Phononic band gaps by inertial amplification mechanisms in periodic composite sandwich beam with lattice truss cores

Journal Pre-proofs Phononic band gaps by inertial amplification mechanisms in periodic composite sandwich beam with lattice truss cores Jingru Li, Peng Yang, Sheng Li PII: DOI: Reference:

S0263-8223(19)32680-7 https://doi.org/10.1016/j.compstruct.2019.111458 COST 111458

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Composite Structures

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18 July 2019 17 September 2019

Please cite this article as: Li, J., Yang, P., Li, S., Phononic band gaps by inertial amplification mechanisms in periodic composite sandwich beam with lattice truss cores, Composite Structures (2019), doi: https://doi.org/10.1016/ j.compstruct.2019.111458

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Phononic band gaps by inertial amplification mechanisms in periodic composite sandwich beam with lattice truss cores Jingru Li1，Peng Yang1 and Sheng Li2,3 1. Mechanical and Electrical Engineering College, Hainan University, Haikou, 570228, P. R. China 2. State Key Laboratory of Structural Analysis for Industrial Equipment, School of Naval Architecture, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, P. R. China 3. Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, P. R. China

Abstract In order to acquire multiple and wider stop bands in periodic sandwich lattice, this work applies the inertial amplification mechanisms to the composite sandwich beam with pyramidal truss cores and presents a theoretical study on the wave propagation characteristics of the proposed sandwich lattice. An analytical model based on the plane wave expansion method and the Galerkin method is developed to investigate the dispersion relation of transverse waves propagating through the sandwich beam, whose validity is verified by the numerical simulations of spatial decaying performance predicted by the finite element method. Numerical results of complex band structures show that the proposed sandwich beam can obtain multiple stop bands and possess much wider bandwidth. Employing the developed model, the effects of varying attached substructure, the layout of truss members and the material damping on the attenuation performance are discussed to show the flexibility and to obtain thorough understanding of the system. Keywords: Sandwich beam; Inertial amplification; Band gaps; Wave attenuation

1. Introduction Phononic band gaps found in phononic crystals [1-5] or acoustic metamaterials [6-10] are preferable to be applied in designing applications for sound and vibration isolation in recent years. Based on the Floquet-Bloch theory, analytical and numerical methods such as the transfer matrix method (TMM) [11-13], the plane wave expansion method (PWE) [14-16], the multiple scattering theory (MST) [17-19] and the finite element method (FDM) [20, 21] have been used to derive the dispersion relations of elastic waves in periodic structures. The extraordinary property of allowing suppress wave propagation at certain frequency range makes it possible to achieve a broad insulation

band over low frequencies. It is well known that the stop bands are predominantly generated by the Bragg scattering or the localized resonance [22-25]. In regard to the implement of low-frequency Bragg gaps, high density/low modulus materials or large sized unit cells are essential to retain low wave speed or large lattice constant [26], nevertheless the mechanical property is unfavorable to be employed in engineering devices. In contrast to Bragg gaps, the band gap induced by local resonant mechanism is accessible to be obtained at low frequencies, but the bandwidth of which largely depends on the additional mass [27, 28]. To achieve broadband and shift downward frequency gaps, research of specific designs of attached substructures have been thoroughly studied [21, 29-33]. An alternative approach to generate band gaps without sacrificing the stiffness or substantially increasing the overall mass has been proposed by Yilmaz [28, 34] et al., from which the stop bands are created with enhanced effective inertia through amplifying the motion of a small mass [35]. The inertial amplification (IA) mechanisms have been severed as the backbone structural components [26, 35-37] until Frandsen [38] employed the concept to create band gaps in a continuous rod. The authors [39] have adopted the IA mechanism on an infinite elastic beam and obtained an ultra-wide gap at low frequency through the hybridization effects between the inertial amplification mechanism and the underlying beam. However, to acquire an extensive design paradigm for mechanical filters, elastic wave guides, sound and vibration isolators, extra efforts should be paid to conduct the analysis of wave propagation in more complex structures. Compared to single-panel structures, sandwich structures commonly consisting of two face sheets and a designed inner core, exhibit broader application prospects because of the intrinsic mechanical properties [40-43], such as high strength-to-weight ratio, excellent characteristics of energy absorption and low thermal conductivity. Lattice sandwich structure with truss cores appeared as a class of newly developed sandwich structures has attracted intensive attention in recent years for its potential characteristics of periodicity, multifunction and designability. Generally, there are several types of topologies commonly used as periodic truss cores, including tetrahedral core [44], pyramidal core [45] and 3D-kagome core [46], etc. Homogenization technique is a powerful tool to acquire the analytical solutions of vibrating lattice sandwich structures through modeling the complex coupling connections as an equivalent homogenous medium. Researchers have carried out a large amount of studies to investigate the mechanical properties and the dynamics of the lattice truss core of sandwich structures using this approach. Scarpa et al. investigated the manufacturing, mechanical properties and wave

