N Ouisse

Bouhaddi

S Timorian

G Petrone

S De Rosa

F Franco

M Ouisse

N Bouhaddi

Spectral analysis and structural response of periodic and quasi-periodic beams

Keywords: imperfections, quasi-periodic structures, band-gaps, wave finite element, Fibonacci sequence

published or not. The documents may come

Spectral analysis and structural response of periodic and quasi-periodic beams

Introduction

The analysis of the propagation of waves in structures is a fundamental task in many engineering applications. The knowledge of dispersion relations, providing information on the type and nature of propagating waves is of interest for the prediction of forced response, acoustic radiation, non-destructive testing and transmission of structure-borne sound. All these themes are nowadays the subject of many studies in order to improve the vibro-acoustic comfort of passenger carries, bridges, pipelines, and space vehicles.

Wave propagation in simple structures can be investigated through analytical models, exact or approximated. However, this kind of analysis usually involves assumptions and approximations concerning the stress, strain and displacement states of the structure, and always more refined numerical models are required as the frequency increases since the wavelength may become comparable with the cross-section dimensions. For example, if the propagation of bending waves in a beam is investigated, Euler-Bernoulli, Rayleigh, Timoshenko or 3-dimensional elasticitybased theories might be used, depending on the frequency range of interest [START_REF] Graff | Wave Motion in Elastic Solids[END_REF][START_REF] Cremer | Structure-Borne Sound: Structural Vibrations and Sound Radiation at Audio Frequencies[END_REF] .

For complex structures, such as layered (composite and sandwich) beam [START_REF] Di Taranto | Theory of vibratory bending for elastic and viscoelastic layered finitelength beams[END_REF][START_REF] Mead | The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions[END_REF][START_REF] Sokolinsky | Consistent higher-order dynamic equations for soft-core sandwich beams[END_REF] and plate [START_REF] Kurtze | New wall design for high transmission loss or high damping[END_REF][START_REF] Moore | Sound transmission loss characteristics of three layer composite wall constructions[END_REF][START_REF] Dym | Transmission of sound through sandwich panels[END_REF][START_REF] Dym | Transmission loss of damped asymmetric sandwich panels with orthotropic cores[END_REF][START_REF] Nilsson | Wave propagation in and sound transmission through sandwich plates[END_REF][START_REF] Reddy | Mechanics of Laminated Composite Plates and Shells: Theory and Analysis[END_REF][START_REF] Ghinet | The transmission loss of curved laminates and sandwich composite panels[END_REF] , or cylinders [START_REF] Kumar | Dispersion of flexural waves in circular cylindrical shells[END_REF][START_REF] Fuller | The effects of wall discontinuities on the propagation of flexural waves in cylindrical shells[END_REF][START_REF] Karczub | Expressions for direct evaluation of wave number in cylindrical shell vibration studies using the Flugge equations of motion[END_REF] , analytical formulations become quite difficult: beyond the required assumptions and approximations in the models, the resulting dispersion relations are usually transcendental and/or of high order, therefore their resolution is not straightforward or requires symbolic manipulation [START_REF] Karczub | Expressions for direct evaluation of wave number in cylindrical shell vibration studies using the Flugge equations of motion[END_REF][START_REF] Banerjee | Development of an exact dynamic stiffness matrix for free vibration analysis of a twisted Timoshenko beam[END_REF] . For this reason, for the analysis of complex structural components, semi-analytical or numerical methods have been developed for the computation of dispersion curves. However, if the structure under investigation presents characteristics which are periodically repeated in one or more directions, the analysis procedure can take advantage of this property by exploiting the periodicity [START_REF] Brillouin | Wave propagation in periodic structures: electric filters and crystal lattices[END_REF] . A generic structure obtained as an assembly of identical elements, called cells, can be considered as periodic. Several engineering structures Prepared using sagej.cls can be assumed as periodic, starting from simple beams and plates, moving to stiffened plates or car tyres, up to aircraft fuselages, railways, tracks, etc. In this case the study of the wave propagation through the waveguide can be reduced to the analysis of a single cell by applying the periodicity conditions together with continuity of displacements and equilibrium of forces at the interfaces between two consecutive cells (Floquet-Bloch theorem) [START_REF] Collet | Floquet-Bloch decomposition for the computation of dispersion of two-dimensional periodic, damped mechanical systems[END_REF][START_REF] Mead | A general theory of harmonic wave propagation in linear periodic systems with multiple coupling[END_REF][START_REF] Billon | Mechanics and band gaps in hierarchical auxetic rectangular perforated composite metamaterials[END_REF] .

Periodic structures found a big interest in engineering applications because they introduce frequency band effects that can improve the vibroacoustic performances.

In fact, in periodic structures, the impedance mismatch generated by periodic discontinuities in the geometry, acting as a waveguide, and/or in the material, can cause destructive wave interference phenomena over specific frequency bands called "stop band" or "band gaps" [START_REF] Spadoni | Wave propagation and band-gap characteristics of chiral lattices[END_REF] . However, the presence of imperfections (i.e.

defects or irregularity) in the structure, due to the manufacturing process or not exact reconstructions of the boundary conditions for example, lead to the loss of the periodicity of the structure: this can have a noticeable impact on the vibrational and/or acoustic behaviour of the elastic structure.

In this case it is more correct to speak about quasi-periodicity which is the property of a structure that displays irregular periodicity. A quasi-periodic structure can be idealised as repeated substructures which have asymmetric translations in any direction of the Euclidian space. It can be considered as an intermediate case between periodic and random elastic medium [START_REF] Velasco | Elastic waves in quasiperiodic structures[END_REF] . Quasi-periodic behaviour is thus a pattern of recurrence with a component of unpredictability that does not lend itself to a precise measurement. An example of a natural quasi-periodic structure is a quasicrystal. It was discovered in 1981 by Dan Shechtman [START_REF] Shechtman | Metallic phase with long-range orientational order and no translational symmetry[END_REF][START_REF] Bindi | Natural quasicrystals[END_REF] and it is a structural form that are both ordered and non-periodic [START_REF] Banerjee | Development of an exact dynamic stiffness matrix for free vibration analysis of a twisted Timoshenko beam[END_REF] . These structures are characterized by several properties, such as low coefficient of friction and low heat conductivity, just to cite some of them, that made them very attractive and interesting for technological applications, mainly in the fields of crystallography and photonics [START_REF] Shechtman | Metallic phase with long-range orientational order and no translational symmetry[END_REF][START_REF] Bindi | Natural quasicrystals[END_REF][START_REF] Vardeny | Optics of photonic quasicrystals[END_REF][START_REF] Kraus | Quasiperiodicity and topology transcend dimensions[END_REF][START_REF] Kraus | Topological states and adiabatic pumping in quasicrystals[END_REF][START_REF] Ozawa | Synthetic dimensions in integrated photonics: From optical isolation to four-dimensional quantum hall physics[END_REF][START_REF] Kraus | 4-dimensional quantum hall effect in a twodimensional quasicrystal[END_REF] . Quasi-crystals were used as non-stick coating on frying pans and cooking utensils [START_REF] Fikar | Al-Cu-Fe quasicrystalline coatings and composites studied by mechanical spectroscopy[END_REF] and to develop heat insulation, led and new materials able to convert heat to electricity [START_REF] Kalman | The Quasicrystal Laureate[END_REF][START_REF] Dubois | Towards applications of quasicrystals[END_REF][START_REF]Quasicrystals: An introduction to structure, physical properties and applications[END_REF] .

