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International Journal of Applied Mechanics Vol. 7, No. 2 (2015) 1550018 (22 pages) c Imperial College Press DOI: 10.1142/S1758825115500180
Wave Finite Element Method Based on Reduced Model for One-Dimensional Periodic Structures C. W. Zhou∗ , J. P. Lain´e and M. N. Ichchou Laboratoire de Tribologie et Dynamique des Syst` emes ´ Ecole Centrale de Lyon - 36 Avenue Guy de Collongue 69134 Ecully, France ∗
[email protected]
A. M. Zine ´ Institut Camille Jordan, Ecole Centrale de Lyon - 36 Avenue Guy de Collongue, 69134 Ecully, France Received 16 August 2014 Revised 6 February 2015 Accepted 6 February 2015 Published 14 April 2015 In this paper, an efficient numerical approach is proposed to study free and forced vibration of complex one-dimensional (1D) periodic structures. The proposed method combines the advantages of component mode synthesis (CMS) and wave finite element method. It exploits the periodicity of the structure since only one unit cell is modelled. The model reduction based on CMS improves the computational efficiency of unit cell dynamics, avoiding ill-conditioning issues. The selection of reduced modal basis can reveal the influence of local dynamics on global behavior. The effectiveness of the proposed approach is illustrated via numerical examples. Keywords: Wave propagation; wave finite element method; model reduction; periodic structure; stop band.
1. Introduction Spatially periodic structures consisting of identical unit cells possess a variety of interesting dynamic behaviors. Many studies have been conducted concerning periodic structures [Mead, 1996]. A lot of attention has been paid on the existence of stop bands, the frequency range where the propagation of vibration are forbidden [Wang et al., 2004; Li et al., 2013]. At low frequencies, a homogenized model of periodic structure can be found with equivalent dynamic behavior. For higher frequencies, a variety of methods have been developed based on Floquet–Bloch theorem [Floquet, 1883; Bloch, 1929], which ∗ Corresponding
author. 1550018-1
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allows conversion of the study of whole structure into the study of a single unit cell. Among these methods are receptance methods [Mead, 1984], transfer matrix method (TMM) [Lin and McDaniel, 1969] and space-harmonic method [Lee and Kim, 2002]. For complex structure, finite element method (FEM) has been integrated to model the unit cell. Such idea was developed and completed in various studies [Mace et al., 2005; Kharrat et al., 2014] and the method has been named wave finite element method (WFEM). In WFEM, conventional FE software packages can be used for modelling. The dynamic stiffness matrix of the unit cell is built explicitly. This means that structures with complex geometries or material distributions can be analyzed with relative ease. However, for waveguides with complex cross-section, even a single unit cell may contain large number of degrees of freedom (DOFs), which makes the analysis time-consuming red [Hussein et al., 2014]. Therefore, several reduction techniques of WFEM have been proposed. Droz et al. [2014] proposed a reduction formulation to determine the propagating wave in one-dimensional (1D) refined model of a laminated composite beam, solving the eigen problem only at cut-on frequencies, then deduce the solutions for the whole frequency range. Mencik and Ichchou, [2008] developed a substructuring technique in multi-layered systems, as well as a model reduction method to compute the forced response of the waveguide [Mencik, 2012]. Duhamel et al. [2006] gave the formulation for a reduced basis for the displacement of boundary nodes. But few studies have been carried out for the model reduction of internal nodes in WFEM. However, for periodic structures, the internal DOFs could be numerous, which leads to excessive computational time, up to several hours even several days. In this paper, the mode-based reduction — component mode synthesis (CMS) [Thorby, 2008] is applied to project internal physical DOFs onto a fixed boundary mode basis. Then the dynamic stiffness of the reduced model is used in WFEM to study the wave propagation characteristic. The proposed method combines the advantages of CMS and WFEM. First, it speeds up the calculation of unit cell dynamics since only a small set of internal modal DOFs is retained. It makes the dynamic condensation of internal DOFs much easier and avoids the ill-conditioning issues which may result from the inversion of the stiffness matrix. Second, by selecting the local basis of the unit cell, the proposed method can reveal the influence