with ∂z(0, z0) ∂z0 =I (14) since z(0, z0) =z0. Hence, the matrix ∂z(t, z0)/∂z0 at t =T can be obtained by numerically integrating over T the initial-value problem defined by thelinear ordinary differential equations (ODEs) (13) with the initial conditions (14). In addition to the integration of the current solution z(t, x0) of (2), these two methods for computing ∂z(t, z0)/∂z0 require 2n numerical integrations of 2n-dimensional dynamical systems, which may be computationally intensive for large systems. However, equations (13) are linear ODEs and their numerical integration is thus less expensive. The numerical cost can be further reduced if the solution of equations (13) is computed together with the solution of the nonlinear equations of motion in a single numerical simulation[13]. The sensitivity analysis requires only one additional iteration at each time step of the numerical time integration of the current motion to provide the Jacobian matrix. The reduction of the computational cost is therefore significant for large-scale finite element models. In addition, the Jacobian computation by means of the sensitivity analysis is exact. The convergence troubles regarding the chosen perturbations of the finite-difference method are then avoided. Hence, the use of sensitivity analysis to perform the shooting procedure represents a meaningful improvement from a computational point of view. As the monodromy matrix ∂zp(T, zp0)/∂zp0 is computed, its eigenvalues, the Floquet multipliers, are obtained as a by-product, and the stability analysis of the NNM motions can be performed in a straightforward manner. 3.4 Algorithm for NNM Computation The algorithm proposed for the computation of NNM motions is a combination of shooting and pseudo-arclength continuation methods, as shown in Figure 1. It has been implemented in the MATLAB environment. Other features of the algorithm such as the step control, the reduction of the computational burden and the method used for numerical integration of the equations of motion are discussed in[12]. So far, the NNMs have been considered as branches in the continuation space (zp0,T). An appropriate graphical depiction of the NNMs is to represent them in a frequency-energy plot (FEP). This FEP can be computed in a straightforward manner: (i) the conserved total energy is computed from the initial conditions realizing the NNM motion; and (ii) the frequency of the NNM motion is calculated directly from the period. 4 NUMERICAL EXPERIMENT - FULL-SCALE AIRCRAFT The numerical computation of the NNMs of a complex real-world structure is addressed. This structure is the airframe of the Morane-Saulnier Paris aircraft, which is represented in Figure 2. This French jet aircraft was built during the 1950s and was used as a trainer and liaison aircraft. The structural configuration under consideration corresponds to the aircraft without its jet engines and standing on the ground through its three landing gears with deflated tires. For information, general characteristics are listed in Table 1. A specimen of this plane is present in ONERA’s laboratory, and ground vibration tests have exhibited nonlinear behavior in the connection between the wings and external fuel tanks located at the wing tip. As illustrated in Figure 3, this connection consists of bolted attachments. 4.1 Structural Model of the Paris Aircraft 4.1.1 FINITE ELEMENT MODEL OF THE UNDERLYING LINEAR STRUCTURE The linear finite element model of the full-scale aircraft, illustrated in Figure 4, was elaborated from drawings by ONERA [14]. The wings, vertical stabilizer, horizontal tail and fuselage are modeled by means of 2-dimensional elements such as beams and shells. The complete finite element model has more than 80000 DOFs. Three-dimensional spring elements, which take into account the structural flexibility of the tires and landing gears, are used as boundary conditions of the aircraft. At each wing tip, the external fuel tank is connected with front and rear attachments (see Figure 3). In this linear model, these connections between the wings and the fuel tanks are modeled using beam elements. The linear model, originally created in the Nastran 228
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