Tunable Low Frequency Band Gap and Waveguide of Phononic Crystal Plates with Different Filling Ratio

crystals

Article Tunable Low Frequency Band Gap and Waveguide of Phononic Crystal Plates with Different Filling Ratio

Shaobo Zhang, Jiang Liu * , Hongbo Zhang and Shuliang Wang

School of Mechanical & Automobile Engineering, Qingdao University of Technology, No. 777 Jialingjiang Road, Qingdao 266520, China; [email protected] (S.Z.); [email protected] (H.Z.); [email protected] (S.W.) * Correspondence: [email protected]; Tel.: +86-131-6502-8316

Abstract: Aiming at solving the NVH problem in vehicles, a novel composite structure is proposed. The new structure uses a hollow-stub phononic-crystal with filled cylinders (HPFC) plate. Any unit in the plate consists of a lead head, a silicon rubber body, an aluminum base as outer column and an opposite arranged inner pole. The dispersion curves are investigated by numerical simulations and the influences of structural parameters are discussed, including traditional hollow radius, thickness, height ratio, and the new proposed filling ratio. Three new arrays are created and their spectrum maps are calculated. In the dispersion simulation results, new branches are observed. The new branches would move towards lower frequency zone and the band gap width enlarges as the filling ratio decreases. The transmission spectrum results show that the new design can realize three different multiplexing arrays for waveguides and also extend the locally resonant sonic band gap. In summary, the proposed HPFC structure could meet the requirement for noise guiding and filtering. Citation: Zhang, S.; Liu, J.; Compared to a traditional phononic crystal plate, this new composite structure may be more suitable Zhang, H.; Wang, S. Tunable Low for noise reduction in rail or road vehicles. Frequency Band Gap and Waveguide of Phononic Crystal Plates with Keywords: phononic crystal; whispering gallery mode; one-side filled plate; vibration and noise Different Filling Ratio. Crystals 2021, reduction 11, 828. https://doi.org/10.3390/ cryst11070828

Academic Editors: Zhimiao Yan, 1. Introduction Ting Tan and Víctor With the development of phononic crystals (PCs) [1–4], new characteristics of controlling J. Sánchez-Morcillo acoustic/elastic waves have been studied, such as low frequency band gaps [5–7], waveguides [8–10], filters [11,12], sensing [13], and sound focusing lenses [14,15]. Among these Received: 16 June 2021 research topics, the band gap and waveguide are more important for vehicle NVH problems. Accepted: 14 July 2021 Published: 16 July 2021 When the periodic pillar arrays are deposited on the thin PCs, two types of band gaps can be found by selecting appropriate geometric parameters to impede acoustic wave

Publisher’s Note: MDPI stays neutral transmission [16,17]. Based on the wave analysis method, Lv et al. [18] investigated the with regard to jurisdictional claims in propagation characteristics of finite Timoshenko local resonance beams in forced vibration published maps and institutional affil- and periodically attached two-degree-of-freedom force-type resonators. The beams had a iations. wide low frequency band gap, but are very hard to be embedded into the vehicle’s skin structure. Using the Brillouin light scattering experiments, Graczykowski et al. [19] built whispering-gallery modes (WGMs). The models used holes and pillars within a silicone thin square film and introduced a high quality factor to define the lattice. WGMs could guide and filter sound waves in Bragg band gaps with a smaller size compared to the Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. former beams. However, this thin film structure limited low frequency band gaps and This article is an open access article cannot adapt to the change in vehicle drive situations. Liang et al. [20] calculated the distributed under the terms and dispersion curves of Timoshenko beams by a comprehensive with differential quadrature conditions of the Creative Commons method, unconventional matrix-partitioning method, and variable substitution method. Attribution (CC BY) license (https:// The results showed that the first band gap could be widened by increasing the mass ratio creativecommons.org/licenses/by/ or stiffness ratio, or by decreasing the lattice constant. In addition to the three parameters 4.0/). studied by Liang, the thickness or the filling rate of a composite structure may also affect

Crystals 2021, 11, 828. https://doi.org/10.3390/cryst11070828 https://www.mdpi.com/journal/crystals Crystals 2021, 11, x FOR PEER REVIEW 2 of 14

