Vibration based algorithm for crack detection in cantilever beam containing two different types of cracks

In this paper, a simple method for detection of multiple edge cracks in Euler–Bernoulli beams having two different types of cracks is presented based on energy equations. Each crack is modeled as a massless rotational spring using Linear Elastic Fracture Mechanics (LEFM) theory, and a relationship among natural frequencies, crack locations and stiffness of equivalent springs is demonstrated. In the procedure, for detection of m cracks in a beam, 3m equations and natural frequencies of healthy and cracked beam in two different directions are needed as input to the algorithm.     

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.     

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 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.     

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.     

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.     

FREE VIBRATION ANALYSIS OF NON-UNIFORM BEAMS WITH AN ARBITRARY NUMBER OF CRACKS AND CONCENTRATED MASSES

An exact approach for free vibration analysis of a non-uniform beam with an arbitrary number of cracks and concentrated masses is proposed. A model of massless rotational spring is adopted to describe the local flexibility induced by cracks in the beam. Using the fundamental solutions and recurrence formulas developed in this paper, the mode shape function of vibration of a non-uniform beam with an arbitrary number of cracks and concentrated masses can be easily determined. The main advantage of the proposed method is that the eigenvalue equation of a non-uniform beam with any kind of two end supports, any finite number of cracks and concentrated masses can be conveniently determined from a second order determinant. As a consequence, the decrease in the determinant order as compared with previously developed procedures leads to significant savings in the computational effort and cost associated with dynamic analysis of non-uniform beams with cracks. Numerical examples are given to illustrate the proposed method and to study the effect of cracks on the natural frequencies and mode shapes of cracked beams.     

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.     

An analytical method for free vibration analysis of functionally graded beams with edge cracks

In this paper, an analytical method is proposed for solving the free vibration of cracked functionally graded material (FGM) beams with axial loading, rotary inertia and shear deformation. The governing differential equations of motion for an FGM beam are established and the corresponding solutions are found first. The discontinuity of rotation caused by the cracks is simulated by means of the rotational spring model. Based on the transfer matrix method, then the recurrence formula is developed to get the eigenvalue equations of free vibration of FGM beams. The main advantage of the proposed method is that the eigenvalue equation for vibrating beams with an arbitrary number of cracks can be conveniently determined from a third-order determinant. Due to the decrease in the determinant order as compared with previous methods, the developed method is simpler and more convenient to analytically solve the free vibration problem of cracked FGM beams. Moreover, free vibration analyses of the Euler–Bernoulli and Timoshenko beams with any number of cracks can be conducted using the unified procedure based on the developed method. These advantages of the proposed procedure would be more remarkable as the increase of the number of cracks. A comprehensive analysis is conducted to investigate the influences of the location and total number of cracks, material properties, axial load, inertia and end supports on the natural frequencies and vibration mode shapes of FGM beams. The present work may be useful for the design and control of damaged structures.     

Resonance characteristics of high-speed trains passing simply supported bridges

This paper investigates the resonant characteristics of three-dimensional bridges when high-speed trains pass them. Multi-span bridges with high piers and simply supported beams were used in the dynamic finite element analysis. The dominated train frequencies proposed in this study can be clearly seen from the finite element result. To avoid resonance, the dominated train frequencies and the bridge natural frequencies should be as different as possible, especially for the first dominated train frequency and the first bridge natural frequency in each direction. If the two first frequencies are similar, the bridge resonance can be serious. This study also indicates that a suitable axial stiffness between two simple beams can reduce vibrations at a near-resonance condition. The axial stiffness of the continuous railway and the friction of the bearing plate should be enough to obtain this axial stiffness.     

Exact solutions for free vibration of shear-type structures with arbitrary distribution of mass or stiffness.

In this paper, shear-type structures such as frame buildings, etc., are treated as nonuniform shear beams (one-dimensional systems) in free-vibration analysis. The expression for describing the distribution of shear stiffness of a shear beam is arbitrary, and the distribution of mass is expressed as a functional relation with the distribution of shear stiffness, and vice versa. Using appropriate functional transformation, the governing differential equations for free vibration of nonuniform shear beams are reduced to Bessel's equations or ordinary differential equations with constant coefficients for several functional relations. Thus, classes of exact solutions for free vibrations of the shear beam with arbitrary distribution of stiffness or mass are obtained. The effect of taper on natural frequencies of nonuniform beams is investigated. Numerical examples show that the calculated natural frequencies and mode shapes of shear-type structures are in good agreement with the field measured data and those determined by the finite-element method and Ritz method.     

VIBRATIONS OF NON-UNIFORM CONTINUOUS BEAMS UNDER MOVING LOADS

The dynamic behavior of multi-span non-uniform beams transversed by a moving load at a constant and variable velocity is investigated. The continuous beam is modelled using Bernoulli-Euler beam theory. The solution is obtained by using both the modal analysis method and the direct integration method. The natural frequencies and mode shapes used in the solution of this problem are obtained exactly by deriving the exact dynamic stiffness matrices for any polynomial variation of the cross-section along the beam using the exact element method. The mode shapes are expressed as infinite polynomial series. Using the exact mode shapes yields the exact solution for general variation of the beam section in case of constant and variable velocity. Numerical examples are presented in order to demonstrate the accuracy and the effectiveness of the present study, and the results are compared to previously published results.     