propagation characteristics of a lattice sandwich structure with kirigami auxetic pyramidal cores, where the results have shown complex modal behavior due to the unusual deformation mechanism of the lattice [47]. Chen [48] regarded the complex truss core as an equivalent 2D homogeneous continuous plate to obtain the vibration and bulking characteristics. Based on the homogenization model, Lou [49] derived the governing equation of simply supported sandwich beams using Hamilton’s principle and analyzed the free vibration properties of such lattice. Further, optimization works concerning with the influence of geometric and material parameters ware studied by Xu et al. [50]. Employing the classic plate theory (CLPT), first-order shear deformation theory (FSDT), Reddy’s third-order shear deformation theory (TSDT) and Zia-Zag theory, Chen et al. [44] compared the natural frequencies of composite sandwich plates with different truss cores. In addition, the nonlinear effects on the frequency responses were taken into account by Zhang [51]. In the field of soundproof investigations, Shen et al. transformed the pyramidal truss core into a plate on account of the homogenization theory and the equivalent strain energy assumption [52]. The sound transmission loss (STL) was studied theoretically and experimentally. Further, the effects of external mean flow on the STL of three-dimensional lattice sandwich structures were discovered by Fu et al.[46]. However, it should be noteworthy that the long wavelength regime must be guaranteed in above presented cases. Due to the orthotropic and open-pored characteristics, pyramidal truss cored sandwich structures have been regarded as one of the most promising lattice structures [52]. But the dynamic analyses were mainly concentrated on the mechanical property, the vibration loading and the sound insulation performance, the dynamics of band gaps which are useful to prohibit wave transmission were less discussed. Since it is of great significance to improve the wave attenuation performance meanwhile maintain the superior mechanical properties, this paper aims to focus on the propagation characteristics and the enhancement of decaying performance at low-frequencies of transverse waves through an infinite sandwich beam with lattice cores. The inner cores are designed as one-dimensional type of the pyramidal lattice that degenerated from two-dimensional case. Wang et al.[45] have established an accurate model to describe the dynamic properties of the composite sandwich lattice through analyzing the dynamic characteristics of truss members and taking all possible motions between the face-sheet and trusses into account. Using the Euler beam theory, the truss cores can be replaced by the translational and rotational springs. This work adopts this model and adds the IA mechanisms to the composite sandwich beam for acquiring broad isolation band. Following the introduction, the motion of

the IA mechanism is analyzed and the dispersion relation of transverse waves through the composite lattice is derived by the PWE method and the Galerkin method. Then this work gives numerical results of the complex band structure and compares the IA induced gaps to the local resonant gaps. Moreover, the influences of IA attachment, layout of truss cores and damping properties are discussed to gain a flexible system design.

Nomenclature L lt Kz Kr EI EII ρI ρII xr1 xr2 θ Mt G

length of unit cell length of truss member compressive stiffness of truss rotational stiffness of truss Young’s modulus of top beam Young’s modulus of bottom beam density of top beam density of bottom beam location of truss at one side location of truss at the other side amplification angle mass of per truss reciprocal-lattice constant

Et ρt hb bb ht bt A I x1 x2 α η k

Young’s modulus of truss density of top truss length of cross section in base beam width of cross section in base beam length of cross section in truss width of cross section in truss cross-sectional area moment inertial location of IA mechanisms at one side location of IA mechanisms at the other side elevation angle of truss damping loss factor wave vector

2 Theoretical Formulations 2.1 Model descriptions As seen in Fig. 1(a), the sandwich beam considered in this work consists of two parallel elastic beams and lined with pyramidal lattice cores. An idealized configuration of the inertial amplification unit periodically displayed on the surface of the sandwich beam is given in Fig. 1(b). The band structure calculations are performed through modeling a representative unit cell, whose version can be clearly observed from Fig. 1(c). The length of unit cell is denoted as L and the location of the additional IA mechanism is measured by the parameters xr1 and xr2. The truss core holding length lt with an elevation angle α is attached at x=x1 and x=x2=L-x1 of the top beam and at x=x0=L/2 of the bottom beam. In this model, only the force and moment generated by the lattice truss along the out-of-plane direction are considered to have effects on the deformation of the base beams. According to the geometry limitation, the relation of geometrical length between the base beam and the truss member is derived as:

lt   L / 2  x1  cos  

(1)







(a)

(b)

z x

α

xr1

α

lt

lt

O

x1

xr2 x2

L (c) Fig. 1 Schematic diagram of the sandwich beam and the inertial amplification mechanisms: (a) the geometrical model of the pyramidal lattice structure in one dimensional form; (b) the idealized configuration exhibiting infinite periodicity; (c) the unit cell model Several assumptions are made for the derivation of the dynamic motion of the IA attachment. First of all, the vertical and inclined links shown by heavy lines represent the rigid connections in the mechanism. Second, the corners between them are designed as moment-free hinges. At last, a similar hinge is employed at the top connection [38]. These assumptions ensure that no moment can be transferred through the mechanism. In this way, the links have no deformation but move the mass by rigid motion. Moreover, this work ignores the influence of rotational deformation of base beam on the IA mechanisms. A detailed analysis about the dynamic motion is given in the next subsection.

2.2 Dynamic analysis for IA mechanisms Fig. 2 provides the motion of added mass ma in the xoz plane. Due to the out-of-plane excitation exerted on the sandwich beam, the deformation of the attached beam motivates the additional mass to translate along the x and z directions. The motion quantified by z1 and z2 is governed by the

amplification angel θ, the displacement at the attachment w1I , w2I and the constraint forces P1, P2 induced from the interaction between the IA mechanisms and the base sandwich beam. This study is concerned with the transverse wave propagation characteristics, thus in plane extensional vibration is omitted here.

d

d z1

z2 H

l w1I

l

H I 2

w

θ P1

P2

Fig. 2 The illustration of the rigid motion of additional mass ma Comparing the state of static to that of deformed IA mechanisms, the horizontal and vertical motions are found to participate in the following relationship: l 2   d  z1    H  w1  z2  2

2

l 2   d  z1    H  w2  z2  2

(2)

2

where l and H denote the length of rigid connection and the height from the plane with moment free hinges to the added mass in static state. The parameter d represents half length of the horizontal distance between the attachment points. By solving Eq. (2), the motions z1 and z2 are able to be obtained:

z1   2 H  w1  w2   2 z2  w1  w2   w12  w22   4d  z2 



l 2   d  z1   l 2   d  z1   2 H   w1  w2  2

2

(3)

