Passive nonlinear targeted energy transfer

Abstract

Nonlinearity in dynamics and acoustics may be viewed as scattering of energy across frequencies/wavenumbers. This is in contrast with linear systems when no such scattering exists. Motivated by irreversible large-to-small-scale energy transfers in turbulent flows, passive targeted energy transfers (TET) in mechanical and structural systems incorporating intentional strong nonlinearities are considered. Transient or permanent resonance captures are basic mechanisms for inducing TET in such systems, as well as nonlinear energy scattering across scales caused by strongly nonlinear resonance interactions. Certain theoretical concepts are reviewed, and some TET applications are discussed. Specifically, it is shown that the addition of strongly nonlinear local attachments in an otherwise linear dynamical system may induce energy scattering across scales and ‘redistribution' of input energy from large to small scales in the linear modal space, in similarity to energy cascades that occur in turbulent flows. Such effects may be intentionally induced in the design stage and may lead to improved performance, e.g. it terms of vibration and shock isolation or energy harvesting. In addition, a simple mechanical analogue in the form of a nonlinear planar chain of particles composed of linear stiffness elements but exhibiting strong nonlinearity due to kinematic and geometric effects is discussed, exhibiting similar energy scattering across scales in its acoustics. These results demonstrate the efficacy of intentional utilization of strong nonlinearity in design to induce predictable and controlled intense multi-scale energy transfers in the dynamics and acoustics of a broad class of systems and structures, thus achieving performance objectives that would be not possible in classical linear settings.

This article is part of the theme issue ‘Nonlinear energy transfer in dynamical and acoustical systems’.

1. Intentional strong nonlinearity and nonlinear targeted energy transfer

Energy transfer is a central problem in dynamical and acoustical systems, and the main focus in many applications, ranging from vibration isolation concerned with reducing the energy flux into a structure, and vibration absorption seeking to remove energy from a system, to energy harvesting aiming at transferring energy from the mechanical to the electrical domains within a system based on the balance between the energy distribution in the system and the rate at which this energy is harvested. A central feature of linear systems is that energy can be segregated in their linear normal modes, with no possibility of energy transfer or exchange between them. By contrast, while the response of nonlinear systems can be described in terms of the vibration modes identified from a linearized analysis, these modes are no longer decoupled because nonlinearity makes possible for energy transfers between. Under certain conditions [1], however, nonlinear normal modes (NNMs) can be defined for which energy is again segregated, allowing for an approximate description of the system dynamics in terms of isolated (non-resonant) invariant manifolds of the dynamics. Hence, understanding and passively controlling how energy is transferred between coupled systems or NNMs, and across temporal or spatial scales within a system is central to the analysis and design of dynamical and acoustical systems. While inherent mode coupling has been viewed as a disadvantage of nonlinear systems, one can take the view, as done in this work, that such coupling induced by intentional strong nonlinearities can realize dynamical or acoustical behaviour and system performance that is unmatched in linear or even weakly nonlinear settings.

Given that the conventional view is to regard nonlinearity as either unwanted or detrimental—a ‘nuisance' to be accommodated or at most tolerated and ‘designed around'—or as a mere perturbation of the predominantly linear dynamics, the two standard approaches upon which most current methods of nonlinear dynamics and acoustics are based were developed: the linearized and weakly nonlinear approaches. Yet, recent work has revealed the important benefits that strong nonlinearity has to offer over broad areas, namely from turbulent flows [2], fluid–structure interactions [3], vibration and shock isolation [4], and acoustic metamaterials [5], to lattice dynamics [6], energy harvesting [7], blast [8] and acoustic [9,10] mitigation and nanoresonator design [11].

