Vibrations of a textile machine rotor

In this paper the vibrations of a textile machine rotor, whose angular velocity is constant, are analyzed. The function of the rotor is to wind up a band of textile material into a roll. The elastic force in the shaft is assumed to be non-linear. First the free vibrations of this rotor are analyzed analytically and numerically. The results are compared. After that the vibrations in the non-resonant case are analyzed. The solution is found by use of the analytical method of multiple scales. The results for free vibrations and for the non-resonant case are compared.     

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.     

Order tracking signal processing for open rotor acoustics

Counter-rotating open rotor acoustic measurements were processed using a two-shaft Vold–Kalman order tracking filter, providing new insight into the complicated noise generation mechanisms of this type of system. The multi-shaft formulation of the Vold–Kalman filter can determine a time-accurate output of shaft order tones associated with each rotor, even as the rotation rate of the two rotors varies. This is a major improvement over the usual short time Fourier transform method for many applications. It was found that the contribution from each rotor to the individual tones varies strongly as a function of shaft order and operating condition. The order tracking filter is also demonstrated as a robust tool for separating the tonal and broadband components of a signal for which the usual shaft phase averaging methods fail.     

Order domain analysis of speed-dependent friction-induced torque in a brake experiment

A friction-induced forced vibration problem, as excited by the geometric distortions of the brake rotor, is studied in this article. The focus is on the order domain analysis, as the speed-dependent behavior of friction torque is not well understood. First, a new laboratory experiment is constructed to simulate vehicle brake judder in a scientific and yet controlled manner. The variations in pressure and torque are measured as the rotor slows down, and the order domain tracking is used to construct shaft torque vs. speed diagrams. A quasi-linear model of the laboratory experiment is then developed to obtain an analytical solution and to estimate the torque envelope function. A nonlinear model of the laboratory experiment (with a clearance) is also investigated to examine the resonant amplitude growth. Finally, predictions are successfully compared with measurements. Several contributions emerge over the prior literature. In particular, the experimental data clearly show that multiple-orders of the rotor surface distortion profile excite the friction-induced torque, and a clearance in the torsional system controls the resonant amplitude regime. New analytical and numerical solutions provide much insight into the speed-dependent resonant amplitude growth process.     

Stability and transient dynamics of a propeller–shaft system as induced by nonlinear friction acting on bearing–shaft contact interface

This paper investigates the friction-induced instability and the resulting self-excited vibration of a propeller–shaft system supported by water-lubricated rubber bearing. The system under consideration is modeled with an analytical approach by involving the nonlinear interaction among torsional vibrations of the continuous shaft, tangential vibrations of the rubber bearing and the nonlinear friction acting on the bearing–shaft contact interface. A degenerative two-degree-of-freedom analytical model is also reasonably developed to characterize system dynamics. The stability and vibrational characteristics are then determined by the complex eigenvalues analysis together with the quantitative analysis based on the method of multiple scales. A parametric study is conducted to clarify the roles of friction parameters and different vibration modes on instabilities; both the graphic and analytical expressions of instability boundaries are obtained. To capture the nature of self-excited vibrations and validate the stability analysis, the nonlinear formulations are numerically solved to calculate the transient dynamics in time and frequency domains. Analytical and numerical results reveal that the nonlinear coupling significantly affects the system responses and the bearing vibration plays a dominant role in the dynamic behavior of the present system.     

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.     

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.     

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.     

Influence of reflector temperature gradients on bit-error rate in periscope laser communication systems in the presence of detector noise and platform vibrations

We consider the influence of thermal deformations of elliptic reflectors in periscope laser communication terminals on the bit-error rate (BER) taking into account the satellite-platform vibrations and the detector noise. We study the relationship between the average BER and temperature gradients of elliptic reflectors in inter-satellite laser communication systems at different values of the light beam wavelength and truncation rate, and different vibration amplitudes. The average BER increases with increase in the vibration amplitude, and it is noteworthy that the average BER increases with increase in the beam truncation rate, which contradicts the result obtained in [1] without taking noise into account. We find that for some temperature gradients there is an optimum communication wavelength that provides the minimum average BER. We use the back-fixing method for fixing the elliptic reflectors, which has proved to be much less sensitive to temperature gradients compared to the traditional around-fixing method by a press board.     

