VIBRATION-BASED DIAGNOSTICS OF FATIGUE DAMAGE OF BEAM-LIKE STRUCTURES

The analytical investigation of vibration of damaged structures is a complicated problem. This problem may be simplified if a structure can be represented in the form of a beam with corresponding boundary and loading conditions. In this connection, free vibrations of an elastic cantilever Bernoulli-Euler beam with a closing edge transverse crack is considered in the present work as a model of a structure with a fatigue crack. The modelling of bending vibrations of a beam with a closing crack is realized based on the solutions for an intact beam and for a beam with an open crack. The algorithm of consecutive (cycle-by-cycle) calculation of beam mode shapes amplitudes is presented. It is shown that at the instant of crack opening and closing, the growth of the so-called concomitant mode shapes which differ from the initially given mode shape takes place. Moreover, each of the half-cycles is characterized by a non-recurrent set of amplitudes of concomitant modes of vibration and these amplitudes are heavily dependent on the crack depth.The vibration characteristics of damage based on the estimation of non-linear distortions of the displacement, acceleration and strain waves of a cracked beam are investigated, and the comparative evaluation of their sensitivity is carried out.     

IDENTIFICATION OF CRACK LOCATION IN VIBRATING BEAMS FROM CHANGES IN NODE POSITIONS

It is known that the effect of a single crack in an axially vibrating thin rod is to cause the nodes of the mode shapes move toward the crack. This paper is an analytical/experimental investigation of the analogous problem for a thin beam in bending vibration. The monotonicity property linking changes in node position and crack location does not hold in the bending case. The analysis of the direct problem, however, shows that the direction by which nodal points move may be useful for predicting damage location. Analytical results agree well with experimental tests performed on cracked steel beams.     

Damage detection of cracked thick rotating blades by a spatial wavelet based approach

This paper presents a technique for blade damage detection based on spatial wavelet analysis. The wavelet transform is used to analyze spatially distributed signals (e.g. mode shape) of cracked thick rotating blades. First, a finite element model is applied to the vibration of a thick rotating blade with a single edge crack. The effects of transverse shear deformation and rotatory inertia are taken into account. Then the mode shapes of the cracked rotating blade are analyzed by wavelet transformation. The effects of crack locations and sizes on the wavelet coefficients are studied. It is found that the distributions of the wavelet coefficients can identify the crack position of the rotating blades by showing a peak at the position of the crack. Then the signals are analyzed by wavelet transform. It is found that the distributions of the wavelet coefficients can identify the crack position. Assumed measurement errors are added to nth mode shape for evaluating the effect of measurement errors on the capability of detecting crack position. The moving average method is used to process the data with assumed measurement errors. The crack positions can also be identified when there exist assumed measurement errors.     

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.     

Free vibration analysis of a cracked beam by finite element method

In this paper, the natural frequencies and mode shapes of a cracked beam are obtained using the finite element method. An ‘overall additional flexibility matrix’, instead of the ‘local additional flexibility matrix’, is added to the flexibility matrix of the corresponding intact beam element to obtain the total flexibility matrix, and therefore the stiffness matrix. Compared with analytical results, the new stiffness matrix obtained using the overall additional flexibility matrix can give more accurate natural frequencies than those resulted from using the local additional flexibility matrix. All the elements in the overall additional flexibility matrix are computed by 128-point (1D) or (128×128)-point (2D) Gauss quadrature, and then further best fitted using the least-squares method. The explicit form best-fitted formulas agree very well with the numerical integration results, and are very convenient for use and valuable for further reference. In addition, the authors constructed a shape function that can perfectly satisfy the local flexibility conditions at the crack locations, which can give more accurate vibration modes.     

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.     

Crack effects on the in-plane static and dynamic stabilities of a curved beam with an edge crack

The effects of a single-edge crack and its locations on the buckling loads, natural frequencies and dynamic stability of circular curved beams are investigated numerically using the finite element method, based on energy approach. This study consists of three stages, namely static stability (buckling) analysis, vibration analysis and dynamic stability analysis. The governing matrix equations are derived from the standard and cracked curved beam elements combined with the local flexibility concept. Approximation for the displacements using coupled interpolations based on the constant-strain, linear-curvature element (SC) has yielded results with reasonable accuracy. The numerical results obtained from the present finite element model are found to be in good agreement with those, both experimental and analytic, of other researchers in the existing literature. Results show that the reductions in buckling load and natural frequency depend not only on the crack depth and crack position, but also on the related mode shape. Analyses also show that the crack effect on the dynamic stability of the considered curved beam is quite limited.     

An insight into the breathing mechanism of a crack in a rotating shaft

The time history of local flexibilities associated with a breathing crack in a rotating shaft is the concern of this paper. Considering quasi-static approximation, the deflections of a circular cross-section beam presenting a crack of different depths, due to bending or torsion loads are analyzed with the aid of a refined nonlinear contact-finite element procedure in order to predict accurately the time-variant flexibility of the fractured shaft. This method predicts the partial contact of crack surfaces, and it is appropriate to evaluate the instantaneous crack flexibilities. The bending load is applied in several aperture angles, in order to simulate a rotating load on a fixed beam. Results obtained for the rotating beam can then be used for the analysis of cracked, horizontal axis rotors. The effect of friction is also considered in the cracked area. Portions of crack surfaces in contact are predicted, the direct and the cross-coupled flexibility coefficients are calculated by applying energy principles. The numerical results compared with relevant previously published results, show high consistency.     

