Non-linear modal analysis methods for engineering structures

This thesis presents two novel nonlinear modal analysis methods, aimed at the identification of representative engineering structures. The overall objective is to detect, localize, identify and quantify the nonlinearities in large systems, based on nonlinear frequency response functions (FRFs) as input data. The methods are first introduced in a direct-path, by analyzing a general theoretical system. Then, the concepts are extended to tackle a nonlinear identification via the reverse-path of the same methodologies. The nonlinear formulation of this work is based in first-order describing functions, which represent the nonlinearities by amplitudedependent coefficients. This formulation is the basic “engine” of the methods and techniques developed here. For the sake of clarity, the research has been restricted to deal with cubic stiffness and friction damping nonlinearities, although the inclusion of other types should be straightforward, given the generality of the developments. The first direct-path method, the so-called “explicit formulation” (EF), is conducted entirely in the physical domain. This technique manipulates the physical coefficients stored in the system matrices, thus the term “explicit”, yielding the nonlinear FRF at a selected DOF as a closed-form expression, regardless of the system’s size. An optimized version of this method has been validated against real measurements taken from a test rig, and it was found that the nonlinear behaviour was predicted with reasonable accuracy. A reverse path of the “explicit formulation”, REF, was implemented as a nonlinear identification tool. In spite of successful results, it was concluded that the computational cost of this approach was too high to gain acceptance in a practical analysis. Still, the method provides a much needed bridge between a full-size theoretical model and the relatively small number of experimental measurements that may be available. The second main method, operating in a direct-path, is called the “hybrid modal technique” (HMT). It is based on a novel nonlinear modal expansion, which is analogous to existing nonlinear modal superposition techniques. The underlying linear system is expressed in generalized modal coordinates, while the nonlinearities are kept in the physical domain. The use of hybrid coordinates is a central feature, by which the localization of the nonlinearities is fully addressed. A reverse-path of this method, R-HMT, incorporates the successive application of several “standalone” techniques, also developed here, which can be used independently to tackle different aspects of nonlinear modal analysis. When gathered together, the individual techniques provide a robust methodology, able to perform a nonlinear identification within the usual experimental restrictions, while exhibiting high computational efficiency. The type of the nonlinearity can be identified by a newly introduced technique, based in the geometrical “footprint” of the extracted nonlinear component. The localization of the nonlinearities is then achieved by a linear least-squares calculation over a predefined nonlinear region of arbitrary size. This technique provides an unambiguous localization, provided that the analyzed frequency range is a fair representation of the system. Although the nonlinear natural frequencies and modal damping are not explicitly needed for identifying the system or regenerating the responses at some other forcing level, a fast approximation technique (FAT) is introduced, allowing the analytical derivation of these parameters via newly-developed expressions. The FAT establishes links with other nonlinear methods and standard linear modal analysis techniques. I would like to dedicate this thesis to: My daughter, who is on her way right now... Rosca, for loving and supporting me no matter what... My parents, J. Hugo & Delia, for absolutely everything else... Ana Sofia arrived on July 31st, 2004.