To overcome the ill⁃posed least square problem in sensitivity equations

a secondary feedback ridge estimation method was proposed to detect structural damages based on modal flexibility changes.Modal eigenvalues and mode shapes were utilized to assemble the modal flexibility matrix

which was used to establish linear equations for detecting elemental damage severities by modal flexibility changes.During the damage localization stage

damage sites were primarily assessed using the non⁃negative least square method and the feedback ridge technique.Since the damage localization results obtained by the two methods were typically disturbed by different undamaged elements

suspicious damaged elements were comprehensively determined by taking the intersection of the two sets of damage localization results.At the damage quantification stage

the diagonal entries of the regularization matrix corresponding to suspicious elements were decreased and the diagonal entries associated with other elements were augmented.Then the second feedback ridge estimation was conducted to accomplish damage quantification.The numerical analysis of a simply supported beam indicates that the proposed method is more robust in resisting noise

compared with the non⁃negative least square method and feedback ridge estimation technique.Regardless of whether it is damage localization or damage quantification

the proposed method demonstrates higher reliability in detecting accuracy.