Enhanced Damage Detection Using Linear Matrix Inequalities M. Abdalla, K.M. Grigoriadis, and D.C. Zimmerman Department of Mechanical Engineering University of Houston Houston, TX 77204-4 792 ABSTRACT In this work, linear matrix inequality methods are pro- posed to solve damage detection problems in structures. The damage detection problem is formulated as a convex optimization problem involving linear matrix inequality (LMI) constraints. LMI optimization problems have low computational complexity and they can be solved very efficiently using recently developed interior-point meth- ods. Both a matrix update and a parameter update for- mulation of the damage detection problem is provided in terms of LMis. The proposed techniques are applied to detect damage in a 12-DOF vibrating bar model. 1 NOMENCLATURE M- analytic mass matrix K- analytical (healthy) stiffness matrix Kd- damaged stiffness matrix Wdi- measured eigenfrequencies vdi- measured mode shapes nd- measured eigenfrequency matrix Vd- measured mode shape matrix q- generalized coordinates vector 2 INTRODUCTION Techniques for damage detection and health monitor- ing of aerospace, civil and mechanical engineering struc- tures are essential to determine their safety, reliability and operational life. For example, regular monitoring and assessment of the structural integrity of the Space Station Freedom would be necessary to estimate poten- tial damage by micrometeoroids and orbital debris im- pact, space shuttle dockings and fatigue due to nomi- nal loading or accidents. The structural health monitor- ing problem consists of obtaining information about the existence, location and extent of damage in structures using non-destructive methods. Based on experimen- tal modal analysis and signal processing techniques, a promising approach for health monitoring of structures is to monitor and interpret changes on structural dynamic measurements. The extraction of natural frequency and mode shape information of a vibrating structure can be accomplished using modern vibration testing equipment and instrumentation. Modal and structural dynamic 144 data can be utilized for cost-effective health monitor- ing and operational life assessment without a need for dismantling the structure. Most prior work on damage detection of structures has been directed towards the general framework of finite element model (FEM) refinement methods. The moti- vation behind these techniques is the need to refine and "validate" FEM structural models before their accep- tance as accurate models of the structure [1]. Model updating, model correction and health monitoring ap- proaches have attracted a substantial attention in the past few years, as it is evident from several recent review articles [2],[3],[4]. Optimal matrix update methods that seek to determine the system property matrices, such as the stiffness matrix, using measured test data have been used extensively for FEM refinement and damage detec- tion. Health monitoring information can be associated with changes in the system matrices, for example stiff- ness reduction can be attributed to damage. In their work on optimal orthogonalization of measured modes Baruch and Bar Itzhack [1] obtained a closed-form so- lution for the minimum Frobenius norm adjustment to the mass-weighted structural stiffness matrix that in- corporates the measured frequencies and mode shapes. However, the zero/nonzero sparsity pattern of the orig- inal stiffness matrix may be destroyed. Algorithms by Kabe [5], Kammer [6], Smith and Beattie [7] and Smith [8] have been developed recently to preserve the origi- nal stiffness matrix pattern, thereby preserving the orig- inal load paths of the structural model. Although these methods appear to provide satisfactory results for model refinement, their use for damage detection is question- able since damage typically results in localized changes in the system property matrices. Parameter updating methods that are based on element- by-element adjustments of the FE model have been used in the past for model refinement and damage detection in structures based on modal test data. In the para- meter update formulation, the estimated system prop- erty matrices are constructed using the estimates of the structural parameters; hence, the intrinsic FEM connec- tivity properties of the model are preserved. Many pa-