88 Amir K. Bagheri et.al and James [7] employed it for computing invariant manifolds in discrete-time systems, demonstrating computational savings, and Artola et al. [8] showcased its utility in nonlinear optimal control of flexible structures. Finally, Scha¨fer et al. [9] used a variation of the method for nonlinear model predictive control, achieving notable reductions in computational time. This paper builds on these advancements, applying the multiple shooting method to 3D beam formulations in the Special Euclidean Lie GroupSE(3) framework to enhance computational efficiency and accuracy in modelling flexible mechanical structures. The method is used to compute nonlinear normal modes, providing insights into the dynamic behaviour of highly flexible beams under geometric nonlinearities. Multiple Shooting Formulation We denote the result of the time integration of the system’s equations of motion, over an interval of length T and froman initial condition vector x0 as ϕ(x0,T), ϕ: RM×R→RM, (1) We now consider s as a partition of the unit interval such that 0=s0 <s1 <· · · <sm−1 <sm=1, (2) with ∆si =si −si−1, (3) Note that if the unit interval is split into equal partitions, we have ∆s = 1 m for all values of i. Multiple shooting then solves the following nonlinear system of mequations for finding a periodic orbit ψ0 =ϕ(x0, ∆s1T) −x1 =0, ψ1 =ϕ(x1, ∆s2T) −x2 =0, .. . ψm−2 =ϕ(xm−2, ∆sm−1T) −xm−1 =0, ψm−1 =ϕ(xm−1, ∆smT) −x0 =0 (4) This is shown schematically in Figure 1, which depicts a periodic orbit split intompartitions over the period. For the orbit to be periodic, the time integration from the starting point of each partition and over the length of the partition should bring the system to the starting point of the next partition, within the tolerance specified. For the final partition, the time integration should bring the system back to the first partition in order to close the orbit. To find a periodic solution, the sensitivities of 0 1 2 3 −2 −1 ( 0,Δ 1 ) Fig. 1 Schematic illustration of the multiple shooting method.
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