Vibration band gaps for elastic metamaterial rods using wave finite element method

Transcription

1 Vibration band gaps for elastic metamaterial rods using wave finite element method E.D. Nobrega a,, F. Gautier b, A. Pelat b, J.M.C. Dos Santos a a University of Campinas, UNICAMP-FEM-DMC, Rua Mendeleyev, 200, CEP , Campinas, SP, Brazil. b Université du Maine, ENSIM - LAUM, 1 Rue Aristote, Le Mans, France. Abstract Band gaps in elastic metamaterial rods with spatial periodic distribution and periodically attached local resonators are investigated. New techniques to analyze metamaterial systems are using a combination of analytical or numerical method with wave propagation. One of then, called here wave spectral element method (WSEM) consists of combining the spectral element method (SEM) with the Floquet-Bloch s theorem. A modern methodology called wave finite element method (WFEM), developed to calculate dynamic behavior in periodic acoustic and structural systems, utilizes a similar approach where SEM is substituted by the conventional finite element method (FEM). In this paper, it is proposed to use WFEM to calculate band gaps in elastic metamaterial rods with spatial periodic distribution and periodically attached local resonators of multidegree-of-freedom (M-DOF). Simulated examples with band gaps generated by Bragg scattering and local resonators are calculated by WFEM and verified with WSEM, which is used as a reference method. Results are presented in the form of attenuation constant, vibration transmittance and frequency response function (FRF). For all cases, WFEM and WSEM results are in agreement, provided that the number of elements used in WFEM are sufficient to convergence. An experimental test was conducted with a real elastic metamaterial rod, manufactured with plastic in a 3D printer, without local resonance-type effect. Corresponding author. Tel.: address: edilsonnobre@gmail.com (E.D. Nobrega) Preprint submitted to Mechanical Systems and Signal Processing March 10, 2016

2 The experimental results for the metamaterial rod with band gaps generated by Bragg scattering are compared with the simulated ones. Both numerical methods (WSEM and WFEM) can localize the band gap position and width very close to the experimental results. A hybrid approach combining WFEM with the commercial finite element software ANSYS is proposed to model complex metamaterial systems. Two examples illustrating its efficiency and accuracy to model an elastic metamaterial rod unit-cell using 1D simple rod element and 3D solid element are demonstrated and the results present good approximation to the experimental data. Keywords: Elastic metamaterial, Periodic structure, Wave propagation, WFEM. 1. Introduction The engineering studies of periodic structures using wave propagation, began to spread in the mid-70 with Mead s works [1, 2, 3, 4]. In recent decades, new methods have been developed that use the same basic concepts of periodicity together with approximated solution as a way to reduce computational costs and to solve complex engineering models that can not be solved analytically neither numerically using traditional methods [5, 6, 7, 8]. One of them is the wave finite element method (WFEM), which consists of modeling a small slice of elastic waveguide by FEM, to apply periodicity condition with Floquet-Bloch s theorem to obtain a transfer matrix eigenproblem. The solution provides attenuation and wave-modes, from where wave motion amplitudes are obtained. The method have been applied in various types of finite element model, such as beams, thin plates, cylindrical shells, including different material properties, couplings and mediums. More recently, it has been applied as a numerical solution for modeling periodic structures in the mid-frequency range where conventional methods are not efficient. In the last decade applied research in phononic materials has been plentiful. However, some fundamental evaluation from engineering point of view, 2