or stiffness ratio, or by decreasing the lattice constant. In addition to the three parameters studied by Liang, the thickness or the filling rate of a composite structure may also affect the band gap. Although the physical phenomena of the low frequency band gaps in PCs studies provide a potential application for vehicle’s NVH, the crystal lattice’s structure design has to face the challenge from serious radiated noises. Waveguides phenomena of the single or composite structure have also been studied Crystals 2021, 11, 828 by many scholars. By changing the geometry of the pillar, Hsu [21]2 of 13numerically studied the propagation of Lamb waves through a stepped resonator array on a thin plate. How- ever, the relationship between the two components is not clear and how to construct the the bandhigh gap. Althoughfrequency the waveguide physical phenomena by the structure of the low desi frequencygn is not band mentioned gaps in PCs either. Achaoui et al studies provide[22] reported a potential the applicationpropagation for of vehicle’s surface NVH, guided the crystalwaves lattice’s in periodically structure arranged columns design has to face the challenge from serious radiated noises. on a semi-infinite medium. However, only single model waveguide channels were inves- Waveguides phenomena of the single or composite structure have also been studied by manytigated scholars. in By Achaoui’s changing the study. geometry In of the the pillar,above-me Hsu [21ntioned] numerically papers studied [8–10, the21,22], different PC propagationstructures of Lamb are waves designed through ato stepped construct resonator wave arrayguides. on a The thin plate.waveguide However, was usually formed the relationshipthrough between line defects the two or components point defects is not at clear specific and howfrequencies. to construct It has the highbeen proved that these frequencydefects waveguide would by be the determined structure design by unit is not structure mentioned parameters, either. Achaoui for etexample al [22] the radius [2,9] or reportedthe the propagationheight [10,22]. of surface However, guided using waves only in periodically one parameter arranged to columns define on a acomposite structure semi-infinite medium. However, only single model waveguide channels were investigated in Achaoui’smodel study. may In thebring above-mentioned about confusion. papers The [8–10 confus,21,22],ion different regarding PC structures the description are of the rela- designedtionship to construct between waveguides. the substrate The waveguide and wasthe usuallyoscillator formed would through increase line defects the difficulty of wave- or pointguide defects forming. at specific Therefore, frequencies. we It need has been a more proved det thatailed these model defects in calculat would beions and simulations, determinednot by only unit for structure the waveguide, parameters, forbut example also the the band radius gap. [2,9 ] or the height [10,22]. However, usingTo onlyoffer one a satisfied parameter solution to define on a band composite gap structureand waveguide model may calculations bring suitable for ve- about confusion.hicle noises, The confusion a new composite regarding the structure description is proposed of the relationship in Section between 2. A thecrystal subunit in the substrate and the oscillator would increase the difficulty of waveguide forming. Therefore, we need aplate more consists detailed modelof a lead in calculations head, a silicon and simulations, rubber notbody, only an for aluminum the waveguide, base as outer column but also theand band an gap.oppositely arranged inner pole. In order to show more details required by the To offerwaveguide a satisfied and solution gap calculations, on band gap we and define waveguide a dimensionless calculations suitable factor FR for to describe the rel- vehicle noises,ative height a new composite relationship structure for two is proposed components in Section in a2 .lattice. A crystal Simulations subunit in are performed and the platethe consists dispersion of a lead curves head, a siliconand transmission rubber body, an spec aluminumtrum of base hollow-stub as outer column phononic-crystals with and an oppositely arranged inner pole. In order to show more details required by the filled cylinders are given at different FR values along the first irreducible Brillouin zone. waveguide and gap calculations, we define a dimensionless factor FR to describe the relative heightThree relationship new branches for two were components observed in ain lattice. the band Simulations gap simulations. are performed A and peak in the transmis- the dispersionsion spectrum curves and was transmission spotted spectrumcorresponding of hollow-stub to two phononic-crystalsof the branches. with In Section 3, the influ- filled cylindersence of are structural given at different parametersFR values on flatness along the of first dispersion irreducible curve Brillouin and zone.the width of band gaps Three newis branchesanalyzed. were These observed include in the three band gaptraditional simulations. parameters, A peak in thethe transmission radius, the thickness, and the spectrumheight was spotted ratio, correspondingwhich can be tovali twodated of the by branches. other research In Section in3 the, the literature. influence The low frequency of structural parameters on flatness of dispersion curve and the width of band gaps is band gaps in different FR values are calculated and discussed in particular. The results analyzed. These include three traditional parameters, the radius, the thickness, and the height ratio,prove which that can this be validatednew defined by other variable research may in the play literature. a more The important low frequency role in the dispersion band gapsperformance. in different FR Invalues Section are 4, calculated three HPFC and discussed multiplexing in particular. arrays Theare created results with different FR prove thatdistributions. this new defined In the variable simulation may play results a more with important an anti-symmetric role in the dispersion Lamb wave input, variable performance.narrow-band In Section 4frequencies, three HPFC are multiplexing spotted in arrays the aretransmission created with spectrum different FRand the displacement distributions.map. In Finally, the simulation we draw results five with conclusions an anti-symmetric for this Lamb new wave composite input, variable crystal plate in Section 5. narrow-band frequencies are spotted in the transmission spectrum and the displacement map. Finally,Potential we draw mechanism five conclusions and method for this for new FR composite control crystalwith STMf103 plate in Section chips5 are. briefly discussed Potentialin mechanism the supplementary. and methodfor TheFR layoutcontrol of with the STMf103 study is chips illustrated are briefly in discussed Figure 1, with blue dashed in the Supplementarylines dividing Materials. the sections. The layout of the study is illustrated in Figure1, with blue dashed lines dividing the sections.

Figure 1. Diagram of the researchFigure 1. idea. Diagram of the research idea.

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2. The Features of HPFC A novel composite PC structure is created, as shown in Figure 2. The PC plate has a cubic symmetric crystal axis in Cartesian coordinate system. Figure 2a illustrates a unit with two components. Along the z axis, the outer column consists of a lead head, a silicon Crystals 2021, 11, 828 3 of 13 rubber body, and an aluminum base, and the inner pillar has similar but opposite arranged layers. Periodic boundary conditions are applied to each side of the unit element in the (x, y) plane. Within the four boundaries, PC units are homogeneously arrayed. 2. TheTherefore, Features we of name HPFC this new structure a hollow-stub phononic-crystal with filled cylin- dersA novel (HPFC) composite plate. The PC structureremaining is pictures created, in as Figure shown 2 in show Figure the2 .structure The PC plate parameters has a on cubicthe symmetric inner and crystalouter components. axis in Cartesian For the coordinate symbol on system. the inner Figure pillar,2a illustrates‘ri’ is the inner a unit radius withof two the components. stubbed hollow Along column the z plate; axis, the ‘h2’ outer is the column height of consists the lead of ahead lead for head, the apillar; silicon ‘h1’ is rubberthe body,height and of anthe aluminum silicone rubber base, andcylinder; the inner and pillar ‘e’ is hasthe similarthickness but of opposite the aluminum arranged base. layers.The Periodic symbols boundary on the outer conditions cylinder are have applied the same to each definitions, side of the besides unit element its outer in radius the r (x, yand) plane. the Withinlattice dimension the four boundaries, ‘a’. The first PC irreducible units are homogeneously Brillouin zone arrayed.of the square Therefore, lattice is we nameplotted this in newFigure structure 2d. The a calculated hollow-stub parameters phononic-crystal of the material with ﬁlled are shown cylinders in Table (HPFC) 1. plate. The remaining pictures in Figure2 show the structure parameters on the inner

andTable outer 1. components. Physical properties For the of silicone symbol rubber, on the lead, inner and pillar, aluminum ‘ri’ is[17,23]. the inner radius of the stubbed hollow column plate; ‘h ’ is the height of the lead head for the pillar; ‘h ’ is the ρ2 (kg/m3) E (GPa) 1μ height of the silicone rubber cylinder; and ‘e’ is the thickness of the aluminum base. The Silicone rubber 1300 2.15 0.4998 symbols on the outer cylinder have the same deﬁnitions, besides its outer radius r and the lattice dimensionLead ‘a’. The ﬁrst irreducible11344 Brillouin zone of10.73 the square lattice is plotted 0.46 in Figure2d.Aluminum The calculated parameters2702 of the material are shown 70 in Table1. 0.33

(a) (b)

FigureFigure 2. Geometric 2. Geometric parameters parameters of the of HPFC.the HPFC. (a) A (a three-dimensional) A three-dimensional schematic schematic diagram diagram of the of hollowthe hollow cylindrical cylindrical plate. plate. There is a hollow pillar placed on the thin plate; (b) 3D schematic diagram of filling pillar; (c) the integral part of the There is a hollow pillar placed on the thin plate; (b) 3D schematic diagram of ﬁlling pillar; (c) the integral part of the assembly; (d) the first irreducible Brillouin zone of the square lattice. assembly; (d) the ﬁrst irreducible Brillouin zone of the square lattice.

Table 1. Physical properties of silicone rubber, lead, and aluminum [17,23].