Specifically, a radically new paradigm is promoted in this work through which intentional strong nonlinearities induce nonlinear energy transfer phenomena beneficial to the design objectives and enabling heretofore unattainable passive performance. This includes features such as targeted energy transfers (TET) across temporal and spatial scales inducing features of ‘mechanical turbulence' and complexification (randomization) of the regular dynamical or acoustical response, as well as intense nonlinear energy exchanges or multi-scale energy cascades generating passive self-tunability of the nonlinear dynamics and acoustics. An interesting additional feature of the proposed strongly nonlinear designs is that they are passively adaptive to energy and/or frequency, i.e. the dynamical or acoustical properties can adapt to different excitation environments. Moreover, some of the discussed strongly nonlinear configurations have the capacity to manage or guide broadband energy by mimicking the robust and adaptive nonlinear energy transfer mechanisms occurring across different scales in turbulent flows, resulting in enhanced and robust energy guidance without the need for specific resonance conditions. This concept could potentially transform the way energy management in dynamical and acoustical systems is performed from an engineering perspective, having broad impact in a wide range of applications. These include harvesting energy from broadband sources, passively controlling and suppressing instabilities in mechanical systems, and defining preferential directions of energy flux within complex systems, yielding adaptive and efficient passive guidance of energy flow within dynamical and acoustical systems.

Given, however, that strong nonlinearity may also have adverse effects (e.g. unwanted coexisting attractors) it is necessary to carefully study the dynamics and acoustics of systems with inherent or augmented strong nonlinearities in order to predict system parameter and excitation ranges for which the desired benefits are realized robustly. Therefore, critical to the promoted strongly nonlinear approach is the formulation of new nonlinear concepts, methodologies and techniques that enable accurate theoretical predictions of the system responses, as well as reliable experimental validations of the theoretical findings. Accordingly, and given the highly ‘individualistic' nature of nonlinear systems that restricts the unifying dynamical features which are amenable to common analysis, our approach of intentional use of strong nonlinearity for energy transfer and management dictates innovative theoretical and experimental advances to unlock fundamental new nonlinear phenomena over broad and diverse areas and settings. ‘The challenges related to this task are highlighted by the fact that in the proposed strongly nonlinear approach oftentimes no linearization is possible, so conventional analytical techniques based on linear or weakly nonlinear approximations are no longer applicable’.

2. Nonlinear dynamical resonance captures and targeted energy transfers

Perhaps the most common example involving nonlinear TET are those occurring between modes/scales in turbulent fluid flows, whereby energy gets continuously and unidirectionally transferred from large-scale vortices to small-scale ones. From an energy transfer perspective turbulent flows have two very important properties, namely they are robust and almost impossible to suppress, and are capable of dissipating large amounts of energy compared to laminar flows of the same energy. What is the reason for this efficient dissipation in turbulent flows? In a typical turbulent flow, energy is transferred through a nonlinear mechanism over a large number of modes (broadband spectrum) resulting in simultaneous dissipation over a large number of scales—this is a key property for the super ‘dissipation efficiency' of turbulent flows. By contrast, in laminar flows energy is ‘trapped' in a few large-scale modes with limited energy dissipation.

Motivated by such strongly nonlinear energy transfer phenomena occurring in turbulent flows it is reasonable for one to ask the following logical questions:

• (i) Can similar TET from large to small scales be intentionally engineered in dynamical or acoustical systems?

• (ii) If that would be possible, what would be the benefits resulting from such nonlinear designs?

• (iii) And finally, would it be possible for such nonlinear phenomena to be engineered in a safe and robust way through predictable design?

The following exposition aims to initiate a discussion towards addressing these questions.

In a recent series of works it has been theoretically and experimentally shown that the basic concept for implementing passive directional energy transfer is nonlinear TET, i.e. nonlinear directed, irreversible (on the average), transfer of (e.g. vibration, shock or seismic) energy from a primary system, to a set of a priori determined local attachments (termed nonlinear energy sinks (NESs)) where the energy is localized and locally dissipated without backscattering to the primary system. In addition, the NESs through TET are capable of initiating directed large-to-small-scale energy flow from unfavourable system modes/scales to favourable modes/scales according to the specific design objectives. In that context, TET can be construed as the main ingredient of an integrated passive energy management concept in complex applications across temporal/spatial scales.

TET is realized through resonance captures [12] along the intrinsic periodic solutions of the integrated primary system—NES assembly [13,14]. To achieve such resonance captures an NES typically requires two elements: an essentially nonlinear (i.e. nonlinearizable) stiffness and a (usually, linear viscous) damper. The former enables the NES to resonate with any of the linearized modes of the primary subsystem, whereas the latter dissipates the energy transferred through resonant modal interactions.

