Mode Shape Data Research Articles

Detection and localisation of damage in bridge model by spatial wavelet based approach

The objective of current work is to show through modelling of a bridge with damage, the effectiveness of using wavelet transform by means of its capability to detect and localise damage. The spatial data required by the wavelet transform are derived from the mode shape data of the bridge model using (ANSYS 8.0) for undamaged and damaged cases. Damage is induced by reducing stiffness of only 4-6 elements. The wavelet coefficients are derived at various scale indices as to show the effect of local perturbations in coefficients due to difference in mode shapes of damaged and undamaged cases of first few modes to quantify the damage index and location. The Continuous Wavelet Transform (CWT) using complex Gaussian wavelet is used to get the coefficients. It is observed that by using wavelet transform on the modal data as input, damage can be quantified even at very small levels of damages and one can usefully exploit appropriate mode(s) showing larger sensitivity of detection.

Fourier Analysis of Mode Shapes of Damaged Beams

This paper investigates the effect of damage on beams with fixed boundary conditions using Fourier analysis of the mode shapes in spatial domain. A finite element model is used to obtain the mode shapes of a damaged fixed-fixed beam. Then the damaged beams are studied using a spatial Fourier analysis. This approach contrasts with the typical time domain application of Fourier analysis for vibration problems. It is found that damage causes considerable change in the Fourier coefficients of the mode shapes. The Fourier coefficients, especially the higher harmonics, are found to be sensitive to both damage size and location and amplify the changes in the mode shape due to the damage. Therefore, we formulate a damage index in the form of a vector of Fourier coefficients which is robust and unique for a given damage size and damage location. The effect of noise in the mode shape data is considered and it is found that Fourier coefficients provide a useful indication of damage even in the presence of noise. Various damage levels are considered and it is found that higher modes are needed to detect small amount of damage.

Crack detection in simply supported beams without baseline modal parameters by stationary wavelet transform

There are significant changes in the vibration responses of cracked structures when the crack depth is significant in comparison to the depth of the structure. This fact enables the identification of cracks in structures from their vibration response data. However when the crack is relatively small, it is difficult to identify the presence of the crack by a mere observation of the vibration response data. A new approach for crack detection in beam-like structures is presented and applied to cracked simply supported beams in this paper. The approach is based on finding the difference between two sets of detail coefficients obtained by the use of the stationary wavelet transform (SWT) of two sets of mode shape data of the beam-like structure. These two sets of mode shape data, which constitute two new signal series, are obtained and reconstructed from the modal displacement data of a cracked simply supported beam. They represent the left half and the modified right half of the modal data of the simply supported beam. SWT is a redundant transform that doubles the number of input samples at each iteration. It provides a more accurate estimate of the variances at each scale and facilitates the identification of salient features in a signal, especially for recognising noise or signal rupture. It is well known that the mode shape of a beam containing a small crack is apparently a single smooth curve like that of an uncracked beam. However, the mode shape of the cracked beam actually exhibits a local peak or discontinuity in the region of damage. Therefore, the mode shape ‘signal’ of a cracked beam can be approximately considered as that of the uncracked beam contaminated by ‘noise’, which consists of response noise and the additional response due to the crack. Thus, the modal data can be decomposed by SWT into a smooth curve, called the approximation coefficient, and a detail coefficient. The difference of the detail coefficients of the two new signal series includes crack information that is useful for damage detection. The modal responses of the damaged simply supported beams used are computed using the finite element method. For real cases, mode shape data are affected by experimental noise. Therefore, mode shape data with a normally distributed random noise are also studied. The results show that the proposed method has great potential in crack detection of beam-like structures as it does not require the modal parameters of an uncracked beam as a baseline for crack detection. The effects of crack size, depth and location, and the effects of sampling interval are examined.

