EXPERIMENTAL MODAL ANALYSIS AND STRUCTURAL MODELLING OF A GRAND PIANO SOUNDBOARD TO SUPPORT INSTRUMENT DESIGN

Transcription

1 EXPERIMENTAL MODAL ANALYSIS AND STRUCTURAL MODELLING OF A GRAND PIANO SOUNDBOARD TO SUPPORT INSTRUMENT DESIGN R. Corrad 1, P. Fazol 2, S. Mccol 1, F. Rpamont 1, G. Squccarn 1* 1 Department of Mechancal Engneerng Poltecnco d Mlano Va La Masa 1, 2156 Mlano, Italy E-mal: roberto.corrad@polm.t 2 Fazol Panofort Va Ronche 47, 3377 Sacle, Italy E-mal: nfo@fazol.com Keywords: Grand pano soundboard, Expermental modal analyss, Structural modellng, Vbroacoustcs ABSTRACT Ths paper llustrates the frst part of a research project amed at developng and valdatng a numercal model of the vbro-acoustc behavour of a grand pano soundboard, to be used by pano makers as a tool to support the desgn process. Expermental modal analyss results are presented and compared wth correspondng fnte element numercal smulatons, whch nclude modellng of the manufacturng process, to account for the soundboard curvature and the nduced stresses. The fnte element model of a free soundboard before brdge glung has been successfully valdated up to 25Hz, n terms of both mode shapes and egenfrequences. 1 INTRODUCTION From the brth of the pano n 179, when Bartoleomeo Crstofor substtuted the hammers n place of jacks renderng the harpschord capable of graduatons n tone [1], to the begnnng of the XIX century, ths nstrument has been subjected to a lot of emprcal mprovement. Later pano makers started to study and to order expermental studes on the nstrument at the purpose of mprovng ts sound qualty. In 199 a summary of the knowledge n the feld was publshed n the form of Fve Lectures [2]. Several authors dealt wth the most mportant problems nvolved n pano desgn and manufacturng, such as the mechancs, the hammer dynamcs, the soundboard and strngs vbratons. One mportant queston s often mpled n these studes: how do the factors nvolved n pano desgn nfluence the fnal acoustc behavour of the nstrument? In order to provde an answer, several expermental studes were performed by Conkln [3][4][5] and by other authors. H. Suzuk [6] concentrated on modal analyss of a grand pano soundboard and acoustc

2 measurements focused on surface sound ntensty patterns. More recently, the ncreasng computatonal performance of electronc computers made t possble to add one more element n the study of muscal nstruments: the era of numercal smulaton started and challenged the researcher to predct the behavour of the nstrument before buldng t. Ths s what Gordano studed n [7][8][9] and J. Berthaut et al. presented n [1]. In these cases the purpose of the expermental tests was to valdate a numercal model that can predct the behavour of the pano when ts desgn s modfed. Other authors used the computer smulaton to perform parametrc analyss [11][12]. The objectve of ths paper s to present the current development status of a research project amed at developng a vbraton and sound radaton model of a grand pano soundboard, to be used by pano makers as a tool to support the desgn process. The attenton s focused on soundboard vbratons, whch are nvestgated through both expermental modal analyss and FEM smulaton. Ths paper llustrates the adopted step-by-step procedure, based on the dea of performng modal analyss tests at the three most sgnfcant stages of the soundboard manufacturng process: free soundboard before brdge glung, free soundboard after brdge glung, soundboard fnally assembled on the pano structure. The tested soundboard s that of a Fazol F278 grand pano (Fgure 1). The step-by-step approach was adopted for two man reasons. The frst s that the authors are partcularly nterested n nvestgatng how the sngle phases of the pano complex manufacturng process nfluence the vbraton propertes of the soundboard, so as to support the desgn of a grand pano. The second s related to the objectve of smplfyng the development of the soundboard fnte-element model: by performng a step-by-step valdaton, the model s expected to reach ts maxmum relablty and accuracy. To ths end, the development of the fnte element model proceeds n parallel to the expermental tests, by reproducng the soundboard confguraton at the same three manufacturng steps descrbed above. The FEM model takes nto account the taperng as well as the curvature of the board. Pecular to ths approach s that the soundboard curvature s not assumed a pror but t s obtaned by explctly smulatng the manufacturng process. As a sde effect, the presence of the nduced self-stresses s also consdered. Fgure 1. Fazol F278 grand pano In the followng, expermental modal analyss results for each one of the three steps are presented, together wth a descrpton of the features of the fnte element model and a numercal to expermental comparson, n the case of the free soundboard before brdge glung. In order to better understand the reasons for the adopted modellng approach, a bref summary of the soundboard manufacturng process s at frst provded.

