Dynamical Inverse Problems: Theory and Application

Classical vibration theory is concerned with the determination of the response of a given dynamical system to a prescribed input. These are called direct problems in vibration and powerful analytical and numerical methods are available nowadays for their solution. However, when one studies a phenomenon which is governed by the equations of classical dynamics, the application of the model to real life situations often requires the knowledge of constitutive and/or geometrical parameters which in the direct formulation are considered as part of the data, whereas, in practice, they are not completely known or are inaccessible to direct measurements. Therefore, in several areas in applied science and technology, one has to deal with inverse problems in vibration, that is problems in which the roles of the unknowns and the data is reversed, at least in part. For example, one of the basic problems in the direct vibration theory - for infinitesimal undamped free vibrations - is the determination of the natural frequencies and normal modes of the vibrating body, assuming that the stiffness and mass coefficients are known. In the context of inverse theory, on the contrary, one is dealing with the construction of a model of a given type (i.e., a mass-spring system, a string, a beam) that has given eigenproperties. In addition to its applications, the study of inverse problems in vibration has also inherent mathematical interest, since the issues encountered have remarkable features in terms of originality and technical difficulty, when compared with the classical problems of direct vibration theory. In fact, inverse problems do not usually satisfy the Hadamard postulates of well-posedeness, also, in many cases, they are extremely non-linear, even if the direct problem is linear. In most cases, in order to overcome such obstacles, it is impossible to invoke all-purpose, ready made, theoretical procedures. Instead, it is necessary to single out a suitable approach and trade-off with the intrinsic ill-posedeness by using original ideas and a deep use of mathematical methods {}from various areas. Another specific and fundamental aspect of the study of inverse problems in vibration concerns the numerical treatment and the development of ad-hoc strategies for the treatment of ill-conditioned, linear and non-linear problems. Finally, when inverse techniques are applied to the study of real problems, additional obstructions arise because of the complexity of mechanical modelling, the inadequacy of the analytical models used for the interpretation of the experiments, measurement errors and incompleteness of the field data. Therefore, of particular relevance for practical applications is to assess the robustness of the algorithms to measurement errors and to the accuracy of the analytical models used to describe the physical phenomenon. The purpose of the CISM course entitled ``Dynamical Inverse Problems: Theory and Application", held in Udine on May 25-29 2009, was to present a state-of-the-art overview of the general aspects and practical applications of dynamic inverse methods, through the interaction of several topics, ranging {}from classical and advanced inverse problems in vibration, isospectral systems, dynamic methods for structural identification, active vibration control and damage detection, imaging shear stiffness in biological tissues, wave propagation, computational and experimental aspects relevant for engineering problems. The course was addressed to PhD students and researchers in civil and mechanic engineering, applied mathematics, academic and industrial researchers. In the first chapter Gladwell discusses matrix inverse eigenvalue problems. He describes the classical inverse problems for in-line systems, such as discrete models of rods in longitudinal vibration and beams in flexural vibration. He illustrates the theory governing the formation of isospectral systems, and describe how it may be used to construct isospectral finite-element models of membranes. Throughout the chapter, emphasis is placed on ways of choosing data that lead to a realistic system. Morassi in the second chapter describes some classical approaches to the inversion of continuous second-order systems. Attention is focused on uniqueness, stability and existence results for Sturm-Liouville differential operators given in canonical form on a finite interval. A uniqueness result for a fourth order Euler-Bernoulli operator of the bending vibration of a beam is also discussed. The next chapter by R\"{o}hrl presents a method to numerically solve the Sturm-Liouville inverse problem using a least squares approach based on eigenvalue data. The potential and the boundary conditions are estimated {}from two sequences of spectral data in several examples. Theorems show why this approach works particularly well. An introduction to the Boundary Control method (BC-method) for solving inverse problems is presented by Belishev in the fourth chapter. In particular, the one-dimensional version of the BC-method is used for two dynamical inverse problems. The first problem is to recover the potential in a Sturm-Liouville operator describing the transverse vibration of a semi-infinite taut string with constant linear mass density by time-history measurements at the endpoint of the string. The second problem deals with a second-order vectorial dynamical system governing, for example, the longitudinal vibration of two semi-infinite connected beams having constant linear mass densities. An inverse problem is to recover the matrix coefficients of the lower order terms via time-domain measurements at the endpoint of the beam. Connections between the BC-method and asymptotic methods in PDEs, functional analysis, control and system theory, are especially investigated in this chapter. In the fifth chapter Vestroni and Pau introduce dynamic methods for dynamic characterization and damage identification of civil engineering structures. Indeterminacy and difficulties in modal identification and model updating are discussed with reference to several experimental cases of masonry structures. A damage identification procedure in a steel arch with concentrate damage is also presented. Eigenvalue assignment problems in vibration are presented by Mottershead, Tehrani and Ram in the sixth chapter. Procedures are described for pole placement by passive modification and active control using measured receptances. The theoretical basis of the method is described and experimental implementation is explained. The book ends with the lectures by Oberai and Barbone on inverse problems in biomechanical imaging. The authors briefly describe the clinical relevance of these problems and how the measured data is acquired. Attention is focused on two distinct strategies for solving these problems. These include a direct approach that is fast but relies on the availability of complete interior displacement measurements. The other is an iterative approach that is computationally intensive but is able to handle incomplete interior data and is robust to noise.