Curvature mode shape-based damage detection in composite laminated plates

Abstract There is a pressing need to develop effective techniques for structural health monitoring, so that the safety and integrity of the composite structures can be improved. The main objective of this study is to evaluate dynamics-based damage detection techniques for composite laminated plates using smart piezoelectric materials and modern instrumentation like scanning laser vibrometer (SLV). The study comprises of testing an E-glass/epoxy composite plate with an embedded delamination using two different actuator–sensor systems: (1) SLV with PZT (lead–zirconate–titanate) actuators (PZT-SLV), and (2) polyvinylidenefluoride (PVDF) sensors with PZT actuators (PZT-PVDF). The numerical finite element (FE) analysis is also performed to complement the damage detection. The experimental and numerical curvature mode shapes are primarily used to detect the presence, location, and size of the delamination. Three relatively new damage detection algorithms (i.e., simplified gapped smoothing method (GSM), generalized fractal dimension (GFD), and strain energy method (SEM)) are employed to analyze the experimental and numerical curvature mode shape data and uniform load surface (ULS) curvatures. From the damage detection outcomes, it is observed that the PZT-SLV system proves to be more convenient and effective, and it is capable of scanning a large number of points over the entire plate specimens; while the PZT-PVDF system, in which the curvature mode shapes are directly acquired, exhibits good sensitivity to damage. The damage detection algorithms like GSM, GFD and SEM based on the utilization of three consecutive mode curvatures (modes 3–5) and resulting ULS curvature successfully identify the presence, location, and relative size of delamination in the composite plate.

Diagnosis of a portal frame using advanced signal processing of laser vibrometer data

This paper introduces an original, non-destructive diagnosis tool using modal data only. Our objective is twofold: to identify damage in structural elements (beams) and to establish a classification of the connections between elements using no a priori information except the modal data. This paper introduces enhanced wavelet analysis as a tool for damage detection based on mode shape data, using an analytical mode shape fit as the undamaged state. Then we demonstrate that damage severity can be identified using a supervised neural networks approach. To supplement the local diagnosis, boundary conditions are characterised using spatial frequency. Finally, a damaged beam of a wooden portal is analysed using a high-resolution mode shape acquired with a laser Doppler vibrometer. The experimental results confirm the accuracy of the method for detecting damages and classifying boundary conditions.

Determination of damage location in RC beams using mode shape derivatives

This paper describes the determination of the location of damage due to single cracks and due to honeycombs in RC beams using mode shape derivatives from modal testing. The cracks were induced by application of point loads at predetermined locations on the RC beams. The load was increased in stages to obtain different crack heights to simulate the extent and severity of damage. Experimental modal analysis was performed on the beams with cracks prior to and after each load cycle, on a control beam, and beams with honeycombs. The mode shapes and the eigenvectors were used to determine the location of damage. The indicator | λ 4 | was obtained by rearranging the equation for free transverse vibration of a uniform beam, and applying the fourth order centered finite-divided difference formula to the regressed mode shape data. The equation is an eigenvalue problem, and the value of | λ 4 | will be a constant. Differences in the values indicate stiffness change, and the affected region indicates the general area of damage. Analysis of results using | λ 4 | was able to indicate the general region of damage, the exact location being around the center of the region. Curve fitting with Chebyshev series rationals onto the mode shape also highlighted points of high residuals around the region of damage. The proposed algorithm on the mode shape can form the basis of a technique for structural health monitoring of damaged reinforced concrete structures.

Benchmark of Damage Localisation Algorithms Using Mode Shape Data

This paper presents a comparative study of three enhanced signal processing methods to locate damage on mode shape data. The first method called curvature mode shape is used as a reference. The second tool uses wavelet transform and singularity detection theory to locate damage. Finally we introduce the windowed fractal dimension of a signal as a tool to easily measure the local complexity of a signal. Our benchmark aims at comparing the crack detection using optimal spatial sampling under different severity, beam boundary conditions (BCs) and added noise measurements.

Structural damage detection using information fusion technique

In order to precisely identify the multiple damage locations of a structure, an information fusion technique is proposed in this paper. First, the frequency data and the mode shape data are regarded as two different information sources, and local decisions can be obtained using the multiple damage location assurance criterion (MDLAC) method and the frequency change damage detection method (FCDDM), respectively. Then, the local decisions are sent to a fusion centre. In the fusion centre, three main fusion approaches are applied to integrate all the local decisions. Finally, a global decision is acquired. The measurement errors are also taken into account in the fusion process. The numerical example and analysis demonstrate that the identification results of the three information fusion methods are better than those of the MDLAC method and the frequency change damage detection method.

