The NNM computation is carried out by finding the periodic solutions of the governing nonlinear equations of motion (2). In this context, the shooting method is probably the most popular numerical technique. It solves numerically the two-point boundaryvalue problem defined by the periodicity condition H(zp0,T) ≡zp(T, zp0) −zp0 =0 (4) H(z0,T) = z(T, z0) −z0 is called the shooting function and represents the difference between the initial conditions and the system response at timeT. Unlike forced motion, the periodT of the free response is not known a priori. The shooting method consists in finding, in an iterative way, the initial conditions zp0 and the period T that realize a periodic motion. To this end, the method relies on direct numerical time integration and on the Newton-Raphson algorithm. Starting from some assumed initial conditions z (0) p0 , the motion z (0) p (t, z (0) p0 ) at the assumed period T (0) can be obtained by numerical time integration methods (e.g., Runge-Kutta or Newmark schemes). In general, the initial guess (z (0) p0 ,T (0)) does not satisfy the periodicity condition (4). A Newton-Raphson iteration scheme is therefore to be used to correct an initial guess and to converge to the actual solution. The corrections∆z (k) p0 and∆T (k) at iterationk are found by expanding the nonlinear function H z (k) p0 +∆z (k) p0 ,T (k) +∆T(k) =0 (5) in Taylor series and neglecting higher-order terms (H.O.T.). The phase of the periodic solutions is not fixed. If z(t) is a solution of the autonomous system (2), thenz(t+∆t) is geometrically the same solution in state space for any ∆t. Hence, an additional condition, termed the phase condition, has to be specified in order to remove the arbitrariness of the initial conditions. This is discussed in detail in[12]. In summary, an isolated NNM is computed by solving the augmented two-point boundary-value problem defined by F(zp0,T) ≡ H(zp0,T) = 0 h(zp0) = 0 (6) whereh(zp0)=0 is the phase condition. 3.2 Continuation of Periodic Solutions Due to the frequency-energy dependence, the modal parameters of an NNM vary with the total energy. An NNM family, governed by equations (6), therefore traces a curve, termed an NNM branch, in the (2n+1)-dimensional space of initial conditions and period(zp0,T). Starting from the corresponding LNM at low energy, the computation is carried out by finding successive points (zp0,T) of the NNM branch using methods for thenumerical continuationof periodic motions (also calledpath-following methods) [8, 9]. The space(zp0,T) is termed the continuation space. Different methods for numerical continuation have been proposed in the literature. The so-called pseudo-arclength continuation method is used herein. Starting from a known solution (zp0,(j),T(j)), the next periodic solution (zp0,(j+1),T(j+1)) on the branch is computed using a predictor step andacorrector step. Predictor step At stepj, a prediction(˜zp0,(j+1), ˜T(j+1)) of the next solution(zp0,(j+1),T(j+1)) is generated along the tangent vector to the branch at the current point zp0,(j) ˜zp0,(j+1) ˜T(j+1) = zp0,(j) T(j) +s(j) pz,(j) pT,(j) (7) 225
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