of local modes on the global behavior. The approach is referred henceforth as “CWFEM” for condensed WFEM method. The proposed method is a combined wave/mode based approach. As it is wellknown that the vibration of periodic structures can be viewed either in terms of stationary modes or in terms of elastic wave motion, known as “wave-mode duality”. Here in this paper, both descriptions are involved. Modal description is employed on the mesoscopic unit cell level, then the macroscopic structure is considered as a waveguide and the vibration is described with elastic waves. The proposed combination of WFEM with CMS is assured by the wave-mode duality. It allows, when passing from a unit cell to the whole structure, to alternate from the modal description
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to wave description. However, the equivalence between these two descriptions still needs clarification [Lyon and Dejong, 1995; Langley, 1997] despite many previous discussions. The present work shows that by using a reduced modal basis, the proposed CWFEM is able to obtain precisely the wave propagation characteristics. The conclusion can be taken as little contribution in understanding the relation between two descriptions. The paper is organized as follows: In Sec. 2, the proposed “CWFEM” predicts the dispersion relation, stop bands as well as forced response under excitation which is explained in detail. Subsequently, in Sec. 3, the effectiveness of the proposed method is illustrated via both a binary circular beam and a thin-walled beam including warping effect using finely meshed solid elements. Finally, the concluding remarks and perspectives are presented in Sec. 4. 2. The Formulation of Proposed “CWFEM” on One-Dimensional Periodic Structures The 1D periodic structure can be obtained by repeating the unit cell in the propagation direction. Based on Floquet–Bloch theorem, the study of the structure can be converted into the study of a unit cell. However the choice of a unit cell can be various, such as two choices of cell in Fig. 1. It is advantageous to use the element with the minimum number of coupling co-ordinates to decrease the computational size [Mead, 1975]. So it is more convenient to choose Fig. 1(b1) as the unit cell which contains less boundary DOFs. 2.1. Model reduction on the unit cell CMS is a mode-based method for efficient dynamic simulation of complex systems such as periodic structures. One of the most common method is the fixed interface
Fig. 1. (a1) 1st choice of unit cell, (a2) periodic structure considering a1 as unit cell, (b1) 2nd choice of unit cell and (b2) same periodic structure as a2 considering b1 as unit cell. 1550018-3
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method, known as the Craig–Bampton method. Here this method is applied because it is more straightforward with the boundary DOFs expressed using physical coordinates. The equation of motion of the unit cell can be written in the following manner, with the mass and stiffness matrices M and K obtained using conventional FE packages. ˜ MLL ˜ MRL ˜ IL M
˜ LR M ˜ RR M
˜ LI q¨L K ˜ LL M ˜ RI q¨R + K ˜ RL M
˜ LR K ˜ RR K
˜ LI qL F˜L K ˜ RI K qR = F˜R
˜ IR M
˜ II M
˜ IR K
˜ II K
˜ IL K
q¨I
qI
(2.1)
0
where “∼” represents the matrix of the full model before reduction. Co-ordinates qR , qL , qI represent the physical DOFs of the left boundary, right boundary and internal nodes, while their sizes are n, n, nI respectively. The physical DOFs qI are then reformulated to a reduced modal basis of modal DOFs, with generalized coordinate PC , the size of which is nc . Then the Craig–Bampton hybrid co-ordinates are related to the physical co-ordinates using matrix B:
T T
(qL )T (qR )T (qI )T = B (qL )T (qR )T (PC )T (2.2) with
In B = 0
In
ΨL
ΨR
0
0
0.
(2.3)
ΨC
[ΨL ΨR ] is the static boundary mode with −1 ˜ ˜ II KIL , ΨL = −K
−1 ˜ ˜ II KIR . ΨR = −K
(2.4)
Fixed boundary modes ΨI are calculated with qL = qR = 0 ˜ II ]ΨI = 0. ˜ II − ω02 M [K
(2.5)
Matrix ΨC is a reduced basis in ΨI with nI rows and nc columns. The selection of ΨC and a converge study will be discussed in the numerical example later. Then the equation of motion of the reduced model can be written into the following form: ∗ ∗ ∗ ∗ ∗ ∗ MLC KLC q¨L KLL KLR qL FL MLL MLR ∗ ∗ ∗ ∗ ∗ (2.6) MRL MRR MRR q¨R + KRL KRR KRR qR = FR . ∗ ∗ ∗ ∗ ∗ ∗ ¨ PC M M M K K K PC 0 CL
CR
CC
CL
CR
CC
The new mass matrix as well as the stiffness matrix can be written in the following manner: ˜ ∗ ∗ ∗ ˜ LR M ˜ LI MLC MLL M MLL MLR ∗ ∗ ∗ = BT ˜ ˜ RR M ˜ RI (2.7) MRR MRL M B. MRL MRR ∗ ∗ ∗ ˜ IL M ˜ IR M ˜ II MCL MCR MCC M 1550018-4
