Flexural Wave Attenuation in A Periodic Laminated Beam

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1 Aercan Journal of Engneerng Research (AJER) e-issn: p-issn : Volue-5, Issue-6, pp Research Paper Open Access Flexural Wave Attenuaton n A Perodc Lanated Bea Zhwe Guo, Mepng Sheng *,Tng Wang (School Of Marne Scence And Technology, Northwestern Polytechncal Unversty, Chna) ABSTRACT: The flexural wave attenuaton property of a perodc lanated bea s exaned n ths paper. The equaton of oton of a sngle lanated bea s frstly derved by Halton prncple, and then the transfer atrx ethod and Bloch-Floquet boundary condton are appled to deterne the flexural wave band-gaps of an nfnte perodc lanated bea. The vbraton transsson characterstc s studed by fnte eleent ethod (FEM), and the nuercal result shows that, the band-gaps of an nfnte perodc bea fro present odel atch very wth the transttance valleys of a fnte perodc bea fro FEM odel, whch shows good accuracy of the present theoretcal odel. Studes show that, the perodc lanated bea provdes good vbraton attenuaton perforance, wth broad band-gap wdths and strong attenuaton ablty, thus can be used n the vbraton and nose control. The present odel derved n ths paper can also predct the longtudnal wave propagaton property. The Euler odel s also exaned to gve a sple odel for convenent purpose n the Engneerng applcatons. The result shows that the splfed odel can be used n the low frequency bandgap predcton, whle t wll nduce errors n the hgh frequency. Keywords: Flexural wave band-gap, Halton prncple, Transfer atrx ethod, Perodc lanated bea. I. INTRODUCTION Vbraton s a general physcal phenoenon when a echancal structure s excted. Although soe vbratons brng benefcal effect, ost of the vbraton wll nduce dsgustng nose or daage to the echancal equpent. Thus n order to reduce the vbraton, any ethods have been proposed. One of the s usng perodc structure. Perodc structure conssts of perodc dentcal eleents, alternatng n the wave propagaton drecton [1]. It s well known that, the waves n the band-pass propagate whle the waves n the band-gap attenuate, because of the Bragg-scatterng effect [2] or locally resonant effect [3]. Attracted by the great potental of vbraton suppresson, extensve studes about perodc structure have been conducted [1, 4, 5]. As bea-type structure s wdely used n the engneerng applcatons, the structure perodcty was ntroduced n the bea-type structure to reduce the vbraton n extensve studes. Wen [6] studed the flexural wave propagaton of a perodc thn straght bea and dscussed the band-gap property caused by Bragg-scatterng effect. Yu [7] and Xao [8] studed the band-gap property of a bea attached wth perodc sprng-ass systes fro nuercal sulaton and experent, whch gves a physcal explanaton about the band-gap startng frequency and the cut-off frequency. Wen [9] studed the perodc bea attached wth ult-oscllators n order to prove the band-gap perforance. Lu [1] studed the perodc curved bea, and obtaned soe specal property copared wth the perodc straght bea. There are stll a nuber of works dealng wth the perodc bea n varous ponts of vew [11-14]. As the coposte lanated structure provdes hgh stffness-to-weght rato, hgh strength-to-weght rato and any other attractve propertes [15, 16], t s used wdely n the aerospace and shp engneerng. However, ost of the prevous studes about perodc bea are focused on the sngle layer bea, whle the characterstc of perodc lanated bea s not well addressed. In order to solve the vbraton control proble et n lanated structure, the wave attenuaton property of the perodc lanated bea s studed n ths paper. A theoretcal odel of a ult-layered bea s establshed n the paper, whch wll represent the wave propagaton and wave attenuaton perforance. The generally used sandwch structure and the b-layer structure becoe the specal cases of the current odel. The perodc odel presented n ths paper gves theoretcal support of wave attenuaton property n the perodc lanated bea and gves a new sght to the vbraton control of the coposte lanated structure. II. BAND-GAP FORMULATION BASED ON TIMOSHENKO THEORY Fg. 1 has shown the odel of a perodc lanated bea wth layers, alternatng wth A and B. Halton prncple s used to obtan the vbraton equatons of a sngle lanated bea. w w w. a j e r. o r g Page 258

