ID-1204 STRUCTURAL-DYNAMIC AND ACOUSTIC DESIGN OF ANISOTROPIC MULTILAYERED COMPOSITE STRUCTURES

Transcription

1 ID-104 STRUCTURAL-DYNAMIC AND ACOUSTIC DESIGN OF ANISOTROPIC MULTILAYERED COMPOSITE STRUCTURES Werner Huenbach, Carten Holte, Bingquan Zhou, Ola Täger Intitut ür Leichtbau und Kunttotechnik (ILK), Techniche Univerität Dreden, D-0106 Dreden, German Tel.: (+49) , Fax: (+49) SUMMARY: Lightweight tructure made o compoite material are increaingl ued in man indutrial ield or high-technolog application due to their veratile propert proile. Epeciall or dnamicall loaded tructure, a high material damping combined with low contructive weight and adequate tine i required. But in toda' ophiticated application, lightweight tructure will alo have to meet ver high acoutic (low noie) tandard. Thereore, an acoutic anali ha to be included in the deign proce. The tructural-dnamic and acoutic behaviour o aniotropic multilaered compoite tructure can not be decribed b claical model. Here, advanced method have been developed at the ILK, which take into account the pecial mechanical propertie o the ibrematrix compound. In advanced tructural-dnamic and acoutic experimental invetigation uing laer canning vibration intererometer technolog and ound power meaurement within a peciall contructed reverberation chamber, the tructural-dnamic and acoutic propert proile o dierent lightweight material ha been determined. For the principal anali o the acoutic radiation o thee material, additional numerical imulation uing the Finite Element Method (FEM) and the Boundar Element Method (BEM) have been perormed. On the bai o thee extenive invetigation, a vibro-acoutic model wa developed, which take into account the characteritic tructural-dnamic propertie uch a eigenorm and eigenrequencie a well a the important tructure-luid coupling. KEYWORDS: Structural-dnamic, aniotropic material behaviour, ound radiation, lightweight tructure INTRODUCTION Monolithic material like magneium, aluminium or titanium, which are mainl ued in toda' lightweight application, oten reach their limit with repect to the high demand concerning their tructural-dnamic and epeciall acoutic behaviour. The oer high peciic tine and trength, but a relativel low damping, which ma lead to intene acoutic radiation. Here, compoite or compound material open the poibilit to nergeticall ulill the requirement concerning tine and trength a well a damping and acoutic qualit [1]. The tructural-dnamic characteritic o epeciall aniotropic compoite material uch a ibre-reinorced polmer can how a complicated coupling o bending and torion dependent on the ibre orientation, the matrix compoition and the la-up. Thu, claical model can not decribe the vibration and the acoutic radiation o thee tructure. Here, advanced method have been developed, which on the one hand oer the calculation o eigenrequencie and eigenorm a well a material damping o modern lightweight tructure []. On the other hand, the modelling o the acoutic radiation ha been included. Thi vibro-acoutic deign concept provide the poibilit to calculate the eigenorm and the radiated ound power o plate, which are excited b an acoutic ound ource inide a reverberant chamber, dependent on the pecial material propertie. 1

2 STRUCTURAL-DYNAMIC ANALYSIS OF ANISOTROPIC COMPOSITE PLATES In general plate are plane load bearing tructure, which have a thickne h igniicantl maller than the length a and the width b (Fig. 1). Fig. 1: Plate geometr The plate o thickne h i here cut into halve b the midplane. A diplacement in x-, - and z-direction, the quantitie u, v and w are ued, with w being mall in comparion to the thickne o the plate. Thereore, the tructural-mechanic anali can be done uing a linear theor. The eigenrequencie o multilaered aniotropic plate tructure are calculated on the bai o the HAMILTONIAN principal a an extremum principle in elato-dnamic. Auming a harmonic vibration, the kinetic energ i then determined b T 1 u = ρ dv (1) t V with the irt-order time-derivative o the diplacement vector u which decribe a vibration conitent with the boundar condition. The potential energ o the multilaered compoite can be derived rom the deormation energ unctional U + h a b [ 1 ~ ( k ) ~ ( k) ~ ( k ) ~ ( k) Q11 εx + Q1 εxε + Q16 εxγ x + Q ε h 0 0 () ~ ( k) ~ ( k ) ~ ( k) ~ ( k) ~ ( k) + Q6 εγ x + Q66 γ x + Q44 γ z + Q55 γ xz + Q45 γ zγ xz ] ddxd = which oer the poibilit to include a hear-elatic diplacement approach uing hear deormation theor a well a the claical KIRCHHOFF plate theor [3]. For the cae o tead-tate condition, the component o the diplacement vector can be decribed b harmonic unction. Here, the eigenorm are approximated b uing the correponding olution o the beam problem, which ulill the boundar condition [4]. Minimiing the energ unctional lead to a tem o homogeneou linear equation or the