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The generalized force remains the same value as before: ˜ ˜ FL FL FL ˜ T˜ FR = B FR = FR . 0
0
(2.8)
0
Assuming harmonic response, the equation can be written with the dynamic stiffness matrix D∗ = K ∗ − ω 2 M ∗ . ∗ ∗ ∗ DLL DLR qL DLC FL ∗ ∗ ∗ (2.9) DRL DRR DRC qR = FR . ∗ DCL
∗ DCR
∗ DCC
PC
0
From the third line of Eq. (2.9), internal DOFs can be removed using dynamic condensation. Equation (2.9) becomes DLL DLR qL FL = , (2.10) DRL DRR qR FR where ∗ ∗ ∗ −1 ∗ DLL = DLL − DLC (DCC ) DCL , ∗ ∗ ∗ −1 ∗ − DLC (DCC ) DCR , DLR = DLR ∗ ∗ ∗ −1 ∗ − DRC (DCC ) DCL , DRL = DRL
(2.11)
∗ ∗ ∗ −1 ∗ − DRC (DCC ) DCR . DRR = DRR
Vectors in ΨC are normalized with respect to the modal mass matrix, i.e., ˜ II ][ΨC ] = Inc . The modal stiffness matrix is [ΨC ]T [K ˜ II ][ΨC ] = [Ω2 ]. So we [ΨC ]T [M ∗ ∗ 2 ∗ 2 2 have DCC = KCC − ω MCC = Ω − ω Inc , which is a diagonal matrix and of smaller ∗ ∗ ∗ ∗ = KII − ω 2 MII . DCC is much easier to be inverted. The poor condisize than DII ∗ may be avoided. tioned due to the inversion of DII 2.2. Application of WFEM on reduced model of the unit cell After dynamic condensation of the internal modal DOFs, the equation of motion contains only boundary nodes as shown in Eq. (2.10). For unit cell k, Eq. (2.10) can be reformed into the following form: (k)
uR (k) uL
(k) (k) ((qL )T (−FL )T )T
(k)
= SuL ,
(k) uR
(k) (k) ((qR )T (FR )T )T
(2.12)
where = and = represent the left and right state vectors for the unit cell k. S is a symplectic matrix [Zhong and Williams, 1995] with following expression: −1 −1 −DLR DLL −DLR . (2.13) S= −1 −1 −DRL + DRR DLR DLL −DRR DLR 1550018-5
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Free wave propagation characteristics are represented by wavenumbers and wave basis, which are associated to the eigen values and eigen vectors of the following eigen-problem: SΦi = λi Φi ,
|S − λi I2n | = 0,
(2.14)
where I2n represents the identity matrix of size 2n. However, for large numbers of DOFs, typically for 2D cross-sections, direction application of numerical solvers can lead to difficulties because S may be poor conditioned [Mace et al., 2005]. So the eigenvalue problem can be reformulated as follows: [N ]Φ = λ[L]Φ , where Φ = and
[N ] =
0
In
DRL
DRR
qL qR
(2.16)
,
(2.15)
[L] =
In
0
−DLL
−DLR
For the state vector Φ, it is related using matrix L. qL q Φ= = L L = LΦ . q −FL R
.
(2.17)
(2.18)
Other formulations were also suggested, such as the eigenvalue problem in terms of (λ + 1/λ) proposed by Zhong and Williams [1995]. The eigenvalues λi and wavenumbers ki are linked through the relation λi = e−jki dx × dx denotes the length of the unit cell in propagation direction x-axis and j 2 = −1. The direction of the phase velocity of the corresponding waves can be distinguished in the following manner: If |λi | < 1, then the phase propagates in the positive direction. If |λi | > 1, the phase propagates in the negative direction. For the propagative waves in the passing band with |λi | = 1, the propagation direction can be identified using the sign of the (ki ), where (ki ) represents the real part of the complex wavenumber ki . (ki ) > 0 indicates that the phase propagates in the positive x-direction, (ki ) < 0 the phase propagates in the negative x-direction. The matrix Φ of the eigenvectors can be written as follows: + Φq Φ− q , (2.19) Φ= Φ+ Φ− F F where the subscripts q and F refer to the components which corresponds to the + T T T displacements and the forces, respectively; ((Φ+ q ) (ΦF ) ) stands for the wave basis − T T T vectors which propagate in the positive direction, while ((Φ− q ) (ΦF ) ) stands for the wave basis vectors which propagate in the negative direction. Finally, state 1550018-6
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vector uL
(k)
and uR
(k)
uL
of any cell k can be expressed using eigenvectors {Φi }i=1,...,2n : = ΦQ(k) ,
(k)
uR
= ΦQ(k+1)
∀ k ∈ {1, . . . , N }.
(2.20)
The analysis of the dynamic response consists of evaluating a set of amplitudes associated with positive and negative going modes. Q+(k) (k) . (2.21) Q = Q−(k) 2.3. The forced response of the structure by CWFEM The Forced-WFEM formulation are widely employed to study the stationary response of continuous structures [Mencik and Ichchou, 2007; Waki et al., 2009; Renno and Mace, 2010]. For finite periodic structures, the response can be studied by Forced-CWFEM formulation using the wave basis Φ. According to the coupling relations between two consecutive cells k and k − 1 (k) (k−1) (k) (k−1) and −F L = FR , the following relation can be (k ∈ {2, . . . , N }), qL = qR found: (k)
(k−1)
∀ k ∈ {2, . . . , N },
uL = uR
(2.22)
which conducts to (k)
(k−1)
uL = SuL
∀ k ∈ {2, . . . , N }.
