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.     

Crack detection in beams using changes in frequencies and amplitudes of frequency response functions

In recent years, significant efforts have been devoted to developing non-destructive techniques for damage identification in structures. The work reported in this paper is part of an ongoing research on the experimental investigations of the effects of cracks and damages on the integrity of structures, with a view to detect, quantify, and determine their extents and locations. Two sets of aluminum beams were used for this experimental study. Each set consisted of seven beams, the first set had fixed ends, and the second set was simply supported. Cracks were initiated at seven different locations from one end to the other end (along the length of the beam) for each set, with crack depth ratios ranging from 0.1d to 0.7d (d is the beam depth) in steps of 0.1, at each crack location. Measurements of the acceleration frequency responses at seven different points on each beam model were taken using a dual channel frequency analyzer.The damage detection schemes used in this study depended on the measured changes in the first three natural frequencies and the corresponding amplitudes of the measured acceleration frequency response functions.     

CRACK IDENTIFICATION IN VIBRATING BEAMS USING THE ENERGY METHOD

An energy-based numerical model is developed to investigate the influence of cracks on structural dynamic characteristics during the vibration of a beam with open crack(s). Upon the determination of strain energy in the cracked beam, the equivalent bending stiffness over the beam length is computed. The cracked beam is then taken as a continuous system with varying moment of intertia, and equations of transverse vibration are obtained for a rectangular beam containing one or two cracks. Galerkin's method is applied to solve for the frequencies and vibration modes. To identify the crack, the frequency contours with respect to crack depth and location are defined and plotted. The intersection of contours from different modes could be used to identify the crack location and depth.     

A novel method for crack detection in beam-like structures by measurements of natural frequencies

A novel method is proposed for calculating the natural frequencies of a multiple cracked beam and detecting unknown number of multiple cracks from the measured natural frequencies. First, an explicit expression of the natural frequencies through crack parameters is derived as a modification of the Rayleigh quotient for the multiple cracked beams that differ from the earlier ones by including nonlinear terms with respect to crack severity. This expression provides a simple tool for calculating the natural frequencies of the beam with arbitrary number of cracks instead of solving the complicated characteristic equation. The obtained nonlinear expression for natural frequencies in combination with the so-called crack scanning method proposed recently by the authors allowed the development of a novel procedure for consistent identification of unknown amount of cracks in the beam with a limited number of measured natural frequencies. The developed theory has been illustrated and validated by both numerical and experimental results.     

CRACK DETECTION IN BEAM-TYPE STRUCTURES USING FREQUENCY DATA

A practical method to non-destructively locate and estimate size of a crack by using changes in natural frequencies of a structure is presented. First, a crack detection algorithm to locate and size cracks in beam-type structures using a few natural frequencies is outlined. A crack location model and a crack size model are formulated by relating fractional changes in modal energy to changes in natural frequencies due to damage such as cracks or other geometrical changes. Next, the feasibility and practicality of the crack detection scheme are evaluated for several damage scenarios by locating and sizing cracks in test beams for which a few natural frequencies are available. By applying the approach to the test beams, it is observed that crack can be confidently located with a relatively small localization error. It is also observed that crack size can be estimated with a relatively small size error.     

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.     

Identification of embedded horizontal cracks in beams using measured mode shapes

Mode shapes (MSs) have been extensively used to detect structural damage. This paper presents two new non-model-based methods that use measured MSs to identify embedded horizontal cracks in beams. The proposed methods do not require any a priori information of associated undamaged beams, if the beams are geometrically smooth and made of materials that have no stiffness discontinuities. Curvatures and continuous wavelet transforms (CWTs) of differences between a measured MS of a damaged beam and that from a polynomial that fits the MS of the damaged beam are processed to yield a curvature damage index (CDI) and a CWT damage index (CWTDI), respectively, at each measurement point. It is shown that the MS from the polynomial fit can well approximate the measured MS and associated curvature MS of the undamaged beam, provided that the measured MS of the damaged beam is extended beyond boundaries of the beam and the order of the polynomial is properly chosen. The proposed CDIs of a measured MS are presented with multiple resolutions to alleviate adverse effects caused by measurement noise, and cracks can be identified by locating their tips near regions with high values of the CDIs. It is shown that the CWT of a measured MS with the n-th-order Gaussian wavelet function has a shape resembling that of the n-th-order derivative of the MS. The crack tips can also be located using the CWTs of the aforementioned MS differences with second- and third-order Gaussian wavelet functions near peaks and valleys of the resulting CWTDIs, respectively, which are presented with multiple scales. A uniform acrylonitrile butadiene styrene (ABS) cantilever beam with an embedded horizontal crack was constructed to experimentally validate the proposed methods. The elastic modulus of the ABS was determined using experimental modal analysis and model updating. Non-contact operational modal analysis using acoustic excitations and measurements by two laser vibrometers was performed to measure the natural frequencies and MSs of the ABS cantilever beam, and the results compare well with those from the finite element method. Numerical and experimental crack identification can successfully identify the crack by locating its tips.     