ρ (kg/m3) E (GPa) µ Silicone rubber 1300 2.15 0.4998 Lead 11,344 10.73 0.46 Aluminum 2702 70 0.33 Crystals 2021, 11, x FOR PEER REVIEW 4 of 14

In order to describe the height relationship between the pillar and the cylinder, we introduce a new dimensionless factor ‘FR’. It is defined as the ratio of the filled part of the hollow pillar plate to the full-filling pillar. Then, sufficient simulations are performed using finite element method. Typical dispersion curves with different FR values are shown in Figure 3. For all the four subfigures, we set the geometric parameters a = 10 mm, r/a = 0.3, ri/a = 0.145, h2 = 0.4a, h1 = 0.3a and e = 0.05a. Figure 3a depicts the dispersion curve along the boundary of the first irreducible Brillouin zone (as shown in Figure 2d) when FR is set to 100%.The band curves are drawn with magenta points. It can be seen that the fully filled HPFC’s dispersion clearly show three band gaps, plotted in cyan color. The low frequency band gaps are due to the local resonance. The upper two Bragg band gaps are caused by the periodicity of the crystal and the collective scattering effect between the pillars. It should be pointed out that although the hollow cylindrical and the filling pillar have a small height with the same lead material, geometry parameters remain unchanged in this section, except for FR. When we adjust the FR to 70%, 46.67%, and 6.67%, the dispersion curves at different FR values are given in Figure 3b–d. Three new dispersion branches could be observed. We label them as 1, 2, and 3 in red color. The branches are not very clear when FR is above 70%. These new branches are not present in the fully filled state. With the reduction in FR, the new dispersion branches move toward a lower frequency. When the FR = 46.67%, the new dispersion branch at the high-frequency band gap is relatively straight and flat. How- Crystals 2021, 11, 828 ever, due to the interaction between the plate and the mode, the low frequency branching4 of 13 will not produce isolated branches as the Bragg gap. The new branches could offer a possibility to form waveguide structures. When the FR = 6.67% in Figure 3d, the band gap width at the low frequency band is In order to describe the height relationship between the pillar and the cylinder, we 900 Hz larger than the FR = 100%. The new dispersion branches are related to the short introduce a new dimensionless factor ‘FR’. It is defined as the ratio of the filled part of stubs on both sides of the plate. The same resonant eigenmode and the plate’s Lamb the hollow pillar plate to the full-filling pillar. Then, sufficient simulations are performed modes have a strong coupling. It may lead to an increase in the width of the locally reso- using finite element method. Typical dispersion curves with different FR values are shown nant frequency band gap. Therefore, when we solve the vehicle NVH problem, we have in Figure3. to set the structure within the low frequency band gaps.

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(a) (b)

FigureFigure 3. 3.Dispersion Dispersion curves curves of of the the double-pillar double-pillar ﬁllable fillable structure structure along along the the boundary boundary of of the the ﬁrst first irreducible irreducible Brillouin Brillouin zone zone ГГ-х-х.(. (aa)) DispersionDispersion curvecurve whenwhen the FR = 100%; 100%; ( (bb)) dispersion dispersion curve curve when when the the FRFR == 70%; 70%; (c ()c dispersion) dispersion curve curve when when the the FR = 46.67%; (d) dispersion curve when the FR = 6.67%. FR = 46.67%; (d) dispersion curve when the FR = 6.67%.

ForFigure all the4 shows four subfigures,the change in we the set three the geometricnew branches parameters with the afilling = 10 ratio.mm, r/aFigure = 0.3 4a, and b show the displacement field distribution of the single unit at the FR = 46.67%. Figure ri/a = 0.145, h2 = 0.4a, h1 = 0.3a and e = 0.05a. Figure3a depicts the dispersion curve along the3c boundaryshows that of the the branching first irreducible frequency Brillouin would zone nonlinearly (as shown decrease in Figure with2d) whenthe reductionFR is set in tothe 100%.The FR. In addition band curves to three are drawn frequency with points magenta for points. each FR It canvalues, be seen the thatthree the band fully gaps filled at HPFC’sthe FR = dispersion 100% were clearly added show with three dot lines band in gaps, different plotted colors. in cyan Thus color. we can The see low whether frequency the bandfrequency gaps arepoints due located to the in local the resonance.band gaps. TheThese upper frequency two Bragg points band are taken gaps arefrom caused Figure by3c. the They periodicity are the end of thepoint crystal of branch and the 1, 2, collective and 3 at scatteringthe right edge effect marked between with the ‘X’. pillars. The ‘X’ It shouldrepresents be the pointed boundary out in that the although first irreducible the hollow Brillouin cylindrical zone. When and FR the = 20% filling and pillar 30%, have the 1st a smalland 2nd height branching with the points same leadfall on material, the low geometry frequency parameters band gap. remain This means unchanged that the in thisnew section,structure except may forprovideFR. guiding ability in low frequency, compared to the traditional high frequencyWhen wewaveguide. adjust the WeFR performedto 70%, 46.67%, simulation and 6.67%,s from the 10% dispersion to 100% with curves a 10% at different interval. FRIt isvalues interesting are given that in these Figure new3b–d. branches Three newhave dispersion disappeared branches when FR could < 20% be observed.and FR > 70%. We label them as 1, 2, and 3 in red color. The branches are not very clear when FR is above 70%. These new branches are not present in the fully filled state. With the reduction in FR, the new dispersion branches move toward a lower frequency. When the FR = 46.67%, the new dispersion branch at the high-frequency band gap is relatively straight and flat. However,

(a) (b) (c) Figure 4. Displacement field distribution and dispersion branch with different FR. (a) The top view of the displacement field; (b) the main view of the displacement field; (c) The graph of the new dispersion branch X point with the different FR. The three-color dotted line represents the band gap width when the FR = 100%.