3 such as model simulation (analytical, numerical and hybrid) and experimental approaches developed for conventional materials and structures needs to be employed to understanding and explore the metamaterial systems [9]. One of the most attractive characteristic of acoustic and elastic metamaterial is their wave filtering behavior. This provides some frequency ranges known as band gaps or forbidden bands where the waves can not propagate. Band gaps are generated on spatial periodicity of the impedance mismatch domains which produces Bragg scattering effect. Also, locally resonant mechanism [10] provide band gaps at sub-wavelength, which are well below to the Bragg scattering band gaps. In 2006, a theoretical and experimental study of longitudinal wave propagation in a rod structure including periodic local resonators was presented by Wang et al. [11]. They shown that both results produce an asymmetric band gap attenuation which is influenced by local resonator stiffness and mass ratios. Nevertheless, this work was centered on the band gaps conception and its property to attenuate vibration, without explore fully the band gap formation mechanisms. More late, a theoretical and numerical study of locally resonant elastic metamaterial rod system with periodic multi-degree-of-freedom (M-DOF) resonators, was presented by Xiao et al. [12]. The band gap behavior and vibration attenuation performance was analyzed in a more systematic way. A new metamaterial rod model, based on a combination of spectral element method and Floquet- Bloch s theorem was proposed, which will be called here wave spectral element method (WSEM). They provided explicit expressions to predict band edge frequencies, demonstrated that both Bragg- and resonance-type band gaps co-exist in metamaterial rods and that multiple resonance band gaps can be achieved using M-DOF local resonators. These authors had also applied WSEM for band gap investigation in flexural metamaterial beams with local resonators [13]. Recently, Khajehtourian and Hussein [14] presented a study of wave dispersion in a nonlinear elastic metamaterial rod system with periodically attached local resonators. The type of nonlinearity considered is large elastic deformation. The metamaterial rod model is based on a combination of standard trans- 3

4 fer matrix method and Floquet-Bloch s theorem. The results demonstrate that nonlinearities on metamaterial rods can affect band gap position, width, and its type (Bragg scattering or local resonance). They shown that large deformation alone may induce a pair of Bragg- and resonance-type band gaps to merge in a one hybrid and form a combined wide band gap. They also shown that as the wave amplitude increases, the effect of the nonlinearity in the metamaterial rod system is no longer negligible and the error incurred by assuming linear elastic wave propagation theory increases quickly. In this paper, the wave finite element method (WFEM) is applied to analyze elastic metamaterial rods with spatial periodic distribution and periodically attached M-DOFs local resonators. The main purpose of this paper is to demonstrate the accuracy and efficiency of WFEM for modeling metamaterial system. Considering that WSEM is based on the exact analytical solution of rod govern equation and WFEM is based on the approximated finite element solution, it can be said that one spectral element must be equal to infinite elements of FEM. It means that for a uniform metamaterial rod calculated by WSEM the unit-cell dynamic stiffness matrix requires only one dynamic spectral element matrix, while by WFEM the unit-cell dynamic stiffness matrix requires an assembly of as many FEM stiffness and mass element matrices as required to obtain good convergence. Therefore, WSEM will be used here as a reference method to verify the accuracy of WFEM. Clearly, for the uniform metamaterial rod considered in the paper, WSEM is more accurate than WFEM. Nevertheless, WFEM is able to model more complex metamaterial rod system, such as nonuniform rods, which can not be easily modeled using WSEM. Both methods are computationally implemented, evaluated and compared to each other. By using an actual elastic metamaterial waveguide an experimental test was performed. The elastic metamaterial waveguide was originally developed to be used in another research project as a beam-like (flexural waves) and shaftlike (torsional waves) structures with spatial periodic distribution and local resonators. For this work it was configured as a rod-like (longitudinal waves) structure, which maintain the spatial periodic distribution, but the local res- 4

5 onators becomes inactive. The metamaterial rod is manufactured with plastic (Vero White Plus) in a 3D printer with UV curing technology. All simulated results with WSEM and WFEM are compared with the experimental data and some different behaviors and mismatches between numerical and experimental FRFs were found. After a try-and error numerical model updating by varying material property parameters (Young modulus, mass density and internal damping) these differences are minimized, and the numerical models can localize the band gap position and width very close to the experimental results. A hybrid approach combining WFEM with a finite element analysis commercial software ANSYS (Mechanical APDL Release 14.5) is proposed and two examples illustrating its efficiency to model an elastic metamaterial rod unit-cell using a 1D and 3D FEM models are demonstrated. Based on this results it can be extrapolated that WFEM, as compared with WSEM and other analytical and numerical methods, is able to model metamaterial unit-cell using commercial FEM packages, which facilitates greatly its application for real complex engineering structures. 2. Metamaterial Elastic Rod Modeling by WSEM and WFEM 2.1. Metamaterial unit-cell model Let s consider an elastic metamaterial rod with spatial periodic distribution and periodically attached M-DOFs local resonators, as sketched in Fig. 1. The length of each unit-cell is L. For the spatial periodic distribution the band gaps are generated by Bragg scattering, which appears around frequencies governed by the Bragg condition L = n(λ/2)(n = 1, 2,...), where λ is the unit-cell wavelength. The local resonators are represented by series of springs k i and masses m i with i = 1, 2,..., N where N is the number of springs and masses in resonator. The equilibrium equation of the resonator attached at an unit-cell 5

