The identification of the dynamic characteristics of nonlinear systems is of increasing interest in the field of modal testing. In this work an investigation has been carried out into the force-state mapping approach to identification of nonlinear systems proposed by Masri and Caughey. They originally suggested a nonparametric identification technique based on curve fitting the restoring force in terms of the velocity and displacement using two dimensional Chebyshev polynomials. It has been shown that the use of Chebyshev polynomials is unnecessarily restrictive and that a simpler approach based on ordinary polynomials and special functions provides a simpler, faster and more accurate identification for polynomial and nonpolynomial types of nonlinearity. This simpler approach has allowed the iterative identification technique for multi-degree of freedom systems to be simplified and a direct identification approach, which is not subject to bias errors, has been suggested. A new procedure for identifying both the type and location of nonlinear elements in lumped parameter systems has been developed and has yielded encouraging results. The practical implementation of the force-state mapping technique required the force, acceleration, velocity and displacement signals to be available at the same instants of time for each measurement station. In order to minimise the instrumentation required, only the force and acceleration are measured and the remaining signals are estimated by integrating the acceleration. The integration problem has been investigated using several approaches both in the frequency and time domains. An analysis of the sensitivity of the estimated parameters with respect to any amplitude and phase measurement errors has been carried out for single-d.o.f. linear systems. Estimates are shown to be extremely sensitive to phase errors for lightly damped structures. The estimation of the mass or generalised mass and modal matrices required for the identification of single or multi-d.o.f. nonlinear systems respectively, has also been investigated. Initial estimates were obtained using a linear multi-point force appropriation method, normally used for the excitation of normal modes. These estimates were then refined using a new technique based on studying the sensitivity of the mass with respect to the estimated system parameters obtained using a nonlinear model. This sensitivity approach seemed promising since accurate results were obtained. It was also shown that accurate estimates for the modal matrix were not essential for carrying out a force-state mapping identification. Finally, the technique has been applied experimentally to the identification of a cantilevered T-beam structure with stiffness and damping nonlinearity. The cases of two well separated and then two fairly close modes were considered. Reasonable agreement between the behaviour of the nonlinear mathematical model and the structure was achieved considering inaccuracies in the measurement set-up. Conclusions have been drawn and some ideas for future work presented.

Thesis (Ph.D.)--Hong Kong University of Science and Technology, 2007. / Includes bibliographical references (leaves 153-177). Also available in electronic version.

This thesis introduces, develops and applies methods for analysing nonlinear systems with the multiple challenges of time-varying, non-polynomial, uncertain or large-scale proper- ties. Both computational and analytic methods using Lyapunov functions are developed and the methods are applied to a range of examples. Generalised Absolute stability is introduced, which is a method of treating polynomial systems with non polynomial, uncertain or time-varying feedback. Analysis is completed with Sum of Squares programming, and this method extends both the applicability of sum of squares as well as existing absolute stability theory. Perturbation methods for invariant Sum of Squares and Semidefinite programs are introduced, which significantly improves scalability of computations and al- lows sum of squares programming to be used for large scale systems. Finally, invariance principles are introduced for nonlinear, time-varying systems. The concept of trajectories leaving sets uniformly in time is introduced, which allows a non-autonomous version of Barbashin-Krasovskii theorem.