In order to analyze the transmission performance of the plat, we built a row PCs with five HPFC units at FR = 46.67%. The layout and simulation results are shown in Figure 5. Perfect matching layers (PML) are applied to the inlet and outlet of the board to avoid any reflection from the external edges. Periodic boundary conditions are imposed on each side of the cell in the Y direction. The incident wave is an A0 Lamb wave of the plate, propagating along the X direction. It is generated by applying a harmonic shift on the (y, z)

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due to the interaction between the plate and the mode, the low frequency branching will not produce isolated branches as the Bragg gap. The new branches could offer a possibility to form waveguide structures. When the FR = 6.67% in Figure 3d, the band gap width at the low frequency band is 900 Hz(c larger) than the FR = 100%. The new dispersion(d) branches are related to the short stubs on both sides of the plate. The same resonant eigenmode and the plate’s Lamb modes Figure 3. Dispersion curves of the double-pillar fillable structure along the boundary of the first irreducible Brillouin zone have a strong coupling. It may lead to an increase in the width of the locally resonant Г-х. (a) Dispersion curve when the FR = 100%; (b) dispersion curve when the FR = 70%; (c) dispersion curve when the FR = 46.67%; (d) dispersion curvefrequency when the band FR gap.= 6.67%. Therefore, when we solve the vehicle NVH problem, we have to set the structure within the low frequency band gaps. FigureFigure4 shows 4 shows the the change change in thein the three three new new branches branches with with the fillingthe filling ratio. ratio. Figure Figure4a,b 4a showand theb show displacement the displacement field distribution field dist ofribution the single of unit the atsingle the FR unit= 46.67%.at the FRFigure = 46.67%.3c shows Figure that3c theshows branching that the frequency branching would frequency nonlinearly would nonlinearly decrease with decrease the reduction with the in reduction the FR. In in additionthe FR. to In three addition frequency to three points frequency for each pointsFR values, for each the threeFR values, band gapsthe three at the bandFR = gaps 100% at werethe FR added = 100% with were dot added lines in with different dot lines colors. in different Thus we colors. can seeThus whether we can thesee frequencywhether the pointsfrequency located points in the located band gaps. in the These band frequency gaps. These points frequency are taken points from are Figure taken3c. from They Figure are the end point of branch 1, 2, and 3 at the right edge marked with ‘X’. The ‘X’ represents 3c. They are the end point of branch 1, 2, and 3 at the right edge marked with ‘X’. The ‘X’ the boundary in the ﬁrst irreducible Brillouin zone. When FR = 20% and 30%, the 1st represents the boundary in the first irreducible Brillouin zone. When FR = 20% and 30%, the 1st and 2nd branching points fall on the low frequency band gap. This means that the new and 2nd branching points fall on the low frequency band gap. This means that the new structure may provide guiding ability in low frequency, compared to the traditional high structure may provide guiding ability in low frequency, compared to the traditional high frequency waveguide. We performed simulations from 10% to 100% with a 10% interval. It frequency waveguide. We performed simulations from 10% to 100% with a 10% interval. is interesting that these new branches have disappeared when FR < 20% and FR > 70%. It is interesting that these new branches have disappeared when FR < 20% and FR > 70%.

(a) (b) (c)

FigureFigure 4. Displacement4. Displacement field field distribution distribution and and dispersion dispersion branch branch with with different differentFR FR.(a. )(a The) The top top view view of of the the displacement displacement field; (b) the main view of the displacement field; (c) The graph of the new dispersion branch X point with the different field; (b) the main view of the displacement field; (c) The graph of the new dispersion branch X point with the different FR. FR. The three-color dotted line represents the band gap width when the FR = 100%. The three-color dotted line represents the band gap width when the FR = 100%.

InIn order order to to analyze analyze thethe transmissiontransmission perfor performancemance of of the the plat, plat, we we built built a row a row PCs PCs with withfive ﬁve HPFC HPFC units units at FR at =FR 46.67%.= 46.67%. The layout The layout and simulation and simulation results resultsare shown are shownin Figure in 5. FigurePerfect5. Perfectmatching matching layers (PML) layers are (PML) applied are applied to the inlet to the and inlet outlet and of outlet the board of the to board avoid to any avoidreflection any reﬂection from the fromexternal the edges. external Periodic edges. boundary Periodic boundaryconditions conditions are imposed are on imposed each side onof each the cell side in of the the Y cell direction. in the Y Thedirection. incident The wave incident is an A0 wave Lamb is an wave A0 Lamb of the wave plate, of propa- the plate,gating propagating along the alongX direction. the X direction. It is generated It is generated by applying by applying a harmonic a harmonic shift on shiftthe ( ony, z) the (y, z) plane in front of the crystal in Figure2. The transmission spectrum shows that, although some modal conversion occurs at the right exit of the HPFC, the transmitted wave could still maintains its original characteristics. The anti-symmetric Lamb wave excitation produces a narrow pass band transmission along the Г-х direction, labeled as peak A in Figure5b. The frequency of peak A would shift down with the decreasing of the FR. The detailed graphics will not be discussed in this paper. Crystals 2021, 11, x FOR PEER REVIEW 6 of 14

plane in front of the crystal in Figure 2. The transmission spectrum shows that, although some modal conversion occurs at the right exit of the HPFC, the transmitted wave could still maintains its original characteristics. The anti-symmetric Lamb wave excitation produces a narrow pass band transmission along the Г-х direction, labeled as peak A in Figure Crystals 2021, 11, 828 6 of 13 5b. The frequency of peak A would shift down with the decreasing of the FR. The detailed graphics will not be discussed in this paper.

(a) (b) FigureFigure 5. 5. TheThe transmission transmission spectrum spectrum under thethe arrayarray element.element. ( a()a Distribution) Distribution of of structural structural displacement displacement ﬁeld. field. The The top top viewview is isthe the upper upper half half and and the the front front view is the lowerlowerhalf; half; ( b(b)) dispersion dispersion curve curve (left) (left) and and transmission transmission spectrum spectrum (right) (right) of of anti-symmetricanti-symmetric Lamb Lamb wave wave passing passing through hollowhollow pillar pillar PCs PCs with withFR FR= 46.67%.= 46.67%.

