On the amplitude dynamics and crisis in resonant motion of stretched strings

Abstract

An N -mode truncation of the equations governing the resonantly forced nonlinear 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 nonlinear dynamics. By using the amplitude equations, which are a function of the damping and the frequency of excitation, it is shown that to O(e) , only the resonantly forced mode has non-zero amplitude. Both planar (i.e. lying in the planar of forcing) and non-planar constant solutions are studied and amplitude-frequency curves are determined. 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. 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. These results are discussed in the context of the Sil’nikov mechanism near a homoclinic orbit for a saddle-focus. Results from the investigations of the averaged system are interpreted for the truncated string system using the averaging theory and the theory of integral manifolds. Numerical investigations with the single mode truncation of the non-autonomous string system show that there is a good correspondence even between chaotic solutions of the averaged system and those of the original system.

Vibration Dynamics of an Inclined Cable Excited Near Its Second Natural Frequency

“…They found the associated stable periodic dynamics and mention that no recurrent response is expected past the simple case of the Shilnikov bifurcation. Moreover, these authors observed the merging of two reflectionally symmetric orbits into a single symmetric orbit [Bajaj & Johnson, 1992;Johnson & Bajaj, 1989]. Our results (a) are very similar, but they are for an inclined cable, which differs from a horizontal string by the presence of a plane of inclination and the associated specific difference between in-plane and out-of-plane motion -not to mention the differences in parameter values between a string and a bridge cable.…”

“…It is not shown here, but we remark that such an isolated branch was also found for a vibrating horizontal string in [Bajaj & Johnson, 1992;Johnson & Bajaj, 1989]. The bifurcating periodic solutions undergo period-doubling at the curves P D 11 and P D 21 shown in Fig.…”

“…We remark that a Shilnikov homoclinic bifurcation with positive saddle index was found by [Bajaj & Johnson, 1992] in a horizontal vibrating string upon variation of the frequency detuning parameter. They found the associated stable periodic dynamics and mention that no recurrent response is expected past the simple case of the Shilnikov bifurcation.…”

“…In vibrating strings also more complicated motion has been observed. [Miles, 1984] mentions quasi-periodic responses and the possibility of a chaotic response, and [Bajaj & Johnson, 1992;Johnson & Bajaj, 1989] found them in numerical simulations. ] as well as [O'Reilly & Holmes, 1992] found quasi-periodic and chaotic string responses experimentally.…”

“…The nonlinear equation of motion of a string go back to [Kirchhoff, 1883], and the simplest model for the transverse vibration of a string, which involves motion in one plane only, yields a forced damped Duffing equation; see [Narashima, 1968]. The general equations of coupled modes from [Narashima, 1968] are reduced in [Miles, 1984] and in [Bajaj & Johnson, 1992;Johnson & Bajaj, 1989] to a set of four averaged ODEs for the amplitudes of the basic horizontal and the vertical modes, which shares many features with the one developed by O'Reilly and Holmes [O'Reilly & Holmes, 1992] who use a different scaling. In vibrating strings also more complicated motion has been observed.…”