(2.23)
∀ k ∈ {1, . . . , N }.
(2.24)
Equation (2.23) leads to (k)
(1)
uL = S k−1 uL In addition to the relation in cell N :
(N )
uR
(N )
= SuL ,
(2.25)
(1)
(2.26)
we have (N )
uR
= S N uL ,
where S 0 = I2n . Equations (2.24) and (2.26) are projected on the wave basis {Φi }i considering Eq. (2.20). Since it has been assumed that matrix eigenvectors {Φi }i are linearly independent, so Φ is invertible, which leads to: Q(k) = Φ−1 S k−1 ΦQ(1)
∀ k ∈ {1, . . . , N + 1}.
(2.27)
Since λi is the eigenvalue of the matrix S, with the vectors in Φ its corresponding eigenvectors, as shown in Eq. (2.14), Eq. (2.27) can then be written in k−1 Λ 0 Q(1) ∀ k ∈ {1, . . . , N + 1}, (2.28) Q(k) = 0 Λ−1
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where Λ stands for the (n × n) diagonal eigenvalue matrix for wave modes propagating in x positive direction, expressed by Eq. (2.29). λ1 0 . . . 0 0 λ2 . . . 0 Λ= . . (2.29) .. . . .. .. . . . 0 0 . . . λn For the classical Neumann and Dirichlet boundary conditions can be expressed as follows: [0|I]u = F [I|0]u = q
(Neumann),
(2.30a)
(Dirichlet).
(2.30b)
If Neumann boundary condition is set at cell 1 and Dirichlet boundary is set at cell N , then the boundary conditions can be rewritten via the projection of the state vector u onto the wave basis: +(1) −(1) + Φ− = F0 , Φ+ FQ FQ +(N+1) −(N+1) Φ+ + Φ− = qN+1 . q Q q Q
(2.31a) (2.31b)
Using Eq. (2.28), the above equations can be simplified as: +(1) −(1) Φ+ + Φ− = F0 , FQ FQ N +(1) −N −(1) + Φ− Q = qN+1 , Φ+ q Λ Q q Λ
(2.32a) (2.32b)
then the amplitude of the positive going wave modes of the first cell can be deduced in the following manner: + −1 Q+(1) ΦF F0 Φ− F = . (2.33) N −N Q−(1) Φ+ Φ− qN+1 q Λ q Λ Once the amplitude of the wave modes Q(1) is obtained, the wave amplitude as well as physical response at the boundaries of all unit cells can be obtained using Eqs. (2.28) and (2.20). As for the internal nodes of unit cell k, the displacement can be deduced based on Eqs. (2.2) and (2.3): (k)
qI
(k)
= ΨL qL
(k)
+ ΨR qR
(k)
+ ΨC PC .
(2.34)
The constraint modes of the unit cell [ΨL ΨR ] and truncated fixed boundary mode ΨC are shown in Eqs. (2.4) and (2.5). As for modal co-ordinates PC , it can be deduced from Eq. (2.9) that ∗ −1 ∗ ∗ PC = −(DCC ) (DCL qL + DCR qR ).
(2.35)
In this manner, the forced response of the all the nodes (internal and boundary) in finite waveguide can be determined. However, attention should be paid when inverting the matrix in Eq. (2.33) since it may be ill-conditioned. Alternative formulations to avoid numerical issue can be found in the reference [Waki et al., 2009]. 1550018-8
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2.4. Stop band of periodic structures The waves which occur in periodic media, known as Bloch waves [Kittel, 2004], have very unusual dispersion and attenuation characteristics compared to those in continuum media. The periodic waveguide, even a nondissipative one, strongly attenuates waves on bands of the frequency spectrum known as stop bands. Noise control and vibration absorption techniques can be developed based on the stop bands of the periodic structures. As a result, the optimum design and parameter study of the stop band width have attracted a lot of interest among researchers [Yu et al., 2009; Olhoff et al., 2012; Xiao et al., 2013]. According to Mead, wave propagation can occur when ki is a real number [Mead, 1975], since λi + 1/λi = e−jki dx + ejki dx = 2 cos(ki dx ), so in the pass bands we have: −2 ≤ λi + 1/λi ≤ 2.
(2.36)
Therefore the pass bands are bounded by the frequencies at which λi + 1/λi = ±2, λi = ±1. So the stop bands can be calculated easily using the dynamic stiffness matrix. Since the left state vector and the right state vector are related by λ, we have (k) (k) qL qR =λ , (2.37) (k) (k) FR −FL then Eq. (2.6) becomes: ∗ DLL ∗ DRL
∗ DLR ∗ DRR
∗ . DRC λqL = −λFL .