2

(4)

Employing the small displacement assumption, the displacements cannot be comparable to the height H, thus z1 in Eq. (2) is simplified to:

z1   w1  w2  H

 2d    w1  w2  tan 

2

(5)

Further, expanding the expression of z2 using Taylor’s series method and substituting Eq. (5) to Eq. (4), the final solution of z2 is achieved:

z2   w1  w2  2

(6)

Based on the solutions, the kinetic energy and the potential energy can be easily calculated; then the dynamic equation of the IA mechanism is able to be derived using the Lagrange’s equation:

ma m 1I  a  tan 2    1 w 2I  P1 1  tan 2    w  4 4 ma 1  tan 2    w2I  m4a  tan 2    1 w1I  P2 4 From the dynamic equation, the effective mass parameters are defined as me1  me 2 

(7)

ma 1  tan 2    , 4

ma  tan 2    1 . It is obviously noticed that the relative acceleration of sandwich beam at the 4

attachment location generates proportional constraint forces, by which the effective inertia of the host structure can be correspondingly amplified.

2.3 Derivations of wave propagation characteristics in the sandwich lattice The compressive and bending motion of truss cores have been detailed described in Ref.45. This work adopts the model of the sandwich lattice comprising of translational and rotational springs. According to the theory of Euler beam, the compressive stiffness Kz and the rotational stiffness Kr per truss are derived as: Kz 

E Al t

2 t t

sin 2    12 Et I t cos 2    lt3

2 2   E I sin   cos   K r   1  3sin 4    t t  Et At lt  lt 4  

(8)

where, Et is the Young’s modulus of the truss; At and It represent the cross-sectional area and the moment inertia of the truss member with rectangle cross section. The moment and constraint forces generated by the coupling interaction between the top and bottom beams are transferred through the translational and rotational springs. The unit cell model of the sandwich lattice is required to be redrawn and given in Fig. 3.

Fig. 3 The unit cell model of the lattice structure containing translational and rotational springs Then the governing equations to describe the transverse motion of the sandwich lattice can be formulated as: EI I I



j 1,2 M t j 1,2  4 wI   A w w x x R K           R wII I I I I j z 2 x 4 R

 2w  K r   2I R  x  j 1,2

EII I II

x x j

 2 wII  x 2

x  x0

 wI

 2w  K r   2II R  x 

x  x0

  x  x  R  j

    x  x j  R   P1    x  xr1  R   P2    x  xr 2  R   R R x  x0 

j 1,2  4 wII  II    x  x0  R   K z   wI II  M t w   II AII w 4 x R R  j 1,2

x x j

2w  2I x

x x j

x x j

    x  x0  R   

 wII

x  x0

    x  x0  R  

(9a)

(9b)

where, Mt represents the mass of the truss member. The subscripts “I” and “II” denote the top and bottom beam respectively. E, ρ, I and A represent the Young’s modulus, the density, the moment inertia and the cross-sectional area, which are consistent with that stated before. R denotes the periodic repetition of attachments and can be assumed as mL according to the one-dimensional periodicity, where m is an integer ranging from  to  .The PWE method allows the displacement to be expressed as the superposition of plane waves in the reciprocal case: wI  WIG exp  i  k  G  r + it 

(10a)

wII  WIIG exp  i  k  G  r + it 

(10b)

G

G

where G denotes the reciprocal-lattice constant. Due to the one-dimensional periodicity, G, k and r are formulated to: G=2mπ/L, k=k and r=x. Substituting Eq. (10a) and Eq. (10b) into Eq. (9a), and taking advantage of the Galerkin method, the external force induced by translational spring, the constraint force generated from the IA mechanisms and the moment term at the right side of the dynamic equation

for the top beam are transformed to:

 '  iG ' x  iGx K z    WIG  e j WIG e j G'   j 1,2

' '  iGx j ik  x j  x0   e e ' WIIG eiG x0  G  

(11a)

' ' ' '     me1   WIG eiGxr 1 WIG e  iG xr 1   me 2   WIG eiGxr 1 eik  xr 1  xr 2  WIG e  iG xr 2  G' G'    

(11b)

' ' ' '     me1   WIG eiGxr 2 WIG e  iG xr 2   me 2   WIG eiGxr 2 eik  xr 2  xr 1  WIG e  iG xr 1  G' G'    

(11c)

  

 W

I

j 1,2

G

'   iGx j G '  iG x j  G '     WI   k  G  e   k  G WI e  j 1,2 G'    K r   ' '    ik x x    WIG   k  G  eiGx j e  j 0    k  G ' WIIG e  iG x0    j 1,2 G'   

(11d)

The characteristic terms at the left side of Eq. (9a) can also be rewritten to:

 '  iG ' x  iGx 4 L WIG EI I I WIG  k  G   WIG 2  I AI WIG   M t 2     WIG  e j WIG e j G'   j 1,2





   

(12)

In these equations, the time dependence factor exp(iωt) is omitted. Similar technique can also be applied to the bottom beam, each term in Eq. (9b) is ought to be reformulated as:

 ' '   4 L WIIG EII I II WIIG  k  G   WIIG 2  II AII WIIG  M t    WIIG  eiGx0 WIIG e  iG x0   G'   





 ' '   K z  2 WIIG  eiGx0 WIIG e  iG x0   G'   

 W

j 1,2

G II

'  iG ' x  iGx0 ik  x0  x j  e e ' WIG e j G 

 iGx G ' G '  iG ' x   2 WII   k  G  e 0   k  G WII e 0   G'   K r  ' '     WIIG   k  G  eiGx0 eik  x0  x j    k  G ' WIG e  iG x j  j 1,2 G' 

