Stability of Forced Oscillations of a Vibrating String

Abstract

The nonlinear response of a stretched string to a transverse, harmonic excitation in the spectral neighborhood of the first natural mode is considered. Approximate, two-degree-of-freedom solutions of the equations of motion are determined by retaining terms of first and third order in the amplitude of the motion and by retaining only the dominant terms in the Fourier-series representations of the transverse motions in perpendicular planes. Let ω be the forcing frequency. It is found that (a) planar harmonic motion is unstable over a finite frequency interval, say ω1 < ω < ω2, just above the natural frequency; (b) nonplanar harmonic motion is stable for ω1 < ω < ω3, where ω3 > ω2 for sufficiently small damping; (c) the energy associated with the transition from stable planar motion to stable nonplanar motion vanishes at ω = ω1 but is positive for ω1 < ω < ω3. It is predicted that the lower transition between planar and nonplanar motion should occur at ω1 as ω either increases or decreases through ω1, but that the upper transition between nonplanar and planar motion should tend to occur at ω3 for increasing ω and at ω2 for decreasing ω. The effects of additional degrees of freedom, corresponding to higher modes in the Fourier series representation of the motion, are considered in Appendix A.