282 A. delli Carri and D.J. Ewins x^3 dx^2 both x^2 poly5 linear 0.6 1 guesses coherence index Fig. 25.12 Reverse path method, guessing step for characterisation. For coherence estimation, the stiffness nonlinearity is more important than the damping one. When both are included the coherence is maximised 15 20 25 30 1 frequency [Hz] coherence both linear Fig. 25.13 The overall improvement of the coherence function with respect to the linear case is clearly visible 15 20 25 30 10−5 10−4 10−3 10−2 frequency [Hz] amplitude [m/N] raw FRF reconstr FRF true linear FRF Fig. 25.14 The estimated underlying linear system is correct in terms of modal frequency. The damping estimation is much less accurate 25.3.1.3 Quantification The reverse path method was able to reconstruct the underlying linear system and to estimate the order of magnitude of the stiffness nonlinearity correctly. The estimation of the nonlinear damping coefficient proved to be beyond the capability of this method, as can clearly be seen in Fig. 25.14. It is noteworthy to observe that, in spite of the poor linear damping estimation, the method still managed to reconstruct a fair linear FRF with a pronounced peak that retains the underlying linear resonant frequency, whereas the raw FRF exhibits a hardly-noticeable peak due to the heavy damping. Thanks to the demand for a huge quantity of input data (accelerations, velocities, displacements) the SVD method quantified the nonlinear stiffness with a fair accuracy over a limited range of displacement. The nonlinear damping estimation proved to be less accurate (Fig. 25.15). 25.3.2 Case N2 The second numerical case in Fig. 25.16 is represented by three masses, each moving in a single direction in which nonlinearities are placed between the first and the third mass (quadratic spring) and between the second and the third mass (cubic spring). A summary of the system parameters is found in Table 25.3.
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