      

  

(13a)

(13b)

(13c)

In the presented governing equations, the variable G represents an infinite series which leads to a set of infinite equations. An appropriate truncation:  M  m  M , is taken to solve it numerically where M is selected with a compromise between the accuracy and the computational cost. Combining Eq. (11), Eq. (12) and Eq. (13), the simultaneous set of algebraic equation system is assembled as a (4M+2)×(4M+2) matrix form:  D11 D  21

D12   WIG  M11 2   G     D22   WII  0

0   WIG  0    M 22   WIIG  0 

(14)

where the vectors WI and WII contain each coefficient in the plane wave expansion solutions. The matrix elements are: D11  De1  1F   2F  1R   2R D12   1 F   2F  1 R   2R D21  1 F   2F  1 R   2R D22  De 2  2 0F  2 0R

(15)

M11  M e1   c   IA M 22  M e 2   c

The detailed information about the involved matrixes is given in Appendix. A. It is seen that Eq. (14) represents a generalized eigenvalue problem for ω2. The band structures can be obtained by solving the problem for each Bloch wave k. This analysis corresponds to the ω(k) approach which is able to observe the existence of band gaps and study the characteristics of wave propagation within the pass bands. To further verify the generation mechanisms of the band gaps and estimate the spatial attenuation of transverse waves through the sandwich lattice, a consistent eigenvalue problem in the form of k(ω) should be developed. The exponential term exp(iG(xj-x0)) and other exponential terms make it complicated to derive the solution of k directly for given ω based on Eq.(14). Thus the FEM is employed to achieve the complex wavenumbers and predict the decaying performance of transverse waves through the lattice. Based on the finite element model, the forces and moments induced and transferred by the translational and rotational springs are taken into account and the effects of attached IA mechanisms are provided in terms of an added mass matrix. In addition, the results obtained by the PWE and the FEM are able to support the correctness of each other. The discrete matrix equations are given in the following:

K u   2 M u R ub   WI  0      K b   2 M b   WII  0   R bu

(16)

where the components are shown in Appendix. B. The periodic boundary condition on the generalized displacements is defined as:  WIL   I  in     WI    0  W R   0  I    WIIL   I  in     WII    0  W R   0  II   

0 0 0   L  W    I   exp  ikL  0 0    Iin    Y0   Y1  W I W I 0    I  0  0 0 0   L  W    I   exp  ikL  0 0    IIin    Y0   Y1  W II W I 0    II  0 

(17)

 and W  represent the independent structural Vectors composed of generalized displacements W I II DOFs. Matrix I is an identity matrix. Applying the boundary condition on the discrete governing equations, Eq. (16) is transformed to the polynomial form with respect to λ:

  D0sI R ub   D1sI R1ub   D2sI R ub    WIL  0  0 2 2      bu     bu    in     sII sII sII   bu  R1 D1   R 2 D2    WI  0    R 0 D0 

(18)

At last, the complex wave vector can be solved by:  Z1  I

    Z0   W  Z 2 0   W            0 I    , W   WI ; WII  0   W W     

(19)

Once the eigenvalue λ is solved by Eq. (19), the wavenumber k whose real and imaginary components represent the phase and attenuation constants is acquired. In the numerical examples of the following section, the dispersion relation curves both in the ω(k) form and the k(ω) form are presented to study the wave propagation characteristics in the sandwich lattice attached with IA mechanisms.

3. Theoretical studies and parametric design The results of wave propagation reported in the current and following subsections consider the one-dimensional lattice made of two-phase materials and composed of a frame of beam components with rectangular cross sections. Table 1 and Table 2 show the geometrical and material parameters. The above parameters are kept unchanged in the following unless otherwise stated. Since the infinite plane wave series are truncated to finite sums, the convergence should be validated. In this study, the convergence is satisfied when the band diagrams calculated at two successive calculations are within a pre-set error. M is decided as 100 in terms of this principle. To directly examine the band structure, the frequency is normalized:  EI I I   2 fL2    I AI 

  

(20)

where, f=ω/2π.

3.1 Validation of the theoretical model The simulation of the dispersion relations of flexural waves in the periodic sandwich beam without inertial amplification effects is first performed by the PWE and the FEM respectively. The results are given and compared in Fig. 4(a), from which an excellent agreement is obtained between the numerically predicted dispersion curves. Using the proposed analytical approach, the unknown

frequencies are calculated for real wave numbers. Table 1 Geometrical parameters of the sandwich lattice with IA mechanisms Parameters

Descriptions

Values

L

The length of unit cell

0.1 m

θ

The amplification angel

9π/20 rad

α

The elevation angle of truss

π/4 rad

xr1

Location of IA attachment

0m

x1

Location of truss core

0.01 m

hb

Length of rectangular section of base beam

0.005 m

bb

Width of rectangular section of base beam

0.005 m

ht

Length of rectangular section of truss member

0.003 m

bt

Width of rectangular section of truss member

0.003 m

Table 2 Material characteristics of the sandwich lattice with IA mechanisms Parameters

Descriptions

Values

EI

Young’s modulus of face beam

7×1010 Pa

ρI

Density of face beam

2700 kg m-3

υI

Poisson’s ratio of face beam

0.33

EII

Young’s modulus of bottom beam

7×108 Pa

ρII

Density of bottom beam

2700 kg m-3

υII

Poisson’s ratio of bottom beam

0.33

Et

Young’s modulus of truss member

7×1010 Pa

ρt

Density of bottom truss member

2700 kg m-3

υt

Poisson’s ratio of truss member

0.33

In Fig. 4 (a), the bounding of the appearance for local resonant (LR) gaps is marked by hollow ellipse. The flat parts of branches indicating local resonant modes block wave transmission through the sandwich beam lattice, with the cut-on and cut-off frequencies corresponding to the modal frequencies of assigned unit cell models. As shown in Table 3, the bounding frequencies are able to be estimated via solving the eigenvalue problem of the composite beam with simply supported or sliding boundary conditions. These two configurations are defined as the configuration type (a) and the configuration type (b), respectively. For simplicity, only the first three gaps are given. It can be observed that the bandwidths of LR gaps are extremely narrow.