2 Layer Layer 2 Layer 1 Cell A Cell B Cell A Cell B A Eleent n-1 B B 2 A Eleent n Fgure 1: The structure odel of a perodc lanated bea. The sngle lanated bea s shown n Fg. 2. Assung that there s no axal slp between two layers, and all the layers have dentcal transverse dsplaceent w and rotaton. The shearng deforaton and the oent of nerta are consdered n the odel. The paraeters of the th layer are specfed by Young s odule E, shearng odule G, densty, shear coeffcent k, length L, wdth b and thckness h. Thus, the 3 cross-sectonal area s A b h and the oent of nerta s I b h 12, respectvely. The transverse dsplaceent, longtudnal dsplaceent and the rotaton are denoted by w, u and, respectvely. φ z w y h h 2 h 1 L B B 2 u u 2 u 1 Fgure 2: Dsplaceents and densons of the sngle lanated bea. The longtudnal dsplaceent of the th layer s, 1 u u h (1) 1 1 where h h h 2 h. The kneatc energy, stran energy and the work done by external forces can 1 1 j be expressed as, j 2 1 L E A w A u I d x k n L 2 2 E E A u E I k A G w d x p o t 2 1 W N u M Q w Apply Halton prncple, then the partal dfferental equatons can be obtaned as, e L x (2) (3) (4) w w w. a j e r. o rg Page 259

3 k A G w A w k A G E A u A u h E A h A h E A u h A u E I h E A and the external general forces are, I h A 1 k A G k A G w 1 1 N E A u h E A M h E A u 1 1 E I h E A Q k A G k A G w 1 1 (6) j t As the syste s assued n haronc oscllaton, thus w x, t W x e j t, u x, t U x e and j t x, t x e. Substtutng these expresson nto (5) gves, The soluton of (7) can be assued as polynoal equaton, W c W c W c W U d W d W d W e W e W 1 2 (8) x W x W e. Substtutng ths expresson nto (7) gves the followng c c c Sx roots can be obtaned as and by solvng (9). Takng account of (5-9) gves, 6 If we ake T T T T T T T j x j x j x W T e ; U ; j 1 T e j j T e j j j 1 j 1 j 1 N b T e ; M b T e ; Q b T e j x j x j x 1 1 j j 2 j j 3 j j j 1 j 1 j 1 ψ, D U W N M Q L 1, L L L 1, L L L T, we can obtan that, D D T (5) (7) (9) (1) D U W N M Q 1, 1, and Bψ Cψ L (11) where B and C are 6 6 atrxes, and D and D are the state vectors at x and x L. L The transfer atrx ethod s used to calculate the band-gap frequency. As shown n Fg. 3, the unt eleent s consttuted by cell A and cell B, wth length L A and L B. By usng (11), the state vectors of cell A can be expressed as D B ψ and D C ψ, and those of cell B can be expressed as D B ψ and L D C ψ, respectvely. 2 L 2 2 w w w. a j e r. o rg Page 26

4 PBC Cell A CBC Cell B PBC A B B 2 as, L A Fgure 3: The coponents of a unt eleent and the boundary condtons. The contnuous boundary condton (CBC) at the nterface between cell A and cell B can be expressed L B D D 1 L 2 (12) and the perodc boundary condton (PBC) between cell A and cell B can be expressed as, q L e D 1 2 L D (13) where q s the wavenuber along the axl drecton, wth real part representng the wave propagaton property, and the agnary part representng the wave attenuaton property. The lattce constant s L L L. A B Takng account of (12, 13) gves, 1 1 where I s the dentty atrx and Τ B C B C. For a gven frequency f ,we can obtan fro (14) that, ql Τ e I ψ 1 q L Τ f e I (15) The wavenuber q can be obtaned by solvng the above equaton. And then the dsperson curves (frequency vs wavenuber) can be obtaned and the band-gap property s then deterned. The band-gaps of both flexural wave and longtudnal wave can be acqured by (15). As the ephases of ths paper s on the flexural wave, the followng text wll focus on the flexural wave band-gap property. III. NUMERICAL EXAMPLE A sandwch bea s