3.3. Influence Influence of StructuralStructural Parameters Parameters TheThe simulation modelmodel inin thethe previous previous section section has has demonstrated demonstrated that that the the HPFC HPFC can can bebe used used as as a a low low frequency frequency muffler muffler and and a a tunable tunable filter. filter. In In this this section, section, the the effects effects of of dif- ferentdifferent structural structural parameters parameters on the on themodel model are arestudied. studied. The structural The structural parameters parameters include include the hollow radius (ri), the thickness of the plate (e), and the height ratio of the pillar the hollow radius (ri), the thickness of the plate (e), and the height ratio of the pillar (HR = (HR = h2/h1). In addition to the three traditional parameters, the proposed factor FR and h2/h1). In addition to the three traditional parameters, the proposed factor FR and its cor- its correlation with HR are studied more carefully. The original model (ri = 0.145, e = 0.05, relationand HR with= 4/3) HR was are used studied for comparison. more carefully. The structure The original at FR =model 46.67% (r isi = used0.145, to e calculate = 0.05, and HRdispersion = 4/3) was curves. used for comparison. The structure at FR = 46.67% is used to calculate dispersion curves. 3.1. Influence of Hollow Radius 3.1. InfluenceThe hollow of Hollow radius Radius is set as 0.1a, 0.145a (original model), 0.2a, and 0.25a. The results are shownThe in hollow Figure 6radius. We can is seeset thatas 0.1 thea, frequency0.145a (original of the dispersionmodel), 0.2 curvea, and would 0.25a. increase The results arewith shown the radius. in Figure When 6. theWe radius can see was that changed the frequency from 0.1a of to the 0.145a, dispersion the number curve of would band in- creasegaps remains with the unchanged. radius. When The maximumthe radius value was changed of band gap from width 0.1a∆ toT is0.145a, 7.7kHz theappearing number of bandat ri = gaps 0.1a. remainsWhen the unchanged. lattice model The takes maximum larger radius,value of the band dispersion gap width should ΔT payis 7.7kHz more ap- attention to the flatness of the energy bands so that the cyan band gaps are not labeled in pearing at ri = 0.1a. When the lattice model takes larger radius, the dispersion should pay the two figures to the right of Figure6. When r = 0.2a, there are three flatter energy bands more attention to the flatness of the energy bandsi so that the cyan band gaps are not la- than the original model dispersion curve between 20–40 kHz, which are marked with red beled in the two figures to the right of Figure 6. When ri = 0.2a, there are three flatter energy stars. When ri = 0.25a, the three energy bands between 10–20 kHz are the flattest, which bandsare marked than withthe original blue circles. model dispersion curve between 20–40 kHz, which are marked with red stars. When ri = 0.25a, the three energy bands between 10–20 kHz are the flattest, which3.2. Influence are marked of Plate with Thickness blue circles. The effect of plate thickness is discussed afterwards. The thickness is set up as 0.05a (original model), 0.1a, 0.15a, and 0.2a. We analyze the frequency values of the dispersion curves and the flatness of the dispersion branches. Compared with the hollow radius, the thickness has fewer influences on both the frequency and the flatness of the dispersion branch, as shown in Figure7.

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Figure 6. The influence of hollow radius on the dispersion curve of the HPFC.

3.2. Influence of Plate Thickness The effect of plate thickness is discussed afterwards. The thickness is set up as 0.05a (original model), 0.1a, 0.15a, and 0.2a. We analyze the frequency values of the dispersion curves and the flatness of the dispersion branches. Compared with the hollow radius, the thickness has fewer influences on both the frequency and the flatness of the dispersion branch,FigureFigure 6. as 6.The shownThe influence inﬂuence in Figure of of hollow hollow 7. radius radius on on the the dispersion dispersion curve curve of of the the HPFC. HPFC.

FigureFigure 7. The 7. The influence inﬂuence of plate of plate thickness thickness e on e onthe the dispersion dispersion curve curve of ofthe the HPFC. HPFC.

3.3.3.3. Influence Inﬂuence of Pillar of Pillar Height Height Ratio Ratio

TheThe influence influence of ofpillar pillar height height ratio ratio (HR (HR = h =2/h h21/h) is1) discussed. is discussed. We We set set the the pillar pillar height height ratioratio as as1/2, 1/2, 1, 4/3 1, 4/3(the (theoriginal original model), model), and 2. and The 2. simulation The simulation results results show showthat the that fre- the quencyfrequency value valuewould would increase increase with HR with. TheHR dispersion. The dispersion curves at curves HR = at1 haveHR =more 1 have distin- more guisheddistinguished features. features. A 0.02 kHz-wide A 0.02 kHz-wide band gap band around gap around2.95 kHz 2.95 is indicated kHz is indicated by a red by star. a red AFigure flatstar. energy 7. A The flat influenceband energy exists band of plate around exists thickness 10 around kHz e on with 10 the kHz dispersionthis with geometrical this curve geometrical of parameter, the HPFC. parameter, marked with marked a bluewith circle, a blue while circle, for the while dispersion for the dispersionat the other at three the otherHR values, three noHR significantvalues, no variations significant in3.3. variationsband Influence gap inwidth of band Pillar and gap Height dispersion width Ratio and branch dispersion flatness branch were flatness found. were The found. results The of resultsthe above of the above discussion are shown in Figure8. discussionThe influence are shown of in pillar Figure height 8. ratio (HR = h2/h1) is discussed. We set the pillar height ratio3.4. as Influence 1/2, 1, of4/3 Filling (the original Ratio model), and 2. The simulation results show that the frequency value would increase with HR. The dispersion curves at HR = 1 have more distin- The above traditional structure parameters are not sufficient for HPFC’s structure de- guished features. A 0.02 kHz-wide band gap around 2.95 kHz is indicated by a red star. sign. The new proposed FR may play a more important role in the dispersion performance. A flat energy band exists around 10 kHz with this geometrical parameter, marked with a We performed simulations at a 10% interval with FR. The upper and lower boundary blue circle, while for the dispersion at the other three HR values, no significant variations curves of the first band gap are given in Figure9a. The boundary values of the second in band gap width and dispersion branch flatness were found. The results of the above band gap are given in Figure9b. The lower boundary value of the band gap is defined as discussion are shown in Figure 8. Minj and the upper boundary value is defined as Maxj, where j is 1, 2. It is interesting that there are similar two distinguished zones I and II, separated by vertical dotted-lines, so we give the dispersion curves for FR = 40% and FR = 20%. The first and the second band gaps are selected and labeled to show more details.

Crystals 2021, 11, x FOR PEER REVIEW 8 of 14

Figure 8. The influence of pillar height ratio HR on the dispersion curve of the HPFC.

Crystals 2021, 11, 828 3.4. Influence of Filling Ratio 8 of 13 Crystals 2021, 11, x FOR PEER REVIEW 8 of 14 The above traditional structure parameters are not sufficient for HPFC’s structure design. The new proposed FR may play a more important role in the dispersion performance. We performed simulations at a 10% interval with FR. The upper and lower boundary curves of the first band gap are given in Figure 9a. The boundary values of the second band gap are given in Figure 9b. The lower boundary value of the band gap is defined as Minj and the upper boundary value is defined as Maxj, where j is 1, 2. It is interesting that there are similar two distinguished zones I and II, separated by vertical dotted-lines, so we give the dispersion curves for FR = 40% and FR = 20%. The first and the second band gaps are selected and labeled to show more details. Within each zones, the values of 'Maxj', 'Minj' and 'Maxj–Minj' of the first band gap increase with the rise of 'FR'. The second band gap boundary values have similar features to those of the first gaps, except for a decrease when FR = 70. 'Maxj–Minj' for the first band gap has the same trend, but the deviation for the second gap will increase slowly. Alt- hough the reason for the distinguished two zones is still not clear and need further study, the study for band gaps could still provide a theoretical basis for the low frequency sound insulation of the structure. FigureFigure 8. The 8. The influence inﬂuence of of pillar pillar height height ratio ratio HRHR onon the the dispersion curvecurveof of the the HPFC. HPFC.

