Manipulating transverse waves through 1D metamaterial by longitudinal vibrations

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Manipulating transverse waves through 1D metamaterial by longitudinal vibrations Ying-Jing Qian , Qing-Dian Cui , Xiao-Dong Yan , Wei Zhang PII: DOI: Reference:

S0020-7403(19)33052-8 https://doi.org/10.1016/j.ijmecsci.2019.105296 MS 105296

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International Journal of Mechanical Sciences

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16 August 2019 9 October 2019 31 October 2019

Please cite this article as: Ying-Jing Qian , Qing-Dian Cui , Xiao-Dong Yan , Wei Zhang , Manipulating transverse waves through 1D metamaterial by longitudinal vibrations, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105296

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Highlights 

Transverse waves can be manipulated by longitudinal vibrations.



Longitudinal frequency alters band structures by providing pseudo-stiffness.



The longitudinal vibration mainly effects band gaps due to local resonance.

Manipulating transverse waves through 1D metamaterial by longitudinal vibrations Ying-Jing Qian, Qing-Dian Cui, Xiao-Dong Yang, Wei Zhang Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, P. R. China

Abstract: The idea of manipulating transverse waves by longitudinal vibrations has been proposed in this article. The band structures of distributed resonators on an Euler beam undergoing high frequency longitudinal vibrations have been investigated. The supplement of longitudinal vibration alters the transverse wave band structures by providing pseudo-stiffness. The varying band gaps of both Bragg scattering and local resonance have been studied focusing on the contribution of the longitudinal vibration. The analytical results are validated by the finite element numerical method.

Keywords: metamaterial; vibration-tuned waves; band structure; elastic waves; phonons

1. Introduction Metamaterials composed of periodic structures or resonators have gained significant attention in the scientific community owing to their capability of manipulating the propagation of acoustic or elastic waves [1, 2]. The techniques to manipulate the mechanical behavior of metamaterials and their associated functionalities have been investigated widely [3]. The metamaterials show band gaps which allows the ability to

achieve desired filtering properties or to block undesired vibrations. One of the investigation targets of the programmable metamaterials is to manipulate the band gaps by suitable techniques [4, 5] , which leads to tenability in the metamaterial applications, such as wave filtering [6, 7], wave guiding [8-10], and energy harvesting [11, 12]. The band gaps have been found by the mechanism of Bragg scattering, which occur when the distance between periodic cells share the same size with the transmitting waves [6, 13]. The local resonance gave another mechanism to alter the band gaps even for the case when the wave length is much longer than the structure characteristic size [14-17]. Hybrid effect caused by both Bragg scattering and local resonance showed more complicated band structure and yielded further techniques in the design [18]. The modulating techniques are essential based on the two mentioned mechanisms. The band properties, including the range and number of band gaps, can be altered by tuning their parameters such as material properties, boundaries, and geometries [6, 7, 19-22]. To supply mechanical, electrical, magnetic and thermal fields are convenient techniques to tune band gaps [4, 23-29]. Other active materials, such as magnetoelastic, piezoelectric, or piezomagnetic materials, have also been used to tune desired band gaps [13, 29-34]. By altering the geometry or external fields, the above static modulating techniques need additional components to fulfill the tasks. Dynamical modulating techniques provide ways for convenient real-time manipulations of wave in a wide range of frequencies. The essential point to control efficiently the band gaps is to alter the overall stiffness of cells of the metamaterials. Bergamini et al. [35] introduced a phononic crystal

that includes frequency dependent stiffness elements obtained by piezoelectric discs. Yang et al. [36] have proposed the parametric excitation method to modulate the band gap structures of one-dimensional spring-mass system, which inspired the dynamic tenability of band property. The supplement of the high frequency parametric excitation has been verified equivalent to additional pseudo-stiffness and further alter the band structure of the system. The metamaterials usually allow both transverse and longitudinal elastic waves, which show different dispersion relations and give rise to a variety of band structures [37]. In this study, we propose a novel dynamical tuning technique: the longitudinal vibrations are used to alter the equivalent stiffness which control the transverse waves. The 1D beam with distributed resonators present band gaps caused by both Bragg scattering and local resonance, since the periodic resonators alter the geometry of the system. It is found that the longitudinal vibration contributes more on the band gaps caused by Bragg scattering and the band gaps caused by local resonance are hardly affected. The current investigation on tuning waves by vibrations may motivate further research of dynamical manipulations of band structures of metamaterials.

