Frequency Shift Vectors

It is known that incomplete measurements do not affect the estimation of modal frequencies. A solo sensor in the right location can fairly measure all the modal frequencies of the structure. Incomplete measurements result on incomplete mode shapes which consequently affects the assessment of modal flexibility matrix. The core damage localization of this study i.e. DLV is entirely based on the modal flexibility matrix. So if some DOFs are missing from flexibility matrix, it makes DLV insensitive to damaged members associated to those DOFs. The detail of how the missing DOFs are affecting DLV is discussed in chapter 5.6.2 based on the results. Briefly, when the missing DOFs are in a specific part of the frame structure e.g. all the nodes in first floor, DLV is unable to evaluate members that are attached to those DOFs. However, if the damaged member is a column, it is at least able to identify the foot (not the exact member) which is involved in damage, only if the two end of the foot are measured or known (supports). Damaged members that are not associated to unmeasured DOFs can still be detected by DLV in an incomplete measurement. So in such case, DLV can partially evaluate the damage scenario of the frame structure. Although the absent or uncertain part of its results need to be cover with another detection method.

To do this, a frequency based method is proposed to provide additional information to identify possible damaged members located in unmeasured part of the structure. This method, in general, is based on studying the frequency shifts of updated FE model caused by known damage cases and comparing them with the experimental frequency shift caused by unknown damage scenario. With a methodical comparison, it is possible to approximation the damaged member with a reasonable level of confident.

To explain the method, let's use a hypothetical frame structure with a number of unmeasured DOFs. Let's assume that the unmeasured DOFs are associated with 5 members, m1 to m5. Unmeasured member is defined as a member which either one end or both ends of it are linked to an unmeasured node. Furthermore, unmeasured node here is defined as a node which none of its DOFs are measured. This hypothetical structure is subjected to an unknown damage scenario, D. The first 10 modal frequencies and mode shapes of the model are estimated experimentally for undamaged and damage cases. Apparently, only measured DOFs are presented in mode shape vectors and consequently in modal flexibility matrices. As mentioned, these incomplete modal flexibility matrices can still be used to calculate DLVs, but since no DLV is calculated for missing DOFs, the calculated WSI indexes of m1 to m5 are not reliable. However, the estimated WSI of other members are reliable and can be used to evaluate their condition.

In case which the damaged member in damage case D is one of the measured members, it can be identified using DLV. Although if one of the unmeasured members are involved, DLV is unable to detect it. Even if DLV identifies one of the measured members as damaged, there is a chance that D is a multiple damage case and unmeasured members are contributing to it as well. So regardless of the results of DLV,

damage case D cannot be certainly determined without the knowledge of unmeasured members.

The first step to check the possible contribution of unmeasured members in D is to simulate five damage scenarios d1 to d5 by reducing the stiffness of m1 to m5 respectively in updated FE model. The level of stiffness reduction should be set in a way that the overall frequency shifts in FE results do not be significantly different fro m those obtained in the experiment. In other word, the damage severity in FE should not be extensively different from the actual case; however the tolerance is fairly high. This can be achieved by a number of tries and errors.

The modal frequencies of intact FE model and d1 to d5 are then estimated using FE dynamic analysis. It should be reminded that the FE model is updated, so the modal frequencies of undamaged FE and experimental models are adequately similar. The proportion of the frequency shifts relative to undamaged cases are then calculated for d1 to d5 as well as for D and are named fs1 to fs5 and FS, respectively.

Let's imagine that m2 is the sole unknown damaged member which means m2 is responsible for FS. Since FE model is updated, the frequency shifts caused by d2 are supposed to be similar to the frequency shifts caused by D, i.e. FS = fs2. However this statement is extremely ideal. The detected modal frequencies in both FE and experiment are not totally accurate. Although FE model is updated, but there is always a level of error in any simulation. The position and magnitude of stiffness reduction i n FE is just an approximation of the unknown damage case. All these factors and many more are the reason why the frequency shifts in FE and experiment are never the same.

Although the similarity of the two frequency shift vectors is not possible, but there is a valid question; is there any meaningful relationship between the two vectors? Because if

such relationship can be evaluated, it can be used as an indicator to locate the damaged member.