In recent years there is a growing interest in the design possibilities offered by quasi-periodic structures also in the field of structural mechanics. This leads to some modelling issues which will be well analysed due to the impossibility of periodic simplifications, but an adequate design of the quasi-periodicity may offer new vibroacoustic properties to the structure [START_REF] Hou | Acoustic wave propagating in one-dimensional Fibonacci binary composite systems[END_REF][START_REF] Aynaou | Propagation and localization of acoustic waves in Fibonacci phononic circuits[END_REF][START_REF] Chen | Study on band gaps of elastic waves propagating in one-dimensional disordered phononic crystals[END_REF][START_REF] King | Acoustic band gaps in periodically and quasiperiodically modulated waveguides[END_REF] , they also provide experimental verification of the transmission properties of one dimensional phononic crystals based Prepared using sagej.cls on the quasi-periodic Fibonacci and Thue-Morse number sequence. Hou et al. [START_REF] Hou | Acoustic wave propagating in one-dimensional Fibonacci binary composite systems[END_REF] investigated the transmission properties and the frequency spectra of Fibonacci binary composite material with different thickness ratio of two layers. Whereas in this paper Fibonacci series is dedicated for 1D structures i.e. beams and the vibration properties and band structure of their unit cells are investigated. In term of analysis this paper is mainly focused on Frequency Response Functions (FRF) and spectral analysis to study the dynamic behaviour of the structures [START_REF] Wu | Band-gap analysis of a novel lattice with a hierarchical periodicity using the spectral element method.ï¿ 1 2[END_REF][START_REF] Wu | Spectral element method and its application in analysing the vibration band gap properties of two-dimensional square lattices[END_REF][START_REF] Wu | Vibration band-gap properties of threedimensional Kagome lattices using the spectral element method.ï¿ 1 2[END_REF] . Aynaou et al. [START_REF] Aynaou | Propagation and localization of acoustic waves in Fibonacci phononic circuits[END_REF] performed a theoretical investigation on acoustic wave propagation of one-dimensional phononic band gap structures made of slender tube loops pasted together with slender tubes of finite length according to a Fibonacci sequence. In this analysis Aynaou et al, found that besides the existence of extended and forbidden modes, some narrow frequency bands appear in the transmission spectra inside the gaps as defect modes. Similarly, in the results of the current investigation, there are narrow frequency peaks that appears in the frequency bands of the geometrical impedance mismatch case, especially on longitudinal frequency response. Aynaou et al consists a treatment procedure that spatial localisation of the modes lying in the middle of the bands and at their edges is examined by means of local density of states. In the other hand, Chen and Wang 39 studied band gaps of elastic waves propagating in one-dimensional disordered phononic crystals. Similar topological formation of Fibonacci and Thue-Morse are investigated in an experimental observation of the formation of phononic scattering band structure in one-dimensional periodically and quasi-periodically based on the Fibonacci and Thue-Morse number sequences by King and Cox [START_REF] King | Acoustic band gaps in periodically and quasiperiodically modulated waveguides[END_REF] . Gei [START_REF] Gei | Wave propagation in quasiperiodic structures: stop/pass band distribution and prestress effects[END_REF] shows that in the case of axial and flexural vibration for systems based on different Fibonacci sequences, the number of stop/pass bands within a defined range of frequencies changes and follows the Fibonacci recursion rule, by showing also a self-similar pattern. From a design point of view the asymmetrical conditions in quasi-periodic structures can be built by following different sequences, such as: higher order generations of Fibonacci sequence, Thue-Morse, Rudin Shapiro sequences as well as Penrose lattices [START_REF] Barber | Aperiodic Structures in Condensed Matter: Fundamentals and Applications[END_REF] . In this paper the modelling of simple quasi-periodic structures is built with the conventional finite element method (FEM) to fulfil the generation of quasi-periodic patterns since these are based on an asymmetrical distribution of identical cells [START_REF] Timorian | Investigation for the analysis of the vibrations of quasi-periodic structures[END_REF][START_REF] Timorian | Band diagram and forced response analysis of periodic and quasi-periodic panels[END_REF] . Finite, periodic and quasi-periodic structures are thus proposed and compared by using the Fibonacci sequence to investigate about the possibility to have and control useful frequency bands in which the response can be reduced as much as possible. In Section 2 the models and their specific lexicon are presented. Section 3 contains the methods and tools used for the numerical investigations. The main results obtained are commented in Section 4 and finally, some concluding remarks are given in Section 5.

Models and Lexicon

In this paper, quasi-periodic beams with a finite number of cells are analysed. In these models specific sequences like Fibonacci series will be used to generate impedance mismatches in view of the desired degree of quasi-periodicity [START_REF] Becus | Wave propagation in imperfectly periodic structures: a random evolution approach[END_REF][START_REF] Mester | Periodic and near-periodic structures[END_REF] .

The degree of quasi-periodicity might be controlled with mathematical rules that will be introduced in the next section.

Fibonacci Sequence and Nomenclature

The well-known sequence called Fibonacci 49 is a series of integer numbers such that:

S n = S n 1 + S n 2 (1)
For instance the Fibonacci sequence starting with 1 and 2 is 1, 2, 3, 5, 8, 13, 21,....The configuration of the quasi-periodic structures is here carried out by using a sequence of two possible variations according to Fig. 1.

The variations can typically be due to the sections, materials or boundary conditions.

The first cell coincides with S 1 , then the cells can be assembled, forming a sequence defined by a simple integer (order). The S n denote the n-th sequence: Tab.1 shows the number of unit cells necessary to generate a given order of Fibonacci sequence.

Prepared using sagej.cls The numerical models is identified by the order of the Fibonacci sequence, and thus the length of the n-th order sequence, S n , will be greater than S n 1 . As example, Fig. 2 shows a Fibonacci (S 6 ) beam with a sequence of 13 [ABAABABAABAAB] cross-sections.

Cases

In this framework, variations between cells A and B will be obtained through impedance mismatch. Two cases will be considered.

Case M1: impedance mismatch due to geometrical discontinuity, Fig. 3

[case M1];

Case M2: impedance mismatch due to material discontinuity, Fig. 3 [case M2].

The various configurations considered for M1 will be described in the next section.

Without loss of geometry, the material used in case M1 is steel A-36 whose properties are provided in Tab.2

For the typical quasi-periodic structure case M2 the material variation is reported in Tab.2.

Prepared using sagej.cls

Geometrical variations of case M1

Methods and Tools

Two numerical methods are used: the FEM is considered for analysing the Frequency Response Functions (FRF) of the finite beam while the Wave Finite Element Method (WFEM) together with spectral analysis is taken into account for computing the dispersion diagrams.

Finite Element Analysis

The analysis is performed with the conventional FE method: frequency response function (FRF) analysis of damped quasi-periodic beams. The FE analysis is carried out using ANSYS-APDL linked with MATLAB.

The types of elements used are Beam 188, which is a linear 2-node beam element.

Each cell (A and B) are composed of 4 nodes (three beam elements) and each node has three degrees of freedom: longitudinal in the axial direction (x axis), bending in lateral direction (y axis), and torsional rotation around (x axis). The actual distance between each cell is 100 mm.

H kj (⌦) = N X p=1 pk pj m p (! 2 p ⌦ 2 + i2⇠ p ! p ⌦) (2)
where H kj (⌦) is the transfer function, m p the modal mass, ! p the eigenfrequency, ⌦ the forcing frequency; ⇠ p is the modal damping; pk and pj are the components of the p-th eigenvector evaluated at the source and receiver points and N the number of retained eigenmodes. Accordingly, the FRF of the quasi-periodic beams are computed.

These forced response analysis are performed with free-free boundary conditions.

The input force is located at one end of the beam in the transverse direction and the Prepared using sagej.cls

response is computed at the other end of the beam in the same direction.

Spectral analysis of infinite beam (waveguide)

The Floquet-Bloch conditions can be applied to simulate infinite periodic beams that is waveguides. This is classical for perfectly periodic structures. In order to perform spectral analysis on quasi-periodic structures, super unit cells are used.

Super unit cell:

In this work, a super unit cell is a cell hosting every single order of the Fibonacci sequence and it hosts given orders of deterministic quasiperiodic pattern in a single cell. Although it will be repeated in a periodic way, the cell itself has a quasi-periodicity replication inside the super unit cell. In this work, we also consider a second case, called double unit cell, as a reference. This case is perfectly periodic. For illustration, the substructures in Fig. 6 and Fig. 7 are modelled as a super unit cell and double cell respectively. In this example, the super unit cell is defined according to the 6 th order of Fibonacci sequence (type ABAABABAABAAB). The super unit cell is used in the Wave Finite Element Method (WFEM) analysis presented in the next section.

Transfer matrix of super unit cell:

The transfer matrix is a square matrix of even dimension and is function of frequency of the disturbance propagating in the structure [START_REF] Arnold | Lyapunov exponents: proceedings of a conference held in Oberwolfach[END_REF] . For a given structure, each super unit cell composed of n individual cells, has the same transfer matrix [T] such that;

x n = Tx 0 (3)
where x n is the state vector on the left-side of the cell (i.e. on the left-side of the super unit cell), x 0 is the state vector on the right-side of cell 1 (i.e. on the right-side of the super unit cell), and T = T n T n 1 , ...T 1 where T j is the transfer matrix of cell j. In order to obtain the transfer matrices, both mass and stiffness matrices are extracted from APDL-ANSYS.

The dynamic behaviour of the cell number j is described by:

D j q j = f j (4)
where D j ,f j , and q j define respectively the dynamic stiffness matrix, force and displacement vector. The dynamic stiffness matrix writes

D j = ! 2 M j + K j⇤ (5)
with K j⇤ = K j⇤ (1 + i⌘) where M j ,K j , and ⌘ are respectively the mass matrix, the stiffness matrix and the loss factor corresponding to the structural damping. The matrices and vectors are partitioned according to the degrees of freedom: q j L , q j I , and q j R respectively refer to the left-side, internal, and right-side parts of the super unit cell number j. The corresponding terms in the matrices are written

q j L q j I q j R 9 > = > ; = 8 > < > : f j L 0 f j R 9 > = > ; . (6)
The internal degrees of freedom can then be condensed using the second row of Eq.6, in order to retain the analysis to the left and right boundary displacements and forces.

q j I = D j 1 II (D jT LI q j L + D j IR q j R ). (7)
it leads to

" D j LL D j LI D j 1 II D j IL D j T LR D j LI D j 1 II D j IR D j RL D j RI D j 1 II D j IL D j RR D j RI D j 1 II D j IR # ( q j L q j R ) = ( f j L f j R ) . (8)
The reduced dynamic stiffness matrix is written as follow:

" Dj LL Dj LR DjT LR Dj RR # ( q j L q j R ) = ( f j L f j R ) . (9)
One define state vectors for the boundaries of the component j:

u j L = ( q j L f j L ) , u j R = ( q j R f j R ) . ( 10
)
The transfer matrix is hence obtained by reorganising the degrees of freedom according to the state vector:

u j R = T j u j L (11)
where, u k L and u j R are the displacement vector of the right and left component of the unit cell, and T j is the transfer matrix in Eq.11:

T j = 2 4 ˜ D j 1 LR Dj LL ˜ D j 1 LR Dj RL Dj RR Dj 1 LR Dj LL Dj RR Dj 1 LR 3 5 . ( 12
)
The transfer matrix of the super unit cell is then obtained in Eq.3.