Free vibration analysis of beams and frames with multiple cracks for damage detection

The problem of calculating the natural frequencies of beams with multiple cracks and frames with cracked beams is studied. The natural frequencies are obtained using a new method in which a rotational spring model is used to represent the cracks. For beams, dynamic stiffness matrices of order 4 are obtained in a recursive manner, according to the number of cracks, by applying partial Gaussian elimination. The Wittrick–Williams algorithm is used to compute the natural frequencies in the resulting transcendental eigenvalue problem. Published numerical examples for cracked beams are used for validation. The global dynamic stiffness matrix of a frame with multiply cracked members is then assembled. A published two bay frame example is used to evaluate the new method. The effect of changing the location of a crack in a two bay two storey frame is studied numerically, giving insight into the inverse problem of damage detection.     

Crack detection in single-crystalline silicon wafers using impact testing

This paper presents acoustic measurements obtained by mechanically exciting vibratory modes in single-crystalline silicon wafers with hairline periphery cracks of different type and location. The data presented shows a dependence of natural frequencies, peak amplitudes and damping levels of four audio vibration modes in the frequency range up to 1000 Hz on crack type and crack location. Data from defective wafers exhibit lower natural frequencies, higher damping levels, and lower peak amplitudes. The results suggest an impact test method may be useful for solar cell crack detection and quality control.     

Analytical modeling and vibration analysis of internally cracked rectangular plates

This study proposes an analytical model for nonlinear vibrations in a cracked rectangular isotropic plate containing a single and two perpendicular internal cracks located at the center of the plate. The two cracks are in the form of continuous line with each parallel to one of the edges of the plate. The equation of motion for isotropic cracked plate, based on classical plate theory is modified to accommodate the effect of internal cracks using the Line Spring Model. Berger?s formulation for in-plane forces makes the model nonlinear. Galerkin?s method used with three different boundary conditions transforms the equation into time dependent modal functions. The natural frequencies of the cracked plate are calculated for various crack lengths in case of a single crack and for various crack length ratio for the two cracks. The effect of the location of the part through crack(s) along the thickness of the plate on natural frequencies is studied considering appropriate crack compliance coefficients. It is thus deduced that the natural frequencies are maximally affected when the crack(s) are internal crack(s) symmetric about the mid-plane of the plate and are minimally affected when the crack(s) are surface crack(s), for all the three boundary conditions considered. It is also shown that crack parallel to the longer side of the plate affect the vibration characteristics more as compared to crack parallel to the shorter side. Further the application of method of multiple scales gives the nonlinear amplitudes for different aspect ratios of the cracked plate. The analytical results obtained for surface crack(s) are also assessed with FEM results. The FEM formulation is carried out in ANSYS.     

Flexural vibration of non-uniform beams having double-edge breathing cracks

Flexural vibration of non-uniform Rayleigh beams having single-edge and double-edge cracks is presented in this paper. Asymmetric double-edge cracks are formed as thin transverse slots with different depths at the same location of opposite surfaces. The cracks are modelled as breathing since the bending of the beam makes the cracks open and close in accordance with the direction of external moments. The presented crack model is used for single-edge cracks and double-edge cracks having different depth combinations. The energy method is used in the vibration analysis of the cracked beams. The consumed energy caused by the cracks opening and closing is obtained along the beam's length together with the contribution of tensile and compressive stress fields that come into existence during the bending. The total energy is evaluated for the Rayleigh-Ritz approximation method in analysing the vibration of the beam. Examples are presented on simply supported beams having uniform width and cantilever beams which are tapered. Good agreements are obtained when the results from the present method are compared with the results of Chondros et al. and the results of the commercial finite element program, Ansys^©. The effects of breathing in addition to crack depth's asymmetry and crack positions on the natural frequency ratios are presented in graphics.     

