Introduction to a topical issue ‘nonlinear energy transfer in dynamical and acoustical Systems'

Abstract

This topical issue is devoted to recent developments in the broader field of energy transfer across scales in nonlinear dynamical and acoustical systems. Nonlinear energy transfers are common in Nature, with perhaps the most famous example being energy cascading from large to small length scales in turbulent flows. Yet nonlinearity has been traditionally perceived either as an unavoidable nuisance or as an unwelcome design restriction in engineering systems. Nowadays, however, this trend is reversing, with nonlinear phenomena being intensely studied in diverse disciplines. Furthermore, strong nonlinearity is now intentionally used and explored in a variety of mechanical and physical settings, such as granular media, acoustic metamaterials, nonlinear energy sinks, essentially nonlinear and nonlocal lattices, vibro-impact oscillators, vibration and shock isolation systems, nanotechnology, biomimetic systems, microelectronics, energy harvesters and in other applications. This topical issue is an attempt to document in a single volume some of these recent research developments, in order to establish a common basis and provide motivation and incentive for further development. The aim is to discuss and compare theoretical and experimental approaches pursued by research groups in different areas, and describe the state of the art of nonlinear energy transfer phenomena in an as broad as possible range of applications of current interest.

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

1. General scope of the topical issue

In the majority of common designs in mechanical engineering and related fields, dynamical systems operate in linear or weakly nonlinear (quasi-linear) regimes. This choice is easily understandable, because such regimes are deeply understood, are predictable and can be assessable by well-developed methods of analysis [1,2]. These methods often rely on well-established ideas of averaging, multiple-scale expansions and other asymptotic techniques [3–5]. Still, strong (essential) nonlinearities occur in mechanical systems due to various reasons, including clearances, impacts, friction, material, geometric or kinematic nonlinearities, external fields, scale effects and plasticity [6–8]. In many applications such behaviour is necessary, desirable and/or important for operational purposes [6,7]. In others, it is completely unwanted, but still profoundly reveals itself in the dynamics. Cracks in continuous structures are one such example [9–11].

Moreover, the accelerated need to predictively design lighter, faster, smaller and ever-more robust engineering systems with continuously expanding performance envelops, and operating in ever harsher environments dictates a paradigm-shifting approach, in the sense that nonlinear effects and the ensuing cross-scale energy transfers caused by them, cannot be ignored anymore, but rather intentionally used to achieve the design objectives. Among other applications, one can mention targeted energy transfer in essentially nonlinear systems with applications to energy absorption and harvesting [12–24], wave propagation, mitigation, energy redistribution and arrest in granular crystals and in systems characterized as ‘sonic vacua’ (i.e. with zero linear speed of sound) [25–32], dynamic and acoustic non-reciprocity, as well as wave tailoring and control by means of acoustic metamaterials composed essentially nonlinear elements [33–36].

Analysis of the dynamics of essentially nonlinear systems is a major current challenge. A complete picture can be obtained only for extremely rare completely integrable cases [3], yet non-integrability is generic in multi-dimensional nonlinear systems. For other systems, it is sometimes possible to derive exact periodic solutions—examples are nonlinear normal modes [37–39] and discrete breathers in selected models [40–42]. Some information on periodic solutions of a broad variety of systems may be obtained by approximate and numerical methods [43–45], but non-stationary dynamical and acoustical processes are more difficult to model and understand. Yet, these are usually the most interesting and most important for applications involving controlled nonlinear energy transfers, so the predictive analysis of such processes is an active topic of intense current research.

In addition, the idea of intentional use of the strong nonlinearity for passive control of energy transfers in dynamical and acoustical systems has developed over last two decades, transitioning from a limited number of early theoretical models to a diverse theoretical and experimental body of work ranging from vibration and shock isolation, energy harvesting and automotive applications, to acoustic metamaterial design, phononic crystals and nanoresonators. Of special interest is the development of mechanical models that mimic in some sense the intense and robust energy uni-directional energy transfers occurring in Nature, with the most famous being the large-to-small energy cascades occurring in turbulent flows; in that context, the study of dynamical and acoustical systems with features of ‘mechanical turbulence is an area of current interest’. Then, it is desirable to create a kind of common background, framework and language for this rapidly developing field. Accordingly, the Topical Issue aims to bring together researchers active in various aspects of this scientific topic. The papers of this volume were selected in order to provide broad coverage of the field of nonlinear energy transfer, while preserving the logical interconnections between different branches of analysis and applications.

