Supratransmission in a Disordered Nonlinear Periodic Structure

TL;DR

This work analyzes supratransmission in damped, finite, disordered nonlinear periodic structures under harmonic forcing within the linear stop band. It combines transfer-matrix and semi-linear analyses to predict the onset threshold $F_{th}$ and demonstrates that, on average, disorder does not shift the transmission threshold yet reduces transmitted energy by pushing nonlinear waves into the linear pass band. The authors show that transmitted spectra lie in the linear pass band, with ensemble-averaged thresholds robust to disorder but energy transmission sensitive to it, and they derive analytical expressions (e.g., $F_{th}^2=18\alpha^2(p\pm\sqrt{q^3})$) that accurately capture onset in weakly coupled regimes. The findings illuminate the competition between dispersion, nonlinearity, and disorder, provide guidelines for predicting transmission in engineered disordered lattices, and highlight the roles of damping and coupling near pass-band edges.

Abstract

We study the interaction among dispersion, nonlinearity, and disorder effects in the context of wave transmission through a discrete periodic structure, subjected to continuous harmonic excitation in its stop band. We consider a damped nonlinear periodic structure of finite length with disorder. Disorder is introduced throughout the structure by small changes in the stiffness parameters drawn from a uniform statistical distribution. Dispersion effects forbid wave transmission within the stop band of the linear periodic structure. However, nonlinearity leads to supratransmission phenomenon, by which enhanced wave transmission occurs within the stop band of the periodic structure when forced at an amplitude exceeding a certain threshold. The frequency components of the transmitted waves lie within the pass band of the linear structure, where disorder is known to cause Anderson localization. There is therefore a competition between dispersion, nonlinearity, and disorder in the context of supratransmission. We show that supratransmission persists in the presence of disorder. The influence of disorder decreases in general as the forcing frequency moves away from the pass band edge, reminiscent of dispersion effects subsuming disorder effects in linear periodic structures. We compute the dependence of the supratransmission force threshold on nonlinearity and strength of coupling between units. We observe that nonlinear forces are confined to the driven unit for weakly coupled systems. This observation, together with the truncation of higher-order nonlinear terms, permits us to develop closed-form expressions for the supratransmission force threshold. In sum, in the frequency range studied here, disorder does not influence the supratransmission force threshold in the ensemble-average sense, but it does reduce the average transmitted wave energy.

Figures (25)

• Figure 1: The schematic of the periodic structure made of $N$ unit cells. The repeating unit is indicated by the dashed box. The external harmonic force, $\tilde{f}(\tilde{t}\,)$, is applied to the first unit only. Two electromagnets, operated by direct currents, are fixed to the ground under the beam in each unit. The currents are chosen such that the electromagnets have the same polarity facing the beam. We fix $h=d/10$ in this work.
• Figure 2: Decay exponent $\gamma_0$ for an infinitely-long exactly-periodic linear structure. The white and grey backgrounds indicate the pass and stop bands, respectively. When $\zeta>0$, the decay exponent is always positive, even within the pass band. Anderson localization is relevant for frequencies within the pass band, whereas supratransmission occurs within the stop band -- we will discuss these phenomena in subsequent sections.
• Figure 3: Influence of disorder on response localization at $\Omega = 1.12$, near the middle of the pass band. The average response becomes localized to the driven unit ($n=1$) as the value of $D/C$ increases. The strength of coupling is kept constant, $C=0.05$. An ensemble of $10^5$ realizations are used for each non-zero value of $D$. Other system parameters are $N=10$, $\zeta=0.005$ and $\omega_0^2=1.05$.
• Figure 4: Influence of disorder on spatial localization of the mode shapes of the structure. IPR, defined in \ref{['ipr']}, is plotted as a function of $D/C$ for the first and last mode shapes. The insets show the mode shape of a typical realization of disorder within the ensemble at $D/C=0,1,2$. The result for $n=1$ are shown in black and those for $n=N$ are shown in grey.
• Figure 5: Influence of disorder on the linear response of the structure. (a) The average decay exponent $\gamma$, defined in \ref{['xiUNU1']}, is plotted for different values of disorder. The grey area corresponds to the stop band. (b) The average natural frequencies of the first ($\omega_1$) and last ($\omega_N$) modes are plotted as a function of $D/C$. The black circles correspond to $\omega_1$ and grey squares to $\omega_N$. The empty markers correspond to the minimum and maximum values of each natural frequency within the ensemble. The horizontal lines indicate the natural frequencies of the ordered structure.