6 Figure 1: Consecutive unit-cells in metamaterial rod. 110 point is given by: k 1 k lr ω 2 m 0 0 k rl k rr 0 m rr }{{}}{{} K M u 0 u r } {{ } u = F 0 F r } {{ } F or Du = F where D = K ω 2 M, the index l and r represent left and right sides of resonators, m 0, q 0 and F 0 are the mass, displacement and force, respectively at the attachment point between the resonator and the rod unit-cell. The sub-matrices and sub-vectors of the Eq. (1) are given by [12]: k 1 k 1 + k 2 k k 2 k 2 + k 3 k k rl = k T lr = 0, k rr =. 0 k , k N 1 + k N k N k N k N m u 1 F 1 0 m u 2 F 2. m rr = , u r =. and F r = m N 1 0 u N 1 F N m N By considering that there is no external force in resonator (F r u N (1) = 0) the Eq. (1) can be rewritten as, F 0 = D 0 u 0, where D 0 is the resonator dynamics F N, 6

7 stiffness matrix at the attachment point, which can be expressed as: D 0 = k 1 k lr (k rr ω 2 m rr ) 1 k rl (2) The equilibrium equation for the finite rod is similar to Eq. (1). But, its dynamic stiffness matrix is given by D rod = K rod ω 2 M rod, where K rod and M rod are obtained from a numeric or analytical rod model. In this paper two rod models are used: one presented by [12], is formulated with the analytical spectral element method (SEM) and will be used as a reference method; and the other formulated by numeric finite element method (FEM), is proposed here as a new tool to calculate metatmaterial rods. The longitudinal vibration of the finite rod can be modeled by SEM [15], where the spectral dynamic stiffness element matrix is given by, D e cot(βl) csc(βl) rodsem = EAβ, (3) csc(βl) cot(βl) and the spectral dynamic stiffness global matrix is D rodsem = D e rodsem, provided that the structure geometry is uniform. For FEM the dynamic stiffness element matrix is given by, D e rodfem = Ke rodfem ω2 M e rodfem, where K e rodfem = ES L , M e rodfem = ρsl (4) are FEM stiffness and mass element matrix, respectively. The dynamic stiffness global matrix is given by, D rodfem = K rodfem ω 2 M rodfem, where K rodfem and M rodfem are the stiffness and mass global matrix, respectively, which are an assembly of as many stiffness and mass element matrices as required to obtain convergence. as: The D rod matrix can be partitioned and the dynamic equation can be written D ii D il D ir D li D ll D lr D ri D rl D rr u i u l u r = 0 F l F r (5) 7

8 135 where i represents the internal degrees-of-freedom (DOF). From the Eq. (5) the internal displacement can be obtained as: u i = D 1 ii (D ilu l + D ir u r ). (6) 140 Substituting Eq. (6) into Eq. (5) the condensed dynamic stiffness matrix is obtained as, D ll D rl D lr D rr u l u r = where, D ll = D ll D li D 1 ii D il, D rl = D rl D ri D 1 ii D il, D lr = D lr D li D 1 ii D ir, D rr = D rr D ri D 1 ii D ir. By coupling the local resonator with the finite rod model, results: D ll + D 0 D rl D lr D rr u l u = F l F F l F r (7) or D cu c = F c (8) where D c is the dynamic stiffness matrix of the elastic metamaterial unit-cell Periodicity condition In this method the dynamic stiffness matrix of a unit-cell of the whole structure, modeled by WSEM or WFEM is used to apply the periodicity condition in a harmonic disturbance propagating through the structure. Using Floquet- Bloch s theorem periodicity condition results in an eigenvalue/vector problem whose formulation produces the force-displacement relationships. The WFEM has been used for free and forced wave propagation in vibration analysis [6] with applications to one and two-dimensional structural models [7, 16]. In this paper WFEM is extend to model elastic metamaterial rods with spatial periodic dis- tribution and M-DOF local resonators. The SEM have also been used to model one and two-dimensional structural models [15] and was extended as WSEM to model elastic metamaterial rods [12]. Rearranging the Eq.(8) as the Transfer Matrix formulation one produces, u r D 1 = lr D ll F r } {{ } q r D 1 lr } D rl D rr D 1 lr D ll {{ D rr D 1 lr } T u l F l } {{ } q l. (9) 8

