Nonreciprocity in the dynamics of coupled oscillators with nonlinearity, asymmetry, and scale hierarchy

In linear time-invariant dynamical and acoustical systems, reciprocity holds by the Onsager-Casimir principle of microscopic reversibility, and this can be broken only by odd external biases, nonlinearities, or time-dependent properties. A concept is proposed in this work for breaking dynamic reciprocity based on irreversible nonlinear energy transfers from large to small scales in a system with nonlinear hierarchical internal structure, asymmetry, and intentional strong stiffness nonlinearity. The resulting nonreciprocal large-to-small scale energy transfers mimic analogous nonlinear energy transfer cascades that occur in nature (e.g., in turbulent flows), and are caused by the strong frequency-energy dependence of the essentially nonlinear small-scale components of the system considered. The theoretical part of this work is mainly based on action-angle transformations, followed by direct numerical simulations of the resulting system of nonlinear coupled oscillators. The experimental part considers a system with two scales—a linear large-scale oscillator coupled to a small scale by a nonlinear spring—and validates the theoretical findings demonstrating nonreciprocal large-to-small scale energy transfer. The proposed study promotes a paradigm for designing nonreciprocal acoustic materials harnessing strong nonlinearity, which in a future application will be implemented in designing lattices incorporating nonlinear hierarchical internal structures, asymmetry, and scale mixing.

Figure 1

A cell composed of a system of coupled oscillators in series, and incorporating strong stiffness nonlinearity, high asymmetry, and large-to-small scale hierarchy.

Figure 2

Transient responses of the different scales of the cell with $n=5$ of Fig. 1 for (a) excitation $F_1\left(t\right)=0.12\delta \left(t\right)$ applied to LS, and (b) excitation $F_2\left(t\right)=-0.12{\varepsilon }_5^{-1/2}\delta \left(t\right)$ applied to SS5.

Figure 3

A cell with a grounded linear large scale (LS) nonlinearly coupled to a small scale (SS).

Figure 4

Impulsive responses and associated wavelet spectra of the SS response for excitation of (a), (c) the LS loading scenario I, and (b), (d) the SS loading scenario II; the (vertical) frequency axes of the wavelet spectra in (c), (d) are logarithmic.

Figure 5

Percentage of impulsive energy dissipated (%) by the two scales of the cell of Fig. 3 for impulsive excitation of (a) the LS loading scenario I, and (b) the SS loading scenario II; the highlighted region indicates the regime of strong irreversible energy transfer from the LS to the SS.

Figure 6

Actions $I_1\left(t\right), I_2\left(t\right)$ and cosines of the angles $cos{\theta }_1\left(t\right), cos{\theta }_2\left(t\right)$ of the LS and SS components of the cell of Fig. 3 for impulse excitation applied to the (a), (c) LS (loading scenario I), and (b), (d) SS (loading scenario II); the regimes of 1:1 TRC are highlighted in (c), (d).

Figure 7

(a) Two views of the experimental fixture for local nonreciprocity within the cell of Fig. 3; the direction of motion is indicated by the double-sided arrow, and (b) schematic of the cell.

Figure 8

Validation of the identified model by velocity time series reconstructions: (a) applied impulse to LS (loading scenario I), (b) response of the LS, (c) response of the SS.

Figure 9

Experimental nonreciprocity in the fixture of Fig. 7: (a) loading scenario I, 6.9 N impulse applied to LS, (b) loading scenario II, 5.7 N impulse applied to SS; in each case the velocity time series of the LS and SS and their wavelet spectra and FFTs are depicted.

Figure 10

Experimental nonreciprocity in the fixture of Fig. 7: (a) loading scenario I, 23.3 N impulse applied to LS, (b) loading scenario II, 22.5 N impulse applied to SS; in each case the velocity time series of the LS and SS and their wavelet spectra and FFTs are depicted.

Figure 11

Experimental nonreciprocity in the fixture of Fig. 7: (a) loading scenario I, 42.8 N impulse applied to LS, (b) loading scenario II, 45.0 N impulse applied to SS; in each case the velocity time series of the LS and SS and their wavelet spectra and FFTs are depicted.

Figure 12

Instantaneous energy for loading scenarios I and II subject to an impulsive load of nearly equal magnitude. The thin lines with arrows represent the slopes of each curve.