Dispersion analysis:

Periodicity conditions applied on the super unit cells are then written as

u p+1 L = e µ u p L ( 13
)
Prepared using sagej.cls where µ is the propagation constant. Combining Eq.11 and Eq.13,yields to the eigenvalue problem:

T p i = i . (14)
where = e µ .

Depending on the nature of the eigenvalue of [T p ], the waves propagating in a periodic structure are described as travelling waves and attenuating waves which occur in alternating frequency bands known as pass-bands and stop-bands. If the eigenvalues of [T p ] are complex and of the form e ±ikL , k 2 IR the corresponding wave is in a pass-band and the wave travels in the form of e ±ikL , where k is a real wave number, the positive and negative signs indicating left and right travelling waves, respectively.

On the other hand if all eigenvalue of the [T p ] are of the form e ± or e ± +i⇡ , 2 IR, is pure real exponent, the corresponding frequency is in a stop-band and the wave amplitude after travelling n elements are attenuated by the factor e (± n) , in which the real exponent implies attenuated waves [START_REF] Arnold | Lyapunov exponents: proceedings of a conference held in Oberwolfach[END_REF] .The dispersion curves are computed by imposing frequency and computing k according to the given eigenvalue problem. The trend is similar to the flexural waves shown in Fig. 8 when the number of cells is increasing: the band gaps locations do not change, while the depth in the FRF Prepared using sagej.cls is becoming larger. There is also a localised mode which is appearing inside the band gap of longitudinal waves. The effects of dynamic behaviour of the beam-spans when increasing the number of cells following Fibonacci orders is emphasised by this analysis. In practice, these types of beam-spans can be used as a junction filter between structures. The induced vibrational energy transfers through this junction, and it acts as a meta-material filtering property to the elastic waves due to the impedance mismatch in geometry and material.

Frequency response function of finite beams using FEM

Frequency response function of double cell and super unit cell structure using WFEM:

In this part, FRFs of periodic beams are analysed. Two cases are considered, a double cell variation with a span of 13 unit cells having perfect periodic order and a super unit cell Fibonacci 6 th order with 13 unit cells, which has non-symmetric repetition of cells inside periodic super unit cell. The results are discussed with four types of variations in geometry for both cases. The aim of this analysis is to investigate the effect of the geometric variation on width and shift in frequency of band gaps and comparing the two models in order to find a compromised one for vibration control.

The application is based on flexural and longitudinal waves of perfectly periodic models in Fig. 10 and Fig. 11. The analysis is applied for four types of geometrical variations as a test cases.

Both, flexural and longitudinal waves are investigated under FB periodicity condition.

The ranges of frequency of FRF of flexural waves in Fig. 9 is considered at around 10 kHz, whereas for longitudinal waves in Fig. 11 is around 25 kHz, depending on the frequency stop bands zone. More details on geometrical variation of the Types plotted in Fig. 10 and Fig. 11 are reported in Table 3. The response shows an enlargement in frequency stop bands as the height of cross section (B) decreases and the frequency stop bands tends to increase from almost 8 kHz up to around 2.5 kHz.

Similarly, a frequency response function of the same periodic double cell is considered for longitudinal waves to describe the dynamic behaviour of beam in compression conditions. Ultimately, the use of super unit cell has a significant pros compared to the double unit cell. The first point that can be noticed in the FRFs of the Fig. 12 is that there is a tremendous shift of frequency stop bands from lower to higher frequency ranges.

The second point is that there is a frequency stop band appearing in lower frequency ranges bellow 1 kHz, which is not the case in double unit cell approach. In conclusion the characteristics of the beam with Fibonacci series or simply (quasi-periodic beams)

Prepared using sagej.cls The results obtained in this paper shows that the beam with Fibonacci characteristics can improve performances in terms of attenuation level without weight penalty, which can be an asset for meta-materials.

Frequency response function of quasi-periodic beam with geometrical variations of type II:

This test case is influenced by the previous results. As it can bee seen from Fig. 10, Fig. 11 and Fig. 12, that there is an evident band gaps shift and width enlargement while the factor of the cross sections of Tab.3 increased from (1 2.7). Now in this subsection the same procedure is applied to the quasi-periodic beam with the 11 th order of Fibonacci using four types of geometrical variation according to the Tab.3. Fig. 13 shows four curves, each corresponding to the four cross-section types of variations.

Prepared using sagej.cls In conclusion, case M1 shows significant impacts on vibration control of the beam span. The first case, flexural analysis shows that, by increasing the length of Fibonacci orders, the depth of the band gaps grows deeper and it is also emphasised in longitudinal waves. The second part describes width enlargement and shift in frequency of band gaps, in which quasi-periodic super unit cell shows more impact compared to the double unit cell approach.

FRF results for Case M2

The second quasi-periodicity configuration is based on the impedance mismatch of material constituents. The beam is analysed as a continuous span (with no cross section variation). The mass of the system for all orders of Fibonacci is kept the same, while Prepared using sagej.cls The results of the dynamic analysis for flexural waves in Fig. 14 shows similar behaviour as those for the geometrical impedance mismatch in terms of increase of the depth of the band gaps when the order increases.

The main result in this analysis is that material constituent discontinuity does not have a high dynamical influence in the depth growth of frequency stop bands as it is in the cross section variation case. Analysing again the numerical results in comparative way (i.e. without paying attention to the absolute values), the growth of depths in the frequency band gaps is lower than those computed for the variation of cross-sections. flexural waves and also the depth is gradually becoming larger.

The response of the 11 th order of Fibonacci beam in the FRF shows a flat curve in the larger band gaps location (without any higher dynamics), that is due to the longer length of the beam. Beside the flexural and longitudinal (FRFs) of the case M 2, another extra sub-case scenario has been investigated by considering combination of lower and higher sound velocity materials. As there was not a big difference between the sound velocity of (Aluminium 2045-T4) and (Steel A-36), two other materials (copper and magnesium) are also taken into account for the flexural FRFs of the 11 th order of Fibonacci sequence. In conclusion, the dynamic response of quasi-periodic finite span reduces while keeping the four types of variations. It seems from the lexicon that the sound velocity of case M2 with constant cross sections and material variation is simply the ratio between modulus of elasticity and density. In contrary, the sound velocity of case M1

with constant material and cross section variation is the ratio of modulus of elasticity with respect to density multiplied by a factor that is squared of height of the cross Prepared using sagej.cls section. Thus, it explains that M2 is less efficient in terms of change in the impedance mismatch.

5 Spectral analysis of the M1 waveguides

Double cell

The same transfer matrix, extracted for the FRFs of the previous results based on periodic double cell, is used again for dispersion curves computation. Frequencyshift of the stop/pass band positions is quantified, using the real solution of waves [START_REF] Gei | Wave propagation in quasiperiodic structures: stop/pass band distribution and prestress effects[END_REF] . Herein, for quantifying the frequency shift of stop bands, the dispersion curves are plotted considering only the real parts of propagative waves. In Fig.17 In Type III, band gap moves to higher frequencies [1300 1600] Hz. Evaluating the stop band width in four cases, it seems that the stop band width is enlarging after Type I up to Type III, while in Type IV , the width of stop band tends to disappear with a shift to higher frequencies compared to the other types. In fact, it gradually shifts by reducing the height of the second cross section (B) as it reaches the zero impedance mismatch in Type IV .

Super unit cell

Similarly, an important part of analysis had to be taken into account for the spectral analysis of the quasi-periodic embedded super unit cell in periodic infinite systems.

Fig. 18 shows the dispersion curves of 6 th order of Fibonacci. The same technique used for the dispersion of fully periodic double cell is also considered in this system. Again Prepared using sagej.cls the comparison is plotted between Type I, used as reference, and the other Types. Comparing the results, in terms of real part of the wave number, of periodic beam Fig. 17 and quasi-periodic beam Fig. 18 it can be noted that in both cases the stop bands shift to higher frequencies but for the quasi periodic beam it changes quite dramatically.