3 determination o the coeicient o the RITZ approach, which can be written a a characteritic equation o an eigenvalue problem ( M ) a = 0. K λ (3) The calculation o the eigenvalue numericall done b uing tandard algorithm. λ, the eigenrequencie and the eigenvector a i For aniotropic multilaered compoite, the analtical determination o the lo actor d i baed on the concept o complex moduli, which reult in complex parameter in the material law. According to the baic aumption that the complex moduli are tranormed like their elatic correpondancie, a eparation o the reduced complex tinee i poible ~ *( k) ~ '( k ) ~ ''( k ) Q = Q + iq. (4) ij ij Due to the mall value o material damping o ibre-reinorced compoite, the reulting complex characteritic equation o the eigenvalue problem can approximatel be eparated in it real and imaginar part uing the RAYLEIGH quotient, which lead - ater elimination o λ - to the modal lo actor where ' a denote the real part o the complex eigenvector. ij ' T '' ' a K a d =, (5) ' T ' ' a K a EXPERIMENTAL INVESTIGATIONS The determination o the vibro-acoutic propert proile o aniotropic compoite tructure involve experimental invetigation, which have to include tructural-dnamic and acoutic meaurement. The tructural-dnamic meaurement were carried out uing the Laer Scanning Vibration Intererometer (LASVI) technolog which i an advanced meauring tem or the anali o vibrating tructure. Compared with conventional technique uing electromechanical enor, the ma and eigenrequencie o which have negative inluence on the accurac o the meaurement, the LASVI method allow a contactle and highl exact determination o vibrational quantitie and condition. In order to determine the eigenmode o the invetigated plate tructure, a canning tem wa ued, which allow meauring the velocit o dierent urace poition (net point) without mechanicall moving the optical meauring head. Depending on the choen reolution, the number o net point i variable. For the perormed invetigation, a reolution o with 608 meauring point wa ued according to the numerical calculation. The FFT anali o the meaured urace velocit i perormed or ever net point. The ummation o the requenc-dependent data o each point i ued to build the reonance dia - gram o the vibrating plate, which i the bai or the identiication o the eigenrequencie. In Fig. a, the reonance pectrum o the urace velocit o a CFRP plate with a thickne o.0 mm in the requenc interval rom 150 Hz to 50 Hz i hown. 3

4 Fig. : Meaured reonance diagram The acoutic invetigation were carried out within a reverberant chamber, which i a cloed room built b acouticall rigid wall. The ound wave inide the reverberant chamber are relected on all o the rigid wall, o that thi multiple relection and cattering lead to an ideal mixture o wave o dierent requencie. The tanding wave are uperimpoed and the locall varing maxima and minima o the ound preure are approximatel compenated. Thi create a ound ield, which i characteried b an equal ound intenit at ever poition inide the reverberant chamber, a o called diue ound ield. The diue ound ield i then taken a the acoutic excitation o a tet plate, which i clamped inide a rigid wall. Thi wall eparate the reverberant chamber into two room, the o called tranmitter and the receiver hall. Inide the tranmitter hall, an acoutic ound ource create the diue ound ield, wherea inide the receiver hall the radiation o the vibrating plate i the ound ource (Fig. 3a). Fig. 3: Principal drawing o the reverberant chamber Due to the diue ound ield in both o the hall, the energ o the ound wave i approximatel preerved. Thu, a reverberant chamber can be ued to determine the radiated ound power o acouticall excited tructure inide the receiver hall. Thi ound power i an integral and patiall contant phical parameter and thereore a uitable acoutic deign variable. 4

5 The acoutic ound ource i a group o peaker, which are poitioned in the tranmitter hall. The tet plate i ixed in a peciall contructed clamping device, which guarantee reproducible boundar condition. In the receiver hall, the radiated ound preure i meaured b a microphone, which rotate on a pivot mounting in a plane with an inclination o 45. On the bai o the meaured ound preure value, the radiated ound power level i then calculated according to DIN EN 3741 [5]. In order to anale the qualit o the diue ound ield in the tranmitter and the receiver hall, the reverberation time ha to be meaured. Within thi reverberation time, the ound preure level i reduced b 60 db ater the acoutic ound ource i abruptl cut o. Due to the diue ound ield, thi requenc-dependent time period ha to how approximatel the ame behaviour at an poition inide the reverberant chamber (ee Fig. 3b). The requenc anali o the meaured ound power i then done b FFT. For the CFRP plate, Fig. b preent the meaured radia ted ound power. A comparion o the requencdependent behaviour clearl reveal the mechanim o acoutic radiation, which can be een b the coupling o the requenc ditribution o ur ace velocit and radiated ound power. The main ocu o the perormed numerical imulation i on thi important coupling between the tructural-dnamic and acoutic parameter urace velocit and radiated ound power. NUMERICAL CALCULATIONS For a more detailed anali o the vibro-acoutic coupling between the phical parameter urace velocit and radiated ound power, numerical calculation on the bai o the FEM and the BEM have been perormed. Thee calculation were carried out in the worktation pool o the ILK. For the tructural-dnamic anali, mainl the program package I-DEAS and ANSYS were ued and the acoutic anali wa perormed uing the pecial BE Sotware SYSNOISE and I-DEAS VIBRO-ACOUSTICS. The eigenorm calculated b the FEM and the meaured eigenorm b LASVI how a ver good agreement (Fig. 4). Fig. 4: Numericall and experimentall determined eigenorm Thi high accurac o the tructural-dnamic FE calculation i the main bai or the ollowing vibro-acoutic anali uing the BEM. A input data or the BE anali, the tructural FE net i needed. Thi net i then converted into a BE net, which ha the diplacement ield calculated within the harmonic repone anali a a boundar condition. The acoutic quantitie on the radiating urace and on peciall located point (ield point) are computed. 5

