Nonlinear resonance in stretched strings with viscous damping

Abstract

The phenomena of amplitude jumps and nonplanar oscillations, which are characteristic features of non-linear resonance in stretched strings, have been studied in detail, taking the effect of damping into account. Expressions for the resonance frequency and amplitude, and the frequencies and amplitudes at which the jumps occur, have been obtained; and it has been shown that the resonance frequency is not a constant of the system but increases with the driving force. Finally, it is established that, in the presence of damping, (1) nonplanar oscillations exist only if the driving force φ exceeds a certain critical value, φ_u; (2) over a small interval of the driving force φ_j'; >φ> φ_w the nature of the downward jump is qualitatively different from the (normal) downward jump occurring for values of φ greater than φ_j'; (3) if φ, lies in the interval φ_w > φ > φ_j, jump phenomenon is exhibited even though the oscillations are planar at all frequencies; and (4), if φ < φ_j, the response is essentially linear in character. In Appendix A, it has been established that, if φ > φ_w, upward jump in amplitude is always accompanied by a transition to nonplanar motion unless the damping is very heavy.