Band gap behaviour of optimal one-dimensional composite structures with an additive manufactured stiffener

In this work, the banded behaviour of composite one-dimensional structures with an additive manufactured stiffener is examined. A finite element method is used to calculate the stiffness, mass and damping matrices, and periodic structure theory is used to obtain the wave propagation of one-dimensional structures. A multi-disciplinary design optimisation scheme is developed to achieve optimal banded behaviour and structural characteristics of the structures under investigation. Having acquired the optimal solution of the case study, a representative specimen is manufactured using a carbon fibre cured plate and additive manufactured nylon-based material structure. Experimental measurements of the dynamic performance of the hybrid composite structure are conducted using a laser vibrometer and electrodynamic shaker setup to validate the finite element model.


Introduction
Noise and vibration transmission within payload and passenger compartments is a major issue for modern transport vehicles. To ensure the quality of their products, manufacturers in the transport industry are simultaneously trying to optimise the mechanical and the vibroacoustic performance of struc- 5 tural assemblies. It has been demonstrated that judiciously designed periodic structures can induce vibration attenuation and stop-band behaviour in specific frequency ranges (so-called band gaps or stop bands).
Floquet [1] was the first to publish on periodic structures, in which the one-dimensional (1D) Mathieu's equations were studied to predict band gap be- 10 haviour. Floquets work was followed by that of Rayleigh [2], who developed a similar form to Floquet's theorem. During the twentieth century, Mead [3,4], Mace et al. [5] and Langley and Cotoni et al. [6,7] produced mathematical tools based on Brillouin's periodic structure theory (PST) [8]. Using these methods, researchers have the ability to predict the vibroacoustic and dynamic perfor-15 mance of several applications in relatively short times. Application examples are presented with composite panels and shells [9,10], structures with pressurised shells [11], and complex periodic structures [12,13,14,15]. Hussein et al. [16] produced an extensive review of developments in band gap technology.
There are two major mechanisms that have been identified to generate band 20 gap behaviour in periodic structures: Bragg scattering and local resonance.
Bragg scattering is observed when a structure exhibits periodic impedance mismatches and the waves are scattered at the borders of the unit-cell (the part of the structure that is periodically repeated). This scattering can be caused by means of inclusions, and geometrical or material inconsistencies, and leads to the 25 interaction of the reflected waves with the incoming waves. When specific circumstances are met, this interaction causes the partial or complete annihilation of wave propagation [16,17]. It can easily be shown that the frequency at which the band gap is observed depends on the length and the material/geometrical mismatch of the unit cell of the periodic structure. This leads to the need 30 for prohibitively large dimensions to achieve low frequency band gaps. Therefore, researchers' focus was shifted to local resonance [18], where a solid core material with relatively high density is usually preferred, suppressed by an elastically soft material. When this sub-wavelength inclusion/addition resonates, it exhibits behaviour that cancels the propagation of waves, giving rise to effective 35 negative elastic constants or group velocities at certain frequency ranges which are significantly lower than those observed in Bragg scattering. Liu et al. [19] examined the transition between the two band gap production mechanisms and there has been research on coupling of the two mechanisms [20,21]. In this work an optimisation method is developed capable of examining both band gap pro-40 duction mechanisms, and the Bragg scattering mechanism is observed in case study geometry to demonstrate its application.
Structures that exhibit band gap behaviour tend to be of significantly complex geometry and cannot be realised using conventional manufacturing techniques. Therefore, additive manufacturing (AM) technologies attract an in-45 creasing number of researchers [22]. AM eliminates several design limitations and is currently in extensive use in research and industry for small-scale pro- This wide use of AM has led researchers to examine the effect of the property variability of AM on the experimental results [29,30], where a better agreement was obtained between analysis and experimental results by considering uncertainties in the resonators and the host structure. The novelty of the work presented in this paper is:

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• An optimal design of a band gap structure is obtained, so that it serves both as a stiffener and band gap production mechanism, constituting a structural part.
• The optimal design of the structure is obtained by applying a developed computationally efficient unit cell based optimisation scheme.

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• The developed multi-disciplinary design optimisation scheme uses scalarisation for simultaneous mass and vibration minimisation and static stiffness maximisation. Several starting points are used and parametric analysis is completed to evaluate the optimal solution.
More specifically, the multi-disciplinary optimisation of vibration attenuation 95 through band gap and static structural performance of a 1D composite structure with powder bed fusion of polyamide 12 (PA12) material additions is examined.
The structure is modelled using finite element (FE) method and PST is used for predicting its wave propagation characteristics. A constrained non-linear optimisation method is used for the optimal geometric characteristics to be 100 acquired. The optimal solution is realised using a powder bed fusion (laser sintering) method and is tested using a laser vibrometer and electrodynamic shaker setup to validate the FE model.
The paper is organised as follows: in Section 2, the FE and optimisation method are presented. In Section 3, the examined case is presented, along with 105 the results of the analysis. In Section 4, the FE analysis of the case study and the experimental methodology is described and the results are discussed.
Finally, in Section 5, the conclusions of the work are drawn. The PST used in this work is presented in detail elsewhere [6]. A generic structure with 1D periodicity is considered. A unit cell is extracted from the structure (see Fig. 1) and its behaviour modelled with the FE method. A steady-state free harmonic vibration of frequency ω is considered in what follows and all response quantities are represented by complex amplitudes, so that where t is time and the degrees of freedom q can be partitioned into interior where the term ε x = k x L x is referred to as 'phase constant', L x is the periodic element's length in the x direction and k x is the wavenumber. The complete vector of the local degrees of freedom can be ordered The equation of motion for the cell including damping [36] is given by where K, M and C are the stiffness, mass and damping matrices respectively and f is the vector of the nodal forces. Eq. (4) can be used when proportional damping is available [37], where damping is expressed as a linear combination of the mass and stiffness matrices, that is, where α 1 and α 2 are real scalars. In the case of structural damping [9,36], the equation of motion becomes and the structural damping matrix C is given by where n e is the number of the FEs of the unit cell, K i is the stiffness matrix of the i th element and η i is the loss factor of the corresponding element. Eq. (6) can be written as In this work, we do not focus on the accuracy of the model of damping, since the main aim of the wave propagation analysis is to examine the existence of the band gap. In order to write the propagation relation in Eq. (1) in matrix form, we consider transformation matrix R, which is given in the following equation: In this way, we get where In the absence of external forces (free wave propagation), we have where R H denotes the complex conjugate (Hermitian) transpose of R. The resulting homogenous equations in the reduced set of degrees of freedom are then given by and where When a set of phase constants ε x is specified, we get a quadratic eigenvalue problem for Eq. (13), the solution method of which is described elsewhere [38].
Eq. (14) gives a standard eigenvalue problem with eigenvalues λ = ω 2 indicating the frequencies at which a wave can propagate in the structure when a given phase is specified between the edges of the cell. It is noted that in this work, only 115 the real part of the eigenvalues is examined since the target of the examination is the band gap behaviour.

