Metamaterial beam with graded local resonators for broadband vibration suppression
Introduction
In recent years, metamaterials with exotic dynamic behaviors, such as negative effective mass [1], [2], negative effective stiffness and negative refraction [3], [4], have attracted significant research interest. Due to the band gap phenomenon in metamaterials, i.e., in a certain frequency range, wave propagation is forbidden, an important application of metamaterials is for vibration suppression. The band gaps of metamaterials can be easily designed to operate at sub-wavelength frequency ranges [5], [6], [7]. However, the application of conventional metamaterials is often significantly limited, since the band gaps are relatively narrow.
To enlarge the band gaps of metamaterials, various approaches have been proposed. For example, using multiple-degree-of-freedom local resonators can lead to multiple band gaps [6], [8], [9]. This strategy has been implemented for suppressing vibrations of engineering structures, such as rods [10], beams [11], [12], [13], [14] and plates [15]. Attaching multiple independent single-degree-of-freedom local resonators to the main structure can also open multiple band gaps for vibration suppression [16], [17], [18]. Similarly, introducing internal couplings between neighboring local resonators in conventional metamaterials can induce additional band gaps [2], [19], [20], [21]. Hybrid metamaterials with merged Bragg Scattering and local resonance-based band gaps have also been proposed to achieve wider band gaps for broadband vibration suppression [22], [23], [24]. Besides the idea of generating multiple band gaps, some researchers have proposed actively tunable metamaterials for strengthening the broadband performance, typically this tunability is achieved using the piezoelectric shunting technique [14], [25], [26], [27], [28], [29]. The resonance created by the shunt circuit can generate a reaction force and/or moment back onto the mechanical structure through the piezoelectric effect. However, additional power, sensing and feedback control circuits are required for tuning, which increases the implementation complexity. Other strategies to achieve wide band gaps for metamaterials include the use of nonlinearity [30], [31], [32]. In particular, Fang et al. [31] showed that ultra-broad band gaps can be obtained from nonlinear metamaterials by inducing chaotic motions. For the considered metamaterial consisting of Duffing oscillators with the nonlinearity implemented by magnetic forces, the vibration attenuation bandwidth was increased by two orders of magnitude [31].
The research outlined in the above paragraph largely focused on the analysis of uniform metamaterials embedded with identical local resonators. In contrast, Brennan [33] developed a wideband vibration neutralizer that consists of an array of resonators with slightly different natural frequencies. Both theoretical and experimental results demonstrated that the proposed vibration neutralizer could operate over a wide bandwidth. However, the hosting structure for the neutralizer in [33] was assumed to be a single-degree-of-freedom rigid body, and thus not a meta-structure. The concept of using multiple different resonators can also be implemented for metamaterials. Based on this idea, Banerjee et al. [34] investigated a graded one-dimensional metamaterial modelled as a lumped mass-spring system with local resonators of varying natural frequencies. With proper tuning of the frequency spacing between the local resonators, the attenuation bandwidth of this graded metamaterial was increased by 40%, compared to a conventional metamaterial with uniform local resonators. However, the mass-spring model used in [34] is purely theoretical, and in many cases is not able to properly describe the dynamics of real structures, such as beams [35] and plates [36], [37].
In this paper, a metamaterial beam with graded local resonators is proposed for realizing broadband vibration suppression. “Graded” refers to the variation in the natural frequencies of the local resonators. The mass of each local resonator is assumed to be identical, and the spring stiffness of each local resonator is varied to meet the “graded” condition. The spectral element method (SEM) is used to model the graded metamaterial beam. Several figures of merit have been developed for later quantitative analysis. The SEM results are verified with comparisons against a finite element method (FEM). A design strategy is proposed, and criteria towards the tuning of the frequency spacing between local resonators are derived. A parametric study is performed to investigate the effects of the frequency spacing and damping ratio on the vibration suppression performance of the graded metamaterial beam. The derived design criteria are validated by the parametric study results. Finally, to give an example of the practical implementation of the proposed graded metamaterial beam, a piezoelectric shunt circuit is adopted, and a piezoelectric metamaterial beam is developed. The tuning of local resonators can be realized through the manipulation of the shunt circuits. The electric components that constitute the shunt circuits are realized with synthetic circuits for the ease of tuning. Numerical simulation results show that with carefully tuned shunt circuits, the graded piezoelectric metamaterial beam can feature a significantly wider attenuation bandwidth than the conventional counterpart.
Section snippets
Model formulation
Fig. 1(a) shows an infinitely long model of the conventional metamaterial beam with uniform local resonators attached. Fig. 1(b) shows the proposed graded metamaterial beam system with N local resonators attached. The presence of the attached local resonators will be embodied in the continuity conditions for the proposed graded metamaterial beam. Adding a local resonator implies satisfying four equations to guarantee the continuity of displacement, slope, force, and moment at the point of
Design strategy
The band gap width (denoted by ) of the conventional metamaterial beam with attached uniform local resonators (i.e., ) is [40]:
Eq. (25) is obtained assuming an infinitely long undamped model. For a practical metamaterial system containing only a limited number of cells, the effective attenuation frequency range can be narrower than . By intentionally varying the natural frequency spacing , and making each resonator operate in different (but overlapped) frequency ranges, it may be
Effect of the natural frequency spacing δ
This subsection is devoted to investigating the effect of the natural frequency spacing on the vibration suppression performance of the graded metamaterial beam. Without loss of generality, in the following case studies, we consider a metamaterial beam with 6 local resonators attached.
For the graded metamaterial beam with the mass ratio and non-dimensional stiffness of the resonator , Fig. 4(a) presents the evolution of the transmittance pattern in response to variation in . By
Mechanism explanation
The key to realize the proposed graded metamaterial beam with the developed strategy lies in the implementation of the local resonators with readily tunable stiffness and damping ratio. However, it is difficult to achieve this using solely mechanical structures. Using a structure integrated with shunted piezoelectric patches provides a potential solution by forming the local resonance in the electrical circuit. With the piezoelectric effect, an electrical resonance has been proved to have a
Conclusions
In this paper, a graded metamaterial structure consisting of a series of local resonators with different natural frequencies attached to a plain beam has been studied. The spectral element method has been successfully adopted to model the proposed system and verified using the finite element method. Several figures of merit have been defined to enable a quantitative evaluation of the vibration suppression performance of the graded metamaterial from the perspective of the bandwidth of the
CRediT authorship contribution statement
Guobiao Hu: Investigation, Writing - original draft, Formal analysis, Validation, Software. Andrew C. M. Austin: Conceptualization, Methodology, Supervision, Writing - review & editing. Vladislav Sorokin: Conceptualization, Methodology, Supervision, Project administration, Writing - review & editing. Lihua Tang: Supervision, Writing - review & editing.
Acknowledgement
This work is financially supported by the Ideas Day Seed Fund from the Faculty of Engineering of the University of Auckland.
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