Structural Modification Procedure with Mode Shape Expansion for Rib Stiffeners

This paper proposes a procedure for predicting the modal data of a thin plate structure with a rib stiffener even when the rib is attached to any location where mode shapes are not measured. The proposed procedure is based on the structural dynamics modification (SDM) method and the mode shape expansion method is incorporated to estimate the mode shape data at the unmeasured points. In the expansion process, a submodel for the area where the ribs will be attached is used instead of the whole structure model. Numerical results from a plate structure with the two L-shaped ribs show that the proposed procedure produces the accurate predictions of the modal data using the estimated mode shape data at the unmeasured points. In addition, the mode shape data is estimated accurately from the submodel with free boundary condition. Furthermore, the proper number of analysical modes to be combined in the expansion process can be determined from the condition number of the modal matrix of the submodel.

The local vibration modes due to impact on the edge of a viaduct

This paper describes the vibration responses of a cement viaduct model under impulsive force excitation. The frequencies and mode shapes of resonances can control magnitude of the structure-borne noise radiation. A steel hammer is used to excite the cement viaduct model at the centre and at the supporting edge position of the cross-section separately in order to acquire the vibration responses and mode shape data. From such data the global mode and local modes of the cement viaduct model are identified. It is shown that the edge of section supported by the web may have low impedance in the vertical direction. The results of analysis using a Finite Element Method confirmed the experimental findings of the cross-sectional modes of the cement viaduct model. The findings suggested that the vibrations of local modes are of two types: (1) Centre mode – the centre of top panel can move but the edge is fixed. (2) Edge (web) mode – the centre of panel is fixed but the edge (supported by web) can move.

Local damage detection using the two-dimensional gapped smoothing method

This paper presents a procedure for locating variability in structural stiffness. For some types of structure, this variability is directly related to manufacturing defects and/or in-service damage. Unlike many published damage detection methods, the procedure presented here uses only data obtained from the damaged structure. Baseline data and theoretical models of the undamaged structure are not used during the analysis presented here. The procedure locates regions in a structure where the stiffness varies. Providing it is known that the structure, in its undamaged state, is homogeneous with respect to stiffness, the procedure will detect the areas of inhomogeneity that are caused by the incipient damage. For non-homogeneous structures, some knowledge of the structural details (for example, engineering drawings or a baseline test) is required in order to discriminate damage. The procedure is a two-dimensional generalization of a previously published one-dimensional gapped smoothing method, whereby local features in vibration curvature shapes are extracted using a localized curve fit (i.e., smoothing). A variability index is generated for each test point on the structure. Increased variability is due either to structural stiffness features or damage. A statistical treatment of the indices enables discrimination of areas with significant stiffness variability. Providing the damaged areas are sufficiently small compared to the total surface area, their indices will be statistical outliers. The procedure can either analyze mode shape data, or frequency dependent operating displacement shape data. The procedure is demonstrated with a finite element model of a plate, and experiments on composite plates with deliberately induced multiple delaminations. Finally, the method is demonstrated on data taken from a large composite hull structure. In all cases the procedure successfully located the damaged regions.

Structural damage identification of the highway bridge Z24 by FE model updating

The development of a methodology for accurate and reliable condition assessment of civil structures has become very important. The finite element (FE) model updating method provides an efficient, non-destructive, global damage identification technique, which is based on the fact that the modal parameters (eigenfrequencies and mode shapes) of the structure are affected by structural damage. In the FE model the damage is represented by a reduction of the stiffness properties of the elements and can be identified by tuning the FE model to the measured modal parameters. This paper describes an iterative sensitivity based FE model updating method in which the discrepancies in both the eigenfrequencies and unscaled mode shape data obtained from ambient tests are minimized. Furthermore, the paper proposes the use of damage functions to approximate the stiffness distribution, as an efficient approach to reduce the number of unknowns. Additionally the optimization process is made more robust by using the trust region strategy in the implementation of the Gauss–Newton method, which is another original contribution of this work. The combination of the damage function approach with the trust region strategy is a practical alternative to the pure mathematical regularization techniques such as Tikhonov approach. Afterwards the updating procedure is validated with a real application to a prestressed concrete bridge. The damage in the highway bridge is identified by updating the Young's and the shear modulus, whose distribution over the FE model are approximated by piecewise linear functions.