6 Thee ield point are poitioned inide the air volume, which urround the vibrating plate [6]. For the anali o the acoutic behaviour, the value o the ound preure and the ound preure db level, the integrated ound power and the eective urace velocit are o importance. In Fig. 5 the calculated ound preure ditribution on the bai o the given urace velocit ield o a three-laer andwich plate i hown. Fig. 5: Calculated urace velocit and ound preure ditribution o a three-laer andwich plate In order to veri the coupling o urace velocit and radiated ound power, extended FE and BE calculation or dierent material and dierent material damping value were perormed. A an example, Fig. 6 how the requenc ditribution o the averaged urace velocit and the ound power level calculated b the FEM and the BEM or a titanium plate within the requenc interval rom 0 Hz to 1000 Hz. Fig. 6: Frequenc ditribution o the averaged urace velocit and the ound power level According to the reult o the LASVI meaurement and the acoutic invetigation within the reverberant chamber, the numerical calculation uing the FE and BE technique clearl reveal the coupling o the urace velocit and the radiated ound power. Thee two phical quantitie have the ame requenc behaviour, but dierent abolute db value due to the dierent reerence value. Thereore, the coupling o the tructural vibration with the acoutic radiation reult in the coupling o the phical parameter urace velocit and radiated ound power. Thi undamental vibro-acoutic coupling i the bai or the ollowing analtical model. 6

7 MODELLING OF THE SOUND RADIATION The vibro-acoutic behaviour o acouticall excited tructure i characteried b the coupling o the tructural vibration with the urrounding air volume. Thi complex proce o excitation, tructural repone and radiation can onl be analed i the vibration i decribed a a time-dependent olid-borne ound ield. Here, bending vibration o plate repreent a relativel imple model or the creation and propagation o olid-borne ound. In the ollowing, rectangular plate with mall thickne and delection a well a a homogeneou ma ditribution are analed. For the decription o the plate dimenion, a carteian x, z coordinate tem i ued. The x and coordinate open the plane, in which the edge l x and l o the plate are ituated, the thickne h and the delection w are choen to be parallel to the z coordinate (Fig. 7). Fig. 7: Plate under an exterior preure ield p An exterior preure ield normal to the urace p like p iωt = p( e (6) i aumed to create a time-dependent bending wave, which can be written or the delection w in the orm iω t w( = w( e. (7) Thi delection ield w i related to the urace velocit ditribution normal to the plate' x plane b v w = iωw. (8) In term o the choen energ ormulation o thi olid-borne ound ield within an aniotropic plate, the urace velocit v w ha to ati the ollowing variational equation δ Π( ) = 0 (9) v w 7

8 with Π being the total energ in term o Π v ) = U ( v ) T( v ) + W ( v ). (10) ( w w w w Here U v ) denote the deormation energ unctional (ee ()), T v ) the kinetic energ ( w (ee (1)) and W v ) i the potential energ due the exterior preure ield p with ( w ( w l x l W ( v ) = iω p( w( ddx. (11) w 0 0 Beide thi variational ormulation, alo phicall meaningul initial and boundar condition have to be deined in order to ull decribe the urace velocit ield in the plate. The olution o thi elato-dnamic ield problem or a general excitation p can onl be approximated b an appropriate et o unction deined on the whole domain o the elatic bod. A reaonable olution can be ound b ubtituting the general urace velocit ield v w b M R N R c n= 1 with v being the eigenunction o the problem and v = v, (1) R R M N a the order o the RITZ olution, which i ued to determine the eigenunction o the problem. Subtituting the approximation (1) in the variational equation (9) inall ield v = iωv R R M N l l x v p( dd = 1 ρh ( ω ω ) Λ 0 0 (13) with Λ a the norm o the eigenunction v [7]. Thi expanion give the poibilit to calculate the tructural-dnamic repone or a general acoutic excitation p b uing the eigenrequencie and eigenorm o the tructure. The decribed olution method or the tructural-dnamic anali o plate with iotropic material behaviour i implemented into a peciall developed program tem and i the bai or the anali o the radiation proce. The urace velocit ield (13) caue the radiation o ound wave into the air volume. In order to decribe thee ound wave, an idealized model i built. The rectangular plate i aumed to be embedded in a rigid wall o ininite extent, what create two emi-ininite pace. In the negative hal-pace, a contant ound preure p 0 due to a diue ound ield exit. A radiation in thi hal-pace i not calculated. The poitive hal-pace i thought to be the air volume, in which the ound wave propagate. The ound wave are decribed b the phical parameter o the ound particle velocit normal to the rigid wall v z / air and the ound preure p. The radiation o ound wave i caued b the coupling o the olid-borne ound ield v with the air in the poitive hal-pace. An analticall continuou decription o the ound particle velocit on the urace o the wall i thereore onl poible in the orm o a Fourier integral expanion 8