Table 3 Estimation of bounding frequencies of unit cell models without IA effects

Configuration type (a)

Configuration type (b)

w  0, w''  0

w'  0, w'''  0

Band gaps

Normalized Values

Configuration type

Cut-on Frequency: 261.81 Hz

0.2268

(b)

Cut-off Frequency: 309.5 Hz

0.2681

(b)

Cut-on Frequency:1414.77 Hz

1.2255

(b)

Cut-off Frequency:1465.91Hz

1.2698

(b)

Cut-on Frequency:2301.44 Hz

1.9936

(b)

Cut-off Frequency: 2339. 1 Hz

2.0262

(a)

Bounding Frequencies

First (LR)

Second (LR)

Third (LR)

(a) (b) Fig. 4 The complex band diagram of transverse waves in the bare sandwich lattice: (a) the dispersion relation of phase constants, (b) the dispersion relation of attenuation constants Further, to measure the spatial decaying performance of the elastic waves, the imaginary component of the wave number is predicted at given frequencies. Fig. 4(b) provides the dispersion relation of attenuation constants. The locations of stop bands emerged in Fig. 4(b) can be seen close to

that shown in Fig. 4(a). As seen in Fig. 4(b), each gap is appearing as a sharp notch with high amplitude and narrow width, which validates the generation of localized resonant mechanisms. Attentions are focused on the wave propagation over low frequencies, thus the Bragg gap and the LR gap induced at relative higher frequencies are without consideration.

3.2 Phononic band gaps induced by the IA mechanisms Next, the case with IA mechanism attached to the edge of the composite unit cell is examined. The additional mass ma is accounting for 50% of that of the base beams and the amplification angle θ is assumed as 9π/20. Fig. 5 shows the complex band structure composed of dispersion relations of phase constants and attenuation constants. For comparative purpose, the results with no inertial amplification effect are also presented. The most significant feature characterizing the transmission of elastic waves is the appearance of multiple stop bands due to the periodically attached IA mechanisms.

(a) (b) Fig. 5 The complex band diagram of transverse waves in the periodic lattice with IA mechanisms: (a) the dispersion relation of phase constants; (b) the dispersion relation of attenuation constants From the comparison results, broader gaps inhibiting wave transmission are observed by the additional inertial amplification mechanisms. Specific changes on the dispersion relation of transverse waves occur at low frequencies, as seen from the second and the third curves in the band diagram of phase constants, essentially simulate the start of the first IA induced gap. Moreover, it is found that the additional inertial amplification effects reserve resonant modes to constitute the fourth and fifth dispersion curves meanwhile eliminate other mode behaviors, which leads to the emergence of broader stop bands. These resonant frequencies are close to the frequencies located at the third and the forth curves originated from the bare lattice, suggesting that the bounding frequencies dependent on the

intrinsic stiffness and mass properties of the composited system with IA mechanisms are able to be predicted by solving the modal frequencies of the unit cell imposed with appropriate boundary conditions. To demonstrate that, the prediction of the bounding frequencies within IA induced gaps are given in Table 4. In addition, Fig. 5(b) shows the spatial attenuation performance of transverse waves with IA mechanism. The shape of IA induced gap possesses distinctive characteristics and can be seen different from the LR or the Bragg gaps. Further, from Fig. 5(b), the phenomenon that thin pass bands are inserting between the IA induced gaps is observed, which can be explained by the resonant behavior of the sandwich beam with IA mechanisms. These isolated bands segregate the continuous attenuation gaps to several discrete ones. Even though, the complex band diagrams demonstrate extraordinary attenuation properties over a broad frequency range. Table 4 Estimation of bounding frequencies of unit cell models with IA effects

Configuration type (c)

Configuration type (d)

w  0, w''  0

w'  0, w'''  0

Band gaps

Bounding Frequencies (Hz)

Normalized Values

Configuration type

Cut-on Frequency: 851.78

0.7378

(d)

Cut-off Frequency: 1415.94

1.2265

(d)

Cut-on Frequency:1445.38

1.252

(d)

Cut-off Frequency: 2301.44

1.9936

(c)

Cut-on Frequency: 2339.1

2.0262

(c)

Cut-off Frequency: 2887.71

2.5014

(c)

First (IA induced gap)

Second gap)

(IA

induced

Third (IA induced gap)

3.3 Comparisons to sandwich beam attached with local resonators To further demonstrate the capability of the proposed sandwich beam to produce an ultra-wide gap at low frequencies, the spatial attenuation of transverse waves in an infinitely long sandwich beam with periodically spaced spring-mass resonators is calculated and compared to that using the inertial amplification concepts. As seen in Fig. 6, additional resonator is placed at the surface of the top sheet, classic local resonant sandwich beams are constructed by taking both single and multiple resonators into account. To make this a fair comparison, the periodic sandwich beams have same geometrical parameters, material properties and overall mass. Since attention is concentrated on the low-frequency band gap, the resonant frequency Ωr of the spring-mass resonator is turned to fall into the frequency range within the first IA induced gap. Fig. 7 provides the wave attenuation profiles for unit cell models with IA mechanisms and local resonators.