∗ DCL
∗ DCR
∗ DCC
∗ DLC
qL
PC
FL
0
With λ = 1, the equation is reformulated to: ∗ ∗ ∗ ∗ ∗ ∗ qL + DLR + DRL + DRR DLC + DRC DLL ∗ ∗ DCR + DCL
∗ DCC
(2.38)
PC
=
0 0
.
(2.39)
To obtain the nontrivial solution, the determinant of the matrix should be zero. ∗ ∗ ∗ Note that Dij = Kij − ω 2 Mij , so the linear eigen problem can be solved and the eigenvalues ω 2 can be calculated, which corresponds to the bounding frequencies of the stop bands. Along with the bounding frequencies calculated by λ = −1, all the stop bands of the structure can be found using only the dynamic stiffness matrix of the reduced model.
3. Numerical Examples and Discussions The proposed CWFEM is applied on three kinds of binary periodic beam in this section. The binary beam consists of a repetition of section A of length l1 and section B of length l2 , as shown in Fig. 2. A symmetric unit cell is used in CWFEM. 1550018-9
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Fig. 2.
Binary periodic beam and its symmetric unit cell.
Table 1. Materials Epoxy Aluminium Steel
Material proprieties.
Young’s Modulus Shear modulus E (Gpa) G (Gpa) 4.35 77.56 210.6
1.59 28.87 81.0
Density ρ (kg/m3 ) 1180 2730 7780
Sections A and B are of different materials, the materials used in the this section can be found in Table 1. First, the longitudinal waves and flexural waves in a circular beam are studied in Subsecs. 3.1 and 3.2. Subsequently, flexural-torsional coupled waves in a nonsymmetrical thin-walled beam is analyzed in Subsec. 3.3, where the beam is modelled using solid elements which take warping effect in consideration. 3.1. Longitudinal waves in binary periodic beam First, CWFEM is used to study the longitudinal waves in binary periodic beam with Epoxy in Sec. A and Aluminium in Sec. B. Both sections are circular with a radius of 6.44 cm, and the lengths l1 and l2 are both 1 m. The unit cell is divided into 100 elements, so that each wavelength contains at least 10 elements in the frequency range up to 3 kHz. The results obtained by CWFEM is compared with the analytical solution of the wavenumber k given by Tian [Tian et al., 2011]: Z1 Z2 + (3.1) sin(ωt1 ) sin(ωt2 ), cos(ak) = cos(ωt1 ) cos(ωt2 ) − 2Z2 2Z1 where Z = ρc is the characteristic acoustic impedance of a section, c = E/ρ is the wave velocity in the section, and ti = li /c is the time for the wave to pass through the section. 1550018-10
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The first four stop bands: Relative error is compared with f0 . fc (Hz) by CWFEM With n = fmode /fmax
n=1
n = 1.5
n=2
n = 2.5
n=3
372.4 906.9 1087.6 1762.9 1963.1 2439.4 2732.2 2909.3
372.4 904.9 1087.1 1762.9 1963.1 2410.9 2732.2 2895.5
372.4 904.4 1086.7 1762.9 1963.1 2405.9 2732.2 2882.3
372.4 904.3 1086.4 1762.9 1963.1 2404.7 2732.2 2877.0
372.4 904.3 1086.3 1762.9 1963.1 2404.2 2732.2 2875.1
2.9%
1.2%
0.5%
0.24%
0.1%
f0 (Hz) by classical WFEM 372.4 904.2 1086.2 1762.9 1963.1 2403.1 2732.2 2872.6
fa (Hz) analytical result 372.4 903.8 1085.7 1761.6 1961.2 2401.4 2730.3 2868.4
3.1.1. The selection of the modes and the convergence The reduced set of fixed boundary modes Ψc under fmode are conserved in the reduced model. The convergence study on the reduced model is carried out here. The frequency ranges of the stop bands are obtained with different reduced basis, using the formulation in 2.4. fc , the frequencies of the stop bands calculated by CWFEM, are compared with f0 , the frequencies obtained by classical WFEM. The relative errors listed in Table 2 0| . The analytical result fa is given as well, with the sum of the relative equal to |fcf−f 0 errors between the f0 and fa equals to 0.5%. It can be seen from Table 2 that the frequencies fc converge rapidly with n = fmode /fmax × fmax is the maximum frequency studied, which equals to 3000 Hz in this case. Using a reduced basis with n = 3, a total relative error around 1% is obtained. 3.1.2. Dispersion relation of the longitudinal waves The real part of wavenumber k, (k) represents the phase shift per unit length. While the imaginary part of k, (k) represents the attenuation per unit length.The propagating direction of the waves can be deduced using the sign of (k). The dispersion relation is illustrated in Fig. 3, only the positive-going waves with (k) > 0 are presented because the wavenumbers of the positive and negative-going waves are symmetric with respect to the x-axis. Several noteworthy characteristics of dispersion relation can be observed: • At pass band frequencies, the wavenumber is real with (k) = 0. • At stop band frequencies, (k) = 0, which is associated with the exponential attenuation of the wave. • At stop band frequencies, (k) remains constant in the whole band, either (k) = 0 or (k) = ±π/dx . For (k) = ±π/dx , the stop band are bounded with λ = −1, 1550018-11
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Fig. 3. Dispersion relation for the longitudinal waves by different methods: Analytical (-), Classical WFEM (o), CWFEM with fmode = 3fmax (∗).