9 155 where T is the transfer matrix that relates the left state vectors q l with the right state vector q r of the unit-cell. Let s consider now consecutive unit-cells, m and m + 1, in the structure shown in Fig. 1. The displacement continuity condition, u (m) r = u (m+1) l and the force balance F (m) r = F (m+1) l, produces q (m) r = q (m+1) l. By substituting in the Eq. (9), produces: q l (m+1) = Tq l (m). (10) 160 The Floquet-Bloch s theorem for wave propagation in an infinite periodic system [2] applied to consecutive unit-cells, generates: u (m+1) l = e µ u (m) l F (m+1) l = e µ F (m) l (11) where µ = iβl is the attenuation constant and β is the wavenumber [12]. Substituting Eq. (11) in Eq. (10) and rearranging, one obtains: u (m) l T F (m) = u (m) eµ l l F (m) or Tq l = e µ q l (12) l From the Eq. (9) it is observed that T depends on the D 1 lr. However, even if the matrix D 1 lr is invertible, numerical ill-conditioning is likely to occur. To avoid this, a representation by displacement vector alone is preferable [17]. By rewriting the state vectors as a function of the left and right displacement, produces: q l = I n 0 D ll D lr } {{ } L u l u r } {{ } w e q r = 0 I n D rl D rr } {{ } N u l u r } {{ } w, (13) where matrices L and N are invertible if D lr is invertible and T = NL 1. Thus, the Eq. (12) involving the symplectic matrix T can now be replaced by the generalized eigenequation, e µ Lw = Nw (14) 9

10 175 where e µ is the eigenvalue and w is the displacement vector associated to the unit-cell. It can also be shown that e µ and Lw are eigenvalues and eigenvectors of Eq. (14), respectively [17]. Therefore the 2n eigenvalues of Eq. (14) when ordered appropriately can be subdivided into two groups: e µj 1, j = 1, 2,..., n, which corresponds to the waves traveling to the right and e µj 1, j = 1, 2,..., n which corresponds to waves traveling to the left. Then, the state vectors q (m) for a finite structure can be expressed as [18]: q (m) = j Φ j Q (m+1) j = j Φ j e ikjd Q m j, with m = 1, 2, 3,..., N c, N c+1, 180 (15) where N c is the cell number, β j is the wavenumber and Φ j = Lw j is the eigenvector or waveform. Q (m+1) j and Q (m) j the unit-cell boundary conditions m + 1 and m, respectively. are the wave vector amplitudes at The vibration transmittance can be calculated by V t = u (Nc) q (1) = {u (1) l F (1) l } T, q (Nc) = {u (Nc) l r /u (1) l F (Nc) l } T and q (Nc+1) = {u (Nc) r, where F (Nc) r } T. 3. Numerical Verification of WFEM To verify the WFEM and demonstrate its capacity to calculate band gaps in a elastic metamaterial rod with spatial periodic distribution and periodically attached M-DOFs local resonators, simulated examples are performed. In order to facilitate a comparison between results from WFEM and WSEM, the simulated examples are the same as used by Xiao [12]. Elastic system consisting of a uniform rod with periodically attached single (S-DOF) and multi (M-DOF) degree-of-freedom local resonators are evaluated. The simulated metamaterial rod geometric parameters and material properties are summarized in Table 1. Table 2 presents the local resonator parameters used to calculate the three examples used to verify WFEM method. For all simulated examples the local resonator mass at the attachment point are included and its value is m 0 = kg. Also, a structural hysteretic damping (η) was include as a complex Young s modulus, E c = E[1 + iη]. 10

