This chapter deals with vibration testing, which aims to identify inclusions, cracks, or shape changes in a structure by measuring its modal characteristics. The measured eigenparameters are related to the defect or damage location, orientation, and size. The chapter derives asymptotic formulas for eigenvalue perturbations due to small inclusions, cracks, and shape deformations. The main ingredients in deriving the results are the integral equations and the theory of meromorphic operator-valued functions. Using integral representations of solutions to the harmonic oscillatory linear elastic equation, this problem is reduced to the study of characteristic values of integral operators in the complex planes. The chapter focuses on three kinds of elastic inclusions: holes, hard inclusions, and soft inclusions.
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