Vibration damage detection of a Timoshenko beam by spatial wavelet based approach

This paper presents a technique for structure damage detection based on spatial wavelet analysis. The wavelet transform is used to analyze the mode shape of a Timoshenko beam. First, the mode shapes of the Timoshenko beam containing a transverse crack are obtained. The crack is represented as a rotational spring. Then these spatially distributed signals are analyzed by wavelet transformation. It is observed that distributions of the wavelet coefficients can identify the crack position of Timoshenko beam by showing a peak at the position of the crack. It is also demonstrated that the crack position can be detected by this method even though the crack is very small. Assumed measurement errors are added to the mode shape for evaluating the effect of measurement errors on the capability of detecting crack position. The moving average method is used to process the data with assumed measurement errors. The crack positions can also be identified when there exist assumed measurement errors.     

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.     

Bending vibrations of wedge beams with any number of point masses

The natural frequencies and mode shapes of beams with constant width and linearly tapered depth (or thickness) carrying any number of point masses at arbitrary positions along the length of the beams were investigated using the Euler-Bernoulli equation. Use of the closed-form (exact) solutions for the natural frequencies and mode shapes of the unconstrained single-tapered beam (without carrying any point masses) and incorporation of the expansion theorem, the equation of motion for the associated constrained beam (carrying any point masses) were derived. Solution of the last equation will yield the desired natural frequencies and mode shapes of the constrained single-tapered beam. The bending vibrations of a single-tapered beam with six kinds of boundary conditions were investigated. Comparison with the existing literature or the traditional finite element method results reveals that the adopted approach has excellent accuracy and simple algorithm.     

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.     

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.     

Meshless formulation using NURBS basis functions for eigenfrequency changes of beam having multiple open-cracks

This paper presents a meshless formulation using non-uniform rational B-spline (NURBS) basis functions, and its applications to evaluate natural frequencies of a beam having multiple open-cracks. Node-based NURBS basis functions are used to construct the approximation function. The characteristic differentiability of the NURBS basis functions allows it to represent a function having specific degrees of smoothness and/or discontinuity. The discontinuity can be incorporated simply by assigning multiple knots at those locations. Hence, it can yield exact solutions having interior discontinuous derivatives. These advantages of NURBS are well known, and have been used extensively in graphical approximation of geometrical surfaces. However, it is seldom used in other engineering applications. To model the multiple open-cracks in a beam, quartic NURBS basis functions are employed and quadruplicate knots are assigned at the crack locations. Hence, it is capable to model the abrupt changes of slope (the first derivative of displacement) across a crack. In the present applications, additional equivalent massless rotational springs are inserted at the crack locations to represent the local flexibility caused by the cracks. As such, the cracked beam can be treated in the usual manner as a continuous beam. By adopting the meshless Petrov–Galerkin formulation, a generalized stiffness matrix for the cracked beam can be derived. Compared to the conventional finite element method, the present method does not require a finite element mesh for the purposes of interpolation and numerical integration. The advantages and effectiveness of the present method is illustrated in solving the eigenfrequencies of a beam having multiple open-cracks of different depths.     

An approximate method for determining the static deflection and natural frequency of a cracked beam

This paper provides an approximate method to determine the stiffness and the fundamental frequency of a cracked beam. The cracked beam is first represented as an un-cracked beam with equivalent reduced sections around the cracks. The effect of the cracks is explained, visualised and quantified using the equivalence concept developed for stepped beams with periodically variable cross-sections. Then an alternative expression of the improved Rayleigh method is provided to calculate the natural frequencies of a beam with a variable stiffness distribution along its length. As the method is insensitive to the assumed mode shapes, it avoids the difficulty in choosing appropriate mode shapes and yields accurate results. This is shown using several examples to compare the results determined using the proposed method and the Finite Element method (FEM). The method greatly simplifies the calculation of cracked beams with complicated configurations, such as a beam with several cracks, a cracked beam with concentrated masses, a beam with cracks close to each other, and a beam with periodically distributed cracks.     

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.     

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.     

Vibration analysis of a new curved beam element

The common practice in developing a locking-free curved beam element is to ensure that its interpolation functions of displacement explicitly satisfy the inextensible bending mode condition for the membrane locking-free instead of the rigid body modes. In this paper, we study the impact of this practice on the dynamic characteristics of a finite element by conducting vibration analysis using our newly developed three-node locking-free curved beam element. In this case, the inextensible bending mode condition is satisfied explicitly, while the rigid body modes are satisfied implicitly to 4th-order accuracy. Numerical and experimental examples show that with the newly developed curved beam element, developed by using the implicit representation of a rigid body mode condition, it is possible to recover the rigid body modes of curved beams with low and medium slenderness ratios. This is even true for cases involving a half-circular element and the vibration of the curved beam is predicted with high accuracy.     

On the growth rate of bending induced edge cracks in acoustically excited panels

Parts of an aircraft structure may be made to vibrate as a result of acoustic waves generated by various aircraft noise sources impinging on the structure. The stresses associated with this acoustically induced vibration may be sufficiently large to result in fatigue failure of portions of the structure. If acoustically induced fatigue cracks occur in the stiffened skin structures widely used in aircraft construction they may initiate in the skin panels near the stiffener attachment points. The initiation and subsequent propagation of these cracks at the panel edges is primarily due to the bending stresses arising from the out-of-plane vibration of the individual skin panels.The emphasis of the work described in this paper is on examining the growth rate of edge cracks in acoustically excited panels. A single panel with an edge crack is considered and this structural element is modelled as a flat plate clamped on three edges and part of the fourth. The crack is represented by the unclamped part of the fourth edge. Fracture mechanics principles are used to predict the crack growth rates associated with the first two modes of vibration of the edge cracked panel. The crack tip stress intensity factors associated with these panel modes are estimated by a technique based on finding the nominal bending stresses at the crack tips. The nominal bending stresses are in turn found from mode shapes determined by the Rayleigh Principle. The validity of the various assumptions is assessed by comparing the predicted crack growth rates with measured growth rates in panels representative of those used in aircraft construction.     

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.