Lower band gaps of longitudinal wave in a one-dimensional periodic rod by exploiting geometrical nonlinearity

https://doi.org/10.1016/j.ymssp.2019.02.008Get rights and content

Highlights

  • Geometric nonlinearity is used to form a negative-stiffness (NS) mechanism.

  • The stiffness of resonator can be reduced to any low values by NS mechanism.

  • A very low band gap of longitudinal wave can be created by HSLDS resonator.

  • A sufficient number of unit cells are needed to obtain good wave attenuations.

Abstract

In this paper, a local resonant (LR) rod with high-static-low-dynamic-stiffness (HSLDS) resonators is proposed to create a very low-frequency band gap for longitudinal wave propagation along the rod. The HSLDS resonator is devised by employing geometrical nonlinearity, and attached onto a periodic rod composed of rigid frames and rubbers to construct a LR rod. To reveal the band gaps, the LR rod is modeled as a lumped mass-spring chain. The effects of damping and nonlinearity of the HSLDS resonator on the dispersion relation is studied analytically by the Harmonic Balance method. The analytical results indicate that the damping mainly affects the width and depth of the band gap, while the nonlinearity can influence the central frequency and width of the band gap. In addition, both multi-body dynamic analyses and numerical simulations are conducted to predict longitudinal wave propagation along the LR rod, and thus to validate the very low-frequency band gap. The results show that the periodic rod with HSLDS resonators can create a very low-frequency band gap for longitudinal waves propagating along the rod.

Introduction

Frequency band generated by periodic structures presents means for wave attenuation. Phononic crystal or sonic metamaterials are examples of such structures, and the frequency band called band gap or stop band. Generally, there are two mechanisms for opening a band gap, namely Bragg scattering (BS) [1] and local resonance (LR) [2]. However, the band structure and wave attenuating feature caused by these two mechanisms are visibly different from each other. Specifically, the center frequency of the BS band gap is related to the wave velocity and lattice constant of a periodic structure, but that of the LR band gap is only dependent on the resonant frequency. Thus, the LR band gap can be formed at a much lower frequency than the BS band gap, considering the practical dimension of a periodic structure.

There are numerous studies about LR band gaps, as reviewed by Hussein et al. [3], but the literature review in this section does not try to be exhaustive, and only gives a brief review on the contributions to lowering LR band gaps and the works related to nonlinearity.

In order to form LR band gaps at low frequencies, many attempts to construct novel resonators have been made, such as in the form of a mass-spring device [4], [5], [6], [7], a continuum beam [8], [9], an inertial amplification mechanism [10], piezoelectric patches [11], [12], electroactive polymer layers [13], a cylindrical tungsten pillar [14] and a ball coated with a soft material [2]. Especially, the simple mass-spring resonator was modified to broaden the band gap or to achieve multiple band gaps, such as lateral local resonators [15], a multi-stage resonators [16], a local resonator with multi-oscillator [6] and a force and moment resonator [17]. The results of these studies show that the local resonator can successfully create band gaps in low frequency range. However, because of the restrictive space and load-supporting capability of the resonator, it might be impossible to design a local resonator with ultra-large mass and ultra-low stiffness in a traditional way, and thus it is still a challenge for phononic control in the ultra-low frequency domain [3].

Fortunately, a high-static-low-dynamic stiffness (HSLDS) can be realized by exploiting geometrical nonlinearity [18], [19], [20], [21], [22], [23], [24], [25]. In our previous works [26], [27], a local resonator with HSLDS was proposed to create a band gap of bending wave in very low frequency region along an Euler-Bernoulli beam. The HSLDS property was obtained by using the negative-stiffness (NS) mechanism to neutralize the positive stiffness partially, so that the residual stiffness of the oscillator can be tuned towards zero. Therefore, the local resonator containing the NS mechanism is a promising solution for shifting the band gap from a high frequency region to a low one.

Nevertheless, the HSLDS has nonlinear attributes related to displacements and geometry. When the resonator undergoes large-amplitude oscillation, the effects of nonlinearity on the dispersions and band gaps cannot be ignored. In such a situation, the characteristics of nonlinear wave propagating along the sonic periodic structures should be investigated, which would provide valuable guidance for the application of nonlinear local resonators.

The dispersion features of a one-dimensional chain attached with nonlinear resonators have been studied widely. Lazarov and Jensen [28] revealed the band structure of a chain with local resonators possessing cubic nonlinearity. Fang et al. [29] attached oscillators with cubic stiffness to a chain to construct a nonlinear acoustic metamaterials and studied its dispersion feature. Chakraborty and Mallik [30] analyzed the wave propagation characteristics in a nonlinear periodic chain by using the perturbation approach. Manktelow et al. [31] studied the weakly nonlinear wave interactions in a periodic structure by using perturbation method of multiple time scales and proposed several potential applications of nonlinear wave interactions. Rothos and Vakakis [32] presented the dynamic interactions of travelling waves propagating in a linear chain with local essentially nonlinear attachments. Khajehtourian and Hussein [33] studied the band gap opened by a one-dimensional (1D) nonlinear elastic metamaterial with spring-mass resonators and analyzed the effect of nonlinearity on the band gap.