In fact, comparing the same band gap for the two beams it can be noted that the band gap in quasi-periodic beam is almost 2 times larger than periodic one. In view to be more clear in Fig. 19 a comparison between band gaps of periodic and quasi periodic beam, at same cross section ratio, is provided. The horizontal axis of the plot shows an average frequency which corresponds to the band gaps interval f over the overall selected range of frequency f in each Fig. 17 and Fig. 18 respectively. The plot has two different curves, the blue one corresponds to the fully periodic beam with double cell including the geometrical cross-section variation, whereas the red curve corresponds to the quasi-periodic beam with the 6 th order of Fibonacci and including those four types of geometrical cross section variation effects.The band gap width of quasi periodic beam is higher then the periodic one up to ratio 2. After that the band gap width Prepared using sagej.cls of periodic increases by increasing the ratio, while the band gap of quasi periodic decreases. This is more evident in Fig. 21 where the real and imaginary parts of the wavenumber are plotted. This plot is much visible and it can be noted that the number of stop bands for both curves are the same, but the width and the depth of the bands gaps for Fibonacci and Thu-Morse models is different. For instance in the frequency

Conclusions

The structural response of periodic and quasi-periodic beams are investigated. These beams are modelled using deterministic approach. Periodic beams are made of two different cells in terms of mass and cross section dimensions. Whereas quasi-periodic beams has the same configuration of cells made of these two different cells but the replication of periodic cells are not perfect and it follows Fibonacci sequence pattern.

Prepared using sagej.cls frequency band gaps.

Overall the quasi-periodic structures with geometrical impedance mismatch have an efficient impact on reducing the response in lower frequency regimes compared to strictly periodic counterparts. Although, it does not show large widths in lower frequency stop bands, but a small degree of geometrical impedance change, can shift the stop bands drastically compared to the strictly periodic spans. The results obtained have also shown that the quasi-periodicity can improve performance since attenuation, in given frequency range, can be obtained without weight penalty, which can be an asset for lightweight structures.

Highlights and contribution of the study 1. Forced response of quasi-periodic beams with increasing the order of Fibonacci, shows rapid growth in the depth displacement amplitude of the band gaps. For instance geometrical variation of T ypeI and material variation of (Steel-Aluminuim).

4. Spectral analysis of four types of geometrical variations in identical super-unitcells and double unit cells/patterns are analysed. The significants of periodic beam composed of cells whose properties follow Fibonacci sequence of a 6 th order and cells whose properties follow a perfect periodic orders are studied.

The results shows larger ( f ) for a unit cell with composition of cells following Fibonacci sequence.

Marie Curie grant agreement No 675441 and EUR EIPHI (ANR 17-EURE-0002) project.

4. 1 15 4. 3

 1153 FRF results for Case M1-Type II . . . . . . . . . . . . . . . . . . . 13 4.2 Frequency response function of double cell and super unit cell structure using WFEM: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency response function of quasi-periodic beam with geometrical variations of type II: . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.4 FRF results for Case M2 . . . . . . . . . . . . . . . . . . . . . . . . 19

Figure 1 .

 1 Figure 1. Configurations of quasi-periodic beam following a Fibonacci pattern 22 .

Figure 2 .

 2 Figure 2. Fibonacci configuration of 6 th order

Figure 3 .

 3 Figure 3. Configuration of discontinuities: (M1) and (M2).

A

  comparison for the case M1 by keeping constant the mass of the sum of the A and B cells is considered. The aim is to find the most efficient geometrical variation behaviour of unit cells (A) and (B) for vibration control. The factors prescribed in Tab.3 are the ratio of the length of section edges. The four configuration types are displayed in Fig.4. The configuration Type IV has no impedance mismatch, and Type II will be first analysed as a reference.

Figure 4 .

 4 Figure 4. Comparison of case M1 by keeping constant the mass and the length of sum of the A and B cells (the beams have square cross sections).

Figure 5 .

 5 Figure 5. Schematic diagram of beam with 6 th order of Fibonacci for numerical analysis.

Figure 6 .

 6 Figure 6. Periodic wave-guide (Super unit cell with 6 th order of the Fibonacci sequence).

Figure 7 .

 7 Figure 7. Periodic wave-guide (reference double unit cell).

FRFs of finite structures

  are analysed in two subsections according to the cases named M1 and M2.The FRFs are plotted in wider frequency ranges of 10 kHz for flexural and 25 kHz for axial vibration. The frequency range was chosen for the first 22 natural frequencies. A modal analysis is taken into consideration to investigate the number of elements per wavelength in order to fulfil a criterion for a sufficiently accurate numerical modal analysis. The boundary condition is free-free and a frequency range up to 10kHz for flexural and 25 kHz for axial vibration is chosen because the main target is on the first 22 natural frequencies. The model is meshed by 1D 2-node beam element type 188. Mesh setting with two different beam elements (188 2 node and 189 3 node) with the number of elements per wavelength 4, 5, and 10 are checked. The span of the beam is 1300 mm and 39 elements per wave length was more accurate with beam element 188 2 node to converge the exact natural frequency which is 9700 Hz. The strategy to compute the harmonic response function of periodic beams is carried out in MATLAB. A script is created to define finite element model of the periodic double cell with an arrangement of 13 unit cells. The boundary condition is considered as free-free and a white noise (i.e. harmonic force of 1N from 0 10 kHz with a Prepared using sagej.cls bandwidth of 10 Hz is applied to one end of the beam. The white noise is applied in vertical z and horizontal x direction, respectively for flexural and longitudinal vibrations.4.1 FRF results for Case M1-Type IIA first analysis of M1-Type II is performed, considering increasing orders of the Fibonacci sequence (4 th , 5 th , 6 th , 7 th , 8 th , 9 th , 10 th , and 11 th orders). It is reminded that the structure is not periodic, and that an increase in Fibonacci order is associated to an increase in the length of the beam. All results are carried out in the frequency range [0 10000] Hz. The results in Fig.8 shows multiple stop bands which stay coherent from one order to the other, for instance, around [600 900] Hz, [1900 2300] Hz and a larger frequency stop band around [2900 4000] Hz.

Figure 8 .

 8 Figure 8. FRFs of the flexural waves for given orders of Fibonacci beam with geometrical variation

Figure 9 .

 9 Figure 9. FRFs of the longitudinal waves for given orders of Fibonacci beam with geometrical variation

Fig. 11 shows

 11 Fig.11 shows an enlargement of frequency stop bands into the left and right sides of the frequency ranges. The result shows an enlargement of frequency stop bands compared to continuous beam type IV, characterised by no ranges of free wave propagation, to almost 15kHz range covered by stop bands.

Figure 10 .

 10 Figure 10. WFEM frequency response function of periodic double cell beam for flexural waves

Figure 11 .

 11 Figure 11. WFEM frequency response function of periodic double cell beam for longitudinal waves

Figure 12 .

 12 Figure 12. WFEM FRF for flexural waves of super unit cell with 6 th order of Fibonacci sequence

Figure 13 .

 13 Figure 13. FRFs of the flexural waves for 11 th order of Fibonacci beam with geometrical variation

Fig. 15 is

 15 Fig.15 is dedicated to the longitudinal waves of the case M 2, where it shows a change in the location of band gaps compared to the flexural waves. The ranges of band gaps can be observed in two locations between [4400 4800] Hz and [6700 10000] Hz. It has much better influences regarding the locations of the band gaps compared to the

Figure 14 .

 14 Figure 14. FRFs of the flexural waves for given orders of Fibonacci beam with material variation.

Fig. 16 shows

 16 Fig.16shows three curves, each corresponding to the material combination of case M2. Steel A-36 is selected as a constant material (cross-section (A)) and the copper,

Figure 15 .

 15 Figure 15. FRFs of the longitudinal waves for given orders of Fibonacci beam with material variation.

Figure 16 .

 16 Figure 16. FRFs of the 11 th order of Fibonacci beam with material variation.

  , three types of cross-section variation for double cell periodicity is plotted by fixing type I. The investigated frequency range is zoomed to [0 2] kHz for periodic double cell in order to visualise precisely the shift of band gaps. There is one main observation: the shift in frequency. Concerning the first, again the band gaps move to higher frequencies at increasing the height of cross section B. The frequency stop bands in the first types is around [650 850] Hz, whereas for Type II it shifts to higher frequencies at around [1000 1350] Hz with a little width compared to first type.

Figure 17 .

 17 Figure 17. WFEM frequency response function of periodic double cell beam for flexural waves

Figure 18 .

 18 Figure 18. Dispersion curve of super unit cell with 6 th order of Fibonacci sequence

Figure 19 .

 19 Figure 19. Spectral behaviour of band gaps for periodic and quasi-periodic embedded beams

Figure 20 .

 20 Figure 20. FRFs of the Fibonacci and Thue-Morse beams with the material variations.

[2000 2500 ]

 2500 Hz the stop bands of the Thue-Morse model outperforms, both in width and in depth, the one obtained by Fibonacci one. In the frequency range [3000 4000] Hz the behaviour is vice versa.Prepared using sagej.cls

Figure 21 .

 21 Figure 21. Dispersion curve of the Fibonacci and Thue-Morse beam based on flexural waves.

2 . 3 .

 23 Forced response studies of four types of geometrical variations in identical super-unit-cells and double unit cells/patterns shows three aspects: i) change in extension/enlargement of frequency band gaps, ii) shift of frequency band gaps to lower frequency range and iii) an increase in the depth of amplitude of frequency band gaps. Frequency response function of a quasi-periodic beam of 11 th order with four types of geometrical and material variations gives an efficient results containing enlarging the wide of band gaps towards left and right of frequency ranges.

Table 1 .

 1 Example of number of cells according to Fibonacci orders.Fibonacci orders function of number of cells Orders 4 th 5 th 6 th 7 th 8 th 9 th 10 th 11 th

	Number of Cells	5	8	13	21	34	55	89	144

Table 2 .

