Mode Shape Curvature Research Articles

DAMAGE DETECTION IN LARGE STRUCTURES USING MODES SHAPES AND ITS DERIVATIVES

Synopsis: Many techniques for structural damage detection using modal parameters have been developed over few years because the modal parameters are easily obtained from free, forced and ambient conditions. Many of the techniques have shown that mode shape curvature is efficient to localize damage in continuous system. Focusing on the damage detection/localization of typical Indian Railway bridges, in this paper, a numerical model of an existing steel truss Indian Railway Bridge (discrete system) was developed to check the validity of the mathematical model for continuous shear beam. Numerical model was based on the design drawings of the structure and Eigen value analysis was performed to extract modal parameters. Damage was simulated by modifying the member properties of particular truss members. Results demonstrate that change in fundamental mode shape due damage to a member is an excellent indicator of damage localization because it is discontinuous at damage location. Change in higher derivatives (slope and curvature) of the fundamental mode shape is found to have improved sensitivity in damage localization. These results are similar to those of the shear beams. However, the damage signature is different. Further studies are needed to understand such signatures for meaningful application of the method for existing structures.

Damage detection for plate-like structures using generalized curvature mode shape method

Mode shape curvature-based method was originally developed to identify the damage in one-dimensional beam-like structures. In this paper, the method is generalized to fit for damage identification of plate-like structures, using two-dimensional mode shape curvatures. The modified method only requires the first few curvature mode shapes of the plate before and after damage, or only need modal information of the damaged plate if gapped smoothing technique is employed. To evaluate the effectiveness of the proposed method, numerical simulations considering various plate damage scenarios in terms of damage location, damage severity and multiple damages have been conducted, and experimental modal analysis on a steel plate with clamped and free edges is also performed to correlate with the numerical results. The final results show that the proposed method works well to detect the damage in a simple manner. For damaged structures, irregularities in their modal characterizations are induced at the places where local damages exist. Gapped smoothing technique takes advantage of smoothing functions and elevates the suggest method to be a reference-free approach. In the present study, a brief discussion has been performed and suggestions have been given concerning different smoothing functions used in the gapped smoothing technique.

Effect of damage on the vibration modal of a novel three-dimensional and four-directional braided composite T-beam

The effect of damage on the modal properties of a three-dimensional (3Dim) and four-directional (4Dir) braided composite T-beam was investigated by means of modal experiments and finite element analysis (FEA). The experimental tests on these damaged T-beams showed alteration of the vibration mode response. The damage obviously changed the mode shape curvatures. The influence of the damage upon the modal shape curvatures of the T-beam could be used to detect and localize the damage. Finite element modeling (FEM) of the damaged T-beam revealed that the damage affected the natural frequency of the T-beam but did not affect its modal shape.

Damage Detection in Laminated Composite Beams, Plates and Shells Using Dynamic Analysis

Vibration-based technique to detect damage in laminated composite beams, rectangular plates and cylindrical shells is presented in this paper.A parameter called damage indicator calculated based on mode shape curvature isused in this studyto detect the location and size of small damages accurately in laminated composite structures. Through numerical analysis of laminated compositecantilevered beam, plate and cylindrical shell models with edge crack as damage, the absolute change inthe damage indicator is localized in the region of damage. Thechange in damage indicatorincreases withincreasing size of damage. Thisinformationis obtained by considering two cases of damage sizes (case-1 and case-2)in the structures. Finite element methodbased commercial analysis package ANSYSis used to obtain thenormalized displacement mode shapesof the three models both for intact and damaged states and then the damage indicator is calculated from the mode shapes data.The numerical analysis to detect damage is followed by validation by experimental modal testing.

Modal parametric changes in a steel bridge with retrofitting

This paper presents the status improvement of an old damaged deck type rural road steel truss bridge through the modal parametric study after partial retrofitting. The dynamic and static tests on bridge were carried out as in damaged state and after partial retrofitting. The dynamic testing on the steel bridge was carried out using accelerometers under similar environmental conditions with same speed of the moving vehicle. The comparison of the modal parameters i.e., frequency, mode shape mode shape curvature, modal strain energy, along with the deflection parameter are studied with respect to structural analytical model parameters. The status up gradation for the upper and downstream truss obtained was different due to differential level of damage in the bridge. Also after retrofitting the structural elemental behavior obtained was not same as desired. The damage level obtained through static tests carried out using total station indicated further retrofitting requirement.