11 Table 1: Simulated metamaterial rod geometric parameters and material properties. Geometry/Property Value Unit-cell length (L) 0.05 m Cross section area (S) 50 x 10 6 m 2 Number of unit-cells (N c ) 8 Young s modulus (E) 1.5 x Pa Mass density (ρ) 1200 kg/m 3 Structural damping (η) 0.02 Table 2: Local resonator parameters. Resonator Stiffness [N/m] Mass [kg] N. Freq. [Hz] Example DOF s k 1 k 2 m 1 m 2 f 1 f 2 1 Single 5.120x Single 2.302x Two 1.079x x Fig. 2 shows attenuation constant (left-hand side) and vibration transmittance (right-hand side) for the metamaterial rod model of: (a) Example 1, S- DOF local resonator with f 1 = 1650 Hz; (b) Example 2, S-DOF local resonator with f 1 = 3500 Hz; and (c) Example 3, 2-DOFs local resonator with f 1 = 1650 Hz and f 2 = 3500 Hz. It should be considered that blue solid line represent WSEM and red dot line represent WFEM results. Fig. 2a show the attenuation constant and the vibration transmittance of the example 1 calculated by WSEM and WFEM methods. It can be seen that both curves are in good agreement and also present the same band gaps (position and width). It must be noticed that for all examples WFEM requires 50 internal DOFs in order to obtain the nearest result to the WSEM. Fig. 2b also shows the attenuation constant and vibration transmittance calculated by WSEM and WFEM for the example 2, and the results are similar to that obtained in the first example. However, the band gap obtained is wider than the previous one. This result confirms the 11

12 findings of [12] that local resonator with higher resonance frequency can enlarge the band gap. (a) S-DOF, f 1 = 1.65 khz (b) S-DOF, f 1 = 3.5 khz (c) 2-DOFs, f 1 = 1.65, f 2 = 3.5 khz Figure 2: Attenuation constant (left-hand side) and Vibration transmittance (right-hand side) showing the band gaps for three examples of metameterial rod calculated by WSEM and WFEM methods The example 3 (Fig. 2c) consider a 2-DOF resonator were its parameters were chosen to obtain the first and second natural frequencies (f 1 and f 2 ) with the same values as used in the natural frequency of the S-DOF example 1 and 2, and an equivalent total mass m 1 +m 2 = kg. Then, the 2-DOF resonators are tuned with f 1 = 1650 Hz and f 2 = 3500 Hz. It can be seen that attenuation constant and vibration transmittance shows two resonance band gaps around the local resonator natural frequencies. As compared to the S-DOF local resonators the 2-DOF presents two band gaps around the two chosen resonance frequencies. However, the band gap for the S-DOF with f 1 = 3500 Hz (example 2) generates 12

13 a wider frequency band. For all simulated examples the comparative results prove that the WFEM can be used as a good approximated solution for the metamaterial models. In a structural dynamic testing is a common practice to measure the data in the form of a frequency response function (FRF). Then it is very convenient to use this type of response function to show the metamaterial band gaps. Fig. 3 shows point receptance FRFs including band gaps calculated by WSEM and WFEM methods for a metamaterial rod system including: (a) S-DOF local resonator with f 1 = 1650 Hz (Example 1); (b) 2-DOF s local resonator with f 1 = 1650Hz and f 2 = 3500Hz (Example 3); (c) S-DOF local resonator with f 2 = 3500 Hz for an extended frequency band up to 10 khz; and (d) S-DOF local resonator with f 2 = 3500Hz zoomed between khz. (a) S-DOF, f 1 = 1.65 khz (b) 2-DOF s, f 1 = 1.65, f 2 = 3.5 khz Receptance [db] Frequency [khz] (c) S-DOF, f 1 = 3.5 khz, extended up to 10kHz (d) S-DOF, f 2 = 3.5 khz, zoom 6.5 to 10 khz Figure 3: Point Receptance FRF showing the band gaps for the examples of metameterial rod calculated by WSEM and WFEM methods. 235 It can be seen that for all examples evaluated both methods are in good agreement and the band gaps can be also identified easily in the FRFs. Fig. 3d emphasizes the mismatch between WFEM and WSEM as the frequency range increases and the requirement for FEM-based methods of at least

