116 L. Renson et al. Fig. 12.1 (a) Picture of the Wing-Engine structure. (b) FRF of the structure at 0.25 V ( ) and 1.25 V ( ) Fig. 12.2 (a) Qualitative restoring force measured at the right ( ) and left ( ) nonlinear connections. (b) Wavelet transform analysis of the sine-sweep data collected at a high level of excitation Figure 12.1b presents the frequency response functions (FRFs) of the system collected at 0.25 V ( ) and 1.25 V ( ). The structure presents four modes between 10 and 90 Hz. The FRF measured at high level shows clear signs of nonlinear distortions. The peaks corresponding to the first two modes shift towards higher frequencies and the whole FRF is altered by nonlinear stochastic distortions. The first and second modes have mode shapes in which the two masses move out-of- and in-phase respectively and are the most affected by nonlinearity. As such, sine-sweep data were also collected at different excitation levels around these two modes. The sweeps were performed at a linear rate of 5 Hz per minute between 10 and 25 Hz. The restoring force surface (RFS) method was applied to the time series around the first mode in order to characterise the form of the nonlinearity. The nonlinear stiffness curve obtained for each connection is smooth as shown in Fig. 12.2a. The time-frequency content of the high-level sine sweep data were also analyzed using the wavelet transform, revealing the presence of a 5:1 modal interaction between the first and fourth modes of the structure (see Fig. 12.2b). The NPS method comprises two main steps. The first one consists of processing the acquired broadband input and output data to derive an experimental model of the structure. To this end, a frequency-domain subspace identification (FNSI) method is used [5], although any method applicable to broadband data can be employed. The second step of the identification methodology is to extract the NNMs from the identified model using numerical continuation techniques [6]. The broadband input is a pseudo-random signal (multisine with random phases) defined between 10 and 90 Hz in order to encompass the four modes of interest and the fundamental harmonics of the first two modes. The random signal is repeated 15 times and the last ten periods are considered for averaging. The structure response is recorded through 11 accelerometers. The model provided by FNSI is a discrete-time state-space model of the form xkC1 DAxk CBe.yk; uk/ yk DCxk CDe.yk; uk/ (12.1) where the matrices (A; B; C; D) are the state, extended input, output and direct feedthrough matrices, respectively. The vectors x, y, and uare the states, outputs, and external inputs of the system, respectively. The vector e, termed the extended
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