and associated with a single wavelength. As to (k) = 0, the stop band frequencies are bounded with λ = 1 and are associated with nonoscillating and exponential attenuated waves. • (k) is limited in the first Brillouin zone [Brillouin, 1953], which is between [−π/dx , π/dx ]. As shown in Fig. 3, there exists four stop bands between 0 Hz and 3 kHz, the same as the ones given in Table 2. The dispersion relation issued from CWFEM is compared to those calculated with the classical WFEM and analytical method. With a reduced modal basis with fmode ≥ 3fmax , CWFEM is able to predict the dispersion relation with high precision. 3.1.3. Forced response of the beam In the given formulation of Forced-CWFEM in 2.3, the length of the waveguide is no longer infinite and should be specified as well as the boundary conditions. The beam consisting of 10 identical unit cells is clamped at one end and is subjected to harmonic axial loading on the other end. The frequency response at the point of excitation is shown in Fig. 4 and a zoom between 2200 Hz and 3000 Hz is given in Fig. 5. It can be seen that the response via the Forced-CWFEM corresponds quite well to that obtained using FEM. The size of the FE model is 1010, while that of the CWFEM is 12 which leads to a significant decrease of computation time. The peaks in the forced response correspond to the resonances at natural frequencies. The frequency ranges, where no resonance occurs are identical with the stop bands determined in the dispersion relation. Here if boundary condition is changed from clamped to free boundary, same frequency ranges without resonance are identified. The stop bands identified from the forced response function of a finite waveguides should be independent of the unit cells number and boundary condition. 1550018-12
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Fig. 4. The response at excitation point by FEM (-), Forced-WFEM (o), Forced-CWFEM with fmode = 3 fmax (∗).
Magnitude of displacement(dB)
−150 −200 −250 −300 −350 −400 2200
2300
2400
2500
2600
2700
2800
2900
3000
Frequency (Hz)
Fig. 5. A magnified view of the response at excitation point by FEM (-), Forced-WFEM (o), Forced-CWFEM with fmode = 3 fmax (∗).
3.2. Flexural waves in binary periodic beam The study of bending waves in binary periodic beam is carried out in this subsection. The materials and the geometry parameters remain the same as in the longitudinal case except that the radius of the two beams is set to 2.5 cm. The model of Euler– Bernoulli beam is used since the length of each beam section is much smaller than the height of each section and that the effects of shear and rotational inertia can be ignored. 3.2.1. Dispersion relation of the bending waves The dispersion relation of the positive-going bending waves calculated with different methods are given in Fig. 6. The characteristics of dispersion relation mentioned in the longitudinal waves are also observed in the bending ones. In addition, an evanescent wave at all frequency range with (k) = 0 and (k) = 0 is noted in Fig. 6, named “rapidly attenuating waves”. The CWFEM with fmode = 3 fmax and 1550018-13
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Fig. 6. Dispersion relation for the bending waves by different methods: TMM (-), Classical WFEM (o), CWFEM with fmode = 3 fmax (∗), CWFEM with fmode = 80 fmax ().
fmode = 80 fmax are applied to calculate the dispersion relation. The results are compared to the ones obtained by TMM [Lin and McDaniel, 1969] and classical WFEM. The CWFEM with fmode = 3 fmax is able to predict all the wavenumbers except the one of rapidly attenuating wave. More internal modes until fmode = 80 fmax are needed to determine the rapidly attenuating waves. In the frequency domain studied, the unit cell is divided into 80 elements of 160 DOFs. It contains at least 25 elements each wavelength. In the reduced model by CWFEM with fmode = 3 fmax , the number of DOFs is decreased from 160 to 16. For reduced model with fmode = 80 fmax, the number of DOFs is less than 50. The size of the reduced model is not more than one third of the full model.
3.2.2. Forced response of the beam The CWFEM method is applied to calculate the forced response of a binary periodic beam with 10 periods. A harmonic force perpendicular to the beam with an amplitude of 100 N is applied at the left extremity of the 1st cell, and the beam is clamped at the right extremity of the last cell. The response at the right extremity of the 3rd cell is illustrated in Fig. 7. The results calculated by FEM and ForcedWFEM are employed as references. Compared to the forced response in precedent example, two characteristics should be noted: • The result by CWFEM in Fig. 7 is applied with fmode = 3 fmax. The reduced model in CWFEM which fails to determine the wavenumbers of the rapidly attenuating waves is able to obtain the forced response of the beam. An explanation is proposed: Since the attenuating waves decay rapidly with distance, they have negligible contribution to the forced response of the structure. 1550018-14
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Fig. 7. The response at right extremity of the 3rd cell by FEM (-), Forced-WFEM (o), ForcedCWFEM with fmode = 3 fmax (∗).