In the context of dynamics, a typical single-degree-of-freedom NES attached to an N-degree-of-freedom linear system (with mass, viscous damping and stiffness matrices, M, C and K, respectively) is depicted in figure 1. The primary system possessing a set of natural frequencies, , can be excited by various external narrowband or broadband loading such as impulsive loads, periodic or random excitations, fluid–structure interactions, self-excitations, etc. One seeks to (passively) eliminate unwanted disturbances generated in the primary system from such excitations by attaching a single or a set of NESs. Because the NES does not possess any preferential resonance frequency (because it has no linear stiffness component), it can generate a countably infinite number of nonlinear resonance capture conditions [4,12] with the modes of the primary system, through relations of the form, , with m and n primes, where ω_NES is the instantaneous oscillation frequency of the NES. When such conditions occur, intense, unidirectional energy exchanges can occur from the resonating mode of the primary system to the NES, with energy being dissipated by the NES damper. As the total energy decreases, self-detuning of the dynamics occurs leading to escape from resonance capture and elimination of TET. For applications of TET the reader is referred to (e.g. [4,15–21]).

Figure 1.

Schematic of a primary dynamical system under excitation with an attached NES. (Online version in colour.)

The previous scenario describes passive nonlinear energy absorption to, and local dissipation by the NES of unwanted energy of the primary system. However, as shown in [22–24] the NES can also act as nonlinear energy scatterer of unwanted energy across scales, in similar fashion to directional turbulent energy transfers. Indeed, recent findings demonstrate the efficacy of intentionally implementing NES-induced nonlinear energy transfers in structural systems for passive nonlinear energy management. In [8] and [24] blast mitigation of a large-scale (10 ton), nine-story structure augmented by a set of intentionally strongly nonlinear local attachments was demonstrated. This structure (cf. figure 2a) was tailor-built for blast testing, and was augmented by a passive blast mitigation system of six essentially nonlinear energy sinks—NESs on the upper two floors (cf. figure 2b). Two NESs possessed vibro-impact (VI) nonlinearities caused by collisions of rigid blocks with end barriers, whereas four NESs had smooth, essentially nonlinear (purely cubic) stiffnesses caused by the deformation of prismatic elastomeric bumpers.

Figure 2.

Experimental test structure for blast mitigation research: (a) on the USACE CERL shaker table and (b) the system of six NESs attached to the upper floors [8,24]. (Online version in colour.)

Computational studies and experimental tests demonstrated the efficacy of the NESs to rapidly redistribute shock and blast energy within the modal space of the structure by inducing energy scattering from low-to-high frequencies, resulting in rapid attenuation of the response even at early times (i.e. starting from the first cycle). In figure 3, the NES-induced low-to-high-frequency energy transfers for shaker impulsive tests at the USACE CERL shaker facility in Champaign, IL, are depicted; dashed lines indicate the leading-order modal frequencies. In these plots, the wavelet spectra of the ninth floor accelerations are presented for two cases: (i) when the NESs are locked (figure 3a), i.e. the NESs being prevented from functioning and contributing only through mass effects in the structural response; and (ii) when the NESs are unlocked (figure 3b), i.e. being fully functional and inducing strongly nonlinear effects in the response. Note the intense and broadband energy scattering when the NESs are functional (unlocked), leading to rapid and nearly complete elimination of the structural response after nearly 5 s. This is in contrast with the linear response when the NESs are locked and prevented from functioning, where the shock energy is confined mainly in the two lower modes.

Figure 3.

USACE CERL shock test—nonlinear energy transfers from low-to-high structural modes in the wavelet spectra of the ninth floor structural response for (a) locked NESs and (b) unlocked NESs [8,24]. (Online version in colour.)

The passive nonlinear energy transfer from low-to-high frequencies provides a dual benefit for shock mitigation: first, it is well known that higher-frequency structural modes have smaller amplitudes than lower frequency ones, so the nonlinear energy scattering of shock energy caused by the NESs yields rapid overall structural response attenuation; and, second, the inherent dissipative capacity of the structure itself is enhanced by such nonlinear energy redistribution in the modal space, because, typically higher-frequency structural modes are more efficient energy dissipaters compared to lower frequency ones. It follows that, in addition the beneficial capacity of the NESs to passively absorb and locally dissipate shock energy (as discussed previously), the NESs (especially the VI ones) cause rapid scattering of shock energy from low- to-high-frequency structural modes; this rapid energy redistribution in the frequency domain provides an additional efficient and robust mechanism for rapid response attenuation.

