Mode shape based methods

Many modal analysis techniques are available for the extraction of mode shapes from the data measured in the time domain (Heylen et al., 1997; Ewins, 2000). Damage detection methods are developed for the identification of damage based directly on measured mode shapes or mode shape curvatures.

Two commonly used methods to compare two sets of mode shapes are the Modal Assurance Criterion, MAC (Allemang & Brown, 1982) and the Coordinate Modal Assurance Criterion, COMAC (Lieven & Ewins, 1988).The MAC value can be considered as a measure of the similarity of two mode shapes. A MAC value of 1 is a perfect match and a value of 0 means they are completely dissimilar. Therefore, the reduction of a MAC value may be an indication of damage. Salawu and Williams (1995) test a reinforced concrete bridge before and after repair and observed that the modeshape based MAC method is a more robust technique for damage detection than shifts in natural frequencies.

The COMAC is a parameter pointwise measuring of the difference between two sets of mode shapes and takes a value between 1 and 0. A low COMAC value would indicate discordance at a point and thus can be used as damage location indicator. Frýba & Pirner (2001) apply the COMAC method to check the quality of a repair to a prestressed concrete segment bridge. The analysis confirms that the repaired segment responses tend to be consistent with that of a undamaged segment.

A drawback of many mode shape based methods is the necessity of having measurements from a large number of locations. Khan et al. (2000) use a scanning Laser Doppler Vibrometer (LDV), allowing for a dense grid of measurements, to measure mode shapes in a steel cantilever beam, a steel cantilever plate and concrete beams, containing cracks. Since in thick metal structures defects are detectable only when their depths are more than half of the structure thickness, it may be concluded that, despite the potentiality of the scanning LDV, improvements in reducing the noise interference would be necessary for a successful application to actual field structures.

Araújo dos Santos et al. (2000) describe a damage identification algorithm based on the orthogonality conditions of the mode shape sensitivities. The algorithm is applied to a plate with reduced stiffness to simulate damage. The results are compared with those obtained by using the mode shape sensitivities and are found to be more accurate. The same orthogonality based technique is applied by Ren & De Roeck (2002a; 2002b) to a laboratory scale concrete beam. They conclude that, although mode shape based methods are effective with

simulated data, there are significant difficulties in the application to real structures, due to noise and measurement errors, mode shape expansion of incomplete measurements and accurate modelling of test structures.

Pandey et al. (1991) are the first through analytical and FEM beam models, to show that the curvature modeshape can be a successful parameter for identifying and locating damage. The curvature modeshapes are calculated using a central difference approximation from the simulated displacement modeshapes. Furthermore, the authors underline that the changes in the curvature mode shapes increase with increasing size of damage and this increase, more pronounced than that related to the changes in the displacements of the mode shapes, can be used to calibrate damage severity.

Modal curvature based methods are applied also on measured data to identify damage. Rathcliffe & Bagaria (1998) use a gapped smoothing method to successfully locate a delamination in an experimental composite beam. The curvature shape is calculated using Laplace’s difference equations from the displacement mode shape and is then locally smoothed using a gapped polynomial at each point. Through the difference between the curvature and the polynomial at each point, the damage index is defined: the largest index indicates the location of the delamination. Wahab and De Roeck (1999) apply successfully a curvature-based method to the Z24 Bridge in Switzerland. They introduce a damage indicator named the curvature damage factor, equal to the difference in curvature before and after damage averaged over a number of modes. They conclude that the use of modal curvature to locate damage in civil engineering structures seems promising.

The number of modal curvatures useable in damage identification routines is, naturally, limited to the available number of displacement mode shapes. In an effort to increase the amount of data available for input into damage identification routines, Sampaio et al. (1999) extend the curvature approach to all frequencies in the measurement range by using FRF data. This method is tested with data related to an intentionally damaged bridge and it is found to have higher performance than the curvature method.

In conclusion, despite the weak points of measuring mode shapes (mode shapes measurement accuracy is lower than that of the natural frequencies, see Farrar et al., 1997 and Doebling et al., 1997, and measured vibration modes are often incomplete so that they have to be expanded with a consequent increase of measurements errors) mode shape based methods, since they contain more damage information, are more robust than natural frequency based methods to detect, locate and calibrate damage.

Researchers have investigated other methods, such as that using operational deflection shapes (Schwarz & Richardson, 1999; Pai & Young, 2001; Waldron et al., 2002), which are very similar to mode shapes, and other methods based on more complex formulations involving the use of natural frequencies and mode

shapes (modal strain energy based methods, see section 1.3.2.4, dynamically measured flexibility based methods, see section 1.3.2.5, and model updating based methods, see section 1.3.2.7).