Internal resonance in a nonlinear two-degree-of-freedom system. (2nd report. Summed-and-differential and ultra-summed-and-differential harmonic oscillations with no coincidence of resonance points in a system whose natural frequencies are at the ratio of 1:2).

Abstract

The influences of the internal resonance on the summed-and differential harmonic oscillation [p1+p2] and the ultra-summed-and-differential harmonic oscillation [(p1+p2)/2] in a nonlinear two-degree-of-freedom system whose natural frequencies, p1 and p2, are at the ratio of 1:2 are discussed. The resonance points of these oscillations do not coincide with those of other kinds of oscillations. These are the oscillations whose coexisting components do not have the frequency ratio of an integer when the relationship of internal resonance does not hold. In this paper, we investigated the relating nonlinear components, the variations in the shape of resonance curves, the change in the stability of the stationary solutions, the occurrence of the almost periodic oscillations, and the occurrence of new stationary oscillations by theoretical analysis. In the results, it is clarified that, due to the influence of the internal resonance, (a) other kinds of nonlinear components become influential, (b) the stationary oscillations which are observed in a system with no internal resonance change to almost periodic oscillations, and (c) stationary oscillations whose vibration components are at the frequency ratio of an integer become apparent.