STRUCTURAL DAMAGE DETECTION BASED ON A MICRO-GENETIC ALGORITHM USING INCOMPLETE AND NOISY MODAL TEST DATA

This paper describes a procedure for detecting structural damage based on a micro-genetic algorithm using incomplete and noisy modal test data. As the number of sensors used to measure modal data is normally small when compared with the degrees of freedom of the finite element model of the structure, the incomplete mode shape data are first expanded to match with all degrees of freedom of the finite element model under consideration. The elemental energy quotient difference is then employed to locate the damage domain approximately. Finally, a micro-genetic algorithm is used to quantify the damage extent by minimizing the errors between the measured data and numerical results. The process may be either of single-level or implemented through two-level search strategies. The study has covered the use of frequencies only and the combined use of both frequencies and mode shapes. The proposed method is applied to a single-span simply supported beam and a three-span continuous beam with multiple damage locations. In the study, the modal test data are simulated numerically using the finite element method. The measurement errors of modal data are simulated by superimposing random noise with appropriate magnitudes. The effectiveness of using frequencies and both frequencies and mode shapes as the data for quantification of damage extent are examined. The effects of incomplete and noisy modal test data on the accuracy of damage detection are also discussed.

Damage identification of structures with uncertain frequency and mode shape data

AbstractA statistical method with combined uncertain frequency and mode shape data for structural damage identification is proposed. By comparing the measured vibration data before damage or analytical finite element model of the intact structure with those measured after damage, the finite element model is updated so that its vibration characteristic changes are equal to the changes in the measured data as closely as possible. The effects of uncertainties in both the measured vibration data and finite element model are considered as random variables in model updating. The statistical variations of the updated finite element model are derived with perturbation method and Monte Carlo technique. The probabilities of damage existence in the structural members are then defined. The proposed method is applied to a laboratory tested steel cantilever beam and frame structure. The results show that all the damages are identified correctly with high probabilities of damage existence. Discussions are also made on the applicability of the method when no measurement data of intact structure are available. Copyright © 2002 John Wiley & Sons, Ltd.

DAMAGE DETECTION USING A MODIFIED LAPLACIAN OPERATOR ON MODE SHAPE DATA

Localized damage to a structure affects its dynamic properties, and much work has been undertaken investigating the variation of natural frequencies with damage. However, use of mode shape data has seen much less effort. This paper develops and presents a technique for identifying the location of structural damage in a beam. The procedure operates solely on the mode shape from the damaged structure, and does not requirea prioriknowledge of the undamaged structure. The procedure is developed using a one-dimensional finite element model of a beam, and demonstrated by experiment. When damage is severe (a localized thickness reduction of more than 10%), applying a finite difference approximation of Laplace's differential operator to the mode shape successfully identifies the location of the damage. However, when damage is less severe, further processing of the Laplacian output is required before the location can be determined. This post-processing enables the location of thickness reductions of less than 0·5% to be identified. The procedure is best suited to the mode shape obtained from the fundamental natural frequency. The mode shapes from higher natural frequencies can be used to verify the location of damage, but they are not as sensitive as the lower modes.

V-R model predictions of averaged radiation from a violin compared with spatial average of bridge force hammer-excited direct radiation

Mechanical modal analysis data for a violin furnish two essential components for the vibration-radiation (V-R) model [G. Bissinger, J. Acoust. Soc. Am. 97, 3154–3164 (1995)] to reliably predict the averaged radiative properties of a bowed string instrument from its measured mechanical properties, viz., (1) how strongly a particular corpus mode is excited at the bridge and (2) normal mode frequency and shape data for boundary element acoustic radiation calculations of the radiation efficiency for each corpus normal mode. Combining both of these elements in the V-R model gives predictions of a spatially averaged acoustic radiation for a violin. These were compared with similarly spatially averaged measurements of direct radiation from a violin excited by force-hammer impact at the bridge using a rotating semicircular (r=30 cm) seven-microphone array in a small anechoic chamber covering almost an entire sphere around the violin. The V-R model calculations for each mode compared quite well in general to the measured average radiation.