The previously mentioned benefits were clearly demonstrated in the blast tests performed at the USACE GSL facility in Vicksburg, MS (cf. figure 4). In figure 4b, the response of the third-floor response is considered, from which it is deduced that when the NESs are unlocked there is rapid blast mitigation, even starting from the first cycle of the response. Moreover, it was demonstrated that due to its strong nonlinearity the NES-based mitigation system is passively adaptive to shock, blast or seismic excitations with varying frequency and energy contents [22,24].

Figure 4.

USACE GSL blast test: (a) low-intensity blast excitation and (b) third-floor response for NESs locked and unlocked [8,24]. (Online version in colour.)

These findings demonstrate the efficacy of passive energy management by means of intense nonlinear TET between modes in dynamical systems with intentional strong nonlinearities. In the next section, additional applications of TET in strongly nonlinear acoustical systems and nonlinear acoustic metamaterials are discussed. It is shown that strong stiffness nonlinearities yield interesting nonlinear acoustical phenomena, including energy redistributions in the frequency/wavenumber space, similar to the ones discussed in the previous structural example.

3. Targeted energy transfers in nonlinear acoustics

In recent works, it was shown that there are general classes of acoustical metamaterial systems which, because of geometric nonlinearity, become completely devoid of their linear acoustics and behave as nonlinear sonic vacua [25,26]. These are highly degenerate systems with essentially nonlinear (i.e. nonlinearizable) acoustics and zero speed of sound in the sense of classical acoustics. They lack any linear or even linearized resonance spectra but exhibit strongly nonlinear phenomena as discussed below. This class of systems has the capacity of highly intense, passive energy redistributions in the frequency/wavenumber domain, and, hence, TET.

A first class of sonic vacua is one-dimensional (1D) ordered uncompressed granular chains with strongly nonlinear local interactions. A granular medium is defined as an assemblage of interacting granules (or beads) of solid material, either packed (e.g. piles of sand grains or compacted sand bags), sparsely dispersed (e.g. dispersed granules in a fluid, scattered interplanetary particles or planetary rings) or spatially periodic (e.g. one- or multi-dimensional periodic arrays of granules in contact). Applications range from solid-state physics and crystal lattice dynamics, to soil dynamics in geophysics and granular media in the area of metamaterials. The study of the dynamics and acoustics of granular media poses distinct challenges due to their highly discontinuous and nonlinear nature, exhibiting the properties of all three states of matter, that is solids, liquids and gases [27]. The properties of these media are highly adaptive to external environmental stimulations, and so they are considered as a basis for developing new types of nonlinear acoustic metamaterials with unprecedented passively adaptive properties and highly tunable dynamics and acoustics [5,28–30]. Studies of granular media can be divided into two basic categories: granular flows focusing on assemblies of interacting granules with no particular initial order, e.g. [31], and ordered granular media dealing with one- or multi-dimensional ordered arrays or networks of interacting granules, e.g. [25,26,29,32–34]. For example [26] considers homogeneous chains of spherical, linearly elastic granules in contact, which in the long-wavelength approximation [35] generates the following nonlinear sonic vacuum, which is of infinite extent, non-smooth and caused by local interactions between granules:

3.1

In (3.1) u(x, t) is axial displacement, x and t spatial and temporal independent variables, c a real parameter and H(·) the Heaviside function (indicating possibility for separation between granules in the absence of compressive forces). In the absence of a linear term in ∂²u/∂x² in (3.1), the effective speed of sound (in the context of linear acoustics) depends on, and is tunable with energy. The strongly nonlinear acoustics in (3.1) is due to Hertzian granular interactions, whereas non-smooth effects are caused by granule separations and collisions. For an overview of recent results in the area of one- and higher-dimensional ordered granular media refer to [36].

