On nonlinear oscillations of a thin bar

Summary Based on one of the simplest mathematical model of a solid, nonlinear interactions between waves in a rectilinear bar are investigated, in order to reveal and display a number of dynamic properties inherent not only to the bar, but also to most weakly nonlinear mechanical systems with internal resonances. The presence of internal resonances in the bar is twofold. Firstly, there exists a slow periodic energy exchange between the longitudinal and the two quasi-harmonic bending waves involved in the resonant triad due to the phase matching, secondly, triple-frequency envelope solitons can be created from the resonant triad with the same modal state. The paper investigates the evolution of waves in the bar with the aim to classify the elementary type of wave triplet resonant interactions and define their existence and coesistence areas.The research described here has been made possible in part by Grant N R9B000 from the International Science Foundation. The authors would like to thank Professor G.A. Maugin for having sent copies of his papers, in particular [23], as well as for his permanent interest in our work.