Combining nonlinear vibration absorbers and the Acoustic Black Hole for passive broadband flexural vibration mitigation

https://doi.org/10.1016/j.ijnonlinmec.2020.103558Get rights and content

Highlights

  • The combinations of nonlinear vibration absorbers to an Acoustic Black Hole beam for the broadband mitigation of flexural vibrations is addressed.

  • Four different solutions are investigated: the addition of a tuned mass damper, a nonlinear energy sink, a bi-stable energy sink and a vibro-impact.

  • Numerical results show that all methods are able to offer broadband vibration mitigation.

  • Detailed parametric study with guidelines for the optimal tuning of different absorbers are given.

Abstract

The Acoustic Black Hole (ABH) effect refers to a special vibration damping technique adapted to thin-walled structures such as beams or plates. It usually consists of a local decrease of the structure thickness profile, associated to a thin viscoelastic coating placed in the area of minimum thickness. It has been shown that such structural design acts as an efficient vibration damper in the high frequency range, but not at low frequencies. This paper investigates how different types of vibration absorbers, linear and nonlinear, added to the primary system can improve the low frequency performance of a beam ABH termination. In particular, the conjugated effects of the Acoustic Black Hole effect and a Tuned Mass Damper (TMD), a Nonlinear Energy Sink (NES), a bi-stable NES (BNES), and a vibro-impact ABH (VI-ABH) are investigated. Forced response to random excitation are computed in the time domain using a modal approach combined with an energy conserving numerical scheme. Frequency indicators are defined to characterize and compare the performance of all solutions. The simulation results clearly show that all the proposed methods are able to damp efficiently the flexural vibrations in a broadband manner. The optimal tuning of each proposed solution is then investigated through a thorough parametric study, showing how to optimize the efficiency of each solution. In particular, TMD and VI-ABH show a slight dependence on vibration amplitude, while the performance of NES and BNES have a peak of efficiency for moderate amplitudes.

Introduction

The Acoustic Black Hole effect (ABH) is a technique for passively mitigating structural vibrations in beams and plates. It usually consists of a local decrease of the structure thickness profile, associated to a thin viscoelastic coating placed in the area of minimum thickness. It has been shown that such structural design acts as an efficient vibration damper in the high frequency range [1] because of a wave trap effect induced by the flexural waves properties in thin-walled structures with variable thickness. Because of its potentiality for improving the performances of mechanical structures, and especially in light-weight structures, the scientific literature on this strategy has grown rapidly in recent years. Numerous studies have been realized on experimental and numerical viewpoints, and have confirmed the effectiveness of an ABH for passive control of vibration in the high frequency range.A detailed review of the theory and applications is provided in [2].

ABH techniques have been developed to reduce structural vibrations and the resulting structure-borne noise, especially in light-weight structures. Thanks to the decreasing thickness, the vibration field is strongly trapped into the tapered edge, where the damping properties of the added layer ensures a fast energy decay, counterbalancing the effect of a final non-zero thickness. Theoretical and numerical works have been conducted to model the low reaction coefficient for flexural waves in beam, resulting from this mechanism [3], [4], to address the improvement on modal damping ratio and modal overlap factor [5], to optimize the design of configurations and damping layers [6], [7], [8], and to interpret the ABH effect using the critical coupling concept [9]. Experimental evidence of ABH effect using a variety of beam-like and plate-like structures are also numerous [6], [10], [11], [12], [13]. Recent analytical advances for the exact solutions of ABHs in beam structures are shown in [14], and the numerical and experimental contributions that aims at optimizing the design of ABHs for vibration suppression have been discussed in [15], [16], [17], [18], [19] for various structures. In addition, applications of ABH to other areas, including elastic metastructures [20], energy harvesting [21], vibro-impact systems [22] and cochlear systems, have also been investigated.

However, there is also a known drawback regarding the ABH strategy: namely, it is generally inefficient in the low-frequency range [5], [23]. In some recent contributions, it has been rigorously demonstrated that a cut-on frequency exists, below which the ABH may lose its effectiveness [24], [25]. Although in the aforementioned linear ABH structures, the cut-on frequency could be somehow reduced through certain optimizations, it is unfortunately unavoidable in any implementation. To overcome this drawback, the idea of combining an ABH with another vibration damping device might be a desirable way for producing broadband vibration mitigation.