(a) (b) Fig. 6 Unit cell models attached with local resonators: (a) attached with one spring-mass resonator; (b) attached with multiple spring-mass resonators

(a) (b) Fig. 7 Comparisons of spatial attenuation performance induced by attached IA mechanism and local resonators: (a) compared to the case with one spring mass resonator; (b) compared to the case with multiple spring mass resonators

The additional local resonant mechanism indeed achieves larger gap depth, however, the bandwidth of which is narrower than gap induced by IA mechanisms. The comparison of average attenuation constant over the stop band further certifies the efficiency on enhancing broadband attenuation performance by amplifying the effective inertial. In the local resonant sandwich lattice, the average decaying coefficient is 0.5252 and in the lattice with IA effects, it is 0.6253, definitely showing the ability exhibited by IA mechanisms to generate wide low-frequency gaps in the composite structures. Besides, the comparison to the multiple stop bands generated by multiple placed mass-spring resonators indicates that even more than one sharp attenuation peaks can be obtained, the bandwidths around the resonance frequencies are too narrow to be coalesced in this way. In other words, the IA induced gap obtained in the sandwich beam has superiority of suppressing transverse wave propagation over wide low-frequency range.

Fig. 8 The comparison results of vibration transmittance of finite sandwich lattice with IA mechanisms and local resonators In real-world applications, periodic structures possessing finite size are of more practical meaning. To evaluate the behavior of wave propagating through finite sandwich beams, the vibration transmittance is calculated, as well as to validate the width and depth of resulting low frequency gaps. Two cases are considered: finite periodic sandwich beam with IA mechanisms and that with spring-mass resonators. Each of them is designed to be composed of 20 unit cells. A harmonic out-of-plane force with unit amplitude is applied at the input node i.e., left edge, of the sandwich beam, the displacement field is solved using the finite element method. The flexural vibration transmittance Tra is obtained through the out-to-input, i.e., the left to right edge, displacement ratio. In this subsection,

the mass ratio between the added one and the overall beam is 0.25. Fig. 8 gives the comparison results of vibration transmission loss curves. A broad insulation band is found due to the IA effects in the finite periodic sandwich beam although the additional mass is reduced to a smaller value. Furthermore, the bandwidth compared to the local resonant case shows less dependence of added mass on decaying the propagation for low-frequency transverse waves.

3.4 Parametric studies and design From subsection 3.3, it is recognized that the additional mass of IA mechanisms is influential to the transverse wave propagation properties. Except for this factor, the variations of IA attachment, the geometry of truss cores and the damping component-all cause impacts on the spatial attenuation characteristics of transverse waves. This section is presented to discuss the effects in detail.

3.4.1 Effects of additional mass ma and angle θ

(a) (b) Fig. 9 The attenuation performance with varying parameters: (a) additional mass ma; (b) amplification angle θ In this subsection, the attenuation profiles are calculated with respect to varying parameters ma and θ. Fig. 9(a) and Fig. 9(b) show the influence of ma and θ on the attenuation performance of transverse waves of the infinite sandwich beam including IA effects. Several horizontal lines distributed on the map seem to split the entire band diagram, in essence denote the local resonant phenomenon occurred in the sandwich lattice. On one hand, as seen from Fig. 9(a), when the mass ratio is increased, the number of the stop band over the interest frequency range is increased and the first IA induced gap shifts to lower frequencies. However, the locations of the second and third IA induced gaps stay the

same, showing independent of the variation of the additional mass ma. Correspondingly, the total bandwidth of IA induced gaps is broadened. Meanwhile the amplitude of attenuation peak is decreased with increasing mass ratio. On the other hand, from Fig. 9(b), it is noted that the effective inertia is not monotonically varying by increasing angle θ. The bandwidth is narrowed at first and then broadened by bigger θ. In addition, the spatial attenuation ability can be controlled to prohibit wave propagating from high to low frequencies via amplifying angle θ.

3.4.2 Effects of IA location Variety configurations can be constructed by varying the location of IA mechanisms on the surface of the face sheet. Here the parameter Xr=xr1/L is given to measure the location of IA mechanisms displayed at the unit cell. The influence of attachment on the band gaps is discovered by calculating attenuation dispersion curves with different values of Xr. Fig. 10 shows the variations of bounding frequencies and attenuation performances for multiple stop bands, where Xr is assumed as 0, 0.1, 0.2 and 0.4 respectively. The lowest IA induced gap is degenerated to localized gap and gradually moving to higher frequencies with increasing Xr. This is because narrower distance of the attachment points gives smaller level ratio which generates less inertial force to the elastic beam. These various profiles obviously indicate that the proposed composite beam has the ability to tailor spatial attenuation of transverse waves by varying the location of attached substructure.

(a) (b) (c) (d) Fig. 10 Attenuation profiles in the case of Xr: (a) Xr=0; (b) Xr=0.1; (c) Xr=0.2; (d) Xr=0.4

3.4.3 Effects of truss cores This subsection aims to examine the influence of the layout of pyramidal cores on the attenuation properties. With a constant unit cell length, the elevation angle of the truss α dominates the geometry of pyramidal cores. From Eq. (1), the length and the mass of the truss member are definitely amplified by

decreasing angle α. Fig. 11 exhibits the attenuation profiles under different elevation angles. The first local resonant gap can be seen independent of the variation of the truss member, but the IA induced gaps are shifting towards lower frequencies due to the denser cores with declining α.

