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.     

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.     

Vibration response of misaligned rotors

Misalignment is one of the common faults observed in rotors. Effect of misalignment on vibration response of coupled rotors is investigated in the present study. The coupled rotor system is modelled using Timoshenko beam elements with all six dof. An experimental approach is proposed for the first time for determination of magnitude and harmonic nature of the misalignment excitation. Misalignment effect at coupling location of rotor FE model is simulated using nodal force vector. The force vector is found using misalignment coupling stiffness matrix, derived from experimental data and applied misalignment between the two rotors. Steady-state vibration response is studied for sub-critical speeds. Effect of the types of misalignment (parallel and angular) on the vibration behaviour of the coupled rotor is examined. Along with lateral vibrations, axial and torsional vibrations are also investigated and nature of the vibration response is also examined. It has been found that the misalignment couples vibrations in bending, longitudinal and torsional modes. Some diagnostic features in the fast Fourier transform (FFT) of torsional and longitudinal response related to parallel and angular misalignment have been revealed. Full spectra and orbit plots are effectively used to reveal the unique nature of misalignment fault leading to reliable misalignment diagnostic information, not clearly brought out by earlier studies.     

Coupled transverse and axial vibratory behaviour of cracked beam with end mass and rotary inertia

In this paper the vibrational behaviour of a cracked cantilever beam carrying end mass and rotary inertia is investigated. The transverse and axial vibrations of the beam are coupled through the crack model. The values of the ratio between the cracked and uncracked beam natural frequencies, the frequency ratio, are examined and are shown to follow well-defined trends with respect to the crack parameters and end mass and rotary inertia. However, the coupling between the transverse and axial vibrations is shown to be weak for the first two modes for moderate values of crack depth ratio. High crack depth ratios appear to increase the coupling effects. Low aspect ratios are expected to show strong coupling effects and further investigation is recommended using Timoshenko beam theory.     

Generalised modal stability of inclined cables subjected to support excitations

Parametric excitation is of concern for cables such as on cable-stayed bridges, whereby small amplitude end motion can lead to large, potentially damaging, cable vibrations. Previous identification of the stability boundaries for the onset of such vibrations has considered only a single mode of the cable, ignoring non-linear coupling between modes, or has been limited to special cases. Here multiple cable modes in both planes are included, with support excitation close to any natural frequency. Cable inclination, sag, parametric and direct excitation and nonlinearities, including modal coupling, are included. The only significant limitation is that the sag is small. The method of scaling and averaging is used to find the steady-state amplitude of the directly excited mode and, in the presence of this response, to define stability boundaries of other modes excited parametrically or through nonlinear modal coupling. It is found that the directly excited response significantly modifies the stability boundaries compared to previous simplified solutions. The analysis is validated by a series of experimental tests, which also identified another nonlinear mechanism which caused significant cable vibrations at twice the excitation frequency in certain conditions. This new mechanism is explained through a refinement of the analysis.     

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.     

The use of roving discs and orthogonal natural frequencies for crack identification and location in rotors

A variety of approaches that have been developed for the identification and localisation of cracks in a rotor system, which exploit natural frequencies, require a finite element model to obtain the natural frequencies of the intact rotor as baseline data. In fact, such approaches can give erroneous results about the location and depth of a crack if an inaccurate finite element model is used to represent an uncracked model. A new approach for the identification and localisation of cracks in rotor systems, which does not require the use of the natural frequencies of an intact rotor as a baseline data, is presented in this paper. The approach, named orthogonal natural frequencies (ONFs), is based only on the natural frequencies of the non-rotating cracked rotor in the two lateral bending vibration x–z and y–z planes. The approach uses the cracked natural frequencies in the horizontal x–z plane as the reference data instead of the intact natural frequencies. Also, a roving disc is traversed along the rotor in order to enhance the dynamics of the rotor at the cracked locations. At each spatial location of the roving disc, the two ONFs of the rotor–disc system are determined from which the corresponding ONF ratio is computed. The ONF ratios are normalised by the maximum ONF ratio to obtain normalised orthogonal natural frequency curves (NONFCs). The non-rotating cracked rotor is simulated by the finite element method using the Bernoulli–Euler beam theory. The unique characteristics of the proposed approach are the sharp, notched peaks at the crack locations but rounded peaks at non-cracked locations. These features facilitate the unambiguous identification and locations of cracks in rotors. The effects of crack depth, crack location, and mass of a roving disc are investigated. The results show that the proposed method has a great potential in the identification and localisation of cracks in a non-rotating cracked rotor.     

