CHAPTER 3 DAMAGE DIAGNOSIS OF STRUCTURES VIA NEURAL
3.2 Damage Detection Using The UFN Model
Based on recent developments in measuring and data analyzing techniques, modal data (such as natural frequencies and mode shapes) of a structural system can easily be obtained through utilizing system identification procedure. Therefore, the damage detection approach has been developed on the basis of the available natural frequencies and mode shapes of the structures.
3.2.1 Index for Damage Localization
For an undamaged structure, the modal characteristics are described by the following eigenvalue equation:
N
i i
i ] 0 for 1,...,
[K−λMφ = = (3.1) where λi is the ith modal eigenvalue which presents the square of the natural frequency of the structure; φi is the ith eigenvector which presents the mode shape of the structure; K and M are symmetric system stiffness and mass matrices, respectively.
Generally, the damage of a structure is assumed to be the reduction of stiffness but not the loss of mass in structural elements, then the eigenvalue equation for such a damaged structure becomes
0 ) ](
) (
)
[(K−∆K − λi −∆λi M φi −∆φi = (3.2) Assume the system stiffness matrix is the combination of individual member stiffness matrices. The change in stiffness matrix due to damage then be expressed as
∑
=where k is the individual stiffness matrix for the eth element; Ne d is the total number of damaged elements in the structure; and αe, which is within the range between 0 and 1, is the coefficient defining a fractional change of the eth elemental stiffness matrix. Therefore, the index, α , which is damage extent-dependent, makes estimation on the damage extent and the suffix, e, which is damage location-dependent, offers the information about the location of the damage. In the case of αe =0, the eth structural element is not damaged. When αe =1, in contrast, it means that the eth structural element is totally damaged. Accordingly, the problems of locating the damage site and evaluating the damage extent are focus on identifying the index e and computing the corresponding value of αe.
Expand equation (3.2) and neglect the higher order terms of ∆ yields
=0
Pre-multiply equation (3.4) with φiT, the change in eigenvalue is then expressed by
i
This equation expresses the relationship between the structural damage and the change in eigenvalue of the damaged structure. The eigenvalue change is direct proportion to the extent of damage. It is seen that the change in eigenvalue is damage location-dependent (the index, e) as well as damage extent-dependent (the index, α ).
Subsequently, the relationship between the structural damage and the change in eigenvector is derived. Pre-multiply equation (3.4) with the transpose of the jth eigenvector, φjT, and use the relationship, φjTK =λjφjTM, which leads to the following equation:
where ∆ is assumed to be a linear combination of the mode shapes [2], i.e. φi
Substitute equation (3.7) into equation (3.6), and introduce the orthogonal property, equation (3.6) is rearranged as
j
Impose equation (3.8) onto equation (3.6), the expression show the change in ith eigenvector of the system.
Again substituting ∆ in the above equation with equation (3.3) yields K
j
This equation, as equation (3.6), also shows that the change in eigenvector is damage location-dependent as well as damage extent-dependent. It is clear that equations (3.6) and (3.10) show the expression of changes in modal values and vectors, respectively. The changes in modal values and vectors are direct proportion to the stiffness change.
Finally, suppose single damage or multiple damages with similar severity (i.e. all αe, e=1~Nd, are identical) exist in the structure. With this assumption, the expression for the change in the ith modal vector divided by the changes in the jth modal value (i.e. divide equation (3.10) by equation (3.6)), termed Damage Localization Feature (DLF) in this work, can be used as an indicator for identifying the location of structural damage.
The location of damage to a structure is dependent only on the ratio of change in modal vectors and modal values, and can be identified by matching the measured damage localization
feature and the analytical damage localization feature. This kind of problem solving process may be categorized as the technique of pattern recognizing. And the unsupervised neural network model had been widely applied and approved an efficient tool for the problem of pattern recognition [67].
3.2.2 UFN for the Damage Detection of Structures
In the studies of damage detection that based on certain damage indices or features, two main approaches are usually adopted to deal with the detection or diagnosis process. One computed the discrepancy between the measured (or real) damage index and the FEM-based analytical damage index for all potential damage states to a structure. The case with the smallest discrepancy represents the current state for the structure [68, 69]. The other optimizes the specified objective function in which the measured information is included to search for the possible damage state [70]. Accordingly, no matter what approach is adopted, the key point of damage detection is how to rapidly and correctly identify the possible damage state according to the measured data. Therefore, one can establish the damage features for every possible damage state via the analytical FEM. When the measured damage feature is available from measurement, the damage state can then be identified through finding the same or most similar damage features.
In most previous methods, the damage case with the smallest discrepancy between the measured and analytical damage features is selected to be the possible damage state on the structure.
However, the identification of damage state basing on certain measured damage features is an inverse problem; two similar but different damage scenarios could possibly result in similar measured damage features. The relationship from the damage features to the damage state should be fuzzy but not crisp. Therefore, the damage cases with sufficient degree of ‘similarity’ between the measured and analytical damage features are selected as candidates to identify the damage state on the structure.
Note that, the Damage Localization Feature (DLF) was derived based on two assumptions:
first, the higher order terms of ∆ in equation (3.2) were neglected; second, the damage extents
for multiple damages were identical when imposing equation (3.3) on equations (3.6). A consequence was made that the damage location is depended only on
j
matter what the damage extents are,
j i
λ φ
∆
∆ is invariant for the same damage class (i.e. different
damage extent but same damage location). However, basing on the aforementioned two assumptions, the actual computed values,
j i
λ φ
∆
∆ , will no longer be identical for a specific damage
class. For example, the respectively computed values,
j
story with 10% and 20% damage extent will lead to a discrepancy between each other. The higher the difference in damage extent is, the more the discrepancy. Meanwhile, for the example of multiple damages, such as the damage occurred at the 1st and 2nd story with 10% and 20%
damage extent, the computed
j i
λ φ
∆
∆ will also be different to that of the damage occurred at the
same stories but with 20% and 10% damage extent. Even though, one can find out from the example that the DLF is still an effective feature for determining the damage location.
Accordingly, the process of using DLF to find the damage location is more like pattern recognition than functional mapping. Consequently, instead of the most utilized supervised neural network (which is powerful for the functional mapping problems) in the related studies on damage detection or health motoring, this study employs an unsupervised-typed neural network model, the Unsupervised Fuzzy Neural Network (UFN) reasoning model, to implement the damage localization process.
Together with the theories of DLF and the UFN reasoning model introduced in section 2.3, this study makes use of the DLF as the input variables and the existence of the damage site as the output vector for the UFN. Basing on the analytical model, the Analytic Damage Localization Feature (ADLF) for various possible damage cases can be calculated in advance to construct an ADLF instance base. With proper deployment of sensors, the vibration signals of the structure can be easily measured through ambient, free, or forced vibration tests, and the modal parameters can also be generated through the ANNSI model. When the modal parameters of the structure are
Localization Feature (MDLF) with the ADLF through the UFN reasoning.
3.2.3 Input-Output Patterns for the Neural Network
For the UFN, the ADLF is treated as input variable of the neural network. Moreover, the output vector for the UFN represents the condition of the structural elements. Herein, binary value is adopted to represent the condition of the structural element. If the element is damaged, the value is set to be 1 to the associate element; otherwise, the value is set to be 0 to indicate an undamaged element. An example is presented in the next section to examine the feasibility of the proposed approach.