Enhancing bandwidth of metamaterial plate with linear and nonlinear passive absorbers

https://doi.org/10.1016/j.ijnonlinmec.2021.103769Get rights and content

Highlights

  • Passive linear and nonlinear vibration absorbers are used to obtain metamaterial vibrating plate.

  • 77% reduction in the vibration amplitude of the plate is observed, while the total added mass is lower than 10%.

  • Sufficient bandwidth is observed and nonlinear hardening or softening are removed.

  • Sensitivity of the metamaterial plate with changing the parameters is minimized.

Abstract

In this work, passive vibration absorbers are used to obtain a metamaterial vibrating plate. For this purpose, a set of linear and nonlinear passive absorbers are used. The plate is modeled based on Mindlin’s theory in which the effects of rotary inertia and shear deformation are considered. The metamaterial plate is considered to be both linear and nonlinear, with a set of linear, nonlinear, and viscous absorbers (Maxwell and Maxwell–Voigt). By optimizing the absorber parameters, a noticeable reduction in the vibration amplitude of the plate is obtained, without imparting a significant mass to the original plate. The effects of mass, stiffness, damping of the absorber as well as the number of absorbers are also investigated so that the sensitivity of the behavior of the system with changing the parameters is minimized. The effect of absorbers at lower frequencies is the main focus of this work. There is more than 77% reduction in the vibration amplitude of the plate, while the total added mass is lower than 10%, which is much less than the values given in similar works. Also, the nonlinear effects of the plate have been greatly reduced and no new resonances with sufficient bandwidth have emerged, and linearization of the nonlinear plate with absorber is achieved. Finally, the performances of the used absorbers are compared.

Introduction

Metamaterial (derived from the Greek word for “beyond”) is an engineering material with features that do not normally exist in nature [1]. A prototype of this material was made by David Smith at the University of California in 2000 with a combination of fine rods and a set of metal rings and similar materials, which had features like inverted Doppler effect, Cherenkov radiation, and Snell’s law. Metamaterials usually have properties such as negative refractive index, negative Poisson’s ratio, negative effective density, reverse Cherenkov radiation, reverse Doppler effect, and reverse Snell’s law, etc. The unusual properties of these materials have led them to be used in various fields such as optical filter fields, medical devices, earthquake protection structures, etc. [2]. Metastructure is a structure that is made of metamaterials or exhibits behaviors that are contrary to natural species.

Metastructures are man-made structures whose properties are more related to the geometry and composition of every single cell than to the properties of each component. Recently, the construction of metastructure with modern human methods in three dimensions is underway. Research on metastructure first began in the optical field and then quickly expanded to the acoustic and mechanical fields. Metastructure is a type of composite, but the difference is that they can achieve at least one unnatural property not found in nature, due to topographic optimization rather than the composition of their materials [3].

Vibration control is done by the vibration absorbers to prevent the damages caused by undesirable vibrations. The concept of metastructure as the vibration absorber is mainly based on the local resonance system connected to the main system. Generally, the studies on the metastructures are concentrated on the characteristics of wave propagation [4] and the studies performed on acoustic metastructures. Pai et al. [5] investigated the metastructural acoustic beam with multi-frequency vibration absorbers. They presented modeling, analysis, and design of a metamaterial beam consisting of an anisotropic beam with two absorbers to increase the bandwidth of the transverse elastic waves. The beam is modeled by the first-order shear deformation theory.

Nouh et al. [6] investigated the reduction of vibrations and the crack bandgap of a set of internal resonance cells. Modal interaction, frequency, and bandgap of this type of beam were predicted by finite element and validated for excitations between 10 and 5000 Hz frequencies. They found that the designed metastructure would reduce vibrations at lower frequencies than beams of the same size and weight. Cselyuszka et al. [7] investigated the resonant frequency of a one-dimensional acoustic resonator with negative density and unit cell. Zhu et al. [8] investigated a metastructural elastic beam containing mass and spring cells to control the vibrations. The beam is modeled according to Timoshenko’s theory, and an experimental examination of the theoretical model is carried out.

Wang et al. [9] proposed an acoustic metamaterial plate with a negative Young module of elasticity consisting of two horizontal mass and spring systems with a vertically connected spring to prevent the propagation of a bending wave. They found that damping had little effect on the bandgap, but could smooth out the frequency response. They also examined the effect of geometric parameters and found the effect of the horizontal and vertical distance ratio of the subsystem on the bandgap. Jing et al. [10] examined asymmetric acoustic transmission and effective dynamic parameters and vibrational performance of graded beams. The studied graded beam consists of unit cells of different scattering factors. Effective dynamic parameters in cells are determined by the scattering equations and energy band. They calculated the scattering equations and energy band structures of every single cell using the finite element method (FEM), who’s designed graded beam allows for asymmetric acoustic transmission over a wide range of frequencies. The beams are made of bromide polymathic methacrylate (PMMA) and steel, and finally, the propagation of the longitudinal wave using FEM was investigated. They showed that the effective density and modulus of elasticity of the beam depend on the frequency and gradient factor.