2. Contributions

While suggesting and selecting invited papers for this Topical Issue, the editors tried, from one side, to keep a balance between ‘review flavour’ of the contributions, and reports on new results and developments. Besides, they attempted to ensure decent representation of all major modes of scientific endeavour: analytic developments, numerical simulations, fundamental experiments and application-oriented explorations. Of course, the ultimate judgement whether these attempts were successful or not is left to the readers.

Part of the contributions is devoted to dynamics of the essentially nonlinear systems with low numbers of degrees of freedom (low-d.f.). The review of Vakakis [46] addresses recent developments in the problem of targeted energy transfer in mechanical systems with nonlinear energy sinks. The work by Perchikov & Gendelman [47] presents a brief review on dynamics of low-d.f. Hamiltonian systems with isolated resonances, and explores a peculiar relationship between exact and averaged dynamics in coupled strongly nonlinear oscillators. Cho et al. [48] review the state of the art and present novel findings concerning observations and applications of nonlinear dynamic phenomena and energy transfers in the rapidly developing field of micro- and nanomechanical resonators. The work of Kerschen and co-workers [49] presents important and novel applications of meticulously designed nonlinear elements in the more than a century-old problem of vibration absorption.

Most contributions of this Issue, however, go beyond the low-d.f. models and deal with the dynamics of nonlinear chains and lattices. Such choice is completely understandable, because these systems, from one side, exhibit a plethora of intriguing phenomena, absent in low-d.f. systems. From the other side, one-dimensional chains and lattices are still simple and tractable enough to gain deep insight with reasonable analytic, numeric and experimental efforts.

The contribution by Nesterenko [50] is devoted to solitary waves in granular chains and lattices. It presents and describes the main stages in the establishment of what is currently a very active research area, namely the strongly nonlinear wave dynamics of discrete systems. The review addresses fundamental mathematical and physical problems related to the validity of the continuum approximation at spatial scales close to the characteristic scale of physical systems.

The work by Porter and co-workers [51] addresses complicated bifurcation patterns observed in quasi-periodic granular chains. Localization of deformations in nonlinear elastic lattices is discussed in the insightful paper of Ruzzene et al. [52]. Acceleration and deceleration of nonlinear impulse waves in functionally graded materials is demonstrated in the work of Yang and co-workers [53]. James [54] offers a brief, but exhaustive review of the topic, and also communicates some new insights on travelling breathers and solitary waves in lattices with generic strongly nonlinear interactions.

Two further contributions remain in the framework of one-dimensional connectivity, but treat the models where the motion of the component element occurs in multi-dimensional space. Quite naturally, this complication yields a multitude of interesting and important new phenomena, but also makes the analysis substantially more challenging. The contribution by Sminrov & Manevitch [55] addresses a lattice system with nonlocal nonlinearity, with dynamics characterized by zero sound velocity. Such peculiarity is not uncommon for the systems treated by other contributions in this Issue, e.g. the uncompressed granular chains examined in other works reveal similar properties. However, this work deals with more common systems, and the effects of global nonlinearity and sonic vacuum are obtained in appropriate asymptotic limits, due to the special choice of boundary conditions. The work of Starosvetsky [56] describes new possibilities for energy redirection and transfer that arise due to the use of internal rotating elements both in low-d.f. systems and in lattices. Experimental paper of Serra-Garcia et al. [57] addresses a novel concept of frequency down-conversion in magnetic lattices with defects.

Finally, the contribution of Sapsis [58] does not rely on specific low-d.f. or lattice models, but rather takes advantage of the statistical approach to address energy transfers in more generic dynamical systems, and discusses possibilities for prediction and quantification of extreme events.

The Editors of this Topical Issue reveal their deep gratitude to all authors who agreed to take part in this important endeavour, and to Mr. Bailey Fallon and the entire Editorial team of the Philosophical Transactions, who made the completion of this Issue possible.

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Competing interests

We declare we have no competing interests.

Funding

We received no funding for this study.