3 2 THE SOUNDBOARD MANUFACTURING PROCESS The pano soundboard s composed of three elements: the board, consstng n varable wdth spruce planks wth a thckness of approxmately 1mm; the rbs, slender wood beams wth a roughly square cross secton made of the same wood or a smlar one; the brdge, whch s the element provdng vbraton transmsson from the strngs to the board; dfferent types of wood are employed n brdge constructon, lke maple, mahogany or even boxwood. The planks are glued together edge-to-edge to make a panel wth the needed wdth. The plank axes are nclned wth respect to the keyboard of an angle of approxmately 5 degrees. At ths stage the obtaned panel possesses a stffness n the drecton of the gran whch can be also 4 tmes greater than the one n the cross-gran drecton. For ths reason, several rbs are glued on the bottom surface of the board wth a gran drecton normal to that of the planks. Conkln [4] already dscussed that the global stffness of such a wood rbbed panel s domnated by the gran drecton stffness of the rbs and of the spruce strps and that the behavour of the soundboard s almost ndependent of both cross-gran stffness. Egenvalue analyss based on the fnte element model developed by the authors (see secton 4) confrmed ths result: the senstvty of the soundboard natural frequences to cross-gran stffness of both planks and rbs materals s almost neglgble. Snce glung the rbs on the board strongly nfluences the fnal performance of the soundboard, ths process s based on accurate manufacturng steps, whch ensure hgh standard product qualty. Rb Board Mould (a) (b) (c) Fgure 2. The rbs glung process A scheme of the rbs glung process s presented n Fgure 2. Frst, the glue s placed between the rbs and the board whch les on a rgd mould wth upper concavty (Fgure 2-a). Then the rbs are loaded and pushed down untl the back sde of the board completely rests on the surface of the mould; durng ths step the rbs and the board are not ted together (Fgure 2- b). The rbs reman loaded untl glue bondng s completed, and then are fnally unloaded. Durng the release phase, rbs and board are bonded together and cannot slde: at the end of the sprng back (Fgure 2-c) the soundboard exhbts both curvature (the so called crown ) and resdual stresses. The man reason for the curvature s that t helps the board to sustan the load of the strngs and mproves the acoustc qualty of the Pano. The name step 1 descrbes the tests performed on the soundboard at ths manufacturng stage. Afterwards the brdge s glued on the board surface opposte to the rbs (step 2) and later on s fnshed off by proflng ts boundares and by planng down to a specfc profle the rbs extremtes. Now the soundboard s ready to be glued onto the nternal rm, thus creatng what s commonly named the casework ( case ) of the pano, whch s seasoned for several months and s fnally glued on the external rm (step 3, see Fgure 6). Ths rather complcated manufacturng process has been optmzed over the centures to

4 guarantee an exquste tone qualty to the pano; ts nfluence was frst studed n [13], at the very begnnng of the scentfc nvestgaton of the pano vbroacoustc behavour. 3 EXPERIMENTAL MODAL ANALYSIS 3.1 Expermental set up Expermental modal analyss tests were performed at the three steps of the manufacturng process mentoned before. In partcular, at the Fazol factory n Sacle, the soundboard at step 1 (soundboard before brdge glung) and step 2 (soundboard after brdge glung) was tested by suspendng t on a steel cable (Fgure 3 and Fgure 5). The cable length was chosen so as to set the system frst natural frequency below 2 Hz. Ths choce allowed the sound board rgd moton and ts flexble vbraton modes to be decoupled. The board was nstrumented wth 1 pezoelectrc unaxal accelerometers (senstvty 1 mv/g, range 1g, 1kHz bandwdth). The tests were carred out usng two dfferent mpact hammers. Frst a large mpact hammer wth a soft tp (BIG-S n the followng) allowed an nvestgaton of the low-md frequency range ( 1kHz). Then the mpact tests were repeated usng a smaller hammer wth an harder tp (SMALL-H), to excte the hgh frequences, up to about 5kHz. The tests were repeated for two forcng ponts (labelled 49 and 5 n Fgure 4.) and sx dfferent confguratons, coverng the 6 accelerometer postons, n order to create a 2x2 mm equally spaced grd on the board. Analogous tests were performed for the step 3, at the Department of Mechancal Engneerng of Poltecnco d Mlano. In ths case, due to the system hgher weght, the case was lad on three supports n the boundary regon, so as to weakly affect the board vbraton behavour. Durng step 3, n addton to the accelerometers on the 2x2 mm grd, a laser vbrometer was used to scan a 1x1mm grd (black dots n Fgure 12): the vbrometer scanned the grd nodes for a gven number of mpact samples, measurng ts vbraton speed. The collected data were post processed to obtan the Frequency Response Functon (FRF or e α ) defned as the averaged complex rato between the acceleraton of the j-th pont and the nput force, appled to the k-th pont, as a functon of frequency. At least ten hammer hts are averaged. Fgure 3. Step 1 of the expermental modal analyss: free soundboard before brdge glung Fgure 4. Measurement grd used durng all the tests (step1-2-3).

