Abstract
The paper discusses the nonlinear free dynamics of an orbiting string satellite system. The focus is on the transversal oscillations, which are governed by two partial integro-differential equations in two transversal displacement components with quadratic nonlinearities. The system is weakly nonlinear but in practice works in conditions of simultaneous internal resonance. The investigation focuses on nonstationary motions arising from perturbed steady-state nonplanar oscillations. A four-mode model is used to study the problem: two modes are necessary to describe the basic oscillation and at least two other modes are involved in the resonance phenomena when the motion is perturbed. The multiple time scales method is used to obtain the equations that govern the amplitude and phase modulations. For increasing levels of system energy, fundamental and bifurcated paths of fixed points of the seven first-order differential equations are determined and their stability is investigated. The trajectories of motion of periodically modulated amplitude solutions and their stability are also studied. A model with a higher number of modes is used to evaluate the accuracy of the stability analysis of two-mode nonplanar oscillations perturbed by a two-mode disturbance.
Sommario
Nel presente lavoro si studia la dinamica libera nonlineare di un sistema filo-satellite. L'attenzione è rivolta alle oscillazioni trasversali, governate da due equazioni integro-differenziali, con nonlinearità quadratiche, nelle due componenti di spostamento. Il sistema è debolmente nonlineare ma praticamente lavora in condizioni di risonanza interna. Lo studio è concentrato sui moti nonstazionari generati da perturbazioni delle oscillazioni stazionarie spaziali. Per studiare il problema è stato sviluppato un modello con quattro modi: due sono necessari per descrivere il moto base mentre almeno altri due sono interessati dai fenomeni di risonanza interna quando il moto viene perturbato. Per ottenere le equazioni nelle ampiezze e fasi è stato utilizzato il metodo delle scale multiple. Del sistema di sette equazioni differenziali del primo ordine ottenuto, sono stati studiati i percorsi fondamentali di equilibrio e i rami biforcati, prendendo come parametro il livello di energia totale. È stata inoltre esaminata la stabilità di questi rami. Sono state studiate le traiettorie dei moti periodicamente modulati e la loro stabilità. Infine, è stato utilizzato un modello con un numero più alto di modi per valutare l'accuratezza dell' analisi di stabilità delle oscillazioni bimodali spaziali, nella quale la perturbazione è stata descritta da due soli modi.
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Di Egidio, A., Luongo, A. & Vestroni, F. Nonstationary nonplanar free motions of an orbiting string with multiple internal resonances. Meccanica 31, 363–381 (1996). https://doi.org/10.1007/BF00426996
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DOI: https://doi.org/10.1007/BF00426996