CORNER STRESS SINGULARITY EFFECTS ON THE VIBRATION OF RHOMBIC PLATES WITH COMBINATIONS OF CLAMPED AND SIMPLY SUPPORTED EDGES

An accurate method is presented for flexural vibrations of rhombic plates having all combinations of clamped and simply supported edge conditions. A specific feature here is that the analysis explicitly considers the bending stress singularities that occur in the two opposite, clamped–hingedand/or hinged–hinged, corners having obtuse angles in rhombic plates. The strength of these singularities increases significantly as the obtuse angles at the clamped–hinged and/or hinged–hinged corners increases. Stationary condition of single-field, classical thin-plate Lagrangian functional are derived using the Ritz method. Assumed transverse displacements are constructed from a hybrid set of (i) admissible and mathematically complete algebraic polynomials, and (ii) comparison functions (termed here “corner functions”) which account for both the kinematic boundary condition and the bending stress singularities at the obtuse clamped–hinged and/or hinged–hinged corners. Extensive convergence studies demonstrate that the corner functions accelerate the convergence of solutions, and that these function are required if accurate solutions are to be obtained for highly skewed plates. Accurate non-dimensional frequencies and normalized contours of the vibratory transverse displacement are presented for rhombic plates having a large enough skew angle of 75 ° (i.e., obtuse corner angles of 165 °), so that the significant influence of the corner stress singularities may be clearly understood. Accurate solutions for isosceles triangular plates with various combinations of clamped-hinged edges are also available from the frequency and mode shape data presented. The upper bound frequency solutions derived from the presented. The upper bound frequency solutions derived from the present analysis are shown to improve upon the existing upper bound results in the published literature.

Updating model parameters by adding an imagined stiffness to the structure

A mass addition technique for structural parameters updating has recently been described by the authors. It uses eigenvalues of the structure, before and after it is perturbed by adding mass, to adjust selected parameters by sensitivity analysis. The technique avoids the use of noisy mode shape data and overcomes the problem of a non-unique set of parameters when eigenvalues alone are used. However, it is not suitable for a large structure because of the difficulty of adding the necessarily large perturbing mass to the existing structure. This paper presents an alternative technique. An "imagined" stiffness is added to the structure and the FRF of this perturbed structure is obtained from the measured FRF of the original structure by a simple structural modification technique. The eigenvalues of the original and perturbed structures are obtained from the measured and constructed FRFs respectively. These eigenvalues, together with the eigenvalues predicted from an analytical model of the structure, are used to adjust the structural parameters by sensitivity analysis. The technique is demonstrated by a simulated example and by an experiment on an H-frame.

Signal Separation Method for Tower Mode Shape Measurement

Vibration shapes of the fundamental modes of eight-legged, K-braced, offshore towers undergo significant changes when important braces are damaged. These changes, measurable at deck-level, provide valuable clues for detecting damage. Because of the small difference between vibration frequencies of the two fundamental translational modes in these towers, the mode shapes are difficult to measure. A simple method is presented which solves this difficulty. The method uses transient acceleration response records that can be produced by different excitation procedures, and no knowledge of the input that causes the excitation is necessary. The finite Fourier transform of such records is examined, and its important properties are pointed out for low damping. Based on these properties, procedures are set forth by which mode shape data can be retrieved from closely spaced peaks in the spectra. The retrieval is done by separating these peaks from each other. Applications of the method in a numerical experiment, and in a physical experiment on a 1/50th scale offshore platform model, are presented. Application of the method to full-scale towers is discussed.

A comparison of optical and numerical methods for analyzing structural vibrations

An optical holographic interferometry method has been used to determine the resonant frequencies, mode shapes, and peak displacement amplitudes of an irregular, clamped, vibrating plate. The plate was driven by an electromagnetic shaker with swept sine wave execution and resonances were observed by time holographic interferometry. Mode shapes at several resonant frequencies were then photographed by a time‐average holographic interferometry method. The observed resonant shapes were compared with predictions obtained by two finite element programs (GIFTS and NASTRAN). Facilities required for each method of solution and procedures for collecting and interpreting mode shape data will be described. Applications of the optical and numerical methods to problems involving damped vibrating structures will be discussed. [Research supported by DTNSRDC and ONR.]