Optimisation
The optimisation method (see Fig. 3) that is employed to calculate the optimal structure design is described below. The set of parameters can be expressed as where n is the number of parameters examined. The design values may be considered to be constrained (e.g.  geometrical characteristic of the stiffener or the plate itself, such as thickness, that affects the considered variables. The objective function for the specific structure is where a i , b i , c i and d i are the coefficients given by the design cost functions. Adjusting these coefficients, the designers are given the ability to apply polynomial curves to available cost data forming the cost functions of the optimisation, as can be seen in the next section (see Fig. 5). Additionally, this type of objective function renders its differentiability and the calculation of its partial gradients efficient. Higher order polynomial or exponential fitting functions may be applied without loss of accuracy. The gradient of F (p) can, therefore, be computed as By definition, we have b g = f 2 − f 1 , b g m f = f1+f2 2 and f i = ωi 2π , λ i = ω 2 i , where f 1 and f 2 are the frequencies at the lower and upper bound of the band gap. Thus, ∂b g m f ∂β i = 1 2 Taking advantage of the chain rule of differentiation, we get: where the partial derivatives of the eigenvalues are calculated based on the following techniques, given elsewhere [39,40]: where x = q = q I q L T is the eigenvector. In this work, the partial derivatives of K, M and C are calculated considering a perturbation of 0.1% of the examined parameter β i , a percentage that was the result of a convergence analysis.
Thus, we have: where K p , M p and C p are the perturbed stiffness, mass and damping matrices, β i,p is the perturbed considered parameter and β i,0 the unperturbed parameter.
The mass, stiffness and damping matrices M, K and C of the unit cell can also be formed by assembling the local mass, stiffness and damping matrices of individual FEs, which can be calculated analytically [41]. It is, therefore, evident that when the expressions of the partial derivatives for every local mass, stiffness and damping matrix ∂m ∂βi , ∂k ∂βi and ∂c ∂βi are known, then the expressions for the global matrices ∂M ∂βi , ∂K ∂βi and ∂C ∂βi can be derived simply by assembling the expressions of the local matrices together. Therefore, we have: The calculation of ∂b s ∂βi , ∂m ∂βi depends on the geometric and material properties of the case study for which the optimisation method is applied. A constrained 120 nonlinear interior point optimisation algorithm using Newton's method [42] is employed to compute the optimal parameter vector p that minimises F (p).
It should be noted that the optimisation used in this work leads to results in significantly shorter times compared to those that examine the whole finite periodic structure, since it only needs the unit cell's wave propagation char-

Case study
The band gap behaviour of a thin composite beam with AM stiffener is examined and optimised using the scheme described in section 2.2. In this case, the Bragg scattering mechanism is targeted and the absolute band gap is examined. A thin plate of carbon fibre is used as a substrate structure and the stiffener is 135 manufactured using the powder bed fusion method and PA12 material with an EOS Formiga P100 machine. The stiffener is cemented to the plate using a twopart epoxy adhesive (Permabond ET500). The parameters that are examined are the thickness of the stiffener t st and the length of the top of the wedge-like shape of the stiffener l top (see Table 1 and Fig. 4). The mechanical properties of 140 the materials are given in Table 2      When the method is tailored for a specific application, the cost functions can be formulated by the designers accordingly. In this case, the cost functions chosen where the gradient of F (p) is The optimisation is calculated using several starting points with steps of 2 mm 145 and together with the parametric analysis results we consider this as the optimal design (see Fig. 8). The optimal values of the parameters are found to be:  The geometry resulting from the optimisation process is manufactured and 200 dynamically tested (Fig. 12) to validate the FE analysis results. A 5-cell beam is preferred, due to the limitation of the mass of the specimen that the shaker can excite and on the AM machine production size. It is noted that the exper-  As can be seen in Fig. 13, the points' number is indicative of the unit cell's repetition.

Concluding remarks 225
In this work, an efficient multi-objective scheme using scalarisation is presented for optimising the static, dynamic and mass characteristics of a onedimensional composite structure having an additive manufactured stiffener. Additionally, an experimental evaluation of the FE model of the case study is presented. The presented computational scheme gives the ability to adapt the 230 optimisation criteria to specific applications. Furthermore, a parametric analysis is completed offering a valuable insight of the stop band behaviour of the structure with regards to the parameters.

Appendix B. Undamped system
The homogenous equation in the reduced set of degrees of freedom for undamped system is given by The partial derivative of the eigenvalue is For the global matrices we have: