Cross-scale energy transfer of chaotic oscillator chain in stiffness-dominated range

The investigations on the dynamics of hierarchical oscillator chains provide basic theories for the fields of acoustic metamaterial and passive guidance of energy flow. In the present study, the characteristics of stiffness-dominated energy transfer of a reduced cross-scale oscillator chain that generates chaotic vibrations are studied. The semi-analytical solutions are obtained by the complexification-averaging method and the least-square method for thoroughly searching the branches of responses. Then, the energy transfer patterns of the oscillator chain are analyzed based on Runge–Kutta method. Moreover, chaotic responses are determined using the displacements and Lyapunov exponents of the system. The average energy and normalized energy flux are defined to evaluate the energy distribution in each oscillator and the energy transfer between oscillators, respectively. The semi-analytical results show that the oscillator chain produces complex unstable branches of response around the resonance frequency of the linear oscillator. It is found that the detached higher branches of response of the chain are similar to that of a system consisting of one linear oscillator and one cubic oscillator. When chaotic responses occur, the energy transfer patterns that reflected by average energy and normalized energy flux are similar in the same branch of response for different excitation frequencies. However, energy transfer patterns in different branches of response are quite different even at the same excitation frequency, and the higher branch generates the lower energy transfer. Furthermore, analyses of the input energy and the scale between cubic oscillators demonstrate that variations of the two parameters slightly affect the normalized energy flux.