continuation of NNM motions. The objective is to demonstrate that the numerical computation of the NNMs of complex real-world structures is then within reach. The application considered in this study is the airframe of the Morane-Saulnier Paris aircraft, whose ground vibration tests have exhibited some nonlinear structural behaviors. 2 NONLINEAR NORMAL MODES (NNMS) A detailed description of NNMs and their fundamental properties (e.g., frequency-energy dependence, bifurcations and stability) is given in[1−3]. For completeness, the two main definitions of an NNM are briefly reviewed in this section. The free response of discrete conservative mechanical systems with n degrees of freedom (DOFs) is considered, assuming that continuous systems (e.g., beams, shells or plates) have been spatially discretized using the finite element method. The equations of motion are Mx¨(t)+Kx(t)+fnl {x(t), x˙(t)} =0 (1) where Mis the mass matrix; Kis the stiffness matrix; x, x˙ and x¨ are the displacement, velocity and acceleration vectors, respectively; fnl is the nonlinear restoring force vector. There exist two main definitions of an NNM in the literature due to Rosenberg and Shaw and Pierre: 1. Targeting a straightforward nonlinear extension of the linear normal mode (LNM) concept, Rosenberg defined an NNM motionasa vibration in unison of the system (i.e., a synchronous periodic oscillation). 2. To provide an extension of the NNM concept to damped systems, Shaw and Pierre defined an NNM as a two-dimensional invariant manifold in phase space. Such a manifold is invariant under the flow (i.e., orbits that start out in the manifold remain in it for all time), which generalizes the invariance property of LNMs to nonlinear systems. At first glance, Rosenberg’s definition may appear restrictive in two cases. Firstly, it cannot be easily extended to nonconservative systems. However, the damped dynamics can often be interpreted based on the topological structure of the NNMs of the underlying conservative system[3]. Secondly, in the presence of internal resonances, the NNM motion is no longer synchronous, but it is still periodic. In the present study, an NNM motion is therefore defined as a(non-necessarily synchronous) periodic motionof the conservative mechanical system (1). As we will show, this extended definition is particularly attractive when targeting a numerical computation of the NNMs. It enables the nonlinear modes to be effectively computed using algorithms for the continuation of periodic solutions. 3 NUMERICAL COMPUTATION OF NNMS The numerical method proposed here for the NNM computation relies on two main techniques, namely a shooting technique and the pseudo-arclength continuation method. A detailed description of the algorithm is given in[12]. 3.1 Shooting Method The equations of motion of system (1) can be recast into state space form z˙ =g(z) (2) wherez = [x∗ x˙∗]∗ is the2n-dimensional state vector, and star denotes the transpose operation, and g(z)= x˙ −M−1 [Kx+fnl(x, x˙)] (3) is the vector field. The solution of this dynamical system for initial conditionsz(0) =z0 = [x ∗ 0 x˙ ∗ 0] ∗ is written as z(t) =z(t, z0) in order to exhibit the dependence on the initial conditions, z(0, z0) = z0. A solution zp(t, zp0) is a periodic solution of the autonomous system (2) if zp(t, zp0)=zp(t +T, zp0), whereT is the minimal period. 224
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