9 1 v ) + + ( ikxx ik z/ air z = 0) = v = v( k k e e dk xdk 4π v ( ( k x, k and the Fourier tranorm ) ( + + ik ikxx v( k, k ) = v e e dxd. x (14) (15) Inide the ull poitive hal-pace z 0, the ound particle velocit i aumed to propagate a a plane wave in the z -direction due to the plane ound ource and the ininite air volume v ikzz iω t z/ air = vz/ air e e (16) with v ( ikxx ik = vz/ air z = 0) = v = v( k k ) e e dkx dk. 4π z/ air( (17) The calculation o the ound preure i then perormed uing the baic relation between the ound particle velocit and the ound preure v z / air 1 = iωρ 0 p z (18) a a plane wave o the orm ωρ 0 ikzz iωt p = vz/ air e e. (19) k z With thee analtical equation or v z / air and p, the radiated ound ield i completel determined and the ound power, which i tranmitted through a plane z = z 0 in the poitive hal-pace, can be calculated. Auming a zero energ lo in the reverberant chamber, the radiated ound power inide the receiver hall can be calculated olving the integral P 1 Re + + = p( z = z 0, vz / air z = z0, ddx. (0) Inerting equation (16) and (19) tranorm the integral into the Fourier integral and ield P ( v( k, k ) v ( k, k ) dk dk + + ωρ 0 x x = 8 x π k0 k x k (. (1) Thi equation include the Fourier tranorm v ( ( k x, k ) o the olid-borne ound ield, which i related to the urace velocit ditribution (13) in the vibrating plate. Thereore, the developed analtical vibro-acoutic model how the cloe coupling o the urace velocit with the radiated ound power. 9

10 Exemplaril Fig. 8 preent the reult, derived rom thi vibro-acoutic model and rom the tructural-dnamic and acoutic meaurement. In Fig. 8a, the irt reonance peak o the averaged urace velocit or a rectangular aluminium plate ( l x = 860 mm, l = 560 mm, h = 1.97 mm) i hown, which on the one hand i calculated on the bai o the excited urace velocit ditribution v (13) b v = 1 l l x l x l 0 0 ρ hv v ddx () and on the other hand i directl meaured b the LASVI technique. A comparion o the experimentall determined and analticall calculated (1) ound power i given in Fig. 8b. For both the average urace velocit and the radiated ound power a good agreement o the theoretical and experimental data ha been achieved. Fig. 8: Comparion o calculated and meaured tructural-dnamic and acoutic quantitie (material: aluminium allo h = 1.97 mm) REFERENCES 1. Huenbach, W.; Holte, C.; Göhler, W.; Seiart, K., Application o polmer matrix compoite material or tructural component o mall atellite, Proceeding o the International Conerence on Spacecrat Structure, 1996, Vol., , Noordwijk. Huenbach, W.; Holte, C., Structural-dnamic deign o ibre-reinorced tructure, Proceeding o the19 th International SAMPE EUROPE Conerence, 1998, Pari 3. Holte, C., Zum Dämpungverhalten aniotroper Faerverbundtrukturen, Diertation TU Clauthal, Huenbach, W.; Holte, C., Peier G., Dnamical Characteritic o Fibre Compoite, Proceeding o the 1 th International Conerence on Compoite Material, 1999, Pari 5. DIN EN 3741, Ermittlung der Schallleitungpegel von Geräuchquellen, Deutche Faung EN 3471, Huenbach, W.; Holte, C.; Lepper, M.; Täger, O., Inluence o material damping on the dnamic and acoutic behaviour o lightweight tructure, Proceeding o the 0 th International SAMPE EUROPE Conerence, 1999, Pari 7. Cremer, L.; Heckel, M., Körperchall, Springer