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ABSTRACT Structural damage diagnosis using modal data

Many existing structures, which were constructed several decades ago, are still in service, and many have deteriorated. For example, it has been pointed out that nearly 1/3–1/2 of America’s infrastructure, such as bridges, railways and school buildings, is structurally deficient and needs to be repaired [1]. Therefore, the early detection, monitoring and analysis of a damaged structure is vital for its safe performance. While there are many techniques and approaches presented in the nondestructive evaluation (NDE) of structural systems, existing damage identification methods can be categorized into dynamic and static identification methods, using corresponding data. The purpose is to adjust the parameters of numerical or analytical models to match analytical and measured data.

Structural damage is considered to be changes in structural parameters that adversely affect performance. Damage may also be defined as any deviation in the original geometrics of structures or material properties that may cause undesirable stresses, displacements or vibrations in the structure. These weaknesses and deviations may be due to cracks, loose bolts, broken welds, corrosion, fatigue, etc.

During the past few decades, dynamic identification techniques have been developed more maturely compared with the static approach, and the corresponding literature is quite extensive. The basic premise of a vibration-based damage detection method is that changes in physical properties will cause changes in the measured dynamic response of the structure or modal parameters. These changes provide a feature to evaluate the structural state. Detailed literature reviews have been provided by Doebling et al. [2], Stubbs et al. [3] and Mottershead and Friswell [4].

The modal-based model updating technique relies on modal characteristics data obtained from an experimental modal analysis extracted from the measured FRF data indirectly. Modal based methods attempt to correlate the changes in natural frequencies, mode shapes, mode shape curvature or frequency response functions with the occurrence of structural damage. Thus, these methods are developed essentially as applications of traditional experimental modal analysis procedures [2,3,5]. Natural frequencies are the global properties of the structure and, thus, they can be measured at a few locations or even at one point [6]. Furthermore, a number of methods, based on mode shapes, for the damaged structure are proposed, such as sensitivity (perturbation) methods [7,8], modal force error methods [3,9,10], modal residual methods [11,12] and techniques using genetic algorithms and neural networks [13,14].

A main category of model updating methods use structural responses in the frequency domain to tune structural models. In this class of model updating methods, FE models are updated in view of the fully damped response along a frequency axis. The main advantage of FRF methods is that the amount of available test data is not limited to a few identified eigenvalues and eigenvectors, and FEM updating can be performed using many more data points [15,16]. It must be noted that although FRF methods provide model updating techniques with more data, modal data represent more information about structural conditions using less data points.

The main difficulties lie in uncertainties in FE modeling and errors related to modal testing [17]. Uncertainties in the FE model exist due to inaccurate physical parameters, non-ideal boundary conditions and structural non-linear properties. With respect to modal testing, measurement noise is inevitable and the maximum number of measurement locations is limited. Moreover, it has not been possible until now to measure some degrees of freedom, such as rotational and internal. Therefore, the number of equations in model updating is usually smaller than that of the unknown parameters of the model. Hence, it is an underdetermined problem and a small error may cause a large deviation in the results [18].

Model reduction or data expansion is a challenge to overcome incomplete measurements. Lim [19] proposed a systematic method that provides precise identification of damage locations and extent, when exact measured modes at every finite element DOF are used. Also, a procedure was presented to perform damage detection with inaccurate and incomplete measured modes. Sanayei et al. [20] use natural frequencies and associated mode shapes measured at a selected subset of Degrees Of Freedom (DOF) for stiffness and mass parameter estimation, through a condensation algorithm. A model based damage identification method is proposed by Ren and De Roeck [21,22] to predict the location and severity of damage, based on the work done by Araújo dos Santos et al. [23] (1998). In this work, it was demonstrated that multiplying damaged eigenvalue equations with damaged or undamaged modes provides more equations than the strain energy-based method to guarantee damage localization.

Also, to achieve a linear sensitivity equation of modal parameter sensitivity, Chen and Bicanic [7] expressed the change of any mode shape as a linear combination of the original eigenvectors of the intact structure. Mode shape participation factors are a function of measured natural frequencies of a damaged structure, the stiffness matrix perturbation and the changes of mode shape itself. The number of unknowns is equal to the number of structural parameters, plus the number of measured natural frequencies multiplied by the number of mode shapes, which increases the number of unknowns in the optimization problem.

As a drawback of FEM-update techniques, the requirement of reducing FEM degrees of freedom or extending measured modal parameters may result in the loss of physical interpretability and errors, due to the stiffness diffusion that smears the damage-induced localized changes in the stiffness matrix into the entire stiffness matrix. Using incomplete acquired data, Pothisiri and Hjelmstad [24] introduced a method by rearranging the degrees of freedom. Bakhtiari-Nejad et al. [25] proposed a damage detection method using incomplete measured mode shapes, by assuming one of the modal displacements to be equal to one and, then, deriving a set of equations that related the modal displacement of the damaged structure to changes in structural parameters. Furthermore, they extended their previous work and presented a diagnostic algorithm, based on the damage equation method of Ren and De Roeck [21]. The authors applied the damage equations, using the incomplete mode shape data [26].

Furthermore, most modal based model updating techniques are forced to use a number of equations less than the number of unknown parameters. Therefore, proposing various types of sensitivity equation, using the same input data, will improve the robustness of model updating algorithms against measurements errors.

In the present work, the unmeasured part of the modal displacement (eigenvector) of a structure is expressed as a function of the measured part of the mode shape, frequency (eigenvalue), structural parameters and mass matrix of the structure. An element level damage equation was characterized using the mode shapes of an intact structure and the partially measured mode shapes of a damaged structure. An optimization criterion is used to solve equations for estimating the structural parameters. Noise in the measurement is simulated by adding a proportional random error to the exact data obtained from the finite element model of the damaged structure and the issue of selection of measurement locations is investigated.