Free vibrations of spatial Timoshenko arches

This paper addresses the evaluation of the exact natural frequencies and vibration modes of structures obtained by assemblage of plane circular arched Timoshenko beams. The exact dynamic stiffness matrix of the single circular arch, in which both the in-plane and out-of-plane motions are taken into account, is derived in an useful dimensionless form by revisiting the mathematical approach already adopted by Howson and Jemah (1999 [18]), for the in plane and the out-of-plan natural frequencies of curved Timoshenko beams. The knowledge of the exact dynamic stiffness matrix of the single arch makes the direct evaluation of the exact global dynamic stiffness matrix of spatial arch structures possible. Furthermore, it allows the exact evaluation of the frequencies and the corresponding vibration modes, for the distributed parameter model, through the application of the Wittrick and Williams algorithm. Consistently with the dimensionless form proposed in the derivation of the equations of motion and the dynamic stiffness matrix, an original and extensive parametric analysis on the in-plane and out-of-plane dynamic behaviour of the single arch, for a wide range of structural and geometrical dimensionless parameters, has been performed. Moreover, some numerical applications, relative to the evaluation of exact frequencies and the corresponding mode shapes in spatial arched structures, are reported. The exact solution has been numerically validated by comparing the results with those obtained by a refined finite element simulation.     

THE DYNAMIC STIFFNESS MATRIX METHOD IN FORCED VIBRATION ANALYSIS OF MULTIPLE-CRACKED BEAM

The dynamic behaviour of a beam with numerous transverse cracks is studied. Based on the equivalent rotational spring model of crack and the transfer matrix for beam, the dynamic stiffness matrix method has been developed for spectral analysis of forced vibration of a multiple cracked beam. As a particular case, when the excitation frequency is close to zero, the solution for static response of beam with an arbitrary number of cracks has been obtained exactly in an analytical form. In general case, the effect of crack number and depth on the dynamic response of beam was analyzed numerically.     

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.     

Dynamic stiffness method for inplane free vibration of rotating beams including Coriolis effects

The paper addresses the in-plane free vibration analysis of rotating beams using an exact dynamic stiffness method. The analysis includes the Coriolis effects in the free vibratory motion as well as the effects of an arbitrary hub radius and an outboard force. The investigation focuses on the formulation of the frequency dependent dynamic stiffness matrix to perform exact modal analysis of rotating beams or beam assemblies. The governing differential equations of motion, derived from Hamilton's principle, are solved using the Frobenius method. Natural boundary conditions resulting from the Hamiltonian formulation enable expressions for nodal forces to be obtained in terms of arbitrary constants. The dynamic stiffness matrix is developed by relating the amplitudes of the nodal forces to those of the corresponding responses, thereby eliminating the arbitrary constants. Then the natural frequencies and mode shapes follow from the application of the Wittrick–Williams algorithm. Numerical results for an individual rotating beam for cantilever boundary condition are given and some results are validated. The influences of Coriolis effects, rotational speed and hub radius on the natural frequencies and mode shapes are illustrated.     

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.     

Coupled bending and torsional vibration of axially loaded Bernoulli-Euler beams including warping effects

The dynamic transfer matrix method for determining natural frequencies and mode shapes of the bending-torsion coupled vibration of axially loaded thin-walled beams with monosymmetrical cross sections is developed by using a general solution of the governing differential equations of motion based on Bernoulli-Euler beam theory. This method takes into account the effect of warping stiffness and gives allowance to the presence of axial force. The dynamic transfer matrix is derived in detail. Two illustrative examples on the application of the present theory are given for bending-torsion coupled beams with thin-walled open cross sections. The effects of axial load and warping stiffness on coupled bending-torsional frequencies are discussed. Compared with those available in the literature, numerical results demonstrate the accuracy and effectiveness of the proposed method.     

FREE VIBRATIONS OF SELF-STRAINED ASSEMBLIES OF BEAMS

This paper examines the natural frequencies and modes of transverse vibration of two simple redundant systems comprising straight uniform Euler-Bernoulli beams in which there are internal self-balancing axial loads (e.g., loads due to non-uniform thermal strains). The simplest system consists of two parallel beams joined at their ends and the other is a 6-beam rectangular plane frame. Symmetric mode vibration normal to the plane of the frame is studied. Transcendental frequency equations are established for the different systems. Computed frequencies and modes are presented which show the effect of (1) varying the axial loads over a wide range, up to and beyond the values which cause individual members to buckle (2) pinning or fixing the beam joints (3) varying the relative flexural stiffness of the component beams. When the internal axial loads first cause any one of the component beams to buckle, the fundamental frequency of the whole system vanishes. The critical axial loads required for this are determined. A simple criterion has been identified to predict whether a small increase from zero in the axial compressive load in any one member causes the natural frequencies of the whole system to rise or fall. It is shown that this depends on the relative flexural stiffnesses and buckling loads of the different members. Computed modes of vibration show that when the axial modes reach their critical values, the buckled beam(s) distort with large amplitudes while the unbuckled beam(s) move either as rigid bodies or with bending which decays rapidly from the ends to a near-rigid-body movement over the central part of the beam. The modes of the systems with fixed joints change very little (if at all) with changing axial load, except when the load is close to the value which maximizes or minimizes the frequency. In a narrow range around this load the mode changes rapidly. The results provide an explanation for some computed results (as yet unpublished) for the flexural modes and frequencies of flat plates with non-uniform thermal stress distributions.     

Vibration and stability of elastically supported multi-span beams under conservative and non-conservative loads

This paper deals with the vibration and stability of multi-span beams elastically supported against translation and rotation at several intermediate points as well as both ends. The beam is subjected to an axial or tangential load at the ends. The problem is studied on the basis of the Timoshenko beam theory. The influence of the support stiffness on the natural frequencies and the divergence and flutter instability loads are studied in detail.