28 Assessment and Validation of Nonlinear Identification Techniques Using Simulated Numerical and Real Measured Data 287 m1 m2 f1(t) x1(t) x2(t) p · g(x2) Fig. 28.2 A system with grounded nonlinearity described by a nonlinear operator g(.) and coefficient p g(·) g(·) x1(t) x2(t) x2(t) x2(t) H−1 11 P H12 H11 H−1 21 P H22 H21 f1(t) f1(t) Fig. 28.3 The 2DOF system is broken into two MISO analyses To explain the reverse path technique better, the simple 2DOF system presented in Fig. 28.2 is used. This system can be modeled in the frequency domain as: X1 X2 D H11 H12 H21 H22 F1 P F.g.x2// (28.1) The matrix formulation can then be expanded in the equations: X1 H 1 11 C H12 H11 P F.g.x2// DF1 X2 H 1 21 C H22 H21 P F.g.x2// DF1 (28.2) Each of these equations can be rearranged in a reverse-path fashion with the forces at the output, actually forming a set of MISO analyses, one per DOF (Fig. 28.3). The inputs of the MISO analyses consist of a single measured DOF and all the nonlinearities present in the system. If the location or the type of nonlinearity is unknown, one can feed guesses into the system and use the multiple coherence function (28.3) as an index for the goodness of estimation. The multiple coherence function is a linear relationship that measures the causality between one output and all the input signals. As the standard coherence, it ranges between 0 (no correlation) and 1 (the output is completely caused by the input). The coherence function for a nonlinear system will always be less than unity because of the linear nature of the coherence operator. 2 D GFXG 1 XXGH FX GFF (28.3) As long as the guesses are good, the multiple coherence function will continue to improve over the frequency range and eventually it will be maximised when all the nonlinearities have been characterised and localised (Fig. 28.11). The guessing process could be totally blindfolded, iterating over previously defined locations and nonlinear characteristics, but it also permits the user to exploit any knowledge of where or what type of nonlinearity might be present. Once the best coherence has been achieved, the selected guesses can be used to quantify their coefficients. The Reverse Path method needs time histories of forces, displacements and velocities acquired using a broadband excitation. Time histories are needed because they have to pass through the nonlinear operator before they get transformed into the frequency domain and fed to the system. Displacements are used to construct stiffness-based nonlinearities and
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