A first idea is to use a tuned mass damper (TMD) [26], [27], which consists in a linear device added to the beam and tuned to one resonant mode. Based on the classical results presented in [26], it is known that the main parameter optimization criteria follows analytical results in order to properly tune the stiffness and damping coefficients of the linear absorber. Effectiveness of a TMD in the case of linear single and multi degrees of systems has been proven to be a very reliable passive mitigation device in a large number of contexts [28]. Thus, once a TMD is attached to an ABH beam and tuned to one of its resonant mode in the low frequency range, where the ABH effect is ineffective, a reduction on the resonance peaks could hence be awaited [29], and the average performance of the ABH in the low frequency could be improved.

The known major drawback of a TMD is that it is only effective in the neighbourhood of a single frequency. To overcome this limitation and realize a more broadband vibration suppression, using a purely nonlinear vibration absorber such as an NES has become nowadays another classical method in order to obtain targeted energy transfer [30]. Unlike the TMD, the NES has no eigenfrequency and can tune to any frequency content displayed by the primary vibrating structure, thus achieving broadband vibration suppression [31], [32]. From this sense, an appropriately designed NES can outperform linear absorbers in many applications [33], [34]. However, since it relies on a nonlinear mechanism, the NES needs a minimal vibration energy in order to launch the targeted energy transfer, a drawback known as the energy barrier. Besides the classical cubic nonlinearity, there are also many other methods for realizing the nonlinear restoring force in the NES [35], such as piecewise nonlinearity, nonlinear damping [36], [37], and vibro-impact dynamics [38], [39]. Also, different experimental designs have been proposed, see e.g. [40], [41], [42], [43], [44], [45], [46].

Recent studies have considered the case of a negative linear stiffness to create a configuration of a Bistable Nonlinear Energy Sink (BNES), for which in particular has been shown that the minimal energy required to activate the energy transfer is reduced [47], [48], [49]. In this paper we thus want to use an NES or BNES to control more modes of the ABH below its cut-on frequency, and compare its performance to a TMD.

The third idea is to rely on a Vibro-Impact ABH (VI-ABH), recently introduced in [22], in order to enhance the low-frequency efficiency. Adding contact points on which the structure will impact during its vibration, the VI-ABH encompasses a non-smooth nonlinear effect creating fast transfers of energy to the high-frequency range where it is most efficiently damped. Note that such a strong nonlinearity is found to be much more efficient than geometrical nonlinearity, which has also been employed with the same idea to improve ABH in the low frequency range [50]. Vibro-Impact technique, although being different in nature from the addition of a vibration absorber, is contrasted to the other proposed approaches, in order to offer a comparative view of the improvements one can awaited.

This paper is organized as follows. Section 2 is devoted to the modelling of the considered systems, with a special emphasis on ABH beam and the implementation of added vibration absorber in the dynamics. The numerical results are then presented and compared in Section 3, where a thorough parametric study addresses the problem of optimizing the tuning of each vibration absorber in order to enhance the low-frequency efficacy. Section 4 concludes the article.

Section snippets

Modelling

The structures considered in this study are composed of a main ABH beam together with added vibration absorbers. Also, the case of a VI-ABH is contrasted in order to give more insight to the results with cross-comparisons between different solutions. Fig. 1 depicts the cases under study, where Fig. 1(a) refers to an ABH beam with a vibration absorber (be it linear or nonlinear). The second configuration under study, the VI-ABH, is illustrated in Fig. 1(b). In this section, the focus is first

Parameters of the ABH beam and linear performance

A typical ABH beam, made in aluminium, and whose design is similar to the one of experimental beam demonstrators presented in [5], is selected. The parameters are listed in Table 1. Note that in the first line, A uniform beam of constant thickness is also defined, and will be used as a reference in order to highlight the ABH effect.

The input mobility defined as the ratio between the velocity and the input force spectra at xF, is shown in Fig. 2 for both the reference (naked) beam and the ABH

Conclusion

In this contribution, the effect of adding a single vibration absorber to an ABH beam, in order to achieve a broadband vibration mitigation, is discussed. Three different vibration absorbers have been contrasted : a TMD, and two nonlinear absorbers, a NES and a BNES. The proposed solutions have also been compared to a Vibro-Impact ABH (VI-ABH), which is another strategy proposed to enhance the low-frequency efficiency of an ABH by adding contact points to generate shocks. The numerical results

CRediT authorship contribution statement

Haiqin Li: Methodology, Investigation, Writing - original draft. Cyril Touzé: Conceptualization, Supervision, Writing - review & editing. Adrien Pelat: Supervision, Writing - review & editing. François Gautier: Supervision, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (62)

  • BowyerE.P. et al.