(a) (b) (c) (d) Fig. 11 Attenuation profiles against different elevation angle α: (a) π/12; (b) π/6; (c) π/4; (d) π/3

3.4.4 Effects of material damping Damping effects play a significant role in shaping the dynamic structural responses. When the damping is present in materials, each solution obtained at given frequencies becomes complex, which indicates that the original pass bands acquire spatial attenuation ability. However, the attenuation constant of pass bands in general cannot be comparable to that of stop bands. Damping existing in different constitutive components has unique influence on the wave propagation characteristics. To reveal the work mechanisms, four cases are considered: material damping is present in the top beam, bottom beam, truss cores and all structural constitutions respectively. The damping loss factors are kept to a constant: 0.01.

(a) (b) (c) (d) Fig. 12 The influence of damping constituents on the spatial attenuation performance From the comparison results, it is observed that the existence of damping in the bottom beam has most noticeable changes on the attenuation profiles. The usage of the damping material in the bottom

beam significantly weakens the local resonant behavior that separates IA induced gaps, thus an ultra-broad continuous stop band can be found in Fig. 12(b). In addition, due to the presence of damping materials, the amplitude of local resonant gap is lowered while the spatial attenuation performance at higher pass bands is improved, which can be seen from Fig. 12(a) and Fig. 12(c) . The combined effects are illustrated in Fig. 12 (d). One can notice that the stop bands are connecting with the pass bands to increase the prohibition of wave transmission in the proposed sandwich lattice. Further, to determine the degree of wave propagation being affected by various levels of the damping effects, the variations of attenuation performance with different damping loss factors (η=0.005, 0.01, 0.05 and 0.1) are predicted and shown in Fig. 13. From Fig. 13(a) to Fig. 13(d), the boundary distinguishing the pass band and the stop band becomes more and more obscure with increasing η. Meanwhile, the local resonance phenomenon is weakened and the IA induced gap is significantly broadened. When the damping loss factor has sufficient large value, such as 0.05 or 0.1 in this case, the dispersion relation curves of attenuation constants are intensively smoothed out by diminishing sharp amplitude as observed in Fig.13(c) and Fig. 13(d).

(a) (b) (c) (d) Fig. 13 The influence of different damping levels on the spatial attenuation performance: (a) η=0.005; (b) η=0.01; (c) η=0.05; (d) η=0.1

4. Conclusions This work extends the amplified inertial concept to the sandwich lattice for obtaining wide and multiple stop bands that have significant beneficial effects on isolating vibration and sound. The main object is focused on the complex band structures of transverse waves propagating through the infinite periodic sandwich beam with lattice truss cores. Theoretical models for acquiring the dispersion relation of the phase constants of transmitted waves in the sandwich lattice are developed by the PWE method and the Galerkin method. Concerning with the spatial decaying performance, this study adopts

the FEM to examine the attenuation properties with respect to varying frequencies numerically. The results of complex band structures show the accuracy of the proposed analytical approach. The band diagrams demonstrate the ability of additional IA mechanisms to produce multiple and wide band gaps, especially strengthen the wave suppression performance over low-frequency range. Moreover, this work compares the wave attenuation characteristics inside IA induced gaps to that of LR gap generated by adding local resonators which are commonly used to gain low-frequency gaps. Numerical results show that although the LR gap can achieve higher amplitude of the attenuation constant at the resonance frequency, the average decaying performance is inferior to the IA induced gap. Further, the calculation of vibration transmittance of finite sandwich beams composed of IA and LR based unit cells confirms that the finite lattice with IA mechanisms can exhibit less dependency of additional mass on broadening the isolation frequency band. As the proposed structure has enormous potential to tailor wave propagation properties, this paper conducts a parametric study to discover the influences of the IA attachment, the elevation angle of inner cores and the material damping on the spatial attenuation performance. It is found that the IA induced gaps are moving to low frequencies with the increasing of the additional mass and the amplification angle. Larger value of attached mass can lead to wider low-frequency gaps while the amplification angle shows a more complicated trend on the bandwidth. The influence of the location of the IA attachment on the spatial attenuation property is also examined. Corresponding dispersion relation curves verify the tunability of the band gaps by varying the location of attached IA mechanisms. As the distance between the attachments is reduced, the first IA gap apparently shifts to higher frequencies. Additionally, the geometry of truss member modified by changing the elevation angle has forced the attenuation profiles move to higher frequencies with bigger α. At last, this work discussed the role of damping on the propagation characteristics of the transverse waves from two aspects. First, considerations are concentrated on the individual constituent element made up of the damping material. Comparison results reveal that the bottom beam with damping effects is able to eliminate the dynamic behavior of local resonant bands thus generates an ultra-wide stop band for prohibiting wave propagation. Moreover, in the present of the damping material of the top beam or the truss cores, the pass bands are transformed to exhibit spatial attenuation ability. Second, since different levels of damping shape the dispersion curves of attenuation constants to different degrees, the attenuation profiles are calculated with selecting various damping loss factors. It can be seen that

increasing the damping loss factor can broaden the bandwidth of the stop bands and decrease the amplitude of the LR gap.

Acknowledgements The present work is supported by the Scientific Research Foundation of Hainan University, Grant No. KYQD(ZR)1913.

Appendix A. Expressions of the matrix in Eq. (15) Several vectors are defined first to compose the matrixes: P1  e  iG1 x1

e  iG2 x1 ... e  iG2 M 1 x1  ,

P2  e  iG1 x2

e  iG2 x2 ... e  iG2 M 1 x2 

P0  e  iG1 x0 e  iG2 x0 ... e  iG2 M 1 x0  , I r1  e  iG1 xr 1 I r 2  e  iG1 xr 2 e  iG2 xr 2 ... e  iG2 M 1 xr 2 

 k  G2  eiG x  k  G2  eiG x  k  G2  eiG x

Q1   k  G1  e  iG1 x1 Q 2   k  G1  e  iG1 x2 Q 0   k  G1  e  iG1 x0 X10  eiG1 x1 e 

ik  x1  x0 

X01  eiG1 x0 e 

eiG2 x1 e

ik  x1  x0 

ik  x2  x0 

eiG2 x2 e

ik  x0  x1 

eiG2 x0 e

ik  x0  x2 

eiG2 x0 e

X 20  eiG1 x2 e 

X02  eiG1 x0 e 

...