Structural optimization for the avoidance of self-excited vibrations based on analytical models

Self-excited vibrations are a severe problem in many technical applications. In many cases they are caused by friction as for example in disk and drum brakes, clutches, saws and paper calenders. The goal to suppress self-excited vibrations can be reached by active and passive techniques, the latter ones being preferable due to the lower costs. Among design engineers it is known that breaking the symmetries of structures is sometimes helpful to avoid self-excited vibrations. This has been verified from an analytical point of view in a recent paper. The goal of the present paper is to use this analytical insight for a systematic structural optimization of rotors in frictional contact. The first system investigated is a simple discrete model of a rotor in frictional contact. As a continuous example a rotating beam in frictional contact is considered and optimized with respect to its bending stiffness. Finally a brake disk is optimized giving some attention to the feasibility of the modifications for the production process.     

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.     

Scale effects in the bending vibrations of helicopter rotor blades

Scale effects and dynamic similarity in the bending vibrations of helicopter rotor blades are examined by expressing the first three modes of bending vibration of a uniform, conventional rotor blade by a series of Legendre polynomials as suggested by Wilde and others. The natural frequency ratios for these three modes are then determined as functions of a dynamic similarity parameter over the entire range of designs and operating conditions from very flexible, rapidly rotating blades to highly rigid, slowly turning conditions.     

The global responses characteristics of a rotor/stator rubbing system with dry friction effects

Rotor/stator rubbing systems may undertake a number of quite different responses. Recent experiments on rotor/stator rubbing have revealed that two or three different responses may coexist. In this paper the global response characteristics of a general rotor/stator rubbing system, which takes into account the dominant factors in the process of rotor/stator rubbing, especially, the dry friction effect that is mostly neglected in the previous works and is the main factor for the self-excited dry friction backward whirl, are studied. The different solutions of the piecewise nonlinear system are derived and their stability are analyzed to get the existence boundaries of the different responses. An overall picture of the global response characteristics of this model is then obtained by drawing the existence boundaries in a same parameter space. The present results provide good understanding on the coexistence of different rubbing responses observed in tests. Moreover, deeper insight into the types of coexistence of different rubbing responses and their relationship with the system parameters is gained.     

Damping in micro-scale generalized thermoelastic circular plate resonators

The out-of-plane vibrations of a generalized thermoelastic circular plate are studied under different environmental temperature, plate dimensions and boundary conditions. The analytical expressions for thermoelastic damping of vibration and phase velocity of circumferential surface wave modes are obtained. It is noticed that the damping of vibrations and phase velocities of circumferential surface wave modes significantly depend on thermal relaxation time in addition to thermoelastic coupling in circular plates under resonance conditions. The surface conditions also impose significant effects on the vibrations of such resonators. The expressions for displacement and temperature fields in the plate resonator are also derived and obtained. Some numerical results have also been presented for illustration purpose in case of silicon material plate.     

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.     

Coupled horizontal and vertical bending vibrations of a stationary shaft with two cracks

