A probabilistic formulation of the damage detection problem

Abstract

A method is proposed for the probabilistic estimation of structural damage using measured statistical changes in natural frequencies and mode shapes. The variation of the structural integrity of the system (such as natural frequencies and mode shapes) can be interpreted as a variation in the structural parameters of the system (such as mass and stiffness). The relationship between the change in eigenvalues and a set of localized stiffness reduction factors yield a set of underdetermined simultaneous equations derived from a second-order perturbation (by keeping higherorder terms) of the healthy eigenvalue problem. Stochastic expressions for these factors are obtained from a least squares solution using healthy and damaged modal data. These stochastic expressions yield a set of localized probability damage quotients that indicate a confidence level on the existence of damage. The effectiveness of the proposed technique is illustrated using simulated experimental data on a three degree-of-freedom spring-mass system. Graduate Research Assistant, Department of Mechanical Engineering. T Associate Professor, Department of Mechanical Engineering, Member AIAA. Copyright ©1996 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Introduction The Detection, Location, and Estimation (DLE) of structural damage has been the subject of much current research. Many structures, whether on the ground or in space, are often subjected to external hazards. Spacecraft and other large space structures have to endure the hazards of on-orbit existence, assuming they are undamaged during the launch process. Kashangaki provides a concise overview of the on-orbit damage detection and health monitoring process. The expense of maintenance, replacement, and time out of service may be costly. Therefore, it is advantageous to develop methods that can detect, locate, and estimate the extent of damage, if it exists at all. One approach adjusts a finite element model of the healthy structure to obtain the damaged mass and stiffness matrices based upon experimental natural frequencies and mode shapes. Others rely on changes in natural frequencies only (Vandiver, Cawley and Adams, and Hassiotis and Jeong), changes in mode shapes only (Wolff and Richardson^), or both frequency and mode shape changes (Pandey et al.) to indicate the presence of damage. In addition to these, the kinetic and strain energies for each mode may also add information, as considered by Chen and Garba and Kashangaki et al. respectively. Yet another alternative is a probabilistic approach, which examines the eigenvalue problem from a statistical standpoint by considering eigenvalue and eigenvector uncertainty (Collins and Thomson, Hart and Collins, Collins et al., and Hasselman and Hart). This paper is primarily concerned with the probabilistic approach and expands the work conducted by Hassiotis and Jeong. They performed a first-order perturbation (by neglecting higher-order terms) of the healthy eigenvalue problem, yielding the relationship between the variation in the global stiffness and the change in eigenvalues. If the higher order terms are not discarded, the resulting relationship is not that dissimilar to the one they derived. The change in global stiffness is then expressed as a linear sum of the healthy element stiffness submatrices and their corresponding stiffness reduction factors. These reduction factors, first introduced by White and Maytum, were originally intended as a stiffness matrix correction 2626 American Institute of Aeronautics and Astronautics method in the model updating problem. Lim improved the work of White and Maytum, and applied the results to the damage detection field^. These factors indicate the amount of stiffness reduction/increase for each structural element. Stochastic expressions for these coefficients are obtained from a least squares solution of a set of simultaneous linear equations that relate the changes in eigenvalues to those of the element stiffnesses. Stochastic damage stiffnesses are then calculated using these scaling factors, from which a probabilistic comparison is made to determine a probability of damage quotient (PDQ). This PDQ is a confidence indicator on the existence of damage, and can also be employed as a damage threshold indicator. In summary, the OLE problem poses three questions: is there damage?, where is the damage?, and what is the extent of the damage? The probability damage quotients and stiffness reduction factors answer these three questions. The effectiveness of the proposed technique is illustrated using a three degree-of-freedom spring-mass system. Theoretical Development The free vibration eigenvalue problem for an N order undamped dynamical system is given by ([K] -XJM]) { }. = {0} i = l,2,3,...,N (1) where [M] represents the (NxN) symmetric mass matrix, [K] is the (NxN) symmetric stiffness matrix, Xj is the i mode scalar eigenvalue, and {()>}. is the i mode eigenvector or mode shape of size (Nxl). It is tacitly assumed that the eigenvectors are mass normalized. Let us assume that Equation (1) represents the eigenvalue problem of the healthy, pre-damaged system. The eigenvalue problem of the damaged system can similarly be written as [AM] =