14 240 elements/wavelength is violated. In these cases, the internal DOFs in the unitcell must be increased and/or the use of an element with high order interpolation function will minimize the divergences. All numerical computations were made using a MATLAB code. 4. Experimental Validation of Metamaterial Rod Models with Spatial Periodic Distribution 245 A real elastic metamaterial waveguide was used to perform an experimental test. The elastic metamaterial waveguide was designed to be an assembly of parts including two resonator and a waveguide portion (Fig. 4a). It was originally developed to be used in another research project [19] as a metamaterial beam-like (flexural waves) and shaft-like (torsional waves) structures with spatial periodic distribution and local resonators (Fig. 4b). For this work it was (a) (b) Figure 4: Design of metamaterial waveguide: (a) waveguide assembly; (b) local resonator dimensions and physical aspect configured as a rod-like (longitudinal waves) structure, which maintains the spatial periodic distribution, but the local resonators becomes inactive since they are not sensitive to longitudinal movements. Then, the experimental evaluation presented here is relate only to Bragg-type band gap formation in a metamaterial rod. Therefore, the local resonator region was assumed as an empty cavity in the rod and this region was modeled as cross section area change. Fig. 5 shows the views for the original metamaterial unit-cell with local resonators for 14

15 beam/shaft and with the simplification to be used with the rod spatial periodic distribution. The metamaterial rod was built by bonding 10 equal parts of pris- Figure 5: Unit-cell model for the metamaterial: (a) with local resonator for beam/shaft structure; (b) with rod spatial periodic distribution. 260 matic rod (0.24 x 0.08 x 0.02 m) with male and female conic ends. Each part includes two local resonator separated by m and they were fabricated in a 3D printer. The final assembly of metamaterial rod includes also two extra rod parts at the ends without resonators. The actual metamaterial rod geometric parameters and material properties are summarized in Table 3. Table 3: Actual metamaterial rod geometric parameters and material properties. Geometry/Property Value Unit-cell length (L) 0.12 m Periodic part length (L p ) 2.41 m Total length (L t ) 2.88 m Cross section area (S = b h) x m 2 Cross section area at cavity (S r = b h c ) x m 2 Number of unit-cells (N c ) 20 Young s modulus* (E) x 10 9 Pa Mass density* (ρ) 1.3 x 10 3 kg/m 3 Structural damping (η) 0.02 *manufacturing nominal value. 15

16 For the manufacture of the metamaterial rod, some criteria were observed related with the material: a) must have a less important loss angle to reduce the influence of structural damping; b) must be easy to assemble; and c) must be as isotropic as possible. Regards to the manufacturing technology, it was required that the resolution of the printer (the thinnest feasible material layer) was enough to insert metal rods of small dimensions. Based on these requirements the metamaterial rod was manufactured with plastic material (Vero White Plus) in a 3D printer with UV curing technology. Measurement setup consists of hang the metamaterial rod with two rubber bands allowing it to move freely in the longitudinal direction. By using an impact force excitation applied in the right end of metamaterial rod, measurements of acceleration were taking in the left and right ends of metamaterial rod. Inertance point and transfer FRFs are measured with 10 averages, frequency band DC-10.0 khz with the frequency discretization of Hz. The measurement instruments used in the experimental setup are summarized in Table 4. Table 4: Measurement instruments. Instrument Manufacture and Model Sensitivity Measure Range Impulse Hammer PCB 86E mv/n N (peak) Accelerometer PCB 352C mv/g (±5%) 0-10 khz Data Acquisition Oros OR34, 4 ch Fig. 6 shows the experimental setup with the details of impact hammer and accelerometers positions. This figure also shows a typical measured inertance FRF (H 21 ) including the ordinary coherence function. It must be emphasized an interesting behavior of the measured inertance FRF, which presents an oscillatory behavior before the band gap indicating low damping, while after the band gap the curve becomes smooth indicating a high damping behavior. Fig. 7 shows the dispersion curves calculated with the numerical methods (WSEM, WFEM, WFEM-ANSYS, WFEM-ANSYS-3D) where it can be identified the Bragg limit (β = π/l = 26.18), and the band gap position and width for this 16

17 elastic metamaterial rod. Figure 6: Rod measurement setup and a typical measured inertance FRF H21 (blue line) and corresponding ordinary coherence function (green line) Wavenumber( ) [m ] Bragg Limit WSEM WFEM WFEM-ANSYS WFEM-ANSYS-3D ( ) ( ) Frequency [Hz] Figure 7: Dispersion curves calculated by WSEM, WFEM, WFEM-ANSYS and WFEMANSYS-3D. 290 In order to increase the flexibility of WFEM to solve complex engineering models, a hybrid approach using WFEM and a FEM commercial software is 17