Application of higher order finite differences to damage localization in laminated composite plates

The localization of damage in composite plates and other structures is often performed based on mode shape curvatures. These curvatures are usually computed using finite differences. However, finite differences spread and amplify numerical and experimental errors. A technique based on the Ritz method allows to choose an optimal spatial sampling in order to minimize this problem. In the present work we apply this technique, along with the second and fourth order finite differences to compute the mode shape curvatures. To evaluate the successfulness of the damage localizations a quality evaluator is proposed in this paper. The need for the optimal spatial sampling is verified by analyzing two locations of a wide range of damage severities. Damage localizations obtained with second and fourth order finite differences were compared and it was found out that the results are better when one uses the highest order finite difference.

Sparse sensing detection of impact-induced delaminations in composite laminates

The detection of delaminations in composite structures due to impacts is a critical issue for any structural health monitoring (SHM) programme. In this work, an experimental campaign is carried out to investigate the presence of delamination induced in carbon-fibre-reinforced-plastic (CFRP) plates by low velocity impacts (LVI). A reduced number of piezoelectric devices for actuation and sensing purposes are employed. The proposed study starts with the choice of the proper positions of the piezoelectric sensor and actuator through the analysis of the numerical curvature mode shapes of the composite laminated plate. A wavelet packet transform (WPT)-based algorithm is applied on the vibrating response of the plates to extract wavelet-based damage sensitive features. Linear Discriminant Analysis (LDA) is then applied on the extracted damage sensitive features to enhance the performance of the proposed delamination identification procedure. Results show the effectiveness of the developed SHM routine when a reduced number of sensors is either desirable or mandatory.

Natural frequency changes due to damage in composite beams

Transversal cracks in structures affect their stiffness as well as the natural frequency values. This paper presents a research performed to find the way how frequencies of sandwich beams change by the occurrence of damage. The influence of the locally stored energy, for ten transverse vibration modes, on the frequency shifts is derived from a study regarding the effect of stiffness decrease, realized by means of the finite element analysis. The relation between the local value of the bending moment and the frequency drop is exemplified by a concrete case. It is demonstrated that a reference curve representing the damage severity exists whence any frequency shift is derivable in respect to damage depth and location. This curve is obtained, for isotropic and multi-layer beams as well, from the stored energy (i.e. stiffness decrease), and is similar to that attained using the stress intensity factor in fracture mechanics. Also, it is proved that, for a given crack, irrespective to its depth, the frequency drop ratio of any two transverse modes is similar. This permitted separating the effect of damage location from that of its severity and to define a Damage Location Indicator as a sequence of squared of the normalized mode shape curvatures.

Elastic Parameter Identification of Plate Structures Using Modal Response: An ECE Based Approach

AbstractThis article describes an inverse method for the identification of heterogeneous anisotropic elastic parameters of plate structure from an incomplete modal response. Many efforts in this area have shown that derivatives of the displacement mode shapes or curvature mode shapes are much more sensitive to the heterogeneous parameter distribution of the structure when compared with the mode shapes themselves. Thus, the curvature mode shapes are used widely to quantify the material profile. Following this observation, a strategy based on the error in constitutive equation (ECE) function, which naturally incorporates this fact seamlessly into the identification procedure, is proposed. The identification problem is posed as an optimization problem in that the cost function measures the discrepancy in the constitutive equation, which connects kinematically admissible curvature/strains and dynamically admissible couples/stresses. The resulting system becomes an extended system with primary and Lagrangian v...

Impact damage identification in composite skin-stiffener structures based on modal curvatures

The feasibility of a vibration-based damage identification approach for impact damage in two advanced composite skin-stiffener structures is investigated in this paper. Mode shape curvatures combined with the modal strain energy damage index (MSE-DI) algorithm are utilized to identify the damage. Special attention is paid to the effective application of this vibration-based methodology by investigating the relation between the damage location, the structural design, and the dynamic behavior. The performance of a 1D and 2D formulation of the MSE-DI algorithm is compared for several damage scenarios. Experiments demonstrated the capabilities of the MSE-DI algorithm to detect, localize, and roughly quantify the size of barely visible impact damage in advanced composite structures. It is concluded that the method is particularly effective for health monitoring of skin-stiffener connections. The most effective results were obtained by considering the 1D formulation in the direction of the stiffeners for the stiffener mid-section and perpendicular to the stiffeners for the stiffener run-out. The method remained inconclusive in the case of pure skin-related damage. The results obtained in this study contribute to the development of guidelines for vibration-based structural health monitoring of advanced composite skin-stiffener structures. Copyright © 2015 John Wiley & Sons, Ltd.

