Keywords: damage detection, modal strain energy, vibration, structures.

Based on modal strain energy, an improved structural damage detection algorithm is developed in this paper. By dividing the contribution of higher modes into two parts, i.e., the static and dynamic contributions, the static term can be expressed by lower modes and the dynamic term was approximately approached.

According to the structural modal methods, with a change of stiffness and mode shape, it can be assumed that,

where, is the change in the structural stiffness matrix resulting from damage; is the change in the th mode shape resulting from damage; is the total number of elements; is the total number of modes; is the number of low modes; , and are the coefficients defining a fractional change of the stiffness matrix and the mode shape vector.

The total contribution of lower and higher modes can be expressed as follows,

The modal strain energy change (MSEC) of the th element and th mode can be defined as

Substituting equation (61) into equation (62)

Selecting the values of MSE of elements (commonly ), and so damage coefficients of possible damaged elements can be calculated according to the following linear system of equations

Coefficient , can be expressed as

This approach could be applied to detect structural damage location and quantity by using only the first one or two modes.

The three numerical examples with different structures, but the same damaged case, are studied in present. The results illustrate that the present method is simple and effective with satisfactory precision both in locating the place of damage and estimating the magnitude of damage. The following conclusions have been obtained:

1. The damage localization method presented in this paper needs only one or two lower modes in order to obtain satisfactory results.
2. The damage quantification method developed in this paper is practical and simple. The modal truncation error can be ignored even if only one or two lower modes used.
3. In order to obtain the better precision, the coefficient can be taken nonzero, i.e., the influence of dynamic contribution of higher modes is considered.
4. When measured DOFs are incomplete in practice, it can be expanded using a mode shape expansion method.

1

A.K. Pendey, M. Biswas and M.M. Samman, "Damage Detection from Changes in Cuvature Mode Shapes", Journal of Sound and Vibration, 145(2), 1991. doi:10.1016/0022-460X(91)90595-B

2

O.S. Salawn and C. Williams, "Damage Location Using Vibration Mode Shapes", Proc. of the 12th IMAC, 933-939, 1994.

3

N. Stubbs, "Clobal Non-Destructive Damage Evaluation in Solid", International Journal of Analytical and Experimental Modal Analysis, 5(2), 1990.

4

T.Y. Chen and J.A. Garba, "On-Orbit Damage Assessment for Large Space Structures", AIAA Journal, 26(12) , 1998. doi:10.2514/3.10019

5

N.A.J. Lieven and D.J. Ewins, "Error Location and Updating of Finite Element Models Using Singular Value Decomposion", Proc.of the 8th IMAC, 768-773, 1990.

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