Variation in the stability of a rotating blade disk with a local crack defect

The effect of a near root local blade crack on the stability of a grouped blade disk is investigated in this paper. A bladed disk comprised of periodically shrouded blades is used to simulate the coupled periodic structure. The blade crack is modeled using the local flexibility with coupling terms. The mode localization phenomenon introduced by the blade crack on the longitudinal and bending vibrations in the rotating blades has also been considered. Using the Galerkin's method, the imperturbation equations of a bladed disk in which one of the blades is cracked, subject to fluctuations in the rotation speed, can be derived. Employing the multiple scales method, the boundaries of the instability zones in the mistuned turbo blade system are approximated. Numerical results indicate that an additional unstable zone is introduced near the localization frequency and the regions of unstable zones are varied with the crack size and fluctuations in disk speed.     

FUNDAMENTAL FREQUENCY OF CRACKED BEAMS IN BENDING VIBRATIONS: AN ANALYTICAL APPROACH

This paper presents an analytical approach to the fundamental frequency of cracked Euler-Bernoulli beams in bending vibrations. The flexibility influence function method used to solve the problem leads to an eigenvalue problem formulated in integral form. The influence of the crack was represented by an elastic rotational spring connecting the two segments of the beam at the cracked section. In solving the problem, closed-form expressions for the approximated values of the fundamental frequency of cracked Euler-Bernoulli beams in bending vibrations are reached. The results obtained agree with those numerically obtained by the finite element method.     

Three dimensional vibration generators with a single rotational input

This paper presents a novel device capable of generating three-dimensional vibrations with a single-axis of rotation. The device resembles a vibration motor with an eccentric weight, but the weight is allowed to move up and down in parallel to the axis of rotations. While spinning of the eccentric weight causes lateral vibrations, vertical vibrations are excited by superposing a periodic torque on the rotary shaft. The frequency and magnitude of vertical oscillations can be independently regulated. Since the vertical natural frequency is sensitive to rotational speeds, maximum vertical oscillations can be achieved by properly adjusting the rotational speed according to the excitation frequency. Relations between the excitation torque and the vertical shaking force are examined using the frequency response for the linearized system. Numerical simulations on the original nonlinear system are conducted to verify the performance of vertical oscillations.     

A miniaturization of the multi-degree-of-freedom ultrasonic actuator using a small cylinder fixed on a substrate

Multi-degree-of-freedom ultrasonic actuator has been studied for robot arms and multidimensional precision table and so on because of its simple structure, silent operation, and holding force. In this study, we aim to miniaturize multi-degree-of-freedom ultrasonic actuator for fabrication on a substrate. This actuator consists of a stainless steel cylinder and a PZT ring. The cylinder is fixed on a substrate and the PZT ring is glued to the substrate near the cylinder. The 1st longitudinal vibration and the 2nd bending vibration are simultaneously excited in the cylinder to make elliptical motion at the top of the cylinder and a ball rotor placed on the cylinder rotates because of the friction force. Length of the cylinder was decided so as to tune the resonance frequency of the 1st longitudinal vibration to the 2nd bending one. Actuator performances are evaluated experimentally using a 14 mm height and 7 mm diameter stainless steel cylinder with a 0.5 mm thickness PZT ring. The rotation about the cylinder axis is tested using the two orthogonal bending vibrations with 90 degrees phase difference. Also, the rotation about horizontal axes were investigated using the combination of the longitudinal vibration and one of two bending vibrations. We measured the rotation speed of a steel ball and obtained 15.8 rps using a 6 mm diameter ball rotor.     

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.     