2. Model description and the equivalent stiffness As presented in Fig.1a, the Euler-Bernoulli beam is uniformly distributed with 2DOF resonators with mass m0, mass mr, and the inbetween spring stiffness kr. The beam density, cross-sectional area, moment of inertia, and Young’s modulus of the beam

material are respectively , s, I, E. Such a structure with distributed 2DOF resonators may give rise to band gaps caused by both Bragg scattering mechanism and local resonance mechanism.

a. The metamaterial beam

b. The nth cell of the beam Fig. 1 Structure diagram of the metamaterial beam with distributed 2DOF resonators, which cause band gaps by both Bragg scattering and local resonance

The Euler-Bernoulli beam is loaded by periodic longitudinal vibration Acos(Ωt), where Ω is the tuning frequency much higher than the transverse wave frequencies in consideration. It will be found that the equivalent transverse stiffness of the beam will be altered by the higher longitudinal vibration. Since the shear stiffness of the spring is neglected, the spring-mass resonator cannot be affected by the high frequency longitudinal periodic loading. Considering the force between adjacent resonators, the equation governing the transverse deflection of the nth cell beam shown in Fig.1 b can be obtained by the Newton’s second law as

s

 2 un  4un   un   EI  P 0 t 2 x 4 x  x 

(1)

where the transverse deflection of the nth cell is denoted by un, the longitudinal periodic force P due to the high frequency excitation is P   sAΩ 2 cos  Ωt  .

(2)

The governing equation can be written as

 2 un  4 un  2 un 2  K  AΩ cos Ωt 0   t 2 x 4 x 2

(3)

where K=EI/ρs. The transverse deflections of the beam can be considered as a superposition of free transverse vibration in a slow time scale and the fast time scale transverse responses due to the high frequency longitudinal excitations. Hence the deflection can be assumed in a mathematical manner: un  x, t   U n  x, t   n  x, t , 

(4)

where τ=Ωt denote the fast time scale while t is the slow time scale on which we measure the transverse deflections conveying waves. To show the contribution of the fast time vibrations to the slow time waves, the following averaging operator is introduced:

f  t ,  

1 T f  t , d , T 0

where T is the period of the fast vibration. Apparently, the averaging operator linear operator, which allows

(5) is a

f t   f t  ,

d f t  df  t  .  dt dt

(6)

Substituting Eq.(4) to the governing equation (3), applying the averaging operator, subtracting the result to Eq.(3) yield the equations on both slow time and fast time scales:

 2U n  4U n  2n 2  K  AΩ cos   0, t 2 x 4 x 2

(7)

and  2 n 1  2 n 2  2 n K  4 n     2 Ω 2 t 2 Ω t Ω 2 x 4  2U n  2 n  2 n  A cos  A cos   A cos   0. x 2 x 2 x 2

(8)

The excitation amplitude A and the response ξn are assumed small, with the same order of Ω-1. Then, by neglecting the small terms in Eq.(8), the following relation is obtained as

 2n  2U n  2U n  A cos   0   A cos  or . n  2 x 2 x 2

(9)

Substituting Eq.(9) back into Eq.(7) yields

 2U n  1 2 2   4U n K  A Ω  4  0. t 2  2  x

(10)

It is clear that the bending stiffness without high frequency longitudinal excitation has been expanded to a novel equivalent one

EI  EI 

1 ρsA2 Ω 2 . 2

(11)

The additional pseudo-stiffness is proportional to the square of amplitude and frequency. In the following investigation, the frequency will be used as the tuning parameter to manipulate the transverse waves.