Free vibrations of a laminated beam by a microstructure theory

The free vibrations of a laminated beam are considered within the framework of a theory that models the composite beam as a macrohomogeneous beam with microstructure. The beams are assumed to consist of several parallel alternating layers of two homogeneous, isotropic elastic materials. The system of three coupled partial differential equations is solved exactly, and attention is devoted to the determination of natural frequencies of vibration of laminated beams with (i) hinged-hinged ends and (ii) clamped-clamped ends. For the sake of comparison, the same boundary value problems are also solved within the framework of the so-called effective modulus theory, which treats the composite as a transversely isotropic and “fictitiously” homogeneous Timoshenko beam, with effective moduli and density. For relatively long beams, i.e., in the low frequency range, the natural frequencies obtained from the two theories are in excellent agreement, but as the depth-to-length ratio, ζ, increases the microstructure frequencies are observed to be much lower than the effective modulus frequencies, the magnitude of the effect becoming more pronounced with increasing mode number n.     

Surface wave transmission measurements across distributed surface-breaking cracks using air-coupled sensors

Near-field scattering of surface waves by a single surface-breaking crack in solid medium has been well investigated by prior researchers. However, there have been few studies for more realistic problems involving near scattering of surface waves by distributed surface-breaking cracks. One possible reason is complexity caused by the interaction of surface waves between multiple cracks. In this study, interaction of surface waves between two surface-breaking cracks with various crack spacing was investigated. The experimental study was performed on Plexiglas specimens with non-contact sensors (air-coupled sensors, and a laser vibrometer), and compared with numerical simulation results. The effects of crack depth h, spacing a, and the number of cracks N on surface wave transmission were studied. Analyses show that for the very small crack spacing (a/h<0.2), the distributed cracks can be regarded as a single surface-breaking crack. However, for a/h ranging between approximately 1 and 6, transmission coefficient of surface waves is significantly affected by interaction between cracks. The transmission coefficients have the lowest value when a/h is between 2 and 3. When a/h is large (a/h>6), transmission coefficients obtained from experiments, and numerical simulations agree with the theoretical results based on non-interaction crack assumption.     

An anisotropic model for frequency analysis of arterial walls with the wave propagation approach

An anisotropic model for calculating natural frequency of arterial walls is proposed in this paper. The first-order shear deformation theory (FSDT) is used for the arterial walls, and the wave propagation approach is applied that can easily handle the boundary conditions. Results obtained using this model have been evaluated against those available in the literature and the agreement has been found to be good. Experiments were carried out on a natural rubber latex tube. The relative differences of the first four natural frequencies between the experiment and the theory are less than 7%. The variation of the natural frequency of this tube with the longitudinal and circumferential modes m and n is studied which suggests the first four natural frequencies are with n = 1 and m = 1-4. Simulations show that classical Donnell’s, Love’s and beam theories are not suitable for this thick tube while FSDT results closely agree with the experiment. The anisotropy of circumferential elastic modulus on natural frequencies of the tube is analyzed.     

SIMPLIFIED MODELS FOR THE LOCATION OF CRACKS IN BEAM STRUCTURES USING MEASURED VIBRATION DATA

A new simplified approach to modelling cracks in beams undergoing transverse vibration is presented. The modelling approach uses Euler-Bernoulli beam elements with small modifications to the local flexibility in the vicinity of cracks. This crack model is then used to estimate the crack locations and sizes, by minimizing the difference between the measured and predicted natural frequencies via model updating. The uniqueness of the approach is that the simplified crack model allows the location and damage extent to be estimated directly. The simplified crack model may also be used to generate training data for pattern recognition approaches to health monitoring. The proposed method has been illustrated using the experimental data on beam examples.     

Flapwise bending vibration analysis of a rotating tapered cantilever Bernoulli–Euler beam by differential transform method

This paper studies the vibration characteristics of a rotating tapered cantilever Bernoulli–Euler beam with linearly varying rectangular cross-section of area proportional to xⁿ, where n equals to 1 or 2 covers the most practical cases. In this work, the differential transform method (DTM) is used to find the nondimensional natural frequencies of the tapered beam. Numerical results are tabulated for different taper ratios, nondimensional angular velocities and nondimensional hub radius. The effects of the taper ratio, nondimensional angular velocity and nondimensional hub radius are discussed. The accuracy is assured from the convergence of the natural frequencies and from the comparisons made with the studies in the open literature. It is shown that the natural frequencies of a rotating tapered cantilever Bernoulli–Euler beam can be obtained with high accuracy by using DTM.     