consdered n ths nuercal exaple, as t s ost wdely used aong the lanated structures. Thus each eleent s consttuted by sx coponents, ncludng,, A 3,, B 2 and B 3. The ateral propertes of each coponent are shown n Table 1. The lengths and wdths of cell A and cell B are set as L A =.15, L B =.1 and b A =b B =.15. Fro botto to top, the thcknesses are set as h 1 =.3, h 2 =.2 and h 3 =.2. The flexural wave band-gaps are shown n Fg. 4. In the fgure, three band-gaps exst below 2 Hz, and the band-gaps are 55.7~96. Hz, 266.3~837.1 Hz and 159.3~ Hz, wth the band-gap wdths of 4.3 Hz, 57.8 Hz and 466. Hz, respectvely. Ths perodc lanated bea perfors very well n the band-gap property wth the total band-gap of Hz, whch ndcates that, ore than half of the wave below 2 Hz are suppressed. The agnary part of the wavenuber s called decay constant, as t represents the attenuaton ablty of the perodc lanated bea. As shown n Fg. 4(b), the decay constant of the frst band-gap s.11, whch s saller than.57 n the second band-gap and.37 n the thrd band-gap. Resultng that the decay ablty of the frst band-gap s weaker than the second and the thrd band-gap. Also the frst band-gap bandwdth s very narrow, thus the attenuaton perforance of the frst band-gap s nferor to the second and the thrd band-gaps. Therefore, the second band-gap s ost portant n the engneerng applcaton n general vbraton control, because of ts low frequency, strong attenuaton ablty and broad band-gap bandwdth. However, for the low-frequency lne spectru vbraton control, the frst band-gap can also have good perforance, as the target frequency of vbraton control s very narrow. Table 1: Materal propertes of each coponent n an eleent. Materal property A 3 B 2 B 3 Young s odule (GPa) Shear odule (GPa) densty (kg/ 3 ) Shear coeffcent 5/6 5/6 5/6 5/6 5/6 5/6 (14) w w w. a j e r. o rg Page 261

5 dsplaceent () Frequency (Hz) Frequency (Hz) (a) Re(q)L/ (b) I(q)L/ Fgure 4: The band gaps of flexural vbraton wth (a) the real part of noralzed wavenuber and (b) the agnary part of noralzed wavenuber. The present odel s valdated by the FEM analyss wth an eght-eleent fnte perodc lanated bea, whch s shown n Fg. 5. The ateral and geoetry paraeters for a unt eleent are the sae as the paraeters n the above theoretcal odel. The unt haronc dsplaceent exctaton s appled at the left bea end, and the dsplaceent response s easured at the rght bea end. The vbraton response are shown n Fg. 6, where the shaded regon are the band-gaps calculated fro the theoretcal odel. As shown n the fgure, the response valleys of the fnte perodc bea n the FEM odel atch very well wth the band-gaps of the nfnte perodc bea n the theoretcal odel. Thus the present odel has a good perforance n the band-gap predctons. Dsplaceent Exctaton A 3 B 3 B 2 A 3 Eleent 1 Eleent 2-8 B 3 B 2 Fgure 5: The FEM odel of a perodc lanated bea. Dsplaceent response As shown n Fg. 6, the perodc lanated bea behaves as a wave flter, wth the waves n the bandpasses propagatng freely wthout attenuaton and the waves n the band-gaps beng attenuated. Thus the present perodc lanated bea can be used n the vbraton control by desgnng the band-gaps at the frequency range where the vbraton need to be controlled. For an eght-eleent perodc bea, the average attenuaton levels n the frst three band-gaps reach respectvely to 19.3 db, 74.3 db and 3.9 db, whch shows good attenuaton perforance and are very attractve n the vbraton control. A Frequency (Hz) Exctaton Response Fgure 6: The flexural vbraton transsson characterstc of a fnte perodc sandwch bea. (The shadow regon are the band gaps obtaned fro theoretcal odel.) w w w. a j e r. o rg Page 262