3. Manipulating the band structures The transfer matrix method will employed to study the band structures of the transverse waves under longitudinal vibrations. Under local coordinates, the transverse deflection of the nth cell is assumed as U n  x, t   U n  x  eiωt .

(12)

Substituting Eq.(12) into Eq.(10) yields

 4U n  λ 4U n  0 , 4 x

(13)

where U n is the transverse amplitude of the nth cell and λ4  ω2 ρs EI . The solution to Eq.(13) is assumed as U n  x   Cn,1 cos  λx   Cn,2 sin  λx   Cn,3 cosh  λx   Cn,4 sinh  λx 

(14)

where the unknown coefficients Cn,k (k=1, 2, 3, 4) will be determined later. The transverse displacement, rotation angle, bending moment, and shear force of the beam element for the nth cell are respectively, U n  x   U n  x  , θn  x  

U n  x  , x

 2U n  x   3U n  x  M n  x   EI , Q x  EI . n  x 2 x3

(15)

Substituting Eq.(14) into Eq.(15) yields

Un  x   H  x  Cn ,

(16)

where the variables in the state space is Un(x)=[ Un (x) θn(x) Mn(x) Qn(x)]T, the coefficient vector is Cn=[ Cn,1 Cn,2 Cn,3 Cn,4]T, and H(x) is

sin  λx  cosh  λx  sinh  λx    cos  λx     λ sin  λx  λ cos  λx  λ sinh  λx  λ cosh  λx    . H  x    λ2 EI cos  λx   λ2 EI sin  λx  λ2 EI cosh  λx  λ2 EI sinh  λx    3  3 3 3  λ EI sin  λx   λ EI cos  λx  λ EI sinh  λx  λ EI cosh  λx  

(17)

Thus, on the left of the nth cell beam that x=0, the state variables on the local coordinate is Un  0   H  0  Cn .

(18)

On the right of the (n-1)th cell beam that x=d, the state variables on the local coordinate is Un-1  d   H  d  Cn-1 .

(19)

Due to the supplement of the spring-mass resonator, a concentrated shear force is applied between the two cells as shown in Fig. 2, which makes the state variables non-continuous: Un-1  d   Un  0   Qn,r ,

(20)

where Qn,r=[0 0 0 Qn,r]T and Qn,r is the force of the resonator exerted on the beam.

Fig. 2 Transfer relation of state parameter vectors of the two adjacent cells

By studying the 2DOF spring-mass resonator, the harmonic periodic motion equations are





ω2 mrU n,r  kr U n,r  U n  0   0





ω2 m0U n  0   kr U n  0   U n,r  Qn ,r

(21)

where Un,r is the amplitude of the resonator mass and ω is the harmonic frequency. From Eq.(21), the shear force is easily derived as  ω2 mr kr Qn,r   ω2 m0  kr  ω2 mr 

 U n  0  . 

(22)

Substituting Eq.(22) back into Eq.(20) leads to Un-1  d   GUn  0 

(23)

where the matrix G stands for the equivalent dynamic stiffness and can be expressed as 1    2 1  , g  ω2 m  ω mr kr . G 0  1  kr  ω2 mr   1 g

(24)

Further substituting Eqs.Error! Reference source not found. and (19) into Eq.(23) yields the relation between the adjacent cells: Cn  H  0  G 1H  d  Cn-1 , 1

(25)

where the transfer matrix can be defined as D  H  0  G 1H  d  . 1

(26)

Substituting the Block theory Cn=eiqdCn-1, in which q denotes the wave number and d denotes the length of the cell beam, into Eq.(25) yields the final characteristic equation as det  D  eiqd I   0 ,

on which the dispersion relation can be obtained.