A method for determining the location and depth of cracks in double-cracked beams

The influence of two transverse open cracks on the antiresonances of a double cracked cantilever beam is investigated both analytically and experimentally. It is shown that there is a shift in the antiresonances of the cracked beam depending on the location and size of the cracks. These antiresonance changes, complementary with natural frequency changes, can be used as additional information carrier for crack identification in double cracked beams. Experimental results from tests on plexiglas beams damaged at different locations and different magnitudes are found to be in good agreement with theoretical predictions. Based on the results of the present work, an efficient prediction scheme for crack localization and characterization in double cracked beams is proposed.     

Ultrasonic sizing of short surface cracks

This paper presents a method for ultrasonic sizing of surface cracks based on time domain and frequency domain Rayleigh wave near-field analysis. The procedure allows for the entire range of ratio of crack depth to Rayleigh wavelength a/λ to be covered with one single measurement. In the time domain the time-of-flight method was extended to cracks smaller than the wavelength by correlation of the time delay of the transmitted Rayleigh wave with the crack depth. In the frequency domain, the inverse scattering problem was solved by comparison of the measured scattering coefficients and central frequencies of the reflected and transmitted Rayleigh waves with theoretical curves. The sizing procedure was demonstrated experimentally with narrow slots and real fatigue cracks. The out-of-plane displacement component was measured pointwise in the scattered near field by means of laser interferometry. The determination of the scattering parameters in the near field was enabled by a procedure that allows for the Rayleigh wave to be separated from the other modes scattered at the defect. The experimental results showed good accuracy and repeatability down to the smallest available ratio of crack depth to Rayleigh wavelength a/λ = 0.15.     

On the effects of a general form of the initial runout on the oscillation frequencies and critical speeds of a spinning disk

This paper is concerned with the effect of the lack of flatness on the dynamics of spinning disks. Of particular interest is the study of the effect of an unsymmetrical initial runout on the oscillation frequencies and the critical speeds of the spinning disks. It is assumed that a spinning disk is initially warped in an initial asymmetric shape. Using Von Karman's plate theory, the equations of motion are derived in a rotating frame. Taking advantage of the orthogonal properties of the eigenfunctions of a stationary disk, the lateral displacement and the stress function are expressed as their summation. Since these eigenfunctions produce an orthonormal space, any shape and level of initial runout can be written as their summation also. Using Galerkin's method, the equations of motion are discretized and a set of coupled linear equations are found taking into account the effect of an initial arbitrary runout. The numerical results are found to have a good agreement with those obtained using ANSYS. It is found that the natural frequencies of the spinning disk calculated in the space fixed frame increases due to an initial runout. When the initial runout is assumed to be asymmetric, it is found that, due to asymmetric stress distributed in the disk, a frequency split between the backward and forward traveling waves of a given preferential mode of a stationary disk occurs. It is also seen that in some cases the initial runout in the form of the (n,m) mode has the least effect on the natural frequencies of the backward traveling waves of the same mode. As such it is observed that in the case the critical speed the (n,m) mode is less sensitive to an initial runout in the form of the (n,m) mode. To verify the accuracy of the numerical predictions, experiments were conducted on two disks, one which was flat and the other that had an initial nonflatness in the form of the (0,2) mode. The numerical and experimental results indicate a close correspondence.     

Reduced-order modeling for mistuned centrifugal impellers with crack damages

An efficient method for nonlinear vibration analysis of mistuned centrifugal impellers with crack damages is presented. The main objective is to investigate the effects of mistuning and cracks on the vibration features of centrifugal impellers and to explore effective techniques for crack detection. Firstly, in order to reduce the input information needed for component mode synthesis (CMS), the whole model of an impeller is obtained by rotation transformation based on the finite element model of a sector model. Then, a hybrid-interface method of CMS is employed to generate a reduced-order model (ROM) for the cracked impeller. The degrees of freedom on the crack surfaces are retained in the ROM to simulate the crack breathing effects. A novel approach for computing the inversion of large sparse matrix is proposed to save memory space during model order reduction by partitioning the matrix into many smaller blocks. Moreover, to investigate the effects of mistuning and cracks on the resonant frequencies, the bilinear frequency approximation is used to estimate the resonant frequencies of the mistuned impeller with a crack. Additionally, statistical analysis is performed using the Monte Carlo simulation to study the statistical characteristics of the resonant frequencies versus crack length at different mistuning levels. The results show that the most significant effect of mistuning and cracks on the vibration response is the shift and split of the two resonant frequencies with the same nodal diameters. Finally, potential quantitative indicators for detection of crack of centrifugal impellers are discussed.