Frequency–energy plot and targeted energy transfer analysis of coupled bistable nonlinear energy sink with linear oscillator

Abstract

The nonlinear energy sink (NES), which is proven to perform rapid and passive targeted energy transfer (TET), has been employed for vibration mitigation in many primary small- and large-scale structures. Recently, the feature of bistability, in which two nontrivial stable equilibria and one trivial unstable equilibrium exist, is utilized for passive TET in what is known as bistable NES (BNES). The BNES generates a nonlinear force that incorporates negative linear and multiple positive or negative nonlinear stiffness components. In this paper, the BNES is coupled to a linear oscillator (LO) where the dynamic behavior of the resulting LO-BNES system is studied through frequency–energy plots (FEPs), which are generated by analytical approximation using the complexification-averaging method and by numerical continuation techniques. The effect of the length and stiffness of the transverse coupling springs is found to affect the stability and topology of the branches and indicates the importance of the exact physical realization of the system. The rich nonlinear dynamical behavior of the LO-BNES system is also highlighted through the appearance of multiple symmetrical and unsymmetrical in- and out-of-phase backbone branches, especially at low energy levels. The superimposed wavelet frequency spectrums of the LO-BNES response on the FEP have verified the robustness of the TET mechanism where the role of the unsymmetrical NNM backbones in TET is clearly observed.

Appendix

In this section, we discuss the application of the CX-A method to determine the subharmonic periodic motions where the LO vibrates three times faster than the NES, i.e., branches S13 ± . Therefore, two fast frequencies, ω and 3ω, are necessary for accurately modeling the periodic orbits [1]. Accordingly, considering the Hamiltonian version of the system in Eq. (4) in which ${\overline{\lambda }}_{p}=\overline{\lambda }=0$ and ${\omega }_{0}^{2}=1$,, the complexification of the dynamics is firstly performed by introducing four new complex variables

$$ \begin{gathered} \psi_{1} = \dot{x}_{11} + j\omega x_{11} \hfill \\ \psi_{2} = \dot{x}_{21} + j\omega x_{21} \hfill \\ \psi_{3} = \dot{x}_{12} + 3j\omega x_{21} \hfill \\ \psi_{4} = \dot{x}_{22} + 3j\omega x_{22} \hfill \\ \end{gathered} $$

(13)

where $\omega$ is the dominant fast frequency of oscillation. Following the same approach to derive the backbone branches S11 ± , we can express the complex variables defined in (5) in terms of fast oscillations of frequencies, ${\mathrm{e}}^{j\omega t}$ and ${\mathrm{e}}^{3j\omega t}$, modulated by slowly varying complex amplitudes ${\phi }_{i}\left(t\right)$, $i=\mathrm{1,2},\mathrm{3,4}:$

$$ \begin{gathered} \psi_{1,2} \left( t \right) = \phi_{1,2} \left( t \right){\text{e}}^{j\omega t} \hfill \\ \psi_{3,4} \left( t \right) = \phi_{3,4} \left( t \right){\text{e}}^{3j\omega t} \hfill \\ \end{gathered} $$

(14)

By substituting Eqs. (6) and (5) into Eq. (4) and averaging of Eq. (7) with respect to the fast frequencies ω and 3ω, a set of polar representations are introduced as

$$ \begin{gathered} \phi_{1} = A{\text{e}}^{j\alpha } \hfill \\ \phi_{2} = B{\text{e}}^{j\beta } \hfill \\ \phi_{3} = D{\text{e}}^{j\gamma } \hfill \\ \phi_{4} = G{\text{e}}^{j\delta } \hfill \\ \end{gathered} $$

(15)

where $A,B,D$ and $G$ are real amplitudes, and $\alpha , \beta , \gamma$ and $\delta$ are real phases. Following the same approach to derive the backbone branches S11 ± , the real and imaginary parts of the resulting equations can be separately set to zero and stationary conditions can be imposed on the modulation equations. Considering the trivial solution by assuming identity of phases ($\alpha =\beta =\gamma =\delta$), the resulting equations are solved for $A,B,D$ and $G$ for each value of$\omega$. Hence, given that where ${X}_{1}=\frac{A}{\omega }+\frac{D}{3\omega }$ and ${X}_{2}=\frac{B}{\omega }+\frac{G}{3\omega }$, the S13 ± backbone branches can be plotted (Table 2).

The following table provides the initial conditions that have been used for generating the NNMs in Figs. 7, 9 and 11.