(27)

4. Band Structures and FEM Numerical Verification 4.1 Band Structures Based on the dispersion relation, we can determine the band structures of the longitudinally excited system transmitting transverse waves. In Fig. 3a-c, three cases of band structures with different excitation frequencies are presented with the fixed parameters: kr=1, mr=1, m0=1, A=0.01, ρ=1, s=1, E=1, I=1, and d=2. The real value wave number means the pass band, while the pure imaginary value wave number means the stop band. The lower band gap rendered in blue is caused by local resonance and the upper two band gaps rendered in pink is caused by Bragg scattering. The local resonance occurs when the wave frequency is close to the natural frequency of the resonator that ωr  kr mr . It is clear that the band gap caused by Bragg scattering is much wider than that caused by local resonance.

a. Ω=0 Hz

b. Ω=50 Hz

c. Ω=100 Hz Fig. 3 Complex band structure of metamaterial beam at three longitudinal vibration frequencies, where the blue area is the band gap caused by local resonance and pink area caused by Bragg scattering.

It is clear that with the increasing longitudinal vibration frequency, the band gaps caused by Bragg scattering keeps increasing and the gaps are moved upwards, while the band gap caused by the local resonance holds almost the same frequency range. In Fig. 4, the varying band gaps with increasing frequency demonstrate well such trend.

Fig. 4 Relation between band gap range and longitudinal excitation frequency, where the blue area is the band gap caused by local resonance and pink area caused by Bragg scattering.

4.2 FEM Numerical Verification In the following part, the finite element method is performed to verify the band structures. The above discussed system can be designed in a continuous manner as presented in Fig. 5. In the engineering field, the rubber is commonly used to replace the theoretical elastic spring. The thickness of the rubber is 0.002m. The steel is used as the base beam material. The cross section are for the beam is defined as 4 mm2, and moment of inertia is 1.3333 mm2. The lumped masses denoted in blue corresponding to mr and m0 are given no geometry to avoid the involvement of the rotating effect.

Fig.5 Continuous model of metamaterial beam cell with one rubber spring and two lumped masses

The material properties and parameters are given in Table 1, by which the equivalent stiffness of the spring can be obtained for the discrete model.

Table 1 Material parameters and calculation parameters parameters

values

units

Young’s modulus

210

GPa

Density

7850

kg/m3

Poisson’s ratio

0.29

—

Young’s modulus

0.78

MPa

Density

1200

kg/m3

Poisson’s ratio

0.47

—

A

Excitation amplitude

0.0001

m

d

Lattice constant

0.01

m

mr

Upper mass

0.01

kg

m0

Lower mass

0.01

kg

Steel

Rubber

The commercial finite element software Comsol Multiphysics ® is used to extract

the band structures. The theoretical results based on Eq.(27) and finite element numerical results are presented in Fig. 6, which shows a very good agreement. Hence, the theoretical investigation has been validated by the numerical simulations.

a. Ω=0 Hz

b. Ω=4000 Hz

c. Ω=8000 Hz Fig. 6 Band structure of transverse wave via both theoretical and numerical methods with varying longitudinal vibration excitation frequencies

In the numerical simulations, the much higher vibration excitations are adopted since steel beam is used as the base structure. Under such high frequency situation, the band gap caused by local resonance is hardly affected by the longitudinal excitation, while the band gaps caused by Bragg scattering have been changed a lot. Such phenomena can be considered in the further design of programmable metamaterial transmitting elastic waves or vibration isolation structures.

5. Conclusions The idea of manipulating transverse waves by longitudinal vibrations has been proposed. The band structures of Euler beam with distributed resonators undergoing high frequency longitudinal vibrations have been studied. It is found that the longitudinal

frequency may alter the transverse wave band structures by providing pseudo-stiffness to the base beam. The additional pseudo-stiffness is found proportional to the square of the frequency and the amplitude. The contributions of the longitudinal frequency to the band gaps caused by both Bragg scattering and local resonance have been concluded. The band gap caused by Bragg scattering can be more affected by the longitudinal vibrations. The results of spring-mass resonators have been validated by the finite element method on the model of equivalent continuous resonators.

Acknowledgement This work is supported in part by the National Natural Science Foundation of China (Project no. 11972050, 11772009, 11832002), Beijing Natural Science Foundation (Project no. 3172003, 1192002).

Conflict of interest The authors declared that they have no conflicts of interest to this work.

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Graphical abstract