6 IV. SIMPLIFIED MODEL In the Toshenko odel, the shearng deforaton and the oent of nerta are consdered, resultng a very coplcated odel, whch wll nduce an ll-condtoned atrx n soe frequences. Thus hgh resoluton nuercal ethods are needed to solve the above proble. In order to ake t convenent n the engneerng applcaton, the splfed Euler ode s also studed to obtan the band-gap property. And a sple prncple about whch odel should be used at specfc stuaton s gven as a gudelne n ths secton. Each layer s consdered as an Euler-bea, and the ult-layer bea s odeled as an equvalent sngle layer bea by calculatng the equvalent bendng rgdty and the equvalent densty. The spatal force analyss s shown n Fg. 7. M N M N Q M 2 M 1 N 2 N 1 N M+M,x dx dx Q+Q,x dx Fgure 7: The spatal forces analyss sketch of a lanated bea. Fro (1), we can obtan that u u w h ( 2~ ),where w s the dervaton of transverse 1 1 dsplaceent w wth respect to x,and h s the dstance between the dlne of the 1 st layer and the th layer. 1 Fro the relaton of longtudnal dsplaceent and the stran u, we can obtan wh. Thus the 1 1 axal force of each layer can be expressed as, and the axal forces are balanced by Fro (17), we can obtan, 1 N N E A w h 1 1. Substtutng (16) nto the suaton gves, 1 E A w h 1 1 w 1 where E A h E A 1. The total oent of the lanated bea can be expressed as, 1 1 M M N h where M E I w. Substtutng M and (16) nto (19) gves, (16) (17) (18) (19) M E I w (2) where E I E I E A h1 h. Thus the equvalent bendng rgdty EI s deterned. As the equvalent densty can be calculated as A A. Therefore, the equvalent equaton of oton for a lanated bea can be expressed as, 1 1 w w w. a j e r. o rg Page w w E I A 4 2 x t (21) Accordng to the plane wave extenson ethod of a sngle layer bea [6], the band-gap frequency can fnally be deterned.

7 Frequency (Hz) V. PERFORMANCE COMPARISON The flexural wave dsperson curves and the band-gaps of the Toshenko odel and the Euler odel are copared n Fg. 8 and Table 2. As shearng deforaton and oent of nertal are gnored, the Euler odel wll brng soe errors. As shown n Fg. 8, the frst four dsperson curves of the Euler odel nearly concdence wth those of the Toshenko odel. Thus the Euler odel has hgh accuracy n the frst four dsperson curves. Fro the ffth dsperson curve, wth the ncrease curve order, the error of the Euler odel ncreases. The frequency dfference of the two odels reaches as uch as about 1 Hz for the eghth dsperson curve. The band-gap frequences and the band-gap wdths for the two odels are shown n Table 2. We can see that, wth the ncrease of band-gap order, the error ncreases gradually. Fro the frst to the seventh band-gap, the band-gap startng frequency error ncreases fro.1 Hz to 59.8 Hz, and the relatve error ncreases fro.2% to 9.2%; the band-gap cut-off frequency error ncreases fro.7 Hz to 17.9 Hz, and the relatve error ncreases fro.7% to 14.4%. 1 Toshenko odel Euler odel Re(q)L/ Fgure 8: The dsperson curves coparson between two odels. Table 2: The band-gap frequency coparson between two odels. Band-gap Startng frequency/hz Error Relatve Cut-off frequency/hz Error Relatve order Toshenko Euler /Hz Error Toshenko Euler /Hz Error % % % % % % % % % % % % % % Thus we can see, the splfed odel gves accurate results for the low frequency flexural wave bandgaps, and wll brng sgnfcant error when t coes to hgh frequency wave band-gaps. That s because a low frequency s assocated wth a bg wave length, and the effects caused by the shearng deforaton and oent of nerta are relatvely sall, whch can be gnored, thus the Euler odel gves an accurate result. Whle for the hgh frequency wave, the wave length s short, and the shearng deforaton and oent of nerta have sgnfcant effects on the ode frequences and then on the band-gaps. Ignorng these effects gves rse to sgnfcant error. When the effects of shearng deforaton and oent of nerta are not consdered, the equvalent bendng rgdty s enhanced, resultng that the dsperson curve frequences are overestated n the Euler odel, whch can be checked n Fg. 8 and Table 2. Therefore, n the engneerng applcaton of a perodc lanated bea, when the low frequency bandgaps are calculated, the Euler odel can be used because of ts splcty and satsfactory result. However, when the hgh frequency band-gaps are calculated, the Toshenko odel ust be used as t gves ore accurate result. VI. CONCLUSIONS The band-gap characterstc of the perodc lanated bea s exaned wth Halton prncple and transfer atrx ethod. The present theoretcal odel shows hgh accuracy and s valdated by FEM. For the flexural wave, ths perodc structure owns excellent band-gap attenuaton perforance below 2 Hz, wth ore than half of the waves n frequency doan are attenuated. The frst and second band-gap locate at low w w w. a j e r. o rg Page 264