In [37], a 1D granular dimer chain composed of alternating ‘heavy' and ‘light' spherical granules was computationally and experimentally studied. All granules were made of chrome steel (E = 210 GPa, ν = 0.29, ρ = 7810 kg m⁻³), and the radius of the heavy granules was R = 9.525 mm. The radius of the light granules was selected to set various mass ratios for the chain. Nonlinear stress waves in the granular chain were generated by impacting one end of the chain with a striker, which was identical to the heavy particle. For the homogeneous granular chain without internal dissipation (for a discussion of the effects of dissipation refer to [36]) the resulting stress waves are in the form of solitary pulses (the so-called Nesterenco solitary pulses) that propagate unattenuated and undistrorted through the medium [25,26]. Considering the dissipative-less dimer granular chain, the developing stress waves can be either in the form of similar solitary pulses or of rapidly attenuating and disintegrating wave packets. Indeed, depending on the mass ratio between heavy and light granules, countable infinities of resonances [38] and anti-resonances [39] are realized in the granular dimer chain. At resonances there occurs maximum scattering of the propagating stress wave at the interfaces between the heavy and light granules yielding strong attenuation and rapid disintegration of the propagating stress pulse. These resonances can be classified by the ratios of certain characteristic frequencies of the oscillations of the heavy and light granules of the dimer chain [36]. On the contrary, at anti-resonances there occurs negligible scattering of energy of the stress pulse at the granular interfaces, yielding solitary pulses (similar to the Nesterenko solitary pulses in homogeneous granular chains) that propagate unattenuated in the granular dimer chain. Again, classification of these anti-resonances can be made, based on the relative (fast) oscillations of the light granules as they are ‘squeezed' by their neighbouring heavy beads during stress pulse propagation [36,39]. Experimental measurements were performed by a special laser vibrometer technique by the group of Prof. J. Yang at University of Washington, as discussed in [37].

In Figure 5 shows computational and experimental spatio-temporal plots of stress wave propagation in the impulsively excited dimer granular chain. Three different dimers are considered with mass ratios between light and heavy granules yielding an anti-resonance (labelled by AR2), and two resonances (labelled by R1 and R2). In figure 5a,d, a robust solitary pulse is presented corresponding to the anti-resonance AR2. In this antiresonance case, the solitary pulse propagates with almost constant speed without shedding energy at granule interfaces. In sharp contrast to antiresonances, resonances R1 (figure 5b,e) and R2 (cf. figure 5c,f) cause the propagating stress pulse to scatter its energy at granule interfaces, generating oscillating tails, thereby attenuating the leading pulses. In R1, the primary stress pulse loses a considerable fraction of its energy by forming a wave tail oscillating initially in a regular fashion. However, after approximately 80 particles, three leading peaks form a wave packet (see the inset in figure 2b). These pulses exchange energy among them as the wave packet propagates in the granular dimer. This behaviour is attributed to nonlinear beat phenomena, or recurring energy exchanges between nonlinear modes of the tail in the wake of the propagating pulse.

Figure 5.

Computational (a–c) and experimental (d–f) spatio-temporal plots for (a,d) the anti-resonance AR2, (b,e) the resonance R1 and (c,f) the resonance R2 in the granular dimer chain. The insets in (d–f) illustrate comparisons of three successive (heavy–light–heavy) particles' velocity profiles between numerical (solid blue (heavy) and green (light)) and experimental (dashed red) results [37]. (Online version in colour.)

To highlight the nonlinear energy dispersion mechanisms in resonances R1 and R2 in the granular dimer chain, the frequency spectra of the numerical responses of the granule responses (figure 6a), and the wavenumber spectra of a specific granule response versus time (figure 6b,c) are considered. In R1 (figure 3a), a strong high-frequency signal (approx. 9 kHz) appears, and the magnitude of the low-frequency signal decreases as the primary pulse is shedding its energy to the wave tail (around up to 80 granules). This predominant mechanism of energy transfer ceases to exist as soon as the nonlinear beating starts (after approx. 80 granules). This indicates that the mechanical energy carried initially by the low-frequency primary pulse is redistributed to the higher frequency contained in the wave tail due to the nonlinear resonance mechanism, but it is subsequently trapped in the beating mechanism without further substantial decay. Again, there is clear low-to-high-frequency nonlinear scattering of energy in the granular dimer chain, similar to the nonlinear energy scattering results evidenced in the structural dynamics of the previous section. Clearly, the cause for this passive targeted energy transfer is the strong Hertzian nonlinear interactions between granules in compression and the VIs between granules occurring the wave tails in the wake of the propagating primary pulse. The results of this strong energy scattering are evidenced in the strong nonlinear wave dispersion in the spatio-temporal plots of figure 5b,c,e,f.

