On the vibration of stretched strings
An N mode truncation of the equations governing the resonantly forced non-linear motions of a stretched string is studied. The external forcing is restricted to a plane, and is harmonic with the frequency near a linear natural frequency of the string. The method of averaging is used to investigate the weakly non-linear dynamics. The string system is reduced to a coupled set of non-linear ordinary differential equations governing the amplitudes of the N modes. The amplitude equations are functions of the damping and the frequency of excitation. Using these equations it is shown that to $O(\\varepsilon)$, only the resonantly forced mode has nonzero amplitude. The constant solutions of the averaged equations are studied in considerable detail. Both planar (i.e. lying in the planar of forcing) and non-planar solutions are studied and amplitude-frequency curves are determined. For small damping, asymptotic approximations to the frequencies of local maxima, pitchfork bifurcation, and saddle-node bifurcation are obtained. For small enough damping, solutions in the non-planar branch become unstable via a Hopf bifurcation and give rise to a branch of periodic solutions in the amplitude-frequency plane. This branch exhibits several period-doubling bifurcations, but does not directly result in the formation of a chaotic attractor. At lower values of damping, many other branches of periodic solutions exist. A series of bifurcations leads to the formation of chaotic attractors in some of these solution branches. The chaotic attractors are studied using Poincare sections, Lyapunov exponents, and the concept of a fractal dimension. Various types of interactions between the different solutions are found to result in many interesting phenomena including, the formation of a homoclinic orbit and chaos quenching. The results from the investigations of the averaged system are interpreted for the truncated string system using mathematical results from the averaging theory and the theory of integral manifolds coupled with numerical investigations. Numerical investigations with the single mode truncation of the non-autonomous string system show that there is some correspondence even between chaotic solutions of the averaged system and those of the original system. Counterparts to all phenomena found in the averaged system are found in the non-autonomous system.
Bajaj, Purdue University.
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