[5]. Link at al [6] have also presented a practical method for nonlinear identification: the paper is of particular interest for two main reasons: (1) because it presents a practical method for nonlinear identification; and (2) the modal parameters are extracted from the transmissibility function, as opposed to the standard Frequency Response Function (FRF). A method for the identification and quantification of nonlinearities, based on the analysis of measured FRF data, coded as CONCERTO, was recently published by the authors [7]. However, in many practical cases test-pieces are subject to base excitation and therefore only transmissibility data are available. In this paper it is shown that the application of CONCERTO to simulated FRF and transmissibility functions yield the same results. Nonetheless, the method presents some shortfalls which need to be addressed in the future. NONLINEAR IDENTIFICATION METHOD USING FRF DATA As a comprehensive study of the method is presented in [7], only a brief summary is given here. Consider a nonlinear single-degree-of-freedom (SDOF) system of a mass m, with amplitude-dependent damping and/or stiffness - which are the most common classes of nonlinearity in engineering structures - excited by a harmonic force, as depicted in Fig. 1(a). The equation of motion is (1) where X is the amplitude of the response (assumed to be harmonic), c(X) and k(X) are the damping and stiffness functions respectively, F0 is the amplitude of the excitation force and ωis the excitation frequency. Note that a linear system is a special case of Eqn.(1) in which the functions c(X) and k(X) are constants. The identification method discussed in this paper is based on a stepped-sine excitation. Assuming that the system responds at the same frequency as the excitation, the receptance FRF is measured as the ratio between the displacement and the force at steady-state. It is sometimes preferred (but is not strictly necessary) to maintain a constant force level throughout the test, primarily in order to minimise the effect of the forcedrop-out near resonance. However, at any given response amplitude, X, the functions c(X) and k(X) in Eqn.(1) are in effect constants. In other words, it is possible to linearise the system at that specific response amplitude so that the system’s receptance is given by 2 2 2 0 0 ( ) 1 ( , ) ( ) ( ) ( ) ( ) X H X F X X X ω ω ω ω ω ω η = = − + (2) where 0( ) X ω and ( ) X η are the natural frequency and the modal loss factor at that given amplitude. It is important to note that the linearisation must refer to a given value of the amplitude of displacement response and not receptance amplitude. The functions 0( ) X ω and ( ) X η can be extracted from the measured real and imaginary part of the receptance as ( )( ) ( )( ) ( ) ( ) 2 2 2 2 2 1 2 2 1 1 2 1 2 2 1 1 2 2 2 2 1 1 0 2 ( ) R R R R I I I I X R R I I ω ω ω ω ω − − + − − = − + − (3) ( )( ) ( )( ) ( ) ( ) 2 2 2 2 2 1 2 2 1 1 2 1 2 2 1 1 2 2 2 2 1 2 1 ( ) r I I R R R R I I X R R I I ω ω ω ω η ω − − + − − = − − + − (4) ( ) ( ) sin( ) mx c X x k X F tω + + = ɺɺ ɺ 480
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