3.4. Influence of Filling Ratio The above traditional structure parameters are not sufficient for HPFC’s structure design. The new proposed FR may play a more important role in the dispersion performance. We performed simulations at a 10% interval with FR. The upper and lower boundary curves of the first band gap are given in Figure 9a. The boundary values of the second band gap are given in Figure 9b. The lower boundary value of the band gap is defined as Minj and the upper boundary value is defined as Maxj, where j is 1, 2. It is interesting that there are similar two distinguished zones I and II, separated by vertical dotted-lines, so we give the dispersion curves for FR = 40% and FR = 20%. The first and the second band Crystals 2021, 11, x FOR PEER REVIEW 9 of 14 gaps are selected and labeled to show more details. Within each zones, the values of 'Maxj', 'Minj' and 'Maxj–Minj' of the first band gap (a) Theincrease first band with gap the rise of 'FR'. The second band (b gap) The boundary second bandvalues gap have similar features to those of the first gaps, except for a decrease when FR = 70. 'Maxj–Minj' for the first band gap has the same trend, but the deviation for the second gap will increase slowly. Alt- hough the reason for the distinguished two zones is still not clear and need further study, the study for band gaps could still provide a theoretical basis for the low frequency sound insulation of the structure.

FigureFigure 9. Dispersion 9. Dispersion curve curve and andthe value the value of band of band gap gap with with different different FRFR values.values. (a) (Thea) The upper upper and and lower lower boundary boundary differ- encesdifferences of the first of band the first gap. band (b) The gap. upper (b) The and upper lower and boundary lower boundary differe differencesnces of the of second the second band band gap. gap.(c) Band (c) Band gap gap selection for theselection filling forratio the above filling 40%. ratio above(d) Band 40%. gap (d) selection Band gap selectionfor the filling for the ratio filling below ratio below40. 40. (a) The first band gap (b) The second band gap 3.5. Influence of Composite Parameters The FR describes the structure relationship for the two components. A similar parameter HR describes the material relationship. These two dimensionless factors may have a coupling effect on the dispersion performance for HPFC plate, so we investigated them together. The locally resonant band gaps at different FR and HR are simulated as shown in Figure 10. The Figure 10a shows the band gap by varied FR when HR = 4/3, and the results

are shown in Figure 10b when HR = 2. The band gap width ΔT and the band gap median frequency T are selected to quantify the results under different structural parameters. We show the data for different HR and FR in Table 2.

(a) (b) Figure 10. Low frequency band gaps with different parameter structures. (a) Low frequency band gap plots at different FR when HR = 4/3; (b) low frequency band gap plots at different FR when HR = 2.

Table 2. The width and median frequency of the band gap.

Factor 1 Factor 2 ΔT (kHz) T (kHz) FR = 6.67% 3.3540 5.5230 HR = 4/3 FR = 46.67% 0.9480 4.0520 FR = 100% 2.4530 4.9445 FR = 6.67% 2.0490 3.8835 HR = 2 FR = 46.67% 1.9600 8.4660 FR = 100% 1.4570 3.7615

Crystals 2021, 11, x FOR PEER REVIEW 9 of 14

Crystals 2021, 11, 828 9 of 13

Within each zones, the values of ’Maxj’, ’Minj’ and ’Maxj–Minj’ of the first band gap increase with the rise of ’FR’. The second band gap boundary values have similar features to those of the first gaps, except for a decrease when FR = 70. ’Maxj–Minj’ for the first band (c) Dispersion curve at FR = 40% (d) Dispersion curve at FR = 20% gap has the same trend, but the deviation for the second gap will increase slowly. Although Figure 9. Dispersion curvethe and reason the value for of the band distinguished gap with different two FR zones values. is still (a) The not upper clear and and lower need boundary further study, differ- the ences of the first band gap.study (b) The for upper band and gaps lower could boundary still provide differences a theoretical of the second basis band for gap. the (c low) Band frequency gap selection sound for the filling ratio above 40%.insulation (d) Band of gap the selection structure. for the filling ratio below 40.

3.5.3.5. Inﬂuence Influence of of Composite Composite Parameters Parameters TheTheFR FRdescribes describes the the structure structure relationship relationship for for the the two two components. components. AAsimilar similarpa- pa- rameterrameterHR HRdescribes describes the the material material relationship. relationship. TheseThesetwo two dimensionless dimensionlessfactors factors may may havehave a a coupling coupling effect effect on on the the dispersion dispersion performance performance for for HPFC HPFC plate, plate, so so we we investigated investigated themthem together. together. TheThe locally locally resonant resonant band band gaps gaps at at different differentFR FRand andHR HRare are simulated simulated as as shown shown in in FigureFigure 10 10.. The The Figure Figure 10 10aa shows shows the the band band gap gap by by varied variedFR FRwhen whenHR HR= = 4/3, 4/3, and and the the results results areare shown shown in in Figure Figure 10 10bb when whenHR HR= = 2. 2. The The band band gap gap width width∆ ΔTTand and the the band band gap gap median median frequencyfrequencyT Tare are selected selected to to quantify quantify the the results results under under different different structural structural parameters. parameters. We We showshow the the data data for for different differentHR HRand andFR FRin in Table Table2. 2.

(a) (b)

FigureFigure 10. 10.Low Low frequency frequency band band gaps gaps with with different different parameter parameter structures. structures. (a) Low(a) Low frequency frequency band band gap gap plots plots at different at differentFR whenFR whenHR = HR 4/3; = 4/3; (b) low(b) low frequency frequency band band gap ga plotsp plots at different at differentFR when FR whenHR =HR 2. = 2.

TableTable 2. 2.The The width width and and median median frequency frequency of theof the band band gap. gap.