18 proposed. It consists of coupling WFEM with the commercial finite element analysis software ANSYS (Mechanical APDL Release 14.5) to calculate elastic metamaterial rods. The metamaterial rod unit-cell is modeled with ANSYS using an appropriated element type from its element library. Then, using a MAT- LAB code, the dynamic stiffness matrix is calculated and periodicity conditions are applied to obtain the receptance FRFs. Two examples of metamaterial rod unit-cell modeling are performed: one called WFEM-ANSYS using a simple 1D element rod (LINK180), and other called WFEM-ANSYS-3D using a more complex 3D solid element (SOLID185). Fig. 8(a) shows the metamaterial rod unit-cell modeled with ANSYS 1D element rod with 2 nodes and 1 DOF/node. The rod unit-cell was discretized with 40 internal elements, which means 41 nodes and DOFs. Fig. 8(b) shows the metamaterial rod unit-cell modeled with ANSYS 3D solid element with 8 nodes and 3 DOFs/node. The rod unit-cell was discretized with 25 internal elements, which means 200 nodes and 600 DOFs. (a) (b) Figure 8: Metamaterial rod unit-cell modeled with ANSYS: (a) 1D rod element; and (b) 3D solid element. For the first evaluation the simulated receptance FRFs calculated by WSEM, WFEM, WFEM-ANSYS and WFEM-ANSYS-3D using material properties of Table 3 are compared with the experimental FRFs. Fig. 9 illustrates that the FRFs H 11 and H 21 calculated by numerical methods (WSEM, WFEM and WFEM-ANSYS, WFEM-ANSYS-3D) present good agreement among them, but there are some mismatch and different behaviors related with its corresponding 18

19 315 experimental FRFs. Before band gap positions, all numerical FRFs present the same oscillatory behaviors as its corresponding experimental FRFs. The amplitudes of simulated FRFs H 11 do not match very well with its equivalent experimental. However, the amplitudes of simulated FRFs H 21 show better approximation to the experimental. At the band gap positions, all simulated FRFs present similar width as the experimental, but they are shifted to the left of experimental band gap position. After band gap positions, all simulated FRFs return to the oscillatory behaviors, in total disagree with the smooth behavior of corresponding experimental FRFs. Receptance FRF H 11 [db re 1.0 m/n] WSEM -180 WFEM WFEM-ANSYS -190 WFEM-ANSYS-3D Experimental Frequency [Hz] Receptance H 21 [db re 1.0 m/n] WSEM WFEM WFEM-ANSYS WFEM-ANSYS-3D Experimental Frequency [Hz] (a) (b) Figure 9: Simulated FRFs calculated by WSEM, WFEM, WFEM-ANSYS, WFEM-ANSYS- 3D with material properties of Table 3 and experimental FRFs of: (a) point receptance, H 11 and (b) transfer receptance, H This oscillatory response (low damping) before band gap position followed by a smooth response (high damping) after band gap position is a characteristic behavior of the elastic metamaterial rod, which is not very well captured by the numerical models presented here. For the WFEM-ANSYS-3D model the receptance FRF H 21 shows another band gap around 9.0 khz, not included in the others numerical models. It comes from the greater number of wave-modes include in the 3D model as compared with the others 1D models, associated with the uncertainty of the values of material properties (Young modulus and mass density) specified by the plastic manufacturer, which is not guaranteed in 19