Crack as modulator,detector and amplifier in structural health monitoring

A model of a one-dimensional cracked cantilever bar subjected to longitudinal harmonic excitation is used to analyse a nonlinear response as a way to monitor structural health. The effect of the bilinear (nonlinear) character of the crack on the dynamics of the structure is studied. Simulation and experiments were performed to analyse the nonlinear behaviour of the cracked bar. In simulation the nonlinear information is obtained based on a combination of the analytical technique and the Matlab–Simulink computation. From analysis and experiment, it is found that the crack-induced nonlinearity leads to the generation of higher harmonics, whose intensity is a function of a distance from the crack. Side band frequencies were clearly revealed as well. The latter indicate modulation of exciting frequency due to systematic interaction of crack faces. The nonlinear transformation of modulated vibration by crack leads to generation of a low frequency periodic component. Its intensity is proportional to the forced response of the cracked bar at the exciting frequency. The phenomenology revealed can be effective for Structural Health Monitoring.     

Numerical–experimental identification of the most effective dynamic operation mode of a vibration drilling tool for improved cutting performance

This study is concerned with application of numerical–experimental approach for characterizing dynamic behavior of the developed piezoelectrically excited vibration drilling tool with the aim to identify the most effective conditions of tool vibration mode control for improved cutting efficiency. 3D finite element model of the tool was created on the basis of an elastically fixed pre-twisted cantilever (standard twist drill). The model was experimentally verified and used together with tool vibration measurements in order to reveal rich dynamic behavior of the pre-twisted structure, representing a case of parametric vibrations with axial, torsional and transverse natural vibrations accompanied by the additional dynamic effects arising due to the coupling of axial and torsional deflections ((un)twisting). Numerical results combined with extensive data from interferometric, accelerometric, dynamometric and surface roughness measurements allowed to determine critical excitation frequencies and the corresponding vibration modes, which have the largest influence on the performance metrics of the vibration drilling process. The most favorable tool excitation conditions were established: inducing the axial mode of the vibration tool itself through tailoring of driving frequency enables to minimize magnitudes of surface roughness, cutting force and torque. Research results confirm the importance of the tool mode control in enhancing the effectiveness of vibration cutting tools from the viewpoint of structural dynamics.     

Properties study of LiNbO3 lateral field excited device working on thickness extension mode

This paper investigates the properties of thickness extension mode excited by lateral electric field on LiNbO3 by using the extended Christoffel-Bechmann method. It finds that the lateral field excitation coupling factor for amode (quasi-extensional mode) reaches its maximum value of 28% on X-cut LiNbO3. The characteristics of a lateral field excitation device made of X-cut LiNbO3 have been investigated and the lateral field excitation device is used for the design of a high frequency ultrasonic transducer. The time and frequency domain pulse/echo response of the LiNbO3 lateral field excitation ultrasonic transducer is analysed with the modified Krimholtz-Leedom-Matthae model and tested using traditional pulse/echo method. A LiNbO3 lateral field excitation ultrasonic transducer with the centre frequency of 33.44 MHz and the -6 dB bandwidth of 33.8% is acquired, which is in good agreement with the results of the Krimholtz-Leedom-Matthae model. Further analysis suggests that the LiNbO3 lateral field excitation device has great potential in the design of broadband high frequency ultrasonic transducers.     

Mode shapes analysis of a cracked beam and its application for crack detection

In this paper, mode shapes of a cracked beam with a rectangular cross section beam are analysed using finite element method. The 3D beam element is applied for this finite element analysis. The influence of the coupling mechanism between horizontal bending and vertical bending vibrations due to the crack on the mode shapes is investigated. Due to the coupling mechanism the mode shapes of a beam change from plane curves to space curves. Thus, the existence of the crack can be detected based on the mode shapes: when the mode shapes are space curves there is a crack in the beam. Also, when there is a crack, the mode shapes have distortions or sharp changes at the crack position. Thus, the position of the crack can be determined as a position at which the mode shapes exhibit such distortions or sharp changes. While in previous studies using 2D beam element, distortions in the mode shapes caused by a small crack could not be detected, these distortions in the case using the 3D beam element can be amplified and inspected clearly by using the projections of the mode shapes on appropriate planes. The quantitative analysis is also implemented to relate the size and position of the crack with the observed coupled modes. These results can be applied for crack detection of a beam. In this paper, the stiffness matrix of a cracked element obtained from fracture mechanics is presented and numerical simulations of three case studies are provided.     