FactorFactor 1 1 Factor Factor 2 2 ∆ΔTT(kHz) (kHz) TT(kHz) (kHz) FR = 6.67% 3.3540 5.5230 FR = 6.67% 3.3540 5.5230 HRHR= = 4/3 4/3 FRFR= = 46.67% 46.67% 0.9480 0.9480 4.0520 4.0520 FRFR= = 100% 100% 2.4530 2.4530 4.9445 4.9445 FRFR= = 6.67% 6.67% 2.0490 2.0490 3.8835 3.8835 HRHR= = 2 2 FRFR= = 46.67% 46.67% 1.9600 1.9600 8.4660 8.4660 FRFR= = 100% 100% 1.4570 1.4570 3.7615 3.7615

Compared with Figures 10a,b, the lowest T of the band gap exist in the frequency of 3.7615 Hz when FR = 100% and HR = 2 and the widest ∆T exist with the band gap of 3.3540 when FR = 6.67% and HR = 4/3. The band gap width at FR = 6.67% provides a research direction for realizing the low frequency sound insulation and vibration and noise reduction function at a wide frequency. This is very important for the vehicle NVH. All the above discussions provide a direction for the research of locally resonant frequency sound insulation and vibration and noise reduction by different structural parameters. Crystals 2021, 11, x FOR PEER REVIEW 10 of 14

Compared with Figure 10a and Figure 10b, the lowest T of the band gap exist in the frequency of 3.7615 Hz when FR = 100% and HR = 2 and the widest ΔT exist with the band gap of 3.3540 when FR = 6.67% and HR = 4/3. The band gap width at FR = 6.67% provides a research direction for realizing the low frequency sound insulation and vibration and noise reduction function at a wide frequency. This is very important for the vehicle NVH. All the above discussions provide a direction for the research of locally resonant frequency sound insulation and vibration and noise reduction by different structural parameters.

4. Multiplexing Design Based on FR Variation Waveguides enable the transmission of A0Lamb waves at specific frequencies. The Crystals 2021, 11, 828 multiplexing technique could transmit among these different waves in a single10 of 13 channel. The proposed variable FR by itself could define both the waveguide and the physical structure. Therefore, we choose FR as the kernel of the multiplexing for HPFC plate. 4. Multiplexing Design Based on FR Variation 4.1. WaveguidesMultichannel enable Resonator the transmission Multiplexer of A0Lamb waves at specific frequencies. The multiplexingBased on technique the above could analysis transmit of amongdifferent these structural different parameter waves in a models, single channel. a (5 × 5) super ThePCs proposed structure variable is built FRandby simulation itself could results define are both shown the waveguide in Figure and11a. theWe physicalset the periodic- structure.ity condition Therefore, in the we Y choose directionFR as and the PML kernel in of the the multiplexingX direction forof sound HPFC plate.transmission. The 4.1.PCs Multichannel consist of a Resonator row of Multiplexerfull-filling pillars that separate the linear waveguides to prevent significant leakage between them. Waveguides A and B are composed of two rows of hol- Based on the above analysis of different structural parameter models, a (5 × 5) super PCslow structure pillars with is built the and FR simulation = 46.67% resultsand the are FR shown = 50%. in FigureThe other 11a. geometrical We set the periodicity parameters take conditionthe same invalue the Ywithdirection one of and thePML models in the in SectionX direction 3.5 ( ofri = sound 0.145, transmission. e = 0.05, and HR The = 2). The PCsinitial consist displacement of a row of is full-filling set in front pillars of the that PCs separate to generate the linear the waveguides A0 Lamb towave prevent to verify the significanttransmission leakage effect. between them. Waveguides A and B are composed of two rows of hollowAs pillars shown with in the FigureFR = 11a, 46.67% two and narrow the FR bands= 50%. pass The otherthrough geometrical the transmission parameters spectrum takeat the the frequencies same value withof 22.600 one of kHz the modelsand 23.100 in Section kHz. 3.5The( r itransmitted= 0.145, e = 0.05, waves and ofHR waveguides= 2). A Theand initial B corresponding displacement to is settwo in narrow front of thebands PCs are to generate shown thein the A0 displacement Lamb wave to verifyfield distribu- the transmission effect. tion in Figure 11b. Then a multi-channel wavelength multiplexer could be set up.

(a) (b)

FigureFigure 11. 11. SchematicSchematic representationrepresentation of of a multichannela multichannel wavelength wavelength multiplexer. multiplexer. (a) The (a transmission) The transmission spectrum spectrum of anti- of anti- symmetricsymmetric Lamb Lamb waveswaves when when the theFR FRin thein the waveguide waveguideFR(A) FR = 46.67%(A) = 46.67% and FR (B)and = 50%.FR(B) (b =) Displacement50%. (b) Displacement ﬁeld distribution field distribu- tionattwo at two narrow-band narrow-band frequencies frequencies A and B.A and B.

As shown in Figure 11a, two narrow bands pass through the transmission spectrum

at the frequencies of 22.600 kHz and 23.100 kHz. The transmitted waves of waveguides A and B corresponding to two narrow bands are shown in the displacement ﬁeld distribution in Figure 11b. Then a multi-channel wavelength multiplexer could be set up.

4.2. Single-Channel Resonator Multiplexer A single channel multiplexer composed of two alternating hollow pillars with different FR values is plotted in the top of Figure 12a. The FR(C) is 46.67%. The FR(D) is 50%, as the same value with the multi-channel multiplexer. The transmission spectrum is shown in Figure 12a. By a different array arrangement, two narrow bands also appear at 22.801 kHz and 23.801 kHz. This means that two different wavelengths can be transmitted through the same channel. Figure 12b depicts the displacement ﬁeld at frequencies f(C) and f(D), where the enhancement of the ﬁelds inside the hollow pillar can be observed at the two ratios. Crystals 2021, 11, x FOR PEER REVIEW 11 of 14

4.2. Single-Channel Resonator Multiplexer A single channel multiplexer composed of two alternating hollow pillars with different FR values is plotted in the top of Figure 12a. The FR(C) is 46.67%. The FR(D) is 50%, as the same value with the multi-channel multiplexer. The transmission spectrum is shown in Figure 12a. By a different array arrangement, two narrow bands also appear at 22.801 kHz and 23.801 kHz. This means that two different wavelengths can be transmitted through the same channel. Figure 12b depicts the displacement field at frequencies f(C) and f(D), where the enhancement of the fields inside the hollow pillar can be observed at Crystals 2021, 11, 828 11 of 13 the two ratios.

(a) (b)

FigureFigure 12. 12. SchematicSchematic representation of of a a single-channel single-channel wavelength wavelength multiplexer. multiplexer. (a) The (a) transmission The transmission spectrum spectrum of anti- of anti- symmetricsymmetric Lamb Lamb waves whenwhen the theFR FRin thein the waveguide waveguideFR(C) FR =(C) 46.67% = 46.67% and FR and(D) = FR 50%;(D) ( b=) 50%; displacement (b) displacement ﬁeld distribution field distribu- tionat twoat two narrow-band narrow-band frequencies frequencies C and C D. and D.