Figure 6.

Impulsively loaded dimer chain: (a) frequency spectrum of bead velocities for 1 : 1 nonlinear resonance (R1), and (b,c) wavenumber spectra of bead velocities for the R1 and R2 nonlinear resonances, respectively, showing energy cascades and multi-scale energy scattering [37]. (Online version in colour.)

Equally informative are the wavenumber plots of figure 6b,c, revealing how in the resonance cases R1 and R2 the energy at small wavenumbers of the propagating primary pulse is converted to a wavenumber close to π/2 in the wave tail. This length scale corresponds to the length of two sets of heavy and light granules, suggesting the emergence of a periodic travelling wave pattern that is being excited in association with a periodicity of two granules. The wavenumber analysis reveals the excitation of these states in a transient way for resonance R1 until the beating pattern forms. On the contrary, in resonance R2 the pattern is persistent, enabling the continuous targeted energy transfer from the principal propagating pulse to the trailing wave tail. In R2, when the wave tail disintegrates to chaotic oscillations, the length scale also spreads out widely.

These results demonstrate the effects of nonlinear TET on the acoustics of uncompressed ordered granular media, in similarity to TET in the dynamics of discrete oscillators discussed in the previous section. Specifically, irreversible nonlinear energy transfers from low-to-high frequencies/wavenumbers have been presented, yielding strong energy redistribution in the frequency/wavenumber space. This redistribution is caused by the strongly nonlinear (in fact, nonlinearizable) local Hertzian interactions between granules, which generate essentially nonlinear acoustics, with no linear counterparts. It is noted that any applied pre-compression would yield a linear component in the acoustics [36], yet, provided that the pre-compression is small the previous findings would still hold. On the other hand, strong pre-compression would generate weakly nonlinear acoustics and the previous strongly nonlinear phenomena would not expect to hold.

A second class of nonlinear acoustical systems that exhibit inherent strong nonlinearity and gives rise to TET concerns phononic lattices of simple configuration (i.e. composed of linear components), which are ‘transformed' to nonlinear sonic vacua by strong geometric/kinematic nonlinearities. As the simplest example of such a lattice one considers a dissipation-less, in-plane lattice of N − 2 particles of identical mass m connected by linear springs with elastic constants k (unstretched at the horizontal equilibrium), and having fixed boundary conditions (cf. figure 7). Taking into account geometric nonlinearities, the equations of motion are given by Manevitch & Vakakis [6],

3.2

with n = 2, … , N − 1, and z₁ = z_N ≡ 0, y₁ = y_N ≡ 0. In (3.2), y_n and z_n are the transverse and axial displacements of the n − th particle from its equilibrium position, is the tension in the n − th spring (where l_n and denote the unstretched and stretched lengths of the spring, respectively), ϕ_n is the angle between the n − th spring and the horizontal, and without loss of generality we set l_i = l = 1/(N − 1), i = 2, … , N. Expressing ϕ_n and T_n in terms of y_n and z_n, and expanding in Taylor series with respect to (z_n − z_n−1) and (y_n − y_n−1), we consider the limit of small-energy oscillations by introducing the rescalings and the slow time variable τ = ϵ(k/m)^1/2t. Then, (3.2) is expressed as (with (·)′ ≡ d(·)/dτ)

3.3a

and

3.3b

Figure 7.

In-plane lattice giving rise to a sonic vacuum [6].

