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A method is proposed for detecting damage to shear structures by using Support Vector Machine (SVM) and only the first three natural frequencies of the translational modes. This method is able to determine the damage location in any story of a shear building with only two vibration sensors; to obtain modal frequencies, one sensor on the ground detects an input and the other on the roof detects the output. Based on the shifts in the first three natural frequencies, damage location indicators are proposed, and used as new feature vectors for SVM. Simulations of five-story, nine-story and twenty-one-story shear structures and experiments on a five-story steel model were used to test the performance of the proposed method.

Structural health monitoring (SHM) has become a major focus of research in the area of structural dynamics. Many damage detection algorithms based on the modal properties of a structure including the natural frequencies, mode shapes, curvature mode shapes and modal flexibilities have been studied for several decades. However, most algorithms have difficulty identifying the precise location and magnitude of the damage. Moreover, even if such identification is possible, the accuracy and reliability of the properties are not sufficient [

Although frequency shift methods are simple and easy to apply, they have significant practical limitations. For instance, the measurements have to be very precise if the level of damage is not great. In addition, the change in the natural frequency associated with a certain mode in a structure does not provide any spatial information about where the damage occurred, because modal frequencies are global properties of the structure. Salawu [

In 1998, Doebling et al. [

Many methods perform well at frequency-based damage identification in small degrees of freedom. For larger engineering structures, the number of obtained natural frequencies is less than the number of structural elements. This is one of the reasons why frequency change methods have limited abilities to detect damage [

The SVM is a powerful tool for pattern recognition and it seems useful for detecting the location of damage. This requires data from the undamaged and damaged structure successfully train and classify the structure into damaged and undamaged classes. SVM has better generalization performance than other methods of problems of damage detection and location [

Previous studies [11-15] conducted at the Mita Laboratory of Keio University, demonstrated that damage detection methods using SVM worked well in many cases. The scope of this work is to develop an improved method to identify the location of damage by using a limited number of sensors. We know that a natural frequency change associated with a certain mode does not provide spatial information about structural damage, but multiple natural frequency changes can give such information. However, in practical situations, the obtained natural frequencies are usually smaller than the degrees of freedoms. The proposed method only requires the first three natural frequencies. The first three natural frequencies are used to obtain the new damage location indicators. These indicators are the feature vectors for pattern recognition.

The r^{th} natural circular frequency, , of an N-degreeof-freedom (as shown in

where is the first frequency of undamaged structure.

The wave propagation constant mode of safe state can be defined as:

Moreover, the mode of the i^{th} element damaged state was calculated by:

The change due to damage to the i^{th} element, can be obtained from Equation (28) in [

From Equation (5) above, it can be seen that the change, , depends only on the location of damage (i) and the degree of freedom N.

The change in the r^{th} natural circular frequency due to can be found by combining Equations (3)- (5):

The change in the natural frequency can be calculated from Equations (1) and (6):

which is the same as Equation (29b) in [

The above relation showed that the damage was related to the shifts in the natural frequencies. The ratio of the change in the r^{th} and s^{th} frequencies can be used as a pattern depending on the location of the damage:

Equation (8) demonstrates the sensitivity of the change in ratio of the shift of a few natural frequencies to the location of the damage. This should be able to be used as a damage location index.

The sensitivity analysis ideally involves all natural frequencies. However, in some cases, not all of the natural frequencies are available. Therefore, the errors due to incomplete mode parameters should be accounted.

To solve this problem, we define new damage location indicators that have good pattern recognition ability with only the first three natural frequencies. By using the ratio in Equation (8), we defined two damage location indicators:

The changes in and can be used as feature vectors. They may be able to be used as to identify the damage location by using a few lower modal frequencies.

The Support Vector Machine (SVM) is a mechanical learning system that uses a hypothesis space of linear functions in a high dimensional feature space [18,19]. It has been used for structural damage detection because of its ability to form an accurate boundary from a small amount of training data.

The simplest model, Linear SVM (LSVM), works when the data are linearly separable in the original feature space. In most cases, nonlinear classification using the same procedure as in LSVM is possible as a result of introducing nonlinear functions called Kernel functions without having to be conscious of the actual mapping space. This extension to nonlinear feature spaces is called Nonlinear SVM (NSVM) (

To show the feasibility of the proposed method, N-story structures were modeled as one-dimensional lumped mass shear models, as shown in

A reduction in the story stiffness was regarded as damage to the structure. Training feature vectors were generated by assuming three levels of stiffness reduction, 10%, 20%, and 30%, for each story, and the corresponding feature vectors were generated for each damage pattern. A feature vector consisting of zero elements was used to indicate the undamaged structure.

Assuming five levels of stiffness reduction, 8%, 16%,

24%, 32% and 40% for each story, (5 × N) feature vectors were used to verify the proposed method. The first five data corresponded to 8%, 16%, 24%, 32% and 40% stiffness reductions in the first story. The second five data are for the stiffness reduction in the second, third, etc., sets of five data were for the stiffness reductions in the second, third story, and so on. The last data indicated the no damage case.

SVMi denotes a machine that distinguishes the pattern with damage in the i^{th} story from the other patterns.

We considered a five-story structure with the same mass and the same stiffness for all stories: m_{i }= 1000 ton, k_{i }= 2 × 10^{3} MN/m.

The undamaged natural frequencies of the structure were obtained as 2.03, 5.91, 9.32, 11.98 and 13.66 Hz for the 1^{st}, 2^{nd}, 3^{rd}, 4^{th} and 5^{th} modes, respectively.

As mentioned above, a reduction in story stiffness was regarded as damage to the structure. Five damage cases (damage to the 1^{st}, 2^{nd}, 3^{rd}, 4^{th} and 5^{th} stories) were studied.