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 an extended 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 course is 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.

Coordinators: Graham M.L. Gladwell (University of Waterloo, Ontario, Canada) Antonino Morassi (Università di Udine, Italy)