In this paper, a local resonator containing an NS mechanism with geometrical nonlinearity is proposed to filter low-frequency longitudinal waves in a one-dimensional periodic rod. Note that, the NS mechanism is employed to neutralize the stiffness of the positive-stiffness element, and thus to construct an HSLDS resonator in this paper, which is completely different from those works [21], [34], [35] where the NS mechanism is used to construct a bistable resonator with pure negative stiffness. The stiffness of the HSLDS resonator can be tuned to any desired low values by adjusting the NS mechanism provided that the resonator does not undergo large-amplitude oscillations. A lumped mass-spring model is established to reveal dispersion properties and demonstrate band structures theoretically by using the Harmonic Balance method, which are also validated by numerical simulations.

This paper is organized as follows. In Section 2, the static analysis is carried out to show the tunable-low-stiffness feature of the local resonator. Then, the band structure of a longitudinal wave propagating in the LR rod is obtained theoretically and numerically in Section 3. In Section 4, the effects of both the number of unit cells and excitation amplitude on the wave attenuation performance are studied. Finally, some conclusions are drawn in Section 5.

Section snippets

Stiffness feature of the HSLDS local resonator

The physical model of a one-dimension periodic rod with HSLDS local resonators is shown in Fig. 1. All the masses and springs of the HSLDS resonator are installed in the same horizontal plane. Therefore, the gravity of the mass is not considered in the static analysis, which is different from the HSLDS resonator in our previous work [27]. The resonator consists of a mass, two relaxed horizontal springs and two pre-compressed vertical springs. For each spring, one end is connected to the mass,

Equations of motion

In order to analyze the band features of longitudinal waves, the HSLDS-LR periodic rod is modeled as a lumped mass-spring chain, as shown in Fig. 4a. The frame of the resonator is much stiffer than the rubber block, and thus only the stiffness of the rubber block is taken into account to calculate the equivalent stiffness of the connecting spring. The frame of the resonator is simplified as a mass and the rubber block is simplified as a spring with stiffness [36]k=3.6G1+2.22ab2a+bh2abhwhere a, b

Wave attenuations by HSLDS local resonators

In order to find out the wave attenuation performance of the proposed resonator, this section will discuss influences of the number of unit cells and excitation amplitude on the longitudinal wave attenuation in the HSLDS-LR periodic rod.

Conclusions

In this paper, a high-static-low-dynamic-stiffness (HSLDS) local resonator with geometrical nonlinearity is proposed to lower the band gap of longitudinal waves propagating in a one-dimensional periodic rod. The HSLDS-LR periodic rod is modeled as a lumped mass-spring chain to analyze its dispersion relation and reveal the band structure, which is validated by numerical simulations and multi-body dynamic simulations. Several conclusions are drawn as follows.

Firstly and most importantly, the

Acknowledgments

This research work was supported by National Key R&D Program of China (2017YFB1102801), National Natural Science Foundation of China (11572116), Natural Science Foundation of Hunan Province (2016JJ3036) and Hunan Provincial Innovation Foundation for Postgraduate. The first author, Kai Wang, would like to thank the support from the China Scholarship Council (CSC).

References (43)

  • B.S. Lazarov et al.

    Low-frequency band gaps in chains with attached non-linear oscillators

    Int. J. Non. Linear. Mech.

    (2007)
  • G. Chakraborty et al.

    Dynamics of a weakly non-linear periodic chain

    Int. J. Non. Linear. Mech.

    (2001)
  • K.L. Manktelow et al.

    Weakly nonlinear wave interactions in multi-degree of freedom periodic structures

    Wave Motion

    (2014)
  • V.M. Rothos et al.

    Dynamic interactions of traveling waves propagating in a linear chain with an local essentially nonlinear attachment

    Wave Motion

    (2009)
  • M.J. Frazier et al.

    Band gap transmission in periodic bistable mechanical systems

    J. Sound Vib.

    (2017)
  • D.J. Thompson

    A continuous damped vibration absorber to reduce broad-band wave propagation in beams

    J. Sound Vib.

    (2008)
  • Z. Liu et al.

    Locally resonant sonic materials

    Science (80-)

    (2000)
  • M.I. Hussein et al.

    Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook

    Appl. Mech. Rev.

    (2014)
  • S. Zuo et al.

    Studies of band gaps in flexural vibrations of a locally resonant beam with novel multi-oscillator configuration

    J. Vib. Control.

    (2015)
  • M.Y. Wang et al.

    Frequency band structure of locally resonant periodic flexural beams suspended with force–moment resonators

    J. Phys. D. Appl. Phys.

    (2013)
  • X. Wang et al.

    An analysis of flexural wave band gaps of locally resonant beams with continuum beam resonators

    Meccanica

    (2016)
  • Cited by (78)

    • A time domain procedure for the identification of periodic structures

      2024, International Journal of Mechanical Sciences
    View all citing articles on Scopus
    View full text