• Contrary to the longitudinal case, a peak of resonance is observed in the second stop band around 61 Hz. It indicates that the evanescent waves in stop band can generate resonance phenomenon. The phenomenon is studied in the next subsection. 3.2.3. Resonance in stop band The forced response corresponding to the resonance in the stop band is given in Fig. 8, which coincides with a natural mode shape at 61 Hz under the same boundary condition. For free wave propagation to x positive direction at 61 Hz, two evanescent waves were identified in the dispersion curve in Fig. 6: A rapidly attenuated wave (referred as wave I) with purely imaginary wavenumber equalling to −3.6i, a slowly attenuated wave (wave II) with purely imaginary wavenumber equalling to −0.4i. For wave I, the wave amplitude drops to exp(−(−3.6i) × i × 2) = 0.0007 when the wave
Fig. 8. The mode shape with natural frequency at 61 Hz by FEM (-). The response of the beam under harmonic excitation at 61 Hz by forced-CWFEM (∗). The response of the unit cell boundaries by forced-WFEM (•). 1550018-15
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goes through a unit cell. 2 m is the unit cell length. For wave II, the wave amplitude decrease exp(−(−0.4i) × i × 2) = 0.45 when the wave goes through a unit cell. The forced response amplitudes under excitation of 61 Hz at the unit cell boundaries are indicated in Fig. 8. It can be seen that the amplitude drops (1 − 0.26) = 74% from the 1st unit cell to the 2nd unit cell. Starting from the 2nd unit cell, the ratio between the displacements at left boundary of two adjacent unit cells is always 0.45. It indicates that the contribution of the wave I (rapidly attenuated wave) is only limited in the first unit cell. It can be seen that the displacement of the beam is located near the excitation point at the left extremity of the beam. The resonance at 61 Hz can be interpreted as follows: The incident waves due to the excitation at left extremity are reflected at the interface of materials A and B. The resonance then occurs since the reflected waves are in phase with the incident ones. The incident waves are totally reflected at discontinuities of the materials, so they cannot reach the right extremity of the 10-period beam. If the boundary conditions at right extremity are changed from clamped to simply supported, the same mode shape is obtained. Or for a longer beam with more unit cells, the response remains the same for the first 10 unit cells. Many ongoing researches focus only on the stop band of free propagation in the design of periodic structure as vibration isolator. However, the existence of the resonance in stop band indicates that stop band might not always equivalent to low response level. So attention needs to be paid to these resonances. Especially when the excitation is limited to a small band of frequency. If the excitation is distributed in a large band of frequency, the resonance peaks at few frequencies in stop band should not increase significantly the response level. 3.3. Binary periodic and nonsymmetrical thin-walled beam including warping effect Thin-walled beams, such as angles and channel beams are basic structural elements in mechanical, aeronautical and civil engineering. It is demonstrated that the warping effect should be included when it comes to modelling the open section beams. Yu et al. [2009] have studied a binary periodic and asymmetrical thin-walled beam, the model of Bernoulli–Euler beam including warping effect is used with TMM. Their results are used as reference in this paper. The same structure is modelled using finely meshed solid elements. The element Solid185 in ANSYS is used in this study to obtain the dynamic matrix of the unit cell. The beam is binary periodic as shown in Fig. 2 with l1 = l2 = 150 mm, material A is epoxy and material B is steel. The cross-section of the beam is as shown in Fig. 9(a), with geometrical parameters: b1 = 15.5 mm, b2 = 9.5 mm, t1 = 1 mm, h = 16 mm. The unit cell used in CWFEM is shown in Fig. 9(b). The cross-section is in x − y plan, and waves propagate in z-direction. The cross-section is divided into 15 elements which contains 96 DOFs. The unit cell is divided into 90 elements in the 1550018-16
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(a)
(b)
Fig. 9. The thin-walled beam modelled with Solid element. (a) Schema of the cross-section and (b) the unit cell used in CWFEM.
z-direction. The size of the whole unit cell is of 8736 DOFs. This mesh assures the convergence of the natural frequencies below 1 kHz. 3.3.1. Dispersion relation and gain of computation time To study the wave propagation characteristics, the fixed boundary modes under 3 kHz are retained in CWFEM. The result of which is compared with the one obtained by WFEM, it can be seen that CWFEM predicted the same wavenumbers as WFEM. The internal modes with fixed boundary under 3 kHz consist the first 15 modes. So the size of the model is reduced from 8736 DOFs to 207 DOFs, with 15 internal DOFs and 192 boundary DOFs. And computation time decreases from more than 15 hours to less than 7 min, as shown in Table 3. For each frequency, to apply the
−1
Real part of K(m )
10
8
6
4
2
0 0
200
400 600 Frequency
800
1000
Fig. 10. Dispersion relation by different methods: CWFEM with fmode = 3 fmax (-), WFEM on the full model (). 1550018-17
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Comparsion of CWFEM and WFEM.