2 1

2 2

...

2 0

...

 k  G2 M 1  eiG  k  G2 M 1  eiG  k  G2 M 1  eiG

 2 M 1 x2 

2 M 1 x1

2 M 1 x0

... eiG2 M 1 x1 e

 

T

ik  x2  x0 

 

ik  x0  x1 

 

ik  x1  x0 

ik  x2  x0 

... eiG2 M 1 x2 e

ik  x0  x1 

... eiG2 M 1 x0 e

ik  x0  x2 

e  iG2 xr 1 ... e  iG2 M 1 xr 1 

... eiG2 M 1 x0 e

ik  x0  x2 

T

 

T

T

eiG2 xr 1 eik  xr 1  xr 2  ... eiG2 M 1 xr 1 eik  xr 1  xr 2  

S rr12  eiG1 xr 2 eik  xr 2  xr 1 

eiG2 xr 2 eik  xr 2  xr 1  ... eiG2 M 1 xr 2 eik  xr 2  xr 1   ik  x1  x0 

U 20   k  G1  eiG1 x2 e  U 01   k  G1  eiG1 x0 e 

 k  G2  eiG2 x1 e

ik  x1  x0 



T

S rr12  eiG1 xr 1 eik  xr 1  xr 2 

U10   k  G1  eiG1 x1 e 

(A.1)

...

T

 k  G2 M 1  eiG2 M 1x1 e

ik  x2  x0 

 k  G2  eiG x eik  x  x 

...

 k  G2 M 1  eiG

e

ik  x0  x1 

 k  G2  eiG x eik  x  x 

...

 k  G2 M 1  eiG

e

 k  G2  eiG x eik  x  x 

...

 k  G2 M 1  eiG

U 02   k  G1  eiG1 x0 e 

ik  x0  x2 

2 2

2 0

2 0

2

0

0

0

1

2

2 M 1 x2

2 M 1 x0

2 M 1 x0

 

T

ik  x2  x0 

 

ik  x1  x0 

ik  x0  x1 

e

 

ik  x0  x2 

(A.2) T

T

 

T

Based on the vectors, the matrixes can be derived as: Kz  K K K P1 P1 ,  2F  z P2 P2 , 1R  r Q1Q1 ,  2R  r Q2Q 2 L L L L K K K K 1 F  z X10 P0 ,  2F  z X 20 P0 , 1 R  r U10Q 0 ,  2R  r U 20Q 0 L L L L K K K K K K  0F  z P0 P0 ,  0R  r Q0Q 0 , 1 F  z X01P1 ,  2F  z X02 P2 , 1 R  r U 01Q1 ,  2R  r U 02Q 2 L L L L L L M M  c  t  P1 P1  P2 P2  ,  c  t P0 P0 , ,M e1   I AI I, M e 2   II AII I (A.3) 2L L me1  me 2 r 2 me1  me 2 r1  IA   I r1I r1   L  Sr1 I r 2   L  I r 2I r 2   L  Sr 2I r1  L  EI I I (k  G1 4 0 0  0   EII I II (k  G1 4 0 0  0      4 4 0 EI I I (k  G2  0  0 0 EII I II (k  G2  0  0     De1     , De 2                4 4 0 0  0  EII I II (k  G2 M 1   0 0  0  EI I I (k  G2 M 1   1F 

Appendix B. Matrix elements in Eq. (16) Considering the coupling effects and the amplified inertial by additional IA mechanisms, the matrix elements can be expressed by the following components:

K u  K  K z Fu1  K z Fu 2  K r M u1  K r M u 2 ; M u  M  M t 2  Fu1  Fu 2   G m ;

R ub   K z Fu1b  K z Fu 2b  K r M u1b  K r M u 2b ; (B.1)

K b  K  2 K z Fb 0  2 K r M b 0 ; M b  M  M t Fb 0 ; R bu   K z Fbu1  K z Fbu 2  K r M bu1  K r M bu 2 ; where K and M are the global stiffness and mass matrixes for proposed Euler beam. The components composing the element matrixes are:

Pu1  [0 1 0 0 0]T ; Pu 2  [0 0 0 0 1 0]T ; Pu 0  [0  0 1 0 0]T ;

Fu1  Pu1PuT1 ; Fu 2  Pu 2 PuT2 ; Fb 0  Pu 0 PuT0 ;

(B.2)

Fu1b  P P ; Fu 2b = Pu 2 P ; T u1 u 0

T u0

Fbu1  Pu 0 PuT1 ; Fbu 2 = Pu 0 PuT2 ; The location of nonzero values in the vector is consistent with that of structural DOFs of the node attached with springs. Likewise, the moment and inertial terms are defined as:

Yu1  [0 01 0 0 0]T ; Yu 2  [0 0 0 0 0 1 0]T ; Yu 0  [0 0 0 1 0 0]T ;

M u1  Yu1YuT1 ; M u 2  Yu 2 YuT2 ; M b 0  Yu 0 YuT0 ; M u1b  Yu1YuT0 ; M u 2b = Yu 2 YuT0 ; M bu1  Yu 0 YuT1 ; M bu 2 = Yu 0 YuT2 ; 0 0 0 0 0 0            0 me1  0  me 2  0    G m          0  me  0 me  0  2 1           0 0 0 0 0 0   

(B.3)

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