8 frequency regon, thus the perodc lanated bea gves a new ethod to solve the low frequency vbraton control proble. Although the frst band-gap s narrow, t stll has potentals n the lne spectral vbraton control. The present odel derved n ths paper can also gve the longtudnal wave propagaton property, whch lays the foundaton to the future study about the coupled wave propagaton property between the flexural and longtudnal waves. The splfed Euler odel gves accurate result for the frst several dsperson curves, thus can predct low frequency band-gaps n the engneerng applcaton. In the hgh frequency, the Euler odel wll brng sgnfcant error, and the Toshenko odel ust be appled. REFERENCES [1]. D. M. Mead, Wave propagaton n contnuous perodc structures: Research contrbutons fro southapton, , Journal of Sound and Vbraton, 19(3), 1996, [2]. D. Sutter-Wder, S. Deloud and W. Steurer, Predcton of bragg-scatterng-nduced band gaps n phononc quascrystals, Physcal Revew B, 75(9), 27, [3]. Z. Lu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan and P. Sheng, Locally resonant sonc aterals, Scence, 289(5485), 2, [4]. M. I. Hussen, M. J. Leay and M. Ruzzene, Dynacs of phononc aterals and structures: Hstorcal orgns, recent progress, and future outlook, Appled Mechancs Revews, 66(4), 214, 482. [5]. D. Del Vescovo and I. Gorgo, Dynac probles for etaaterals: Revew of exstng odels and deas for further research, Internatonal Journal of Engneerng Scence, 8, 214, [6]. J. Wen, D. Yu, G. Wang, H. Zhao, Y. Lu, X. Dang, Y. Tan, K. Zuo, L. Chen and Y. Zhang, Elastc wave band gaps n flexural vbratons of straght beas, Chnese Journal of Mechancal Engneerng, 41(4), 25, 1-6. [7]. D. Yu, Y. Lu, G. Wang, H. Zhao and J. Qu, Flexural vbraton band gaps n toshenko beas wth locally resonant structures, Journal of Appled Physcs, 1(12), 26, [8]. Y. Xao, J. Wen, D. Yu and X. Wen, Flexural wave propagaton n beas wth perodcally attached vbraton absorbers: Band-gap behavor and band foraton echanss, Journal of Sound and Vbraton, 332(4), 213, [9]. Y. Xao, J. Wen and X. Wen, Flexural wave band gaps n locally resonant thn plates wth perodcally attached sprng ass resonators, Journal of Physcs D: Appled Physcs, 45(19), 212, [1]. S. Lu, S. L, H. Shu, W. Wang, D. Sh, L. Dong, H. Ln and W. Lu, Research on the elastc wave band gaps of curved bea of phononc crystals, Physca B: Condensed Matter, 457, 215, [11]. D. J. Mead and Š. Markuš, Coupled flexural-longtudnal wave oton n a perodc bea, Journal of Sound and Vbraton, 9(1), 1983, [12]. G. Wang, J. Wen and X. Wen, Quas-one-densonal phononc crystals studed usng the proved luped-ass ethod: Applcaton to locally resonant beas wth flexural wave band gap, Physcal Revew B, 71(1), 25, [13]. P. Cartraud and T. Messager, Coputatonal hoogenzaton of perodc bea-lke structures, Internatonal Journal of Solds and Structures, 43(3), 26, [14]. R. Zhu, X. N. Lu, G. K. Hu, C. T. Sun and G. L. Huang, A chral elastc etaateral bea for broadband vbraton suppresson, Journal of Sound and Vbraton, 333(1), 214, [15]. A. P. Mourtz, E. Gellert, P. Burchll and K. Challs, Revew of advanced coposte structures for naval shps and subarnes, Coposte Structures, 53(1), 21, [16]. R. F. Gbson, A revew of recent research on echancs of ultfunctonal coposte aterals and structures, Coposte Structures, 92(12), 21, w w w. a j e r. o rg Page 265