20 the addictive manufacturing process used to make the actual metamaterial rod. In order to reduce the mismatch between experimental and simulated FRFs a try-and-error model updating procedure was performed. By varying material property parameters E and ρ the shift between simulated and experimental band gap position was reduced until the band gaps are almost coincident. These results were founded using the combination of Young s modulus E = 3.2 x 10 9 Pa and mass density ρ = 1,180 kg/m 3. Fig. 10 shows the simulated FRFs (WSEM, WFEM, WFEM-ANSYS, WFEM-ANSYS-3D) obtained with updated parameters and the experimental FRFs (H 11 and H 21 ). Receptance FRF H 11 [db re 1.0 m/n] WSEM WFEM WFEM-ANSYS WFEM-ANSYS-3D Experimental Frequency [Hz] Receptance H 21 [db re 1.0 m/n] WSEM WFEM WFEM-ANSYS WFEM-ANSYS-3D Experimental Frequency [Hz] (a) (b) Figure 10: Simulated FRFs (SEM, WSEM, WFEM, WFEM-ANSYS, WFEM-ANSYS-3D) using updating material property parameters and experimental FRFs of: (a) point receptance, H 11 and (b) transfer receptance H Although the model updating process brings the simulated and experimental FRFs more close for the band gaps, still have different behaviors between experimental and simulated FRFs after the band gaps. A model updating varying the parameter structural damping η was performed without success. Considering that the behavior before and after band gap are strongly different, actually contradictory, an updating process for the whole frequency band of analysis will be not possible. Then, a selective model updating varying η was applied for two frequency band: DC-5.7 khz and khz. Using a try-and-error updating process the best curve fitting for the FRFs was obtained with the structural 20

21 350 damping η = to DC-5.7 khz and η = 0.05 to khz. Fig. 11 shows the comparison between simulated (WSEM, WFEM, WFEM-ANSYS, WFEM- ANSYS-3D) using updating structural damping and experimental results for the measured FRFs (H 1 1 and H 2 1). It can be seen that all numerical FRFs after the band gap change their behavior from oscillatory to smooth as the experimental ones, however still remains significant differences in amplitudes. Receptance FRF H 11 [db re 1.0 m/n] WSEM WFEM WFEM-ANSYS WFEM-ANSYS-3D Experimental Frequency [Hz] Receptance H 21 [db re 1.0 m/n] WSEM WFEM WFEM-ANSYS WFEM-ANSYS-3D Experimental Frequency [Hz] (a) (b) Figure 11: Simulated FRFs (SEM, WSEM, WFEM, WFEM-ANSYS, WFEM-ANSYS-3D) using structural damping and experimental FRFs of: (a) point receptance, H 11 and (b) transfer receptance H Conclusion Band gaps in elastic metamaterial rods with spatial periodic distribution and periodically attached M-DOFs local resonators was presented. A numerical methodology called wave finite element method (WFEM), based on a combination of FEM and Floquet-Bloch s theorem was proposed as an engineering tool to calculate elastic metamaterial rods. Simulated results calculated by WFEM ar presented, verified with the spectral element method (WSEM) solution where it was demonstrate its accuracy and efficiency of modeling elastic metamaterial rods. Three simulated examples with WFEM elastic metamaterial rod system consisting of a uniform rod with periodically attached S-DOF and 2-DOF s resonators were evaluated. For all simulated examples the comparative results 21

22 between WFEM and WSEM prove that WFEM can be used as a good approximated solution for metamaterial rod systems with M-DOFs local resonators. By using the commercial finite element software (ANSYS Mechanical APDL Release 14.5) it was demonstrate an important feature of WFEM, which is its ability to coupling with commercial FEM packages that allows to model complex geometry or nonuniform metamaterial unit-cell. This feature will facilitates the WFEM application to model complex metamaterial systems. However, to obtain good convergence WFEM requires as many elements in the unit-cell model as the frequency range increases. An interesting characteristic of elastic metamaterial rod with spatial periodic distribution observed in the experimental test results, is its behavior of low damping vibration response before band gap and the high damping vibration response after band gap. Finally, this paper presents a comparison between simulated FRFs calculated by WSEM, WFEM and WFEM/ANSYS methods and the experimental FRFs obtained from a real metamaterial rod system. These results shows that all numerical methods presented (WSEM,) and proposed (WFEM, WFEM/ANSYS) can obtain a good agreement with the experimental results and they can localize the position and width of band gap very close to the experimental. Acknowledgements 385 The research leading to this article has been funded by CBM-MA, FAPESP 2015/ , CNPq and CAPES. The support of DMC-FEM-UNICAMP and ENSIM-LAUM-Universite du Maine are also gratefully acknowledged. References 390 [1] D. Mead, Free wave propagation in periodically supported, infinite beams, Journal of Sound and Vibration 11 (2) (1970) doi: doi.org/ /s x(70) [2] D. Mead, A general theory of harmonic wave propagation in linear periodic systems with multiple coupling, Journal of Sound and Vibration 27 (2) 22

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