Frequency–energy plots of steady-state solutions for forced and damped systems, and vibration isolation by nonlinear mode localization

• We consider periodic steady state solutions of forced & damped, nonlinear, coupled oscillators.
• We study the steady periodic responses of a 2-DOF system in the frequency–energy domain.
• Analytical approximations for steady state responses are based on CX-A analysis.
• Effective vibration isolation can be achieved with a strongly nonlinear attachment.

We study the structure of the periodic steady-state solutions of forced and damped strongly nonlinear coupled oscillators in the frequency–energy domain by constructing forced and damped frequency – energy plots (FEPs). Specifically, we analyze the steady periodic responses of a two degree-of-freedom system consisting of a grounded forced linear damped oscillator weakly coupled to a strongly nonlinear attachment under condition of 1:1 resonance. By performing complexification/averaging analysis we develop analytical approximations for strongly nonlinear steady-state responses. As an application, we examine vibration isolation of a harmonically forced linear oscillator by transferring and confining the steady-state vibration energy to the weakly coupled strongly nonlinear attachment, thereby drastically reducing its steady-state response. By comparing the nonlinear steady-state response of the linear oscillator to its corresponding frequency response function in the absence of a nonlinear attachment we demonstrate the efficacy of drastic vibration reduction through steady-state nonlinear targeted energy transfer. Hence, our study has practical implications for the effective passive vibration isolation of forced oscillators.