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 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.     

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.     

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.     

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.     

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.     

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.     

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.     

Free vibration analysis of elastically connected multiple-beams by using the Adomian modified decomposition method

The Adomian modified decomposition method (AMDM) is employed in this paper to investigate the free vibrations of N elastically connected parallel Euler–Bernoulli beams, which are continuously joined by a Winkler-type elastic layer. The proposed AMDM method can be used to analyze the vibration of beam system consisting of an arbitrary number of beams. By using boundary conditions the natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The numerical results for different boundary conditions, beam numbers and the stiffness of the Winkler-type elastic layer are presented. It is shown that the AMDM offers an accurate and effective method of free vibration analysis of multiple-connected beams with arbitrary boundary conditions.     

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.     

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.     

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.     

Free vibration analysis of corner-supported rectangular plates with symmetrically distributed edge beams

Analytical type solutions are obtained for the free vibration frequencies and mode shapes of thin corner-supported rectangular plates with symmetrically distributed reinforcing beams, or strips, attached to the plate edges. The method of superposition is employed. Equations governing reactions at plate-beam interfaces are developed in dimensionless form. The approach is comprehensive in that both lateral and rotational stiffness, and inertia, of the beam are incorporated into the analysis. For illustrative purposes computed eigenvalues and mode shapes are presented for two plate-beam systems of realistic geometries. It is shown that the method is easily extended to cover the case where the edge beams do not have a symmetrical distribution. This appears to be the first comprehensive analytical study of this problem of industrial interest.     

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.     

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.     

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.     

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.     

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.     

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.     

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.