 2 Mechanical properties of quasi-periodic bi-material beam.

Material

Modulus of elasticity (Pa) Poissons Ratio Density (kg • m 3 ) Steel A-36 2 ⇥ 10 11 0.26 Aluminium 2045-T4 73 ⇥ 10 9 0.33 Magnesium 45 ⇥ 10 9 0.35 Copper 110 ⇥ 10 9 0.355

Table 3 .

 3 Sizes of cells A and B as sub-cases for M1.

			Geometrical variation
	Type Cell width [mm] height [mm] Factor
	I	A B	42.00 15.36	42.00 15.36	2.7
	II	A B	40.00 20.00	40.00 20.00	2
	III	A B	38.00 23.58	38.00 23.58	1.6
	IV	A B	31.62 31.62	31.62 31.62	1

Prepared using sagej.cls

Acknowledgements

This paper containing research investigation, carried out in the framework of the VIPER project (VIbroacoustic of PERiodic media). This project has received funding from the European Union's Horizon 2020 research and innovation program under Prepared using sagej.cls

In the first case an analysis of the harmonic response of quasi-periodic beams with increasing orders (length) of Fibonacci sequence with the finite element method is investigated. In this case, beams are made up of cells (constant length) whose cross-section areas and materials properties follow a Fibonacci sequence is studied.

The second one relays on the four types of geometrical variations. The geometrical variations of case M1 are applied in the periodic and quasi-periodic beams of double and super unit cells. In this case while comparing the results, global mass of the beams are kept constant and cross sections are varied.

The last case consist of spectral analysis or the wave propagation behaviour of periodic structures with the WFEM. In this case beams are made up of identical superunit-cells/patterns which are composed of cells whose properties follow Fibonacci sequence of a 6 th order.

Main results

The first results by increasing the orders of a quasi-periodic beam with the impedance mismatch due to geometric variation (cross-section variation) give a clear view of the phenomena. A rapid growth in the depth amplitude of the band gaps by increasing the Fibonacci orders can be noted.

Studies of the geometric variation applied to a quasi periodic beam is extended, in a proper way for the periodicity condition, in the WFEM method to reduce and increase the volume representative of cross-section (A) and cross-section (B) proportionally, while keeping the total mass of the both cells constant. Focused on flexural and longitudinal waves, four types of numerical models are designed for the spectral analysis.

As stated in the last case study conclusion, a quasi-periodic beam with 6th order of Fibonacci sequence is placed in a super unit cell for FB waves analysis. Four types of cross-section variations in the beams are considered. The main results, obtained for 6th order, show that a larger difference between the cross-sections (i.e. cross section (A) is much larger and cross section (B) is much smaller) lead to three main effects: i) change in extension/enlargement of frequency band gaps, ii) shift of frequency band gaps to lower frequency range and iii) an increase in the depth of amplitude of


Hanna Palmér
email: hanna.palmer@lnu.se

Camilla Björklund
email: camilla.bjorklund@ped.gu.se

Toddlers exploring structural elements in play

Keywords: mathematizing, play, structural awareness, toddlers

The focus of this paper is on possibilities for children to discern elements of structure and to develop structural awareness through play. In Sweden, preschool is part of the education system why teaching is to be conducted. At the same time, play is to be promoted and consciously used. In this paper, we explore possibilities for the youngest preschool children (2-3-yearolds) to discern structural elements in a rule-based play. In the example, discerning structural elements becomes of importance for being able to participate proficiently in the play. As such, the example shows that a conscious use of play may promote toddlers' development of structural awareness which in turn is of importance for further mathematical learning and for their participating in the play.

Introduction

Studies show that early mathematics has positive effects on later school performance [START_REF] Duncan | School readiness and later achievement[END_REF][START_REF] Ginsburg | Early mathematics education and how to do it[END_REF] but there is no consensus when it comes to how early mathematics is to be conducted [START_REF] Palmér | Different perspectives on possibledesirableplausible mathematics learning in preschool[END_REF]. A long-time debated question is if teaching is to be integrated with or separated from children's play (Bennet, 2005) and similar to other countries there is a discussion in Sweden regarding the relation between play and teaching. In this discussion, playoriented activities withholding mathematical content sometimes are set as a counter-pole to systematic teaching of a content. At the same time as there are researchers emphasizing a consolidation of play and teaching (Pramling & Pramling Samuelsson, 2011) there are traditions that uphold children's right to play, undisturbed by adults, for the sake of play itself [START_REF] Sundsdal | Til forsvar for barns spontane lek[END_REF].

In this paper we will not immerse in this debate but instead focus on an example where children need to mathematize the idea of structure, to be able to participate proficiently in an ongoing play. Structural awareness has been researched by Mulligan and Mitchelmore (2013a,b) and been shown to be both salient and critical to children's mathematical learning. Mulligan and Mitchelmore (2013b) have made interventions to promote children's structural awareness. These interventions showed that teachers can plan and direct children's attention towards patterns and structures in a developmentally suitable way. In this paper we analyze an example where young children (2-3yearolds), to be able to participate in an ongoing play, need to discern the elements of the structure of the play. Thus, the children in this example are younger than the children in the studies by Mulligan and Mitchelmore (2013b) and the setting is very different. The aim of this paper is to explore if the research on structural awareness by Mulligan and Mitchelmore (2013b) is applicable in this context and how these young children with support of their preschool teacher may discern structural elements in the ongoing play. This is of interest since such a discernment would be of value both for their possibility to participate proficiently in the ongoing play and for their further mathematical learning, as suggested by Mulligan and Mitchelmore (2013a,b).

Teaching and play

In Sweden, preschool is a non-mandatory pedagogical practice available to all children aged one to six years. Preschool is part of the education system why teaching is to be conducted (Education Act 2010:800). At the same time, play is to be promoted and consciously used in relation to children's learning (National Agency for Education, 2010). In the national curriculum it is however not expressed or exemplified how play is to be incorporated with children's learning.

According to [START_REF] Van Oers | Are you sure? Stimulating mathematical thinking during young children's play[END_REF], the relation between teaching, mathematics and play can be seen as either "mathematics made playful" or "mathematizing elements of play" (p.74). Games in which counting, sorting and different mathematical operations are prominent are examples of mathematics made playful while mathematizing elements of play implies that the primary act is play but a teacher introduces mathematical concepts or operations into the play. The teacher's way of acting towards children's play thus have implications for how mathematics is introduced and thereby made meaningful to the children. [START_REF] Walsh | It's teaching ... but not as we know it': Using participatory learning theories to resolve the dilemma of teaching in play-based practice[END_REF] have found three different approaches towards children's play that are considered to be based on the teacher's views on play, learning and teaching. Some teachers do not participate in children's play as they think children learn and develop naturally through play why the participation of preschool teachers would primarily destroy a natural process. An opposite to this is teachers that continually interrupt children's play by asking questions, questions that seldom are meaningful in relation to the play but rather connect to different curriculum content. The third approach includes a balanced commitment to participation in children's play: not entirely child-centered or teacher-led but a balanced mix of reciprocity and listening to children's initiatives and need for support. This third approach was shown to be most successful for children's learning in the study of Walsh et al. A similar approach based on empirical studies is found in so called Developmental Education (van Oers, 2012) which has been further developed and theorized as play-responsive teaching in a recent study by Pramling et al. (in press). Both these theoretical stances emphasize a starting point in children's meaning making and a collaborative structuring (between children and teacher) of activities (for example play) that are enriched with cultural meanings (for example mathematics) in interactive processes.

Teachers who actively take part in children's play with pedagogical intentions may however do this in distinctly different ways which has been observed by [START_REF] Björklund | Teachers' involvement in children's mathematizing -Beyond dichotomization between play and teaching[END_REF]. Some confirm children's direction of interest, which implies that the teacher responds to and confirms the children's initiatives and ideas. A teacher may also contribute by providing strategies, which is a more goal-oriented act of directing the child towards skills or tools that will help him/her master a challenge. Thus, providing strategies can be understood as the teacher providing the child with new strategies to facilitate the activity the child is involved in. If a teacher is situating known concepts, this implies that the teacher is applying a mathematical concept known from another situation into the play, enabling generalization of the concept. Finally, when challenging concept meaning, the teacher highlights or introduces a mathematical problem or concept within the play context where the child/ren need to work together in an intellectual way to solve the problem or to clarify the concept. All of these different ways to actively take part in children's play can be considered as a balanced commitment of participation (c.f. [START_REF] Walsh | It's teaching ... but not as we know it': Using participatory learning theories to resolve the dilemma of teaching in play-based practice[END_REF]) that may extend the child/ren's knowledge and may also add new value to their play.

Pattern and Structure

It is well known that preschool children can both note and construct different kind of patterns. However, it is only more recently that research has shown that children's exploration and construction of patterns may have positive influence on their general learning of mathematics [START_REF] Cross | Mathematics learning in early childhood: Paths toward excellence and equity[END_REF][START_REF] Mulligan | Encouraging mathematical thinking through pattern and structure. An Intervention in the first year of schooling[END_REF]. For example, exploration and construction of patterns offer children opportunities to discover and understand repeating structures, which in turn is fundamental for later understanding of variables, functions and algebra [START_REF] Moss | Knowledge building in mathematics: Supporting collaborative learning in pattern problems[END_REF].

