# Nonlinear Dynamics of an Internally Resonant Base-Isolated Beam under Turbulent Wind Flow  
[PERSON]  
Department of Civil, Construction-Achicultural and Environmental Engineering, University of L'Aquila, 67100 L'Aquila, Italy; [EMAIL_ADDRESS] (S.D.N.); [EMAIL_ADDRESS] (D.Z.)  
[PERSON]  
Correspondence: [EMAIL_ADDRESS]; Tel.: +39-0862-434523  
[PERSON]  
steady [36] and unsteady [37] wind flow, respectively. In particular, in [37] the interaction between self- and parametrically-excited vibrations was first studied. The wind turbulent component was found to reduce the galloping velocity, leading to periodic or quasi-periodic motions. On the other hand, the visco-elastic device provided a beneficial effect, since it increased the galloping velocity and reduced the amplitudes of both limit-cycles and tori. To summarize, calibration of the device visco-elastic parameters must account for several aspects, namely: (i) the purely elastic base-isolation produces a detrimental lowering of the critical wind velocity, making essential the addition of a (linear) dashpot; (ii) in order for the linear dashpot to dissipate energy on the base displacement, the isolation system requires using sufficiently soft spring; (iii) very soft springs, however, should be avoided, to prevent unwanted large motions; (iv) the linear dashpot mainly shifts the critical wind velocity, but does not reduce the amplitude of limit-cycles and tori, for which nonlinear dashpot is also needed.  
In [37] only the internally nonresonant case was analyzed, in which the dynamics of the system was described by just one mode. On the contrary, when internal resonance case occurs, higher-order modes can significantly affect the response [38; 39; 40; 41; 42]. As a few examples, in [24] internally resonant coupled towers under wind effect were analyzed close to a double-Hopf bifurcation, and in [43] the system was extended to the case of isolated bases, tackled in direct approach. In [42], a nonlinear energy sink was used as a passive control device to reduce the response of a nonlinear elastic string, subjected to primary external excitation and in internal resonance conditions.  
Guided by these results, the analysis of [37] is extended in this paper to account for internally resonance. The study is aimed at investigating whether the passive control system, designed when just one mode contributes to the response, is still efficient when two modes are involved in the internal resonance phenomenon. However, an important aspect of modeling needs to be addressed. It is well-known that, based on multiple scale analysis, when just one mode is involved in galloping, the structural nonlinearities, both of geometric and inertial nature, are negligible, since they only affect the phase but not the amplitude modulation of motion. When, in contrast, two or more internally resonant modes contribute to the aeroelastic response, it can be shown that structural nonlinearities do enter to the amplitude equations, by the way of the internal coupling terms. Therefore, at least potentially, structural nonlinearities can play a non-negligible role on nonlinear dynamics, which deserves to be investigated. Accordingly, the model of [37] is extended, in which: (a) the structural nonlinear geometric and inertial effects are fully taken into account; (b) the stiffness of the elastic spring at the base is adjusted to tune the two lower natural frequencies into the 1:3 resonant ratio.  
The paper is organized as follows. In Section 2 the equations of motion of the system are presented (detailed formulation is given in Appendix A) and the modal properties of the associated linear problem are assessed, as well as the internal resonance conditions; in Section 3 the perturbation analysis of the nonlinear problem is addressed; in Section 4 the numerical results are shown; in Section 5 some conclusions are drawn.