Comparative study of performance of neutral axis tracking based damage detection

This paper presents a comparative study of a novel SHM technique for damage isolation. The performance of the Neutral Axis (NA) tracking based damage detection strategy is compared to other popularly used vibration based damage detection methods viz. ECOMAC, Mode Shape Curvature Method and Strain Flexibility Index Method. The sensitivity of the novel method is compared under changing ambient temperature conditions and in the presence of measurement noise. Finite Element Analysis (FEA) of the DTU 10 MW Wind Turbine was conducted to compare the local damage identification capability of each method and the results are presented. Under the conditions examined, the proposed method was found to be robust to ambient condition changes and measurement noise. The damage identification in some is either at par with the methods mentioned in the literature or better under the investigated damage scenarios.

Two-dimensional curvature mode shape method based on wavelets and Teager energy for damage detection in plates

Vibration-based damage detection in plates has been investigated by various methods relying on mode shapes, among which the 2D curvature mode shape is a damage feature attracting much attention of researchers. Unlike the sound understanding of the use of the 1D curvature mode shape for detecting damage in beams, however, use of the 2D curvature mode shape to detect damage in plates is not yet well elucidated, major unresolved issues including lack of clarity about the mechanism of characterizing damage, susceptibility to noise, and insensitivity to sight damage. These deficiencies severely hamper use of the 2D curvature mode shape to portray damage in plates. To deal with these deficiencies, the mechanism of using 2D curvature mode shape to depict damage is analytically clarified in light of thin plate theory. On the basis of this clarification, a synergy between wavelet transform and a Teager energy operator is proposed to tackle the other deficiencies of susceptibility to noise and insensitivity to sight damage, leading an enhanced 2D curvature mode shape. The efficacy of the enhanced 2D curvature mode shape is numerically demonstrated using finite element simulations and experimentally validated through noncontact measurement by a scanning laser vibrometer, whereby its advances of clear mechanism of characterizing damage, robustness against noise, and sensitivity to slight damage are sufficiently corroborated.

A new damage detection indicator for beams based on mode shape data

In this paper, a new damage indicator based on mode shape data is introduced to identify damage in beam structures. In order to construct the indicator proposed, the mode shape, mode shape slope and mode shape curvature of a beam before and after damage are utilized. Mode shape data of the beam are first obtained here using a finite element modeling and then the slope and curvature of mode shape are evaluated via the central finite difference method. In order to assess the robustness of the proposed indicator, two test examples including a simply supported beam and a two-span beam are considered. Numerical results demonstrate that using the proposed indicator, the location of single and multiple damage cases having different characteristics can be accurately determined. Moreover, the indicator shows a better performance when compared with a well-known indicator found in the literature.

Tikhonov's regularization approach in mode shape curvature analysis applied to damage detection

In this paper an on-going research effort aimed at detecting and localizing damage in plate-like structures by using mode shape curvature based damage detection algorithm is described. The proposed damage index uses exclusively mode shape curvature data from the damaged structure. This method was originally developed for beam-like structures. In this paper, the method is generalized to plate-like structures which are characterized by two-dimensional mode shape curvature. To calculate mode shape curvatures from the measured mode shapes, three approaches are proposed: the first one is the well-known central difference approximation, the other two are classical approaches based on Tikhonov's regularization technique with smoothing functional. The applicability and effectiveness of the proposed damage detection algorithms are demonstrated experimentally on an aluminium plate containing mill-cut damage. The validity of the method is assessed by comparing the identification results of the experimental test case to the results obtained from the simulated test case. The modal frequencies and the corresponding mode shapes of the aluminium plate are obtained via finite element models for numerical simulations and by using a scanning laser vibrometer (SLV) for the experimental study.

Numerical and Experimental Study on the Application of Mode Shape Curvature for Damage Detection in Plate-Like Structures

The paper describes on-going research effort at detecting and localizing damage in plate-like structures using mode shape curvature based damage detection algorithm. The proposed damage index uses data on exclusively mode shape curvature from the damaged structure. This method was originally developed for beam-like structures. The article generalizes the method of plate-like structures characterized by two-dimensional mode shape curvature. To examine limitations of the method, several sets of simulated data are applied and the obtained results of the numerical detection of damage are validated by comparing them with the findings of the case of the experimental test. The simulated test cases include the damage of various levels of severity. In order to ascertain the sensitivity of the proposed method for noisy experimental data, numerical mode shapes are corrupted with different levels of random noise. Modal frequencies and corresponding mode shapes of an aluminium plate containing mill-cut damage are obtained via finite element models for numerical simulations and by using a scanning laser vibrometer (SLV) for the experimental study.

