3 Reduction basis We discuss is this section how to form the reduction basis Ψ. We propose a basis of vibration modes calculated around a given equilibrium position ueq enriched by the so-called modal derivatives (MD). The two contributions will be separately discussed. 3.1 Vibration modes Let us consider a static equilibrium position ueq when the applied load is constant and given by f φeq. We can then linearize the system of equations 1 around such configuration assuming that the motion ˜uaroundueq is small, i.e. u=ueq+˜u, ¨u=¨˜u. The linearized dynamic equilibrium equations become: M¨˜u+Keq˜u=f ˜φ(t) (5) where ˜φ(t) is a small load variation from φeq. The tangent stiffness matrix Keq is defined as: Keq = ∂g ∂u u=ueq (6) the associated eigenvalue problem to equation 5 writes: (Keq −ω2 i M) Φi =0, i =1,2,...,N (7) and its solution provides N vibration modes (VMs) Φi and associated vibration frequencies ω2 i . 3.2 Modal derivatives The projection of the governing equations on a reduction basis formed by a reduced set of VMs is a well-known technique for linear structural dynamics. The main advantage of this technique is that the resulting reduced model consists of a system of uncoupled equations that can therefore be solved separately. As discussed in the introduction, several attempts has been made to extend the vibration modes projection for nonlinear analysis. The main limitation of such approach lies in the fact that the vibration basis changes as the configuration of the system changes. It is therefore required to upgrade the basis during the numerical time integration to account for the effect of the nonlinearity. For thin-walled structural applications as the one considered in this contribution, the system is usually characterized by significant nonlinear bending-stretching cou30
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