At the leading-order approximation the rescaled tension is spatially uniform (but slowly varying in time) in all springs, and can be expressed as [6]

3.4

so that the system (3.3b) leads to the following predominantly transverse nonlinear sonic vacuum:

3.5

One notes that the rescaled tension T is a purely quadratic function of the transverse displacements, and that the axial deformations are computed by (3.3a). It follows that (3.5) has an effective ‘nonlinear speed of sound' that is tunable with energy. There are some intriguing aspects regarding (3.5). First, ‘this sonic vacuum was derived in the small-energy limit of an otherwise linear lattice due to geometric nonlinearities’; this feature contrasts with typical nonlinear behaviour where the nonlinear effects are stronger at higher energies. Second, the tension T in (3.4) represents a non-local term that couples the transverse displacement of each particle with that of every other particle; hence, ‘even though the original lattice involves only next-neighbour (local) interactions between particles, the sonic vacuum (3.4) possesses strongly non-local terms that dynamically couple every transverse displacement with all others’. Third, it can be shown [6] that (3.4) admits exactly (N − 2) time-periodic standing waves (or NNMs [1]), with mode shapes given by that are ‘mutually orthogonal and identical to those of the linear chain with constant’ T; all NNMs, with exception of the highest one, are unstable. This NNM instability leads to nonlinear resonance interactions that produce coherent responses with combined stationary and non-stationary parts, designated as ‘pseudo-travelling waves' [6]. These facilitate intense recurrent energy exchanges and cascades between subsets of NNMs. It follows that ‘the sonic vacuum (3.4) possesses a (highly unusual) exact orthogonal nonlinear modal basis’, which, as shown below enables the precise study of TET, and nonlinear energy scattering and cascading in this system. Given the absence of a linear spectrum, the amplitudes and frequencies of the NNMs are tunable with energy leading to passive adaptively of the dynamics and acoustics with energy. Fourth, ‘the boundary conditions play a critical role in the generation of the sonic vacuum’ because they provide the spatially uniform (but slowly varying in time) tension T that is the source of the strongly non-local terms. Finally, a long-wave approximation of (3.5) leads to the continuous sonic vacuum,

3.6

which compared to the granular sonic vacuum (3.1) is finite, it's caused by non-local geometric effects, and is smooth.

The implications of these findings on passive TET in the lattice are significant. In figure 8, a lattice of 50 particles is considered when the 30th (unstable) transverse NNM is excited with the initial small amplitude A = 0.02. The nonlinear energy scattering from low-to-high transverse modes (but not vice versa) is demonstrated by plotting the wavenumber content of the response against normalized time [40]. These results highlight the intense energy scattering occurring in this system. Conceptually the observed nonlinear energy transfers are similar to those depicted in figures 3 and 6. Owing to energy cascading from low-to-high wavenumbers the response of the lattice becomes ‘randomized' or chaotic, attaining features of ‘mechanical turbulence’. This feature is closely related to the instability of the NNMs of the lattice (except for the highest one) which enables the capacity of the sonic vacuum (3.5) for broadband resonance exchanges across spatial and temporal scales [6] and intense energy transfers. Moreover, recent work has revealed that the lattice supports ‘energy explosions' [41] converting abruptly narrowband to broadband particle responses caused by parametric instabilities, as well as accelerating oscillatory fronts caused by strongly non-local interactions [42].

Figure 8.

Simulations of nonlinear low-to-high wavenumber energy cascading in the lattice of 50 particles when the 30th unstable NNM is excited: (a) full system (2) and (b) sonic vacuum (5) [40]. (Online version in colour.)

Finally, interesting nonlinear wave interactions leading to targeted energy transfer between colliding propagating wavefronts are also realized in the lattice (3.2) given its capacity to support both (linearized) axial and (strongly nonlinear) transverse waves. In figure 9, the spatio-temporal evolution of the instantaneous kinetic energy of a lattice of 100 particles, excited at its 50th particle by the transverse impulse F(t) = 0.1δ(t) is depicted. Despite the purely transverse excitation, the geometric coupling between axial and transverse deformations initiates two axial-transverse wave pairs (of which only one pair is shown in figure 9). The axial wave is nearly linear and represents acoustic sound propagation governed approximately by the linear wave equation. The transverse wavepacket is a strongly nonlinear localized travelling pulse with energy-dependent velocity (the wavenumber content of this wavepacket is low, preventing its spreading due to dispersion). At normalized time τ ≈ 100 there occurs almost complete, TET from the (reflected at the left fixed boundary) nearly linear axial wave (whose speed does not depend on the intensity of excitation) to the strongly nonlinear transverse solitary pulse, whose speed increases due its increased energy (this is noted by the change in slope of the transverse pulse after the wave interaction in figure 9). This raises the interesting prospect of ‘nonlinear wave manipulation between different sets of coexisting linear and nonlinear waves’ in this medium by means of geometric or kinematic effects. Such wave manipulation could lead to the design of acoustical systems with capacities for passive wave guidance of different sets of wave packets at preferential directions, and at variable energy-dependent speeds through targeted (irreversible) multi-scale wave energy transfers.