According to Mulligan and Mitchelmore (2013a) awareness of mathematical pattern and structure is both salient and critical to mathematical learning, since effective mathematical reasoning involves the ability to note patterns and structure. They define pattern as involving a predictable regularity and structure as the way in which various elements (for example in a pattern) are organized and related. Based on studies on children aged 4-8 Mulligan and Mitchelmore (a.a) divide children's development of structural awareness into four levels. Prestructural awareness implies children taking notice of properties or features that appeal to them but often the properties or features are not relevant to the underlying mathematical concept. Emergent structural awareness implies that children start to distinguish some regularities and/or structures but they are unable to organize them appropriately. Partial structural awareness implies that children distinguish several patterns and/or structures but they still struggle with organizing and representing them as a whole. The forth level is structural awareness, which implies that the children correctly organize and represent a given structure. In the studies by Mulligan and Mitchelmore (a.a) the children's structural awareness were consistent across tasks and the level of structural awareness correlated with the children's mathematical performance. In later studies, Mulligan and Mitchelmore (a.a) added a fifth level, advanced structural awareness, implying children being able to generalize underlying features between contexts.

Based on the above, Mulligan and Mitchelmore (2013b) have made interventions to promote children's structural awareness. These interventions show that teachers can plan and direct children's attention towards patterns and structures in a developmentally suitable way. When children in their interventions had developed structural awareness they learned basic properties of number, space and measurement more easily and they also made more connections between numerical, measurement and spatial mathematical ideas. There are few studies on structural awareness among preschool children below three years but there are examples of an emerging senses of structure and internal relations expressed when children are allowed to explore materials and activities in play-based manners or with guidance [START_REF] Björklund | Playing with patternsconclusions from a learning study with toddlers[END_REF], relationships that however may not always be based on logic but on the child's own experiences and intuitions [START_REF] Björklund | Klassificering och mönsterförskollärares intentioner och småbarns initiativ[END_REF].

The Study

The empirical material in this paper stems from a larger research project [1] aiming to investigate the teaching-play relation in Swedish preschool practice. Data for analysis consists of teachers' selfcollected video-documentations where the teachers together with their preschool children are engaged in different kinds of play. Only children with written consent from their legal guardians (in accordance with ethical guidelines from the Swedish Research Council, 2017) participate in the video-documented activities. For this particular study, one video-documentation is chosen as it contains a teacher initiated rule-based play where certain acts and roles are necessary to comprise. To participate proficiently in the play it becomes important to discern certain structural elements. Because the play contains repeated parts of actions, it is possible to gain a comprehensive understanding of the complexity of the play and the necessary skills that are needed to conduct the play. Thus, the example is information-orientated aiming to maximize the utility of information from small selection [START_REF] Flyvbjerg | Five misunderstandings about case-study research[END_REF]. In the analysis we explore how the youngest preschool children (2-3-yearolds) discern structural elements in the play. We also explore if Mulligan and Mitchelmore's (2013a) four levels of structural awareness is possible to apply on a situation quite different from the situations in their studies. In their studies, the analysis is based on children's written documentations and the level of structural awareness is set in relation to different mathematical concepts. In this paper the analysis will be based on children's actions in the play and the level of structural awareness is set in relation to children's discernment of the structural elements in the play.

In the following section the example will first be presented divided into 7 episodes. Each episode constitutes a part of the play where the teacher and the children initiate an act and follow it through. These episodes are then analysed through thematic analysis [START_REF] Braun | Using thematic analysis in psychology[END_REF] to identify themes or patterns in the observed activity that are interpreted in accordance with a theoretical framework; in this case the levels of structural awareness (Mulligan and Mitchelmore, 2013a).

Discerning elements of structurean example

Three boys and one girl, aged 2 and 3, are to play hide the key with their preschool teacher. Instead of a key they have a green toy dragon that is to be hidden. It is a new play for the children and to be able to participate proficiently in the play the children need to discern the elements of structure in the play. The video-documentation is ten minutes long and will be presented and discussed through 7 episodes where the children are called boy 1, boy 2, boy 3 and girl 1. Episode 1: The children are standing outside the room while the teacher hides the dragon behind a basket standing under a shelf. She opens the door, sits down and says that she has hidden the toy dragon far down, not high up. She emphasizes down and high by movements with her hand. The children are then encouraged to start looking for the toy dragon. Boy 1 finds the toy dragon before the others have even started their search.

Episode 2: The teacher says that now the child who found the toy dragon is to hide it, that he can hide it far down or high up but he should wait until the others are outside the room. Boy 1 who is to hide the toy dragon is left alone in the room. He is standing in the middle of the room holding the toy dragon in his hand up towards the ceiling. He says "high up". After a little while the teacher opens the door to ask if he is finished.

Teacher:

Can I help you? Boy 1: I hide it up. (stretches his hand with the dragon towards the ceiling) I hide it head. Teacher:

In the ceiling?

The teacher takes the toy dragon and hides it high up (not visible where in the video). The boy opens the door and the other three children instantly run to the previous hiding place. The teacher says to the boy who has hidden the toy dragon not to tell where it is hidden but to tell if it is hidden far down or high up. Boy 1 then looks up and points towards the hiding place.

Episode 3: Boy 2 is in turn to hide the toy dragon and he shows with his hands that he wants to hide it far down. The teacher helps him to find a hiding place behind some big pillows under a table (to the right in Figure 1). After he has opened the door to let the other children in he himself directly walks to the hiding place and points towards the toy dragon. The girl follows him and takes the toy dragon but then boy 2 gets angry. Boy 3 does not take notice of them but instead searches at the first hiding place. While the teacher explains to boy 2 that it is now the girls turn to hide the toy dragon the girl walks out and closes the door. The teacher goes after her telling that she should stay inside because it is now her turn to hide the toy dragon. Then the girl takes the toy dragon and before the other children has had time to leave the room she puts it on same shelf as the first hiding place. Boy 1 and 2 leave the room by themselves and the teacher walks boy 3 out of the room.

Episode 4 When all three boys are outside the room, the girl again hides the toy dragon on the same shelf as the first hiding place. Then she walks to the door. It seems like she is going to go out and close the door but the teacher stops her and asks her to let the other children in and tell them that the toy dragon is hidden quite far down.

Episode 5: Boy 3 is to hide the toy dragon and he chooses the same hiding place as in episode 1 and 4. He himself also "finds" the toy dragon immediately afterwards, before letting the other children into the room. The teacher explains that he instead should let the other children into the room. When he does, they go directly to the hiding place.

Episode 6: Boy 1 says that it is now his turn to hide the toy dragon. The girl immediately goes out of the room and the other two boys follow her. The teacher tells boy 1 that he is to hide the toy dragon in a new place and she emphasizes far down and high up. She also tells him not to tell the others where it is hidden. He hides the toy dragon at a new place under a box standing on the floor. When he lets the other three children into the room they directly run to the previous hiding place.

When the toy dragon is not there they seem to get a little confused. They start to look at other places but find the toy dragon first after some guidance from the teacher.

Episode 7: Boy 2 is about to hide the toy dragon and he chooses the same hiding place as in episode 1, 4 and 5. The teacher tells him not to tell the others where it is hidden but to tell them if they should look far down or high up. When the children are let into the room the boy himself runs to the hiding place and "finds" the toy dragon. As the teacher tries to explain for him that it was the others who were to search for the toy dragon the girl walks outside the door.

Analysis -structural awareness in the play

According to Mulligan and Mitchelmore (2013a) structure is the way in which various elements are organized and related. To make use of a structure the child needs to recognize the internal relationship between the elements. In the "hide the toy dragon play" there are various elements that form the structure of the play that the child has to discern and relate to other necessary elements:

-choose between two options of hiding positions: far down or high up -all but one should leave the room and close the door when the toy dragon is to be hidden -only the one who is to hide the toy dragon can stay in the room -the hiding place should be different from the previous turn -after hiding the toy dragon the door is to be opened and all children gather inside the room -when being let into the room you are to search for the toy dragon -if you have hidden the toy dragon you are not to tell where it is hidden but indicate with spatial instructions "high up" or "down low" -the one who finds the toy dragon is in turn to hide it the next time -all children are to hide the toy dragon once which is not an element of the play but connected to fairness

During the play the children successively distinguish some of the elements above but they are unable to organize them appropriately which is labeled as emergent structural awareness by Mulligan and Mitchelmore (2013a). For example, all children know that they should search for the toy dragon when they enter the room (observed in all episodes). However, mostly they start to search at the same hiding place as the previous turn (episode 2, 3, 5, 6) and boy 3 "finds" the dragon when he is "the hider" (episode 5). Often they also hide the toy dragon at almost the exact same place as the teacher did in episode 1 (episode 3, 4, 5).The element of leaving the room after the toy dragon has been found is not discerned by boy 3. Instead, the teacher takes his hand and walks him to the door each time (for example episode 3). This element is however immediately discerned by the girl who very quickly leaves the room each time the dragon has been found. On the other hand, she does not recognize the specific circumstances of being the "hider" to which this element does not apply, which is shown as she leaves the room when she is the one who found the dragon and thus in turn to enact the hiding element (episode 3). The girl has discerned that one is to leave the room after the toy dragon has been found but has not yet discerned the internal relationship of being a hider and a searcher, a role that in turn is determined by who found the dragon. Thus, one issue that increases the difficulty in discerning the elements of the structure in the play is that the structure's internal relationships change when you are the one who finds the toy dragon. Another expression of this complex relationship is shown when the boys in episode 3 and 7 "finds" the toy dragon they themselves have hidden.

