIFFUSE ACOUSTIC FIELD USING STATISTICAL ENERGY ANALYSIS

Inclusion of non-resonant component is expected to improve the estimation of response of a structure to acoustic excitation. In this paper this is verified experimentally. A typical plate is subjected to acoustic excitation in a reverberant chamber and the acceleration responses are measured. The experimental results match well with the theoretical estimates that are made incorporating the non-resonant component. The results show that if the non-resonant part is not considered, the estimated response is in large error. This is seen in the spatial average response as well as in the response levels for a confidence coefficient of 99%.     

An experimental investigation of dual-resonant and non-resonant responses for vortex-induced vibration of a long slender cylinder

Experimental results of the dual-resonant and non-resonant responses are presented for vortex-induced vibrations(VIV)of a long slender cylinder.The cylinder has a diameter of 10mm and a length of 3.31 m,giving an aspect ratio of 331.The cylinder was towed by a carriage with the velocity up to 1.5 m/s,with the Reynolds number varying from 2500 to 38000.Three different weights were used to provide the initial tension.Dual resonance means that resonance occurs simultaneously in both the cross-flow(CF)and in-line(IL)directions.The experiments were conducted in two stages.At the first stage,dual-resonant dynamic features of the cylinder subjected to vortex-induced excitation were investigated.The features of CF and IL vibration amplitude,motion orbits,phase angle differences,dominant frequencies and mode order numbers are presented.At the second stage of the experiments,particular emphasis was placed on non-resonant dynamic features.The variation of multi-mode modal displacement amplitudes was investigated in detail.     

A built-in force actuator for active control of lateral rotor vibration in cage induction electrical machines

In this paper means of active control of radial rotor vibrations in electrical machines are considered. We examine a built-in force actuator for active generation of force on the machine rotor. The operation of the actuator is based on electromechanical interaction between the rotor and the stator of the machine. The actuator is given a low-order linear state-space model, which is identified by using simulation data obtained from a detailed time-stepping finite element model of the machine. Simulation results obtained by using real machine data and finite element time-stepping method are presented.     

Contact-impact analysis of geared rotor systems

In this paper, a finite element formulation, used to analyze the contact-impact behavior of geared rotor systems coupled with the rotational, lateral, and axial vibrations between gears at high rotational speeds, has been developed. A gear impact element to model the contact-impact behavior between gears has been developed and its numerical method is discussed. A relative displacement measurement idea has been proposed to measure vibration parameter for contrast experiment in high rotational geared system. The equations of motion are derived and solved iteratively during each time increment until the unbalanced force decrease to an acceptable tolerance level. Based on the proposed method, an analysis program, GEARS, has been developed. The contact-impact behavior of geared rotor systems is analyzed especially under high rotational speed condition as numerical examples, which are demonstrated to show the effectiveness of the proposed method.     

Torsional vibrations of non-uniform rotating blades with attachment flexibility

The torsional vibrations of non-uniform pretwisted rotating blades are studied by using finite element methods based on both the Rayleigh-Ritz and Galerkin formulations. The apparent differences between the matrices obtained from these formulations are explained and, as obtained by using three different orders of elements, results are presented for blades with flexibly attached roots and for a non-uniform blade representative of a bearingless rotor. A parametric study is carried out to resolve a controversy regarding the relative importance of certain terms in the equations of motion of pretwisted rotating blades. In Appendix I, an exact solution is presented for the torsional vibrations of flexibly attached rotating blades with piecewise constant inertia and elastic properties, which serves as a benchmark solution for the finite element results.     

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.     

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.     

Theory of the generation of mechanical vibrations by laser radiation in solids containing internal stresses on the basis of the thermoelastic effect

The behavior of the nonsteady deformations in solids containing internal stresses under irradiation by temporally modulated laser radiation is analyzed. In the framework of the nonlinear theory of thermoelasticity a model is proposed for the excitation of mechanical vibrations with allowance for the dependence of the thermoelastic coupling parameter on the initial deformation. For the case of a piezoelectric method of detecting the mechanical vibrations in a uniformly deformed sample, an analytical expression is obtained for the electrical signal taken off from the piezoelement. The behavior of the piezoelectric signal under various conditions is investigated, and the results are compared with the available experimental data and found to be in qualitative agreement. Zh. Tekh. Fiz. 69, 59–63 (July 1999)     

Vibration control by recursive time-delayed acceleration feedback

This paper presents a theoretical basis of time-delayed acceleration feedback control of linear and nonlinear vibrations of mechanical oscillators. The control signal is synthesized by an infinite, weighted sum of the acceleration of the vibrating system measured at equal time intervals in the past. The proposed method is shown to have controlled linear resonant vibrations, low-frequency non-resonant vibrations, primary and 1/3 subharmonic resonances of a forced Duffing oscillator. The concept of an equivalent damping and natural frequency of the system is also introduced. It is shown that a large amount of damping can be produced by appropriately selecting the control parameters. For some combinations of the control parameters, the effective damping factor of the system is shown to be inversely related to the time-delay in the small delay limit. Selection of the optimum control parameters for controlling the forced and free vibrations is discussed. It is shown that forced vibration is best controlled by unity recursive gain and smaller values of the time-delay parameter. However, the transient response can be optimally controlled by suitably selecting the time delay depending upon the gain. The delay values for the optimal forced response may be different from that required for the optimum transient response. When both are important, a suboptimal choice of the delay parameters with unity recursive gain is recommended.