This paper investigates the coupled bending vibrations of a stationary shaft with two cracks. It is known from the literature that, when a crack exists in a shaft, the bending, torsional, and longitudinal vibrations are coupled. This study focuses on the horizontal and vertical planes of a cracked shaft, whose bending vibrations are caused by a vertical excitation, in the clamped end of the model. When the crack orientations are not symmetrical to the vertical plane, a response in the horizontal plane is observed due to the presence of the cracks. The crack orientation is defined by the rotational angle of the crack, a parameter which affects the horizontal response. When more cracks appear in a shaft, then the coupling becomes stronger or weaker depending on the relative crack orientations. It is shown that a double peak appears in the vibration spectrum of a cracked or multi-cracked shaft.Modeling the crack in the traditional manner, as a spring, yields analytical results for the horizontal response as a function of the rotational angle and the depths of the two cracks. A 2×2 compliance matrix, containing two non-diagonal terms (those responsible for the coupling) serves to model the crack. Using the Euler–Bernoulli beam theory, the equations for the natural frequencies and the coupled response of the shaft are defined. The experimental coupled response and eigenfrequency measurements for the corresponding planes are presented. The double peak was also experimentally observed.     

Coupled longitudinal and bending vibrations of a rotating shaft with an open crack

The coupling of longitudinal and bending vibrations of a rotating shaft, due to an open transverse surface crack is investigated. The assumption of the open crack leads to a system with behaviour similar to that of a rotor with dissimilar moments of inertia along two perpendicular directions. The local flexibility due to the presence of the crack can be represented by way of a 6×6 matrix for six degrees of freedom in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling along the directions which are indicated by these terms. Here shear is not considered and three degrees of freedom are used: bending in the two main directions and extension. This leads to a 3×3 stiffness matrix with coupling terms. The undamped free and forced coupled vibration are first considered. The coupling is investigated and the effects of unbalance and gravity are examined. Then damped coupled vibration is considered for free and forced vibration. The existence of coupling between longitudinal and bending vibration due to the crack is a very useful property which, together with the sub-critical resonance due to crack, can form a basis for crack identification in rotating shafts. New and interesting phenomena of coupled transverse and longitudinal motion are presented and discussed.     

Effect of unbalanced rotor whirl on blade vibrations

In this paper, vibration of a bladed unbalanced flexible rotor is studied. The blade that is attached to the disk is considered as a fixed-free Euler-Bernoulli beam. Position of the blade with respect to the eccentric mass is taken into consideration. Coupled equations of the motion of unbalanced rotor and the blades are obtained through Lagrange equations. The dynamic equations have time variant periodic coefficients. Transient vibration analysis showed that rotor acceleration excites the blade vibration with its own natural frequency. While the rotor passes through its own natural frequency (critical speed) the blade vibration is again excited but this time with the rotor natural frequency. Modal behavior of the blades are different for subcritical, supercritical and for critical speed of the rotor. In the subcritical run of the rotor, blades located from 0° to 180° with respect to the eccentric mass are deflected in the negative direction while the rest are deflected in the positive direction. For supercritical run of the rotor, modal behavior of the blades is just the opposite. For critical speed of the rotor, blades located 90° to 270° from the eccentric mass are deflected in the positive direction while the rest of the blades are deflected in the negative direction. Blades have also different deflections. When the deflections of the blades are plotted with respect to their position angle, distribution of the blade deflections has a sinusoidal shape.     

Transforming and separating rotating disk vibrations using a sensor array

This paper presents a method to separate the complex vibration pattern of a rotating disk into simpler entities. The decomposition transforms data measured by an array of sensors into time domain signals representing the contribution of individual modes of vibration. Having performed the decomposition with respect to wavelength, speed and direction of travel, the obtained measurements can be projected onto a rotating body experiencing variable rotational speed relative to the sensors. Unlike previous works, the vibrations here are decomposed into time domain signals that provide better insight into stress levels and fatigue than frequency domain based decompositions. Furthermore, the proposed method works under non-stationary conditions, e.g. under rapid angular acceleration and during transient motions. By exploiting the spatial deployment of sensors, the proposed transformation can produce information about the deformations in the body-fixed or material coordinates which is essential for stress analysis. The main feature of the method is the ability to separate modes of vibration that normally overlap in the frequency domain, to enable better insight into the sources of vibration. The method is demonstrated by analytical, numerical and experimental examples.     

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.