518 T. Rogers et al. Fig. 45.15 FRF generated in the same manner as Fig. 45.12 investigating5 108 <k 3 <1 10 10 Nm 3 The hypothesis that increasing the number of nonlinearities in the system would amplify this effect can be supported by the results shown in Figs. 45.13 and 45.14. With increasing numbers of nonlinearities in the system, the degradation of linear modal structure happens to a greater extent and at a lower value of k3. There is potentially a simple physical interpretation of the low frequency behaviour seen in the FRF plots; the build up of energy around the first natural frequency of the system is representative of the system stiffening to the point of ‘locking up’ with all the masses moving in phase. In all three cases the transition, from structure with many clearly visible peaks to a situation where the structure is washed out, occurs in the range 108 <k 3 <10 10 Nm 3. In order to observe the change in a little more detail, simulations were run with a number of k3 values in this range. The results of these additional runs are shown in Figs. 45.15, 45.16, and 45.17. The closer inspection of the transition away from distinct modal structure shows more clearly the movement of energy towards the ends of the spectrum. The lower frequency modes, except the first, appear to be the first to be washed out by the increasing effect of the nonlinearities. These plots show more clearly the effect of the increased number of nonlinearities in the system. For the cases with 30 or 50 nonlinearities (Figs. 45.15 and 45.16) more remnants of the linear modal structure can be seen even at very high values for k3. In addition to this, the higher frequency peak becomes flatter in the presence of an increased number of nonlinearities. Interestingly, on close inspection, the two surviving peaks at high levels of nonlinearity both show the behaviour of an SdoF Duffing oscillator system using statistical linearisation i.e. with increasing nonlinearity, the peaks shift upward in frequency and down in magnitude. A number of further simulations were run to confirm that the effects here were genuinely an effect of the increased nonlinearity in the system and not artefacts of the solver or the specific realisations of the nonlinear spring positions. These runs considered different realisations of the random spring positions and showed that the results were insensitive to the realisations as long as many springs were added. 45.4 Conclusions The results shown in this paper demonstrate that for a system under random excitation with many nonlinear components, when those components are sufficiently excited, the underlying linear modal structure of the system is largely ‘washed out’ by nonlinear effects. Given a sufficient number of nonlinearities this appears to lead to a simplified statistical order in the
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