Direct method for second-order sensitivity analysis of modal assurance criterion

A Lagrange direct method is proposed to calculate the second-order sensitivity of modal assurance criterion (MAC) values of undamped systems. The eigenvalue problem and normalizations of eigenvectors, which augmented by using some Lagrange multipliers, are used as the constraints of the Lagrange functional. Once the Lagrange multipliers are determined, the sensitivities of MAC values can be evaluated directly. The Lagrange direct method is accurate, efficient and easy to implement. A simply supported beam is utilized to check the accuracy of the proposed method. A frame is adopted to validate the predicting capacity of the first- and second-order sensitivities of MAC values. It is shown that the computational costs of the proposed method can be remarkably reduced in comparison with those of the indirect method without loss of accuracy.