Figure 9.

Nonlinear axial-to-transverse targeted wave energy transfer in the impulsively loaded lattice (only half of the spatio-temporal domain is depicted) [40]. (Online version in colour.)

4. Concluding remarks and forward look

The broad research area described in this paper, namely passively and nonlinearly managing the energy flux in a system through TET, directly impacts a vast array of problems in engineering dynamics and acoustics, including vibration isolation, vibration absorption, energy harvesting, acoustic metamaterials and non-reciprocal acoustical systems. Indeed, the need for passive energy management is ubiquitous across a range of engineering systems and applications. Hence, this area of research promotes a new paradigm for nonlinear energy transfer using intentional strong nonlinearities that goes beyond the current approaches based on resonance. Moreover, robustly designing systems with intentional nonlinearities is tied to uncertainty quantification to uncover levels of performance unavailable to traditional linear or even weakly nonlinear designs. To this end, one aims to develop an integrated nonlinear framework of concepts, methodologies and design tools to induce features of TET and possibly ‘mechanical turbulence' into mechanical systems, mimicking the underlying physics of energy cascading from large-to-small spatial scales in turbulent flows, but in a controlled and predictable way.

By doing so, one aims in robust and efficient energy dissipation or redistribution, and rapid response reduction as energy is irreversibly transferred from the large to the small scale. To achieve this one needs to consider a new class of systems, such as systems with strongly nonlinear attachments or nonlinear sonic vacua discussed in this work, which, due to intentionally designed or geometrically/kinematically induced nonlinearities, become devoid of linear dynamics or acoustics and exhibit strongly nonlinear responses that are passively and adaptively tunable with energy. To this end, there is the need to explore the opportunities and harvest the benefits that intentional strong nonlinearity has to offer, achieving predictive multi-scale TET; wave manipulation and guidance; passively adaptive acoustic filtering; mechanical energy transfer similarity laws analogous to Kolmogorov's 5/3 law for turbulent flows; fundamental understanding of nonlinear energy fluxes using nonlinear stochastic analysis and uncertainty quantification; and design optimization for robust energy transfer and management in the presence of uncertainty and unmodelled dynamics.

The outlined research can be potentially transformative for a wide class of application fields, and its broader impacts are far-reaching. Indeed, the concept of ‘mechanical turbulence' can enable performance advances in areas where energy transfer is critical: From vibration damping and isolation applications to energy harvesting. Furthermore, work on uncertainty quantification and robust design will provide a framework for the future design of systems with strong nonlinearities and robust performance in the presence of noise and parameter variability, capable of managing complexity through strong energy exchanges and irreversible transfers. The overall aim will be to provide a comprehensive demonstration of the capacity and efficacy of designing intentionally nonlinear systems for targeted energy transfer and management, possessing features heretofore unattainable with traditional current linear or weakly nonlinear approaches. This may present a potentially paradigm-shifting approach for designing dynamical and acoustical systems.

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Competing interests

The author has no competing interests.

Funding

The concepts and results reviewed in this work were funded in part by the following Research Grants: DARPA Research Contract HR0011-10-1-0077, ARO MURI grant no. 56150-MS-MUR and NSF Research Grant nos CMMI-1000615 and CMMI-14-638558. Contributors to this work are present and past members, and academic collaborators of the Linear and Nonlinear Dynamics and Vibrations Laboratory (LNDVL) of the University of Illinois at Urbana–Champaign (http://lndvl.mechse.illinois.edu/).