A Comparison Between Frequency and Time Domain Approaches for Determining a Structural Damage Using the Adjoint Method

The present investigation is focused on the solution of a dynamic inverse problem which is concerned with the assessment of damage in structures by means of measured vibration data. Considerable research and effort over the last few decades has taken place in the field of damage detection and damage classification in structures, generating a variety of experimental, numerical and analytical techniques [1, 2]. The inverse problem solution is generally unstable, therefore small perturbations in the input data, like random errors inherent to the measurements used in the analysis, can cause large oscillations on the solution. In order to overcame this difficulty a sort of methods have been propose by using regularization techniques coupled with both deterministic and stochastic approaches. An example of deterministic approach is the conjugate gradient method with the use of the adjoint equation whereas the genetic algorithm method belongs to the class of the stochastic approaches. Both techniques have already been used successfully in inverse heat conduction problems [3]. In this work the inverse vibration problem, i.e. the ill-posed problem, is presented as a well-posed functional form, whose solution is obtained through the use of a hybrid approach which employs the genetic algorithm method coupled with the conjugate gradient method with the adjoint equation, also called variational approach. In addition, two different methodologies have been compared, the one based on frequency domain approach and the other one based on time domain approach. In the first case it has been assumed that the experimental data available are the eigenfrequencies and eigenmodes of the structure, whereas in the second one measurements of displacements of the structure have been assumed as the data available for the analysis. In both cases the damage estimation has been evaluated using noiseless and noisy synthetic experimental data. The idea of this work is to analyze and compare the results obtained with the two formulations (i.e. the time-domain and the frequency-domain formulation) making it possible a better understanding of the inverse method here adopted and its ability to find, locate and quantify a damage in a structure.