Recently, the bistable attachment has been employed as a nonlinear energy sink (NES) for passive targeted energy transfer (TET) from linear structures. The bistable NES (BNES) has been coupled with a linear oscillator (LO) where the resulting LO-BNES system has been studied for passive TET. The nonlinear coupling force between the BNES and the associated LO comprises both negative and nonnegative linear and nonlinear stiffness components. Here, the dynamic behavior of the LO-BNES system on the frequency-energy plot is analyzed. The related FEP plot is obtained via numerical simulation techniques where the wavelet transform is imposed into the FEP for variety of initial conditions and damping content. It is found that the FEP has backbone branches at low energy levels associated with the oscillation of the bistable attachments about one of its stable equilibrium positions where passage through the unstable equilibrium position does not occur.

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