Causality et al. [4] investigated the vibration reduction of metastructural beam with simply supported boundary conditions by linear and nonlinear absorbers. Euler–Bernoulli beam connected to a series of nonlinear mass subsystems acts as vibration absorbers is considered. With linear absorbers 32% and with nonlinear absorbers 46% reduction in the amplitude of oscillation is achieved, and the added absorber mass is more than 10 percent. Reich and Inman [11] examined the vibrations of a one-dimensional metastructure model with an absorber without adding significant mass. Jiao and Alevi [12] studied bending of Nano/micro graphene-reinforced metastructural beam with a periodic lattice pattern in the form of cylinders, ellipses, and hexagons with simply supported boundary conditions based on Euler–Bernoulli theory, using the modified coupled stress theory. They obtained the effective Young modulus of the metamaterial beam using the Young modulus of composite material and the coefficient obtained from the desired patterns.

Wu et al. [13] investigated the effect of thermal stresses on the band structure of metamaterials with the finite element method. Besides, they discussed the side effects of thermal stresses and temperature-dependent material properties on the structure of the Aluminum/Silicon plate. Sagano et al. [14] investigated the mechanical and electromechanical band gaps in metamaterials and metastructures. These bands are created by mechanical and electromechanical resonators. They investigated metastructural hybrid bimorph beams sandwiched with two piezoelectric layers and subjected to bending vibrations. The beam consists of piezoelectric plates and electrode pairs and is connected to mechanical resonators. They showed that for a certain number of resonators, the gaps between mechanical and electromechanical bands do not completely merge.

Meng et al. [15] experimentally and numerically studied the vibrational bandgap of Π-shaped metamaterial beams (called rainbow metamaterials). Beam with fixed–fixed ends is formed of Π-shaped components, to each of which mass resonators are connected. It is shown that the metamaterials with resonators with different attachment distances have a wider bandgap than resonators with the same attachment distances. Sheng et al. [16] experimentally and numerically studied a nonlinear metamaterial acoustic beam. The goal is to reduce the amplitude of vibration and increasing bandwidth at low frequencies by imposing low weight on the host structure. Each nonlinear material cell includes a Duffing oscillator, a flexural resonator, and a Vibro-impact resonator. Finally, they built a sonic metamaterial beam with 28.1% added mass by absorbers. Basta et al. [17] studied the vibrations of a rotating nonlinear beam connected to the absorbing mass, springs, and dampers. Perturbation theory and numerical solutions have been used to solve the equations. They showed that by attaching the absorbers, the vibrations are reduced, and with the absorber closer to the tip of the beam, better results are obtained.

The purpose of this work is to design a metastructural plate in such a way that no resonance with substantial bandwidth is observed or the amplitude of the vibration is reduced as much as possible. For this purpose, optimizing the absorber’s parameters by the desired cost function is used. Also, to reduce the resonance amplitude, the absorbers are designed such that the plate has the necessary statically stiffness. Some constraints are included in the optimization, such that the total mass of absorbers do not exceed 10% of the mass of the plate. In available studies, the mass of absorbers is up to 30% of the host structure, which is not desirable. Many of available studies are concentrated on metastructural beams and plates for wave attenuation, here the metastructural plate is examined for vibrations reduction. The design of nonlinear metastructural plate with linear, nonlinear, and viscous absorbers have been studied in this work. In previous studies the maximum 46%, vibration reduction is achieved, while in this work more than 77% reduction in resonance amplitude is obtained. Also in previous works, vibration reduction in high frequencies is reported, while in this work more attention is given to low frequencies.

In this work, different types of absorbers with linear and nonlinear stiffness and different types of viscous absorbers (Maxwell and Maxwell-Voigt) are used. A comprehensive review of modeling and experimental identification of viscoelastic plate is presented in [18], [19], [20], [21], which presents substantial subjects of this field.

In the present work, a cantilever Mindlin plate under the harmonic excitation force is connected to a set of linear, nonlinear, and viscous absorbers. Linear and nonlinear plate models are studied. The Lagrange–Euler equation and the assumed modes method are also used to find the equation of motion. For an analytical solution, the complex averaging method is used. For solving the obtained nonlinear equations by the complex averaging method, numerical solution by arc length continuation method is used.

In this work, after the introduction, in the second part, the equation of motions for the linear and nonlinear plates with different attached absorbers are presented. Then the optimization method and tuning parameter procedure are discussed. The third section is devoted to present the obtained results in which validation of the equation and convergence studies are examined, and then the performance of different attached absorbers on the reduction of resonance amplitude and increasing bandwidth are given. The performance of different absorbers is compared. The fourth section is devoted to the conclusion and some suggestions.

Section snippets

Modeling and the equations of motion

The schematic of the plate with attached absorbers is shown in Fig. 1. The cantilever plate has length a, width b, thickness H, the flexural rigidity of D=EH312(1v2), mass per unit area m, Young’s modulus E, and mass density of ρ. The plate is excited by the harmonic force with the amplitude f and frequency ω, which is applied at a point (xF,yF).

Results and discussions

In this section, at first, the convergence of solution for linear and then nonlinear plates to increase in the number of assumed modes is examined, then the results obtained by analytical and semi-analytical methods, are compared with those obtained from the time response solution of the equations of motions. The parameter of absorbers is optimized and the effect of these absorbers on the vibration amplitude of the plate is investigated. Finally, the performance of the absorbers is compared. It

Conclusion

In this work, the vibrations of the metastructure clamped plate under harmonic excitation force are investigated. Linear and the nonlinear plate are modeled using Mindlin theory. To solve nonlinear equations, the complex averaging method and the arc length continuation method have been used. A set of linear, nonlinear, and viscous absorbers are connected to the plate and the parameters of the absorber are optimized. Optimization showed that by adjusting the parameters and adding a mass of up to

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 (31)

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