Classical WFEM
Proposed CWFEM
Computation time
Model size
Dynamic condensation: 15 hrs Linear eigen problem: 55 secs Total: > 15 hrs
8736 DOFs
Modal analysis: 356 secs Dynamic condensation: 1 sec Linear eigen problem: 55 secs Total: < 7 mins
Gain
207 DOFs
99%
97.6%
f=538Hz, Real part k=2 m −1
f=85Hz, Real part k=5 m
−1
−4
x 10 2 z (m)
z (m)
0.01
0.005
0
−2 0.02
0 0.02
0.04
0.01 0.02
0.01 y (m)
0.01 0 0
x (m)
0.02
0 0 y (m) −0.01 x (m) −0.02
(a)
(b)
Fig. 11. Wave shapes (Blue), the undeformed cross-section (Red). (a) Longitudinal wave and (b) Flexural–torsional coupled wave.
∗ dynamic condensation, instead of inverting matrix DII , which is not diagonal of ∗ size 8544 × 8544, a diagonal matrix DCC of size 15 × 15 is inverted. CWFEM allows ∗ gaining computation time and avoiding the numerical issue due to inverting DII . Four kinds of wave are identified at low frequencies. The one with the smallest wavenumber corresponds to the longitudinal waves since it is the fastest wave propagating in the structure. Another three waves are all the flexural–torsional coupled waves. The two kinds of wave shapes are given in Fig. 11.
3.3.2. The stop bands of the binary thin-walled beam The dispersion relation predicted by CWFEM is compared with the work of Yu et al. [2009], where only flexural–torsion coupled waves are studied. It can be seen that a good correlation is observed between the two results. However, a considerable discrepancy is observed around 500 Hz, which changes the frequency range of a stop band. The forced response function of a 8-period beam is calculated by FEM in reference [Yu et al., 2009]. It predicts also the frequency range of stop bands where a sharp drop in frequency response occurs. 1550018-18
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Real part of K(m −1)
10
8
6
4
2
0
0
200
400
600
800
1000
Frequency
Fig. 12. Dispersion relation by different methods: TMM with warping effect [Yu et al., 2009] (-), CWFEM with fmode = 3 fmax ().
Table 4. The frequency (Hz) range of stop bands for binary periodic thin-walled beam. Stop bands
Reference (Analytical)
Reference (FEM)
CWFEM
1 2 3
(156, 163) (501, 598) (694, 853)
— (460, 590) (660, 860)
— (460, 592) (680, 846)
It can be seen from Table 4 that almost the same stop bands are identified by FEM and CWFEM, while TMM has a discrepancy around 500 Hz with the two other methods. The error in the analytical model may arise because higher order inertia terms are ignored in Euler–Bernoulli beam model.
4. Conclusions This paper provides an efficient numerical approach to study the free and forced vibration of complex 1D periodic structures. The method proves that with a reduced modal basis, precise wave propagation characteristics can be obtained. The effectiveness of the method is illustrated using three numerical examples. Main conclusions of this work can be drawn as follows: (i) The formulation of the proposed CWFEM is developed and applied in numerical examples. CWFE is able to predict precisely the dispersion relation and forced response compared to classical WFEM and analytical methods. (ii) To ensure the convergence of CWFEM, the internal modes with natural frequency lower than three times the maximum investigated frequency should be retained in the modal basis. 1550018-19
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(iii) CWFEM can speed up the computation of unit cell dynamics (from 15 h to 7 min in the 3rd example). It can also avoid the poor conditioned issues which may occur due to the dynamic condensation. (iii) The frequency ranges without resonance in forced response function of finite waveguides are independent of boundary conditions and the number of unit cells. And they correspond to the stop bands of a infinite waveguides with the same unit cell. However, resonance can exist in the stop bands, while the displacements of the waveguides are located near the boundaries of the systems. The decrease of the computational time due to the model reduction would be very desirable for the optimisation of complex waveguides. Further investigation will address the proposed CWFEM to optimisation of 1D periodic structures or to extend the proposed method to 2D periodic structures. In 2D periodic structures, the dispersion curve is extended to the slowness surface, where the relation between the three parameters (kx , ky , ω) or (kθ , θ, ω) should be considered [Mace and Manconi, 2008].
Acknowledgments The authors’ would like to thank China Scholarship Council (CSC) for the financial support of this work.
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