b'Aerospace | Engineer Innovationinto the studied 2-DOF structural model. In this case, the values of non-linear stiffness is not validated experimentally and the aim of this study will be only related to the method and qualitative results. Figure 6 shows the modified longitudinal coupled system as well as the total restoring force between the two masses. The non-linearity is defined by a loss of approximately 50% of Galileos linear stiffness, KG, for positive relative displacements greater than 0.4 mm. Table 2 shows the mechanical properties of the underlying linearFigure 4: Numerical simulation in case of stable control (Galileo)longitudinal system. The first step of a non-linear approach consists in the harmonic response in presence of non-linearities. The non-linear harmonic response is computed using the Harmonic Balance Method and the results are shown in Figure 8 in function of the acceleration at the base. As expected, the resonance decreases from 29 Hz for linear system to 25.9 Hz for a 1-g base level. This softening phenomenon is due to the decrease of the equivalent stiffness in function of the input energy of the system. In this case, the aim was to show how the linear control could work in the presence of non-linear structures. We choose aFigure 5: Modification of the structural model to account for the loss of stiffness.configuration able to excite the non-linearity within a base acceleration of 0.1 g. To avoid as much as possible controllability issues due to high reactivity, a factor of compression of 12 and a sweep rate of 1 oct/min were also selected. The notching was fixed at 1.65g. If we study the results of spectrum amplitude, as normally done during experimental test rig (Figure 9), we can expect that the control is stable. The control profile is respected and the notching value is not exceeded. But if we consider the time domain results, we can observe that the conclusions are completely different (Figure 10). We see here that the control channel does not follow the control profile and the 2ndFigure 6: Amplitude of the harmonic forced response of Galileo in terms of acceleration using DOF exceeds the notching value. If weSimcenter Samcef Repdyn solver for several levels of base acceleration (black: 0.05 g, red: 0.1 g, perform a FFT of time signal, we canorange: 0.5 g, blue: 1 g)see that the signal has more than one harmonic (Figure 11), which means that in general the control is able to act onlyMass (KG)Stiffness (N/m)Damping (N m/s) Nat frequency (Hz) on the fundamental harmonic. The presence of other harmonics is not at allShaker1,500 2.37 x105 1,88 x103 (5%) 2considered for the Simcenter TestlabGalileo3,500 1.16 x108 25,51 x103 (2%) 29sine control and their presence can have an unexpected effect on the globalTable 2: Underlying longitudinal linear system: physical properties59'