Discussion

Mulligan's and Mitchelmore's (2013b) interventions showed that teachers can plan and direct children's attention towards patterns and structures. The children in our study are younger and the setting is very different why we in the analysis explored possibilities for the youngest preschool children (2-3-yearolds) to discern structural elements based on an ongoing play. We also explored if the four levels of structural awareness is possible to use when analysing a play situation. Even though the teacher's mathematical focus initially seems to be on far down and high up she provides strategies (c.f. [START_REF] Björklund | Teachers' involvement in children's mathematizing -Beyond dichotomization between play and teaching[END_REF] for the children to discern elements of structure. The example illustrates that elements of structure is possible for children to discern within play if a teacher provides strategies for this. The example also indicates that emergent structural awareness starts to develop among children younger than those in the studies by Mulligan and Mitchelmore. The notion of mathematizing implies a process where mathematics becomes of importance when children explore different phenomena [START_REF] Freudenthal | Why to teach mathematics as to be useful?[END_REF]. The example in this paper can be seen as mathematizing structural elements of play [START_REF] Van Oers | Are you sure? Stimulating mathematical thinking during young children's play[END_REF], not in the sense that the teacher tries to introduce mathematical concepts or operations into the play but instead in the sense the teacher tries to make visible mathematical elements of the play. By repeating the play several times children's discernment of important features becomes visible. It also becomes visible how the teacher has an important role in this discernment process. The teacher shows a balanced commitment in the play [START_REF] Walsh | It's teaching ... but not as we know it': Using participatory learning theories to resolve the dilemma of teaching in play-based practice[END_REF], with a mix of reciprocity and listening to children's initiatives and need for support. Considering the emphasis in the Swedish curriculum on play as a way to teach and learn in preschool, this study broadens our knowledge of how discerning structural elements becomes of importance for children to be able to participate proficiently in rule-based play. While Mulligan's and Mitchelmore (2013a,b) have shown that structural awareness is both salient and critical to children's mathematical learning the analysis in this paper indicates that structural awareness may start to develop among children even younger than previously thought and in situations not obviously labelled as mathematical.

[1] The study is funded by the Swedish Institute for Educational Research, grant no. 2016/112.

Figure 1 :

 1 Figure 1: Picture of the room where the play is conducted. The door is in the upper corner.


S Timorian

M Ouisse

N Bouhaddi

S De Rosa

F Franco

BAND DIAGRAM AND FORCED RESPONSE ANALYSIS OF PERIODIC AND QUASI-PERIODIC PANELS

Keywords: Quasi-periodic structures, band diagram, frequency stop-band, forced response, metamaterials

The vast majority of structures introduce symmetry in one or two dimensions, which can be a periodicity in the case of constructing several identical unit cells. Periodic structures have found a big interest in engineering applications because they introduce frequency band effects, due to the impedance mismatch generated by periodic discontinuity in the geometry, which can improve the vibroacoustic performances. However, the presence of imperfections i.e. (defects, irregularity) in the structure, leads to a partial lost of regular periodicity (called quasi-periodic structure) that can have a noticeable impact on the vibrational and/or acoustic behaviour of the elastic system. The tailored irregularity, which is a designed loss of periodicity, can be used, therefore, to improve the dynamic behaviour of structures. In the present paper numerical studies on the vibrational analysis of two-dimensional non-planar periodic and quasi-periodic structures have been performed. The content deals with finite and infinite systems. The finite element models of solid structures focused on the band diagram analysis of the infinite systems and the forced responses of the finite structures. The quasi-periodicity is defined by invoking the Thue-Morse sequence for building the assigned variations (geometry) along the domain of finite element model. The wave characteristics in quasi-periodic panels, present some elements of novelty and could be considered for designing structural filters and controlling the properties of elastic waves.

INTRODUCTION

Periodic metamaterials are one of the central topics in vibration and acoustics domain. A degree of regular periodicity in arrays of engineered metamaterials, always leads to design and create frequency stop bands (i.e. waves cannot freely propagate in these frequency ranges). One of the main features associated with the periodic distribution of mechanical devices is the ability of the system to block the propagation of waves in specific frequency ranges [START_REF] Collet | Floquet-Bloch decomposition for the computation of dispersion of two-dimensional periodic, damped mechanical systems[END_REF]. Such structures behave as filtering systems, since in some ranges of frequencies waves propagate without attenuation (if damping is neglected), while in others decay exponentially [START_REF] Movchan | Quasi-periodicity and multi-scale resonators for the reduction of seismic vibrations in fluid-solid systems[END_REF][START_REF] Carta | Transmission and localisation in ordered and randomly-perturbed structured flexural systems[END_REF]. They are also very well known metamaterials and phononic materials for exhibiting frequency band gaps, that can improve the vibroacoustic response of structures [START_REF] Kook | Investigation of Bandgap Structure in Coupled Acoustic-Mechanical System[END_REF][START_REF] Hussein | Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook[END_REF]. Recent studies show interesting topics on architectured material like phononic crystals, metamaterials, and auxetic sandwich cores [START_REF] Qing-Tian | Wave propagation in sandwich panel with auxetic core[END_REF] to control the propagation of elastic waves. These architectured material rely on two mechanisms for forming band gaps. Phononic crystals can form Bragg type band gaps, but are limited in the low-frequency regimes because their unit cell size scales with wavelength. In contrary metamaterials overcome this size, because they rely on resonances, but the resulting band gaps are narrow [START_REF] Matlack | Composite 3D-printed metastructures for low-frequency and broadband vibration absorption[END_REF][START_REF] Krushynska | Coupling local resonance with Bragg band gaps in single-phase mechanical metamaterials[END_REF].

Numerical analyses lead to the design of nominally perfect periodic structures. However, the presence of imperfections i.e. defects or irregularities in post-manufacturing are highly likely to be occurred in regular periodic designs. Recent studies show that there are effects on random variability due to the additive manufacturing process on the performance of the elastic band gap behaviour in creating a stop band effect in the forced response of the waveguide [START_REF] Fabro | Uncertainty analysis of band gaps for beams with periodically distributed resonators produced by additive manufacturing[END_REF]. On the contrary, the present work is focused on tailored irregularities, which is called (quasi-periodicity), which are considered to have impacts on dynamical behaviour of the structures. The first challenge is how to predict the frequency band gap in infinite quasi-periodic systems. In fact, the design of periodic configurations is usually based on the use of Floquet-Bloch (FB) theorem which considers a single cell with adequate boundary conditions [START_REF] Billon | Design of smart metamaterials for vibration control: extension of Bloch approach to handle finite system boundary conditions[END_REF][START_REF] Hakoda | Using Floquet periodicity to easily calculate dispersion curves and wave structures of homogeneous waveguides[END_REF].

To design an enhanced quasi-periodic structure and analyse the impact of this design in terms of dynamics response, two methods are under investigation. A forced response of the finite quasi-periodic structure and a periodic panel which has commonality in terms of in-plane dimensions. In the second method in order to predict the frequency stop band zones a dispersion analysis of the infinite periodic structure is designed. In this design two meso-scale unit cells, which represent the dispersion analysis of the infinite periodic structures are considered. The predicted dispersion curves can be directly compared with its finite periodic counter parts for frequency band gaps prediction. In the forced response, those two meso-scale cells are combined using Thue-Morse morphism series in order to tailor a quasi-periodic panel. The FRFs of the quasi-periodic panel show effective results compared to strictly periodic panels in terms of band gaps width enlargement and level of attenuation.

MODEL AND LEXICON

Infinite structure

The model consists of a metamaterial waveguide. Metamaterials in the present investigation are an engineered sets of multiple elements to show some filtering properties. The waveguide consists of two different unit cells in meso-scale. The unit cells are made of polymers materials and the mechanical properties are given in Tab.1.

The unit cell is made of two sub-parts, on-board panel, and a resonator. The dimension of the meso-scale wave-guide is detailed in the Fig. 1. The information of the control volume of the both unit cells are taken into account for modelling a lightweight, easy-to-manufacture metamaterial [START_REF] Miniaci | Experimental observation of a large low-frequency band gap in a polymer waveguide[END_REF]. The on-board panel has a thickness of 2mm with an area of 35 × 35 mm 2 . The resonator shape has a height of 17.5mm and a lateral and horizontal span of 30mm. Cell A and cell B have two different shapes with approximated equal control volumes. The wall thickness of the cell A is 4.8mm and the circular hole in cell B has a radius of 8mm. Density (kg • m -3 ) 1050 Poisson's ratio 0.34

Finite structure

In the finite panel two perfectly periodic and two quasi-periodic structure are considered. In the last two models specific Thue-Morse sequences are used to generate impedance mismatches in the view of desired degree of quasi-periodicity. The degree of quasi-periodicity might be controlled with mathematical rules that are introduced in the next section.

