between the mode shapes of the original model x(o) and of the reduced model x(r) is determined using the Modal Assurance Criterion (MAC) MAC= x∗ (o)x(r) 2 x∗ (o) x (o) x∗ (r) x (r) (19) MAC values range from 0 in case of no correlation to 1 for a complete coincidence. In the [0-100Hz] range, MAC values between modes shapes are all greater than0.999 and the maximum relative error on the natural frequencies is 0.2%. It therefore validates the excellent accuracy of the reduced linear model in this frequency range. It is worth noticing that less internal modes are sufficient to ensure such as a correlation in the [0-100Hz] frequency range, which is typically the range of interest for aircrafts. However, a larger number of modes was deliberately chosen for two main reasons. On the one side, it serves to illustrate the ability of the numerical algorithm to deal with the NNM computation of higher-dimensional systems. On the other hand, due to nonlinearity, modes of higher frequencies may interact with lower modes of interest. In nonlinear regimes, higher internal modes should then be necessary to guarantee the accuracy of the model. 4.1.3 NONLINEAR MODEL The existence of a softening nonlinear behavior was evidenced during different vibration tests conducted by ONERA. In particular, FRF measurements reveal the decrease of resonant frequencies with the level of excitation. The connections between the wings and fuel tanks are assumed to cause this observed nonlinear effect. To confirm this hypothesis, both (front and rear) connections of each wing were instrumented and experimental measurements were carried out. Specifically, accelerometers were positioned on both (wing and tank) sides of the connections and two shakers were located at the tanks. This is illustrated in Figure 6 for rear connection. The dynamic behavior of these connections in the vertical direction is investigated using the restoring force surface (RFS) method [16]. By writing Newton’s second law at the wing side of each connection, it follows mc¨xc(t)+fr,c =0 (20) wherefr,c is the restoring force applied to this point. The indexc is related to the connection under consideration (i.e., either the rear or front attachment of the left or right wing). From Equation(20), the restoring force is obtained by fr,c =−mc¨xc(t) (21) Except the multiplicative mass factor mc, the restoring force is then given by the acceleration ¨xc(t). Nevertheless, this total restoring force does not consist only of the internal force related to the connection of interest, but also includes contributions from the linking forces associated to the wing elastic deformation. Provided that these latter do not play a prominent role, the measurement of the acceleration signal may still provide a qualitative insight into the nonlinear part of the restoring force in the connection between the tank and the wing. The aircraft is excited close to the second torsional mode (see Figure 5(c)) using a band-limited swept sine excitation in the vicinity of its corresponding resonant frequency. In Figure 7, the measured acceleration at the wing side is represented in terms of the relative displacement xrel and velocity ˙xrel of the connection obtained by integrating the accelerations on both sides of the attachment. It is given for the rear connections of the right and left wings. A nonlinear softening elastic effect is observed from the evolution of these estimated restoring force surfaces. In particular, the detected behavior has a piecewise characteristic. This is more clearly evidenced by the corresponding stiffness curves also depicted in Figure 7. Softening nonlinearity is typical of bolted connections [17, 18]. Similar nonlinear effect occurs for the front connections, but they participate much less in the considered response. Finally, the deviation between the right and left connections seems to show asymmetry of the connections. Although purely qualitative, the RFS results therefore indicate that the tank connections present a softening stiffness in the vertical direction. As previously mentioned, a model with piecewise characteristic might be consistent with the experimental observations. However, the NNM algorithm, in its present form, cannot handle nonsmooth nonlinearities. Alternatively, linear and negative cubic stiffness terms are one possible manner of describing the observed nonlinear behavior. Indeed, the reconstructed stiffness curve obtained by fitting to the data the mathematical model fr,c =kxrel +k − nlx 3 rel (k − nl <0) (22) is in relatively good agreement. 233
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