Simultaneous Spatial and Temporal Focusing in Nonlinear Microscopy

Simultaneous spatial and temporal focusing (SSTF), when combined with nonlinear microscopy, can improve the axial excitation confinement of wide-field and line-scanning imaging. Because two-photon excited fluorescence depends inversely on the pulse width of the excitation beam, SSTF decreases the background excitation of the sample outside of the focal volume by broadening the pulse width everywhere but at the geometric focus of the objective lens. This review theoretically describes the beam propagation within the sample using Fresnel diffraction in the frequency domain, deriving an analytical expression for the pulse evolution. SSTF can scan the temporal focal plane axially by adjusting the GVD in the excitation beam path. We theoretically define the axial confinement for line-scanning SSTF imaging using a time-domain understanding and conclude that line-scanning SSTF is similar to the temporally-decorrelated multifocal multiphoton imaging technique. Recent experiments on the temporal focusing effect and its axial confinement, as well as the axial scanning of the temporal focus by tuning the GVD, are presented. We further discuss this technique for axial-scanning multiphoton fluorescence fiber probes without any moving parts at the distal end. The temporal focusing effect in SSTF essentially replaces the focusing of one spatial dimension in conventional wide-field and line-scanning imaging. Although the best axial confinement achieved by SSTF cannot surpass that of a regular point-scanning system, this trade-off between spatial and temporal focusing can provide significant advantages in applications such as high-speed imaging and remote axial scanning in an endoscopic fiber probe.     

The vibration signature of chordal cracks in a rotor system including uncertainties

The aim of this paper is to investigate the effects of the presence of a transverse crack in a rotating shaft under uncertain physical parameters in order to obtain some indications that might be useful in detecting the presence of a crack in rotating system. The random dynamic response of the cracked rotor is evaluated by expanding the changing stiffness of the crack (i.e. the breathing mechanism) as a random truncated Fourier series. To avoid the use of the Monte Carlo simulations (MCS), an alternative procedure that is based on a combination of the Harmonic Balance Method and the Stochastic Finite Element Method (SFEM) using the Polynomial Chaos Expansion (PCE) is proposed. So the response of the Fourier components of the cracked rotor is expanded in the polynomial chaoses. The random dynamic response obtained by applying this procedure is compared with that evaluated through numerical integration based on the Harmonic Balance Method and the Monte Carlo simulations.     

Nonlinear modulation technique for NDE with air-coupled ultrasound

The present study is aimed at expanding flexibility and application area of nonlinear acoustic modulation (NAM-) technique by combining the benefits of noncontact ultrasound excitation (remote locating and imaging of defects) with sensitivity of nonlinear methods in a new air-coupled NAM-version. A pair of focused air-coupled transducers was used to generate and receive (high-frequency) longitudinal or flexural waves in plate-like samples. Low-frequency (LF-) vibrations were excited with a shaker or a loudspeaker. Temporal and spectral analysis of the output signal revealed an extremely efficient nonlinear amplitude modulation and multiple frequency side-bands for sound transmission and flexural wave propagation through cracked defects. On the contrary, a negligible modulation was observed for large and medium scale inclusions and material inhomogeneities (linear defects). A new subharmonic mode of the NAM was observed at high excitation levels. It was also shown for the first time that nonlinear vibrations of cracks resulted in radiation of a very high-order harmonics (well above 100) of the driving excitation in air that enabled imaging of cracks remotely by registration their highly nonlinear "acoustic emission" with air-coupled transducers.     

Triply coupled vibrational band gap in a periodic and nonsymmetrical axially loaded thin-walled Bernoulli–Euler beam including the warping effect

The propagation of triply coupled vibrations in a periodic, nonsymmetrical and axially loaded thin-walled Bernoulli–Euler beam composed of two kinds of materials is investigated with the transfer matrix method. The cross-section of the beam lacks symmetrical axes, and bending vibrations in the two perpendicular directions are coupled with torsional vibrations. Furthermore, the effect of warping stiffness is included. The band structures of the periodic beam, both including and excluding the warping effect, are obtained. The frequency response function of the finite periodic beam is simulated with the finite element method. These simulations show large vibration-based attenuation in the frequency range of the gap, as expected. By comparing the band structure of the beam with plane wave expansion method calculations that are available in the literature, one finds that including the warping effect leads to a more accurate simulation. The effects of warping stiffness and axial force on the band structure are also discussed.