Nonreciprocal acoustics and dynamics in the in-plane oscillations of a geometrically nonlinear lattice

We study the dynamics and acoustics of a nonlinear lattice with fixed boundary conditions composed of a finite number of particles coupled by linear springs, undergoing in-plane oscillations. The source of the strongly nonlinearity of this lattice is geometric effects generated by the in-plane stretching of the coupling linear springs. It has been shown that in the limit of low energy the lattice gives rise to a strongly nonlinear acoustic vacuum, which is a medium with zero speed of sound as defined in classical acoustics. The acoustic vacuum possesses strongly nonlocal coupling effects and an orthogonal set of nonlinear standing waves [or nonlinear normal modes (NNMs)] with mode shapes identical to those of the corresponding linear lattice; in contrast to the linear case, however, all NNMs except the one with the highest wavelength are unstable. In addition, the lattice supports two types of waves, namely, nearly linear sound waves (termed “$L$ waves”) corresponding to predominantly axial oscillations of the particles and strongly nonlinear localized propagating pulses (termed “$NL$ pulses”) corresponding to predominantly transverse oscillating wave packets of the particles with localized envelopes. We show the existence of nonlinear nonreciprocity phenomena in the dynamics and acoustics of the lattice. Two opposite cases are examined in the limit of low energy. The first gives rise to nonreciprocal dynamics and corresponds to collective, spatially extended transverse loading of the lattice leading to the excitation of individual, predominantly transverse NNMs, whereas the second case gives rise to nonreciprocal acoutics by considering the response of the lattice to spatially localized, transverse impulse or displacement excitations. We demonstrate intense and recurring energy exchanges between a directly excited NNM and other NNMs with higher wave numbers, so that nonreciprocal energy exchanges from small-to-large wave numbers are established. Moreover, we show the existence of nonreciprocal wave interaction phenomena in the form of irreversible targeted energy transfers from $L$ waves to $NL$ pulses during collisions of these two types of waves. Additional nonreciprocal acoustics are found in the form of complex “cascading processes, as well as nonreciprocal interactions between $L$ waves and stationary discrete breathers. The computational studies confirm the theoretically predicted transition of the lattice dynamics to a low-energy state of nonlinear acoustic vacum with strong nonlocality.

Figure 1

Oscillatory lattice in the plane.

Figure 2

Range of 1:1 resonance interactions between pairs of NNMs for acoustic vacua consisting of (a) $N=10$ and (b) $N=50$ particles.

Figure 3

Slow flow (in cluster coordinates) for the acoustic vacuum with $N=10$, for 1:1 resonance between (a) the ninth and 10th NNMs, (b) the fourth and 10th NNMs, and (c) the fourth and fifth NNMs.

Figure 4

Limiting phase trajectories (LPTs) (shown in bold) for the acoustic vacuum with $N=10$ for (a) 1:1 resonance of the two NNMs with $m=5$ and $k=4$, and (b) absence of 1:1 resonance between NNMs with $m=9$ and $k=4$ [see plot of Fig. 2]; upper plots are in cluster variables $Y_1$ and $Y_2$, and lower plots in modal variables $A_m$ and $A_k$.

Figure 5

Recurrent energy exchanges of maximum intensity corresponding to LPTs between two clusters $Y_1=\left(A_m+A_k/2\right)$ and $Y_2=\left(A_m-A_k/2\right)$ for $N=20,m=20$ and $k=19$ of (a) the acoustic vacuum (2), and (b) the exact lattice (1b).

Figure 6

Nonreciprocal energy transfers from low-to-high wave numbers in the acoustic vacuum (2) following a direct excitation with modal amplitude $A=10.0$ and $\varepsilon ={10}^{-4}$ of (a) the third NNM, (b) the ninth NNM (contour plots of the normalized modal amplitudes $A_p\left({\tau }_0\right)/A,p=1,...,10$ are shown).

Figure 7

Excitation of the third NNM of the acoustic vacuum: (a) Primary excitation of the fifth NNM, and (b) secondary excitation of the 10th NNM; note the time delay between the two modal responses.

Figure 8

Nonreciprocal energy transfers from low-to-high wave numbers following the direct excitation with modal amplitude $\varepsilon A=0.01$ of the 10th NNM: (a) acoustic vacuum (2), and (b) exact lattice (1b) for $N=20$ [contour plots of the modal velocities $A_p^{\prime }\left({\tau }_0\right),p=1,...,N$ are shown].

Figure 9

Nonreciprocal energy transfers from low-to-high wave numbers following the direct excitation with modal amplitude $\varepsilon A=0.02$ of the 30th NNM: (a) acoustic vacuum (2), and (b) exact lattice (1b) for $N=50$ (contour plots of the modal velocities ${\dot{A}}_p\left(t\right),p=1,...,N$ are shown).

Figure 10

Spatiotemporal evolution of the kinetic energy of a lattice with $N=101$ impulsively excited at its middle particle: (a) Full spatial domain showing the two pairs of sonic $L$ waves and nonlinear $NL$ pulses, and (b) detail of the wave interactions over half the spatial domain.

Figure 11

Snapshots of spring tensions (shifted vertically for clarity) in the normalized time period ${\tau }_0\in \left[2,50\right]$ of the response of the impulsively excited lattice corresponding to Fig. 10; note the near uniform and slowly varying state of the spring tension attained between the two $NL$ pulses signifying the transient generation of an acoustic vacuum in that domain.

Figure 12

Spatiotemporal evolution of the kinetic energy of a lattice with $N=101$ excited by a transverse displacement excitation at its middle particle.