Non-Destructive Evaluation Method Based On Dynamic Invariant Stress Resultants

Most of the vibration based damage detection methods are based on changes in frequencies, mode shapes, mode shape curvature, and flexibilities. These methods are limited and typically can only detect the presence and location of damage. Current methods seldom can identify the exact severity of damage to structures. This paper will present research in the development of a new non-destructive evaluation method to identify the existence, location, and severity of damage for structural systems. The method utilizes the concept of invariant stress resultants (ISR). The basic concept of ISR is that at any given cross section the resultant internal force distribution in a structural member is not affected by the inflicted damage. The method utilizes dynamic analysis of the structure to simulate direct measurements of acceleration, velocity and displacement simultaneously. The proposed dynamic ISR method is developed and utilized to detect the damage of corresponding changes in mass, damping and stiffness. The objectives of this research are to develop the basic theory of the dynamic ISR method, apply it to the specific types of structures, and verify the accuracy of the developed theory. Numerical results that demonstrate the application of the method will reflect the advanced sensitivity and accuracy in characterizing multiple damage locations.

Damage Identification by Using a Self-Synchronizing Multipoint Laser Doppler Vibrometer

The vibration-based damage identification method extracts the damage location and severity information from the change of modal properties, such as natural frequency and mode shape. Its performance and accuracy depends on the measurement precision. Laser Doppler vibrometer (LDV) provides a noncontact vibration measurement of high quality, but usually it can only do sampling on a single point. Scanning LDV is normally used to obtain the mode shape with a longer scanning time. In this paper, a damage detection technique is proposed using a self-synchronizing multipoint LDV. Multiple laser beams with various frequency shifts are projected on different points of the object, reflected and interfered with a common reference beam. The interference signal containing synchronized temporal vibration information of multiple spatial points is captured by a single photodetector and can be retrieved in a very short period. Experiments are conducted to measure the natural frequencies and mode shapes of pre- and postcrack cantilever beams. Mode shape curvature is calculated by numerical interpolation and windowed Fourier analysis. The results show that the artificial crack can be identified precisely from the change of natural frequencies and the difference of mode shape curvature squares.

The use of modal curvatures for damage localization in beam-type structures

The localization of stiffness variation in damaged beams through modal curvatures, i.e., second derivative of mode shapes, is studied by exploiting a perturbative solution of the Euler–Bernoulli equation. It is shown that for low order modes the difference between undamaged and damaged modal curvatures has only one distinct peak if the damage is localized in a narrow region. This phenomenon is independent of the presence of experimental noise and of the technique used to reconstruct the curvature mode shapes from the displacement mode shapes. Broader damages cause the modal curvature difference to have several peaks outside the damage region that could result in a false damage localization. The same effect is present at higher modes for both narrow and broad damages. As a result, modal curvatures can be effectively used to localize structural damages only once they have been properly filtered. Here the perturbative solution is used to introduce an effective damage measure able to localize correctly narrow and broad damages and also single and multiple damages cases.

Damage assessment from curvature mode shape using unified particle swarm optimization

A two-step procedure to detect and quantify damages in structures from changes in curvature mode shapes is presented here. In the first step the maximum difference in curvature mode shapes of the undamaged and damaged structure are used for visual identification of the damaged internal-substructure. In the next step, the identified substructures are searched using unified particle swarm optimization technique for exact identification of damage location and amount. Efficiency of the developed procedure is demonstrated using beam like structures. This methodology may be extended for identifying damages in general frame structures.

Damage identification techniques via modal curvature analysis: Overview and comparison

This paper aims to compare several damage identification methods based on the analysis of modal curvature and related quantities (natural frequencies and modal strain energy) by evaluating their performances on the same test case, a damaged Euler–Bernoulli beam. Damage is modelled as a localized and uniform reduction of stiffness so that closed-form expressions of the mode-shape curvatures can be analytically computed and data accuracy, which affects final results, can be controlled. The selected techniques belong to two categories: one includes several methods that need reference data for detecting structural modifications due to damage, the second group, including the modified Laplacian operator and the fractal dimension, avoids the knowledge of the undamaged behavior for issuing a damage diagnosis. To explain better the different performances of the methods, the mathematical formulation has been revised in some cases so as to fit into a common framework where the underlying hypotheses are clearly stated. Because the various damage indexes are calculated on ‘exact’ data, a sensitivity analysis has been carried out with respect to the number of points where curvature information is available, to the position of damage between adjacent points, to the modes involved in the index computation. In this way, this analysis intends to point out comparatively the capability of locating and estimating damage of each method along with some critical issues already present with noiseless data.