On the Internal Resonance in a Nonlinear Two-Degree-of-Freedom System : Forced Vibrations Near the Lower Resonance Point When the Natural Frequencies are in the Ratio 1 : 2

Abstract

A nonlinear multidegree-of-freedom vibratory system is called tuned to internal resonance if several of the corresponding natural frequencies are integral multiples of the lower natural frequencies. In the paper is taken up a typical case of a two-degree-of-freedom system with internal resonance with a higher natural frequency twice the lower natural frequency, having nonlinear spring characteristic of second and third order polynomials of the displacements. Steady forced vibrations are investigated in the vicinity of the lower resonance point of the system. The theoretical analysis indicates that the second order harmonic occurs strongly in addition to the harmonic oscillation with a frequency equal to the excitation frequency; and that almost periodic oscillations occur in a certain region of the excitation frequency. The results of the theoretical analysis are shown to agree with those of an analog-computer analysis.