Modal Analysis for Localization of Harmonic Oscillations in Nonlinear Oscillator Arrays

When a nonlinear oscillator array is harmonically excited, specific oscillators in the array may oscillate with large amplitudes. This is known as the localization phenomenon; however, the reason for localization has not been clarified thus far. Thus, the aim of this study is to elucidate the reason for localization in nonlinear oscillator arrays. We theoretically investigated the behavior of a nonlinear oscillator array, which consists of N Duffing oscillators connected by linear springs under external and harmonic forces. The equations of motion in physical coordinates are transformed into modal equations of motion, which reveal that the array forms an autoparametric system in the modal coordinates when it consists of identical oscillators. The first mode of vibration is directly excited by the external force, whereas the remaining modes are indirectly excited by the nonlinear terms coupled with the first mode. The approximate solutions of the harmonic oscillations were obtained using van der Pol's method. The frequency response curves (FRCs) for both the physical and modal coordinates for N = 2 and 3 are presented. Localization can occur when multiple modes are excited simultaneously. Furthermore, the effects of imperfections in the restoring forces on the responses of the two-Duffing-oscillator array are examined.