Extended modal analysis based on state-space model reveals new structural information

Abstract

Modal frequencies and damping coefficients are the global dynamical properties of a flexible structure provided by the classical modal analysis. They are determined by commercially available programs that look at the system's state matrix eigenvalues. Other matrices of the state-space representation determine the structure's zeros associated with the selected measurement points. This paper discusses the use of this information to determine the physical nature (mass, damping and/or stiffness) and the localization of a parameter which has been changed (e.g., due to structural damage). A lumped-parameter mathematical model is used in system analyses. Experiments are conducted with an aluminum beam. Resulting data support the analyses. INTRODUCTION We developed an approach leading to the determination of the type and localization of structural changes of the flexible structure based on the acceleration time-series, obtained in response to a frequency-rich force excitation, and measured at a sufficient number of locations on the structure. The results obtained came about as a first step in an attempt to tune-up the physical parameters of the structure's ADAMS model (in particular, ADAMS model of a wind turbine) using acceleration time-series obtained at a number of locations on the structure. Our approach has also a potential to be applied for determination and localization of structural changes on a flexible structure in order to evaluate if the new structure dynamics, resulting from those changes, are acceptable so that the safety and performance requirements are still met. To approach this problem, we decided to use a single excitation and, consequently, treat the structure as a single-input multiple-output dynamical system. We represent this system by a linear model whose order is to be determined by the identification algorithm. It is well known that global dynamical properties of such a system, i.e., its modal frequencies and damping coefficients, are determined by the system's eigenvalues. If we put this statement in terms of the system's statespace model, commonly described in terms of the A, B, C, D matrices, we can say that the information about the system's dynamics, provided by the classical modal analysis, is related only to the state matrix A leaving completely unutilized the information contained in the matrices B, C, D. Each row of the matrices C, and D contains elements associated with a particular measurement point and determines the structure's zeros (or roots of the numerator of the transfer function between the excitation input and the acceleration measured) associated with this this point or output on the structure. In other words, the local dynamical properties of the structure are represented by a particular pattern of zeros. The interplay between the global and local dynamical properties of the structure is best visible on the phase frequency response. Consequently, we use the proper identification tools which, first of all, allow us to identify the state-space model. Next, we obtain frequency responses at properly selected measurement points. By augmenting the standard set of modal parameters (provided by the identification algorithm used in this research) with these responses, we obtain a set of information which allows us to solve the problem posed above. We illustrate our approach by analyzing the measurement data obtained from a simple uniform aluminum cantilever beam, as well as from the same structure with a lumped mass added, a Copyright © 1997 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. stiffener added, and both mass and stiffener added. IDENTIFICATION TOOLS Consider a discrete-time state-space model of a continuous-time system with r inputs and m outputs, represented by the following equations: