Resonance-induced energy localization in a weakly dissipative nonlinear chain

This work investigates the emergence of resonance and resonance-induced localization in a weakly dissipative chain of coupled anharmonic oscillators under the influence of a harmonic force applied at one end of the chain. The dynamics of the chain is studied assuming 1:1 (fundamental) resonance, when the response of each nonlinear oscillator has a dominant harmonic component with the frequency close to the frequency of the external excitation. It is shown that weak dissipation in a strongly nonlinear chain may be a key factor preventing large-amplitude resonance. The resulting process in the dissipative system represents resonance-induced large-amplitude oscillations of a part of the chain adjacent to the actuator and escape from resonance of the distant oscillators. Conditions of the emergence of resonance and energy localization are derived. The obtained solutions indicate that the maximal concentration of energy on the excited oscillator arises together with equipartition of energy among the other resonant oscillators. An agreement between the analytical and numerical results is demonstrated.

Figure 1

Parametric thresholds ((3.13)). Points $A$ and $B$ indicate the intersections of the vertical lines ${\varepsilon }_{cr}^{\left(1\right)}\mathrm{and}{\varepsilon }_{cr}$ with the boundary $f_{1\varepsilon }$.

Figure 2

Nonresonant (a) and resonant (b), (c) oscillations of the two-particle nondissipative chains: (a) $(\varepsilon =0.07<{\varepsilon }_{cr},f=0.7)\in D$; (b) $(\varepsilon =0.07<{\varepsilon }_{cr},f=2)\in D_1$; (c) $(\varepsilon =0.13>{\varepsilon }_{cr},f=0.7)\in D_0$; blue solid curve: $u_1$; red dotted curve: $u_2$.

Figure 3

Slowly varying amplitudes $a_1\left(\tau \right),a_2\left(\tau \right)$ for the fast processes $u_1,u_2$ presented in Fig. 2.

Figure 4

Slowly varying amplitudes for the four-particle nondissipative chains; blue solid curve: $a_1$; red dotted curve: $a_2$; greeen dashed curve: $a_3$; black dash dotted curve: $a_4$.

Figure 5

Slowly varying amplitudes for the chains with the parameters near the critical values; blue solid curve: $a_1$; red dotted curve: $a_2$; greeen dashed curve: $a_3$; black dash dotted curve: $a_4$.

Figure 6

Fast responses $u_1,u_2$ of the pair of coupled oscillators with the parameters $\varepsilon =0.13,f=0.7$: (a) $\delta =0.1<{\delta }_2$; (b) $\delta =0.5>{\delta }_1>{\delta }_2$.

Figure 7

Slow processes in the arrays with the parameters $\varepsilon =0.13,f=0.7$ and different coefficients of dissipation: the amplitude (a) and the phase (b) of the resonant chain at $\delta =0.1<{\delta }_2$; the amplitude (c) and the phase (d) of small oscillations at $\delta =0.5>{\delta }_1$.

Figure 8

Capture into resonance and escape from resonance of the oscillators in the two-particle arrays with the parameters $\varepsilon =0.13,f=1$ and different coefficients of dissipation.

Figure 9

Slow amplitudes in the four-particle arrays with the parameters $\varepsilon =0.07,f=2.5$, and different coefficients of dissipation; blue solid curve: $a_1$; red dotted curve: $a_2$; greeen dashed curve: $a_3$; black dash dotted curve: $a_4$.

Figure 10

Slow amplitudes in the four-particle arrays with the parameters $\varepsilon =0.07,f=2$ and different coefficients of dissipation; blue solid curve: $a_1$; red dotted curve: $a_2$; greeen dashed curve: $a_3$; black dash dotted curve: $a_4$.

Figure 11

Oscillator amplitudes in the 12-particle arrays with the parameters $(\varepsilon =0.07,f=2.5)\in D_1$ and different coefficients of dissipation. The notations blue solid curve: $a_1$; red dotted curve: $a_2$; greeen dashed curve: $a_3$; black dash dotted curve: $a_4$ are used in Figs. 11.