Thue-Morse sequence

A Thue-Morse sequence is binary sequence (an infinite sequence of 0s and 1s). The sequence starts with 0 then 01, 0110,01101001, and so on, [START_REF] Allouche | The ubiquitous prouhet-thue-morse sequence[END_REF]. For the simplicity, let us define the Thue-Morse morphism to be the function s from the set of binary sequences to itself by replicating every 0 in a sequence with 01 and every 1 with 10 [14]:

s(x, y) = s(x)s(y) (1)
Eq.1 defines a map s for all strings x, y. Defining the Thue-Morse morphism s(0) = 01, s(1) = 10 leads to the relations for increasing the quasi-periodicity pattern in two directions:

s(0) = 01 = AB s 2 (0) = s(s(0)) = 0110 = ABBA

Periodic sequence

The two periodic panels have a repetition of single cell A for first panel, and single cell B for second panel respectively with the dimensions of 8 × 8 shown in Fig. 3. According to the Fig. 3, the panel has a thickness of 2 mm, and the hallow resonator has a wall thickness of 4.8 mm.

µ U = e -jk y L µ B (2)
where, µ R and µ L are the displacement in the right and left side of the x axis and µ U and µ B are the displacement in the upper and bottom of the y axis. L is the length of reciprocal element, k x and k y are the wavenumers in the x and y directions. Band diagrams are analysed by setting up eigensolver analysis as a parametric sweep using one parameter, k, which varies from 0 to π. The range of the k parameter spans three wave vectors starting from (A to R to S and back to A), covering the edges of the first irreducible Brillouin zone Fig. 4. Two different unit cells are considered for the dispersion analysis. First cell A and second cell B. Both cells have commonalities in terms of mass, volume fraction, material and structural properties in order to be useful periodic tool to be compared with the quasi-periodic counter part. The results predicted by these two cells could lead to design a quasi-periodic lattice, which is a finite panel with a tailored irregularity that is invoked by Thue-Morse morphism deterministic approach. The results of the band diagram is plotted in Fig. 5 There are two complete frequency band gaps in each sub-figure. The first band gaps with small ∆ω = 570 Hz that represents the (resonance band gaps) following the larger Bragg

Finite element analysis

In this part, four types of investigations, on periodic and quasi-periodic panels; specifically, the aim is to apply on a finite structure the results obtained with a infinite periodic sequence. Fig. 6 show three curves; by focusing on the Bragg type band gaps zone, the blue curve with dash lines corresponds to the cell B. It has thinner stop band wide compared to the other curves and there is also a slight shift in the stop bands towards the higher frequencies compared to the two cell A and combined (A, B). Despite of that cell A has a shift of the same band gaps zone towards the lower frequency ranges. Paying attention to the dark green solid curve that corresponds to the combined cell A and cell B, it has wider band gaps compared to both blue dash curve and dotted orange curve almost around 10.5 -14.2 kHz. Also it should be mentioned that in the velocity amplitude response it has the lowest pick in comparison of two other curves. The highest picks could be seen at around 8.25 kHz, and the orange has the highest pick with the amplitude of 20dB. In conclusion of the first part, it could be predicted that the combination of two unit cells in finite quasiperiodic panel (8 × 8 unit cells) is a compromise model and it has impacts on the dynamic behaviour of panel compared to the two other strictly single periodic panels in terms of wide of band gaps and lower attenuation level.

In the second part of the finite element analysis, an FRF of the quasi-periodic structure with (840 × 280 × 2 mm 3 ) is taken into account. This meta-structure shown in Fig. 7 is made of three sub-parts. The middle part is a quasi-periodic panel of combined cell A and cell B and the two other parts in the left and right hand side bare panels with no resonators on it.

Timorian et al

The panel has a cantilever constraint (fixed boundary condition) marked blue in Fig. 7. A unit white spectrum force is applied to the input zone right left hand side bare panel (marked with red arrow) and the response is observed in terms of Root Mean Square (RMS) with the average node points in the input and output zones. Fig. 8 shows FRF response in terms of velocity amplitude in dB. Practically, this type of panel is used as a junction filter between two bare panels. The induced vibration energy transfers through this junction, and it acts as a meta-material filtering property to the elastic waves due to the impedance mismatch in the shape of resonators embedded in the bare panels.

The FRF shows two types of curves, the blue one corresponds to the the average of the 64 nodes in the input zone, where all the elastic waves are confined in the zone and there is no sign of frequency stop bands. The orange curve corresponds to the response of the output zone. The frequency stop bands starts from 10.36 kHz and continues to 13.6 kHz. There are some picks around 12 kHz inside the band gap in shaded area. Three frequencies are chosen inside the shaded area and their operative modes or forced deflections are plotted in Fig. 9.

A similar procedure for the numerical modelling of filtering effects has been done in the paper of Billon et al [START_REF] Billon | Design of smart metamaterials for vibration control: extension of Bloch approach to handle finite system boundary conditions[END_REF]. They basically selected 4 points in the FRFs. Two points are located in the inside the predicted band gap and two outside the band gap. One point inside the band gap does not really reflects good response despite of being in the predicted band gaps. In fact, in the present FRFs only three points are considered, two of them are on the lower and upper bound of the band gaps and the third one is taken in the middle of the attenuation zone. As it can be seen from the results in the paper of Billon et al [START_REF] Billon | Design of smart metamaterials for vibration control: extension of Bloch approach to handle finite system boundary conditions[END_REF], the average squared amplitude velocity around 43.3 kHz which is inside the predicted band gap zone, has visible deformed shapes over -20dB. In contrary, although there are some picks in the middle of band gap in the present result but they are bellow -25dB with less deflections. The operative modes in Fig. 9 shows the deformation of the panel in 10.36 kHz, the start point of band gaps, 11.36 kHz middle of the band gaps, and 13.12 kHz almost at the finishing point of the band gaps. It can be seen that the panel has an efficient filtering effects by using quasi-periodic pattern.

CONCLUSIONS

The first aspect is formulating fundamental procedures to design a quasi-periodic meta-structure using Thue-Morse morphism sequence. The second aspect is dedicated to analysis of the impact of this design in terms of dynamical behaviour. The results of FRFs and dispersion analysis shows that quasi-periodic pattern have impact on dynamic behaviour of panels, especially in terms of isolating unwanted vibrational behaviour all over the metamaterial part of the span, specifically inside the attenuation level. In this analysis, the presented tailored irregularity, which in higher extent is a designed loss of periodicity in finite structure has influential access to extend the attenuation level to lower velocity amplitude-dB and slightly increase the wide of the band gaps. These impact came due to the impedance mismatch of the combined (resonators) embedded on the bare panel. There are some points to be highlighted in the below:

The quasi-periodicity (Thue-Morse morphism) in two directions shows lower frequency band gaps compared to the one direction ones like Fibonacci sequences. Quasi-periodic panel in the present article has an efficient impacts on the vibrational analysis compared to the single periodic panels. In general quasi-periodic pattern is capable of confining the energy in some modes, and in higher extends there are possibilities to shift those localised modes in desired frequency bands.

Figure 1 :

 1 Figure 1: Meso-scale unit cells of infinite periodic panels

s 3

 3 (0) = s(s(s(0)) = 01101001 = ABBABAAB Each 0 and 1 represents cell A and cell B respectively. In the present finite element model only 8 combined unit cells are used, it means that the Thue-Morse morphism map stops at s 3 .Two types of quasi-periodic finite panels are considered. The first one is a pure quasi-periodic panel (8 × 8 unit cells) and the second quasi-periodic panel is designed similar to the first one with an addition of two bare panels without resonators in the right and left sides of the quasi-periodic one. These two bare panels are used as an input zone in the left and output zone in the right side of filtering domain shown in Fig.2.

Figure 2 :

 2 Figure 2: Quasi-periodic panel with the combination of cell A and B

Figure 3 :

 3 Figure 3: Two strictly periodic panels, left with cell A periodicity and right with cell B

Figure 4 :

 4 Figure 4: Diagram of the first irreducible Brillouin zone for a square lattice

Figure 5 :

 5 Figure 5: Band diagram of infinite panels with a representative, left cell A and right cell B

Figure 6 :

 6 Figure 6: FRFs of three finite structures in terms of velocity amplitude in dB response

Figure 7 :

 7 Figure 7: Geometry of meta-structure (filter)

Figure 8 :

 8 Figure 8: FRFs of the meta-structure in terms of RMS of velocity amplitude in dB

Figure 9 :

 9 Figure 9: Operative modes of the meta-structure with three selected frequencies in the upper, lower and middle position of the attenuation level

Table 1 :

 1 Material properties of the metamaterial

Material

Polymer Modulus of elasticity(Pa) 2.32 × 10 9

ACKNOWLEDGEMENTS

This research was supported by the framework of the VIPER project (VIbroacoustic of PERiodic media). This project has received funding from European Union?s Horizon 2020 research and innovation program under Marie Curie grant agreement No 675441 and EUR EIPHI (ANR 17-EURE-0002) project. Thanks to our colleagues who provided contributions with insights and expertise in this research.