    Experimental investigation of damping flexural vibrations in plates containing tapered indentations of power-law profile

    Appl. Acoust.

    (2013)
  • LeeJ.Y. et al.

    Exact solution of euler-bernoulli equation for acoustic black holes via generalized hypergeometric differential equation

    J. Sound Vib.

    (2019)
  • LagnyL. et al.

    Visualization of travelling waves propagating in a plate equipped with 2d abh using wide-field holographic vibrometry

    J. Sound Vib.

    (2019)
  • McCormickC.A. et al.

    Design optimization and performance comparison of three styles of one-dimensional acoustic black hole vibration absorbers

    J. Sound Vib.

    (2020)
  • DengJ. et al.

    Passive constrained viscoelastic layers to improve the efficiency of truncated acoustic black holes in beams

    Mech. Syst. Signal Process.

    (2019)
  • LiH. et al.

    A vibro-impact acoustic black hole for passive damping of flexural beam vibrations

    J. Sound Vib.

    (2019)
  • AkloucheO. et al.

    Scattering of flexural waves by a pit of quadratic profile inserted in an infinite thin plate

    J. Sound Vib.

    (2016)
  • LeeC.L. et al.

    Optimal design theories and applications of tuned mass dampers

    Eng. Struct.

    (2006)
  • KrenkS. et al.

    Tuned mass absorber on a flexible structure

    J. Sound Vib.

    (2014)
  • VakakisA.F. et al.

    Dynamics of linear discrete systems connected to local, essentially non-linear attachments

    J. Sound Vib.

    (2003)
  • StarosvetskyY. et al.

    Vibration absorption in systems with a nonlinear energy sink: nonlinear damping

    J. Sound Vib.

    (2009)
  • GendelmanO.V.

    Analytic treatment of a system with a vibro-impact nonlinear energy sink

    J. Sound Vib.

    (2012)
  • McFarlandD.M. et al.

    Experimental study of non–linear energy pumping occurring at a single fast frequency

    Int. J. Non–linear Mech.

    (2005)
  • PennisiG. et al.

    Design and experimental study of a nonlinear energy sink coupled to an electromagnetic energy harvester

    J. Sound Vib.

    (2018)
  • DenisV. et al.

    Improvement of the acoustic black hole effect by using energy transfer due to geometric nonlinearity

    Int. J. Non-Linear Mech.

    (2017)
  • GoelR.P.

    Transverse vibrations of tapered beams

    J. Sound Vib.

    (1976)
  • IssanchouC. et al.

    A modal-based approach to the nonlinear vibration of strings against a unilateral obstacle: Simulations and experiments in the pointwise case

    J. Sound Vib.

    (2017)
  • SamaniF.S. et al.

    Vibration reduction of beams under successive traveling loads by means of linear and nonlinear dynamic absorbers

    J. Sound Vib.

    (2012)
  • ShepherdM.R. et al.

    Modeling and optimization of acoustic black hole vibration absorbers

    J. Acoust. Soc. Am.

    (2017)
  • FeurtadoP.A. et al.

    An experimental investigation of acoustic black hole dynamics at low, mid, and high frequencies

    J. Vib. Acoust.

    (2016)
  • HookK. et al.

    A parametric study of an acoustic black hole on a beam

    J. Acoust. Soc. Am.

    (2019)
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      For a typical one-dimensional (1D) ABH structure, the flexural waves propagating in the structure can focus and experience little reflection at the ABH tip [50]. Benefitting from this feature, some researchers have successfully accomplished the 1D ABH-enabled vibro-acoustic applications such as vibration mitigation [54–56] and energy harvesting [57,58]. However, few studies have considered the acoustofluidic effects induced by the ABH-enabled focused waves, as well as how to utilize the wave manipulation functions of ABHs for developing new and effective ultrasonic particle manipulation and acoustofluidic devices.

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