4.3. CompactCompact Multiplexer Multiplexer Based Based on Linearon Linear Cavity Cavity A cell with an extremely compact structure is shown in the top of Figure 13a. It consists A cell with an extremely compact structure is shown in the top of Figure 13a. It con- of two rows of hollow pillars with different FR values and surrounded by a row of solid sists of two rows of hollow pillars with different FR values and surrounded by a row of Crystals 2021, 11, x FOR PEER REVIEWpillars (FR = 100%) on each side. The unit has a ﬁnite dimension along the X direction and 12 of 14 issolid periodic pillars in the(FRY =direction. 100%) on The eachFR side.(E) = 46.67%The unit and has the a FRfinite(F) = dimension 50%. The transmission along the X of direction theand anti-symmetric is periodic in Lamb the Y waves direction. is emitted The inFR the(E)X =direction 46.67% and in Figure the FR 13a.(F) It = indicates 50%. The that transmis- twosion narrow of the anti-symmetric bands of the structure Lamb appear waves at is 31.901 emitted kHz in and the 32.401 X direction kHz, and in Figure Figure 1313a.b It indi- depictscates that the displacementtwo narrow ﬁeldbands at of frequencies the structure f(E) and appear f(F). at 31.901 kHz and 32.401 kHz, and Figure 13b depicts the displacement field at frequencies f(E) and f(F).

(a) (b)

FigureFigure 13. The 13. The schematic schematic representation representation of thethe compactcompact wavelength wavelength cavity cavity multiplexer. multiplexer. (a) The (a transmission) The transmission spectrum spectrum of of the anti-symmetricthe anti-symmetric Lamb Lamb wave wave when when FRFR(E) == 46.67%46.67% and and FR(F) FR(F) = 50% = 50% in thein the waveguide; waveguide; (b) displacement (b) displacement ﬁeld distribution field distribution at twoat narrowband two narrowband frequencies, frequencies, E, E, and and F.

Three multiplexed structures are constructed by adjusting the positions of different FR. The results show that the a factor (FR) could be used to define the waveguides and physical structures. It may provide a possibility for the real-time modulation.

5. Discussion A new composite structure is proposed and a dimensionless variable, the filling ratio, is defined to describe the lattice model. Through the calculation and simulations on the transmission spectrum and band gap dispersion curves, five main conclusions can be drawn as follows: 1. The proposed composite structure (HPFC) could guide and filter the sound waves. It may expand the sonic band gap in lower frequencies and realize the waveguide function, validated by the observed three dispersion branches. Compared to other lattices, the oppositely arranged three-layer pillar and hollow stub make it a potential solution for vehicle NVH problems; 2. By introducing the concept of FR into the PCs model description, it helps us to un- derstand how the relationship between the components affects the sonic performances with a similar structure. The frequency of these branches decreases with the reduction in FR. It is interesting that the dispersion branches are mostly flat when FR = 46.67%. When FR = 6.67%, the width of the low frequency band gap is largest. Therefore, when we design a real plate used in vehicles, we may set the structure optimization boundary according these two percentages. 3. The effects of other structural parameters on the dispersion curves of the HPFC are discussed in three aspects: the radius of the central hole(ri), the thickness of the plate(e), and the ratio of the height of the stub(HR). When ri = 0.25a, the new dispersion branch is flatter than ri = 0.145a (the original model). The effect of plate thickness on flatness of dispersion curve and width of the band gap is small. Compared with HR = 4/3 (the original model), the stub height ratio HR = 1 produces a straight branch at the locally resonant frequency band gap and the band gap width is larger; it is interesting that there are similar two distinguished zones I and II. The reason is still not clear and need further study.

Crystals 2021, 11, 828 12 of 13

Three multiplexed structures are constructed by adjusting the positions of different FR. The results show that the a factor (FR) could be used to deﬁne the waveguides and physical structures. It may provide a possibility for the real-time modulation.

5. Discussion A new composite structure is proposed and a dimensionless variable, the filling ratio, is defined to describe the lattice model. Through the calculation and simulations on the transmission spectrum and band gap dispersion curves, five main conclusions can be drawn as follows: 1. The proposed composite structure (HPFC) could guide and filter the sound waves. It may expand the sonic band gap in lower frequencies and realize the waveguide function, validated by the observed three dispersion branches. Compared to other lattices, the oppositely arranged three-layer pillar and hollow stub make it a potential solution for vehicle NVH problems; 2. By introducing the concept of FR into the PCs model description, it helps us to under- stand how the relationship between the components affects the sonic performances with a similar structure. The frequency of these branches decreases with the reduction in FR. It is interesting that the dispersion branches are mostly flat when FR = 46.67%. When FR = 6.67%, the width of the low frequency band gap is largest. Therefore, when we design a real plate used in vehicles, we may set the structure optimization boundary according these two percentages. 3. The effects of other structural parameters on the dispersion curves of the HPFC are discussed in three aspects: the radius of the central hole(ri), the thickness of the plate(e), and the ratio of the height of the stub(HR). When ri = 0.25a, the new dispersion branch is flatter than ri = 0.145a (the original model). The effect of plate thickness on flatness of dispersion curve and width of the band gap is small. Compared with HR = 4/3 (the original model), the stub height ratio HR = 1 produces a straight branch at the locally resonant frequency band gap and the band gap width is larger; it is interesting that there are similar two distinguished zones I and II. The reason is still not clear and need further study. 4. The structural parameters that affect the width of the low frequency band gap and the intermediate frequency value of the band gap are discussed. When HR = 4/3, FR = 6.67%, ∆T = 3.3540, T = 5.5230 kHz. When HR = 2, FR = 6.67%, ∆T = 1.4570, T = 3.7615 kHz. With the same filling ratio, the height ratios show a negative relationship with both the width and the geometric median value of the band gaps. 5. The created three array show different transmission feature for sounds. If we design a control mechanism and actuator properly, we could realize the real time and patterns switchable solution for vehicle noises. This makes the on-vehicle noise control system more flexible for variable driving conditions.

Supplementary Materials: The following are available online at https://www.mdpi.com/article/ 10.3390/cryst11070828/s1, Figure S1: Dispersion curve and the value of band gap with different FR values, Figure S2: The structure diagram for changing the ﬁlling ratio with modular thinking and implement the structure with STM32F103 chip. (a) The picture of the whole structure. (b) The structure of Push board. Author Contributions: For this research, methodology, J.L.; resources, S.Z. and J.L.; writing—original draft, S.Z.; writing—review and editing, S.Z., J.L., H.Z. and S.W. All authors have read and agreed to the published version of the manuscript. Funding: This research work is supported by the National Natural Science Foundation of China (NSFC) under grant No. 51575288. The authors gratefully acknowledge the supporting agency. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Crystals 2021, 11, 828 13 of 13

Data Availability Statement: Data included in the article or Supplementary Materials. The initial data in this study can be found in [1–23]. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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