5 Fgure 5. Step 2 of the expermental modal analyss: free soundboard after brdge glung Fgure 6. Step 3 of the expermental modal analyss: soundboard assembled on the pano structure. Fgure 7 shows an example of FRF measured at step 3, for the node par k=49 and j=22 (Fgure 4), usng dfferent exctaton and measurement devces. The coherence functon γ s 2 calculated as γ = G /( G G ), where G s the power spectrum or the cross spectrum of the xy xx yy nput x (mpact hammer force) and of the output y (measured acceleraton). Ths fgure emphaszes that each hammer s capable of forcng dfferent frequency ranges. Moreover, the good agreement between the blue and the green lnes demonstrates the accuracy of the measurements process. γ α e [m/s 2 /N] <α [rad] Input Force: BIG-S; Output Acceleraton: Laser Input Force: SMALL-H, Output Acceleraton: Accelerometer Freq [Hz] Fgure 7. Example of FRF obtaned usng dfferent exctaton and measurement devces, (force appled on node 49, response measured on node 39) 3.2 Natural modes dentfcaton procedure In the low frequency range, were the overlap factor [14] remans below 3%, modes are clearly spaced and the system shows lmted dampng; e.g. for the free soundboard ths condton s verfed n the range -2Hz and t s possble to effcently apply modal dentfcaton procedures. Startng from the measured FRF functons, an dentfcaton algorthm based on two subsequent steps s appled. A frst tentatve seres of natural frequences ( ω r ) and loss factors ( ξ r ) s dentfed usng the classcal Prony method. In Fgure 8 an example of the results s presented. The poles of the analytcal system whch reproduce the expermental results are plotted whle the order of the

6 system tself s ncreased. The red crcles represent the stable poles defned as the poles that do not change ther frequency and loss factor wth respect to the prevous teraton, wthn a gven tolerance. Blue crcles are stable only wth respect to frequency, whle crosses correspond to nose poles. At the end of ths teratve procedure, the stable values of ω r and ξ r are chosen as startng parameters for a least squares mnmzaton algorthm n a user-defned frequency band (second step). The error functon s defned by: where [ e a e a * ( ) ( )] [ ( ) ( )] (1) ε = α ω α ω α ω α ω A a α ( ω ) = + + ω ω ω ξ ω ω n r M r= n 2 1 r + r r K (2) To mprove the fttng, hgh and low frequency resdues ( K and M ) [15] are added to the resonant modes, to account for the natural frequences out of the dentfcaton frequency band. Fgure 9 shows an example of an dentfed FRF supermposed to the correspondng expermental one. The result of the second dentfcaton step conssts of mproved natural frequency and loss factor estmates and of an array of A values whch can be normalzed to obtan the r-th expermental mode shape shapes are shown. φ e, r r. In secton 3.3 and secton 5, some of the dentfed mode 25 System order Freq [Hz] Fgure 8. Example of stablzaton dagram: red crcles correspond to the stable poles. By followng ths procedure, 4 modes up to 28Hz were dentfed for the step 1 soundboard, 2 modes up to 2Hz for the step 2 soundboard, 1 modes up to 1Hz for the step 3 soundboard. The dentfed loss factors vary from.5 to 1.4% for the frst two steps, from 1 to 2% for the thrd step. When consderng the three dfferent values n the maxmum dentfed natural frequences, t must be ponted out that the three constructon steps correspond to three dfferent structures, n terms of confguraton and especally constrants and dampng. Ths results n three dfferent values of the transton frequency between the low-frequency range, whch s characterzed by ndvdual modal resonances, and the md/hgh-frequency range, whch shows coupled mult-mode response.

7 α 1 Expermental Analytcal <α Freq [Hz] Fgure 9. Comparson between expermental and dentfed FRF (free soundboard before brdge glung): exctaton on node 49, response on node Identfed mode shapes In Fgure 1 and Fgure 11 two of the mode shapes of the step 2 soundboard are reported. Fgure 12 and Fgure 13 show the frst and the thrd mode of the step 3 soundboard, already nvestgated by other authors [1][4][6]. Analogous results for step 1 soundboard wll be shown n the followng (Fgure 18 and Fgure 19) as a comparson wth fnte-element numercal smulatons. Fgure 1. Mode shape, step 2, f=5.5hz. Fgure 11. Mode shape, step 2, f=175.5hz. Fgure 12. Mode shape, step 3, f=45.4hz. Fgure 13. Mode shape, step 3, f=111.8hz.

8 4 THE FINITE ELEMENT MODEL A fnte element model was developed n order to smulate the major manufacturng steps rangng from the bare board to the frame-soundboard assembly. Nonlnear geometrc effects (stress stffenng) are consdered when extractng the natural frequences, n order to take nto account the self-stresses nduced by the constructon procedure. In ths paper only the frst step, namely the glung of the rbs on the board s descrbed. The board under study s made of Norway spruce (Pcea abes) orgnatng from Val d Femme whle the rbs are made of Stka spruce (Pcea stchenss). Both woods were modelled as lnear elastc orthotropc materals, ther nne elastc constants beng taken from Bucur and Keunecke [16][17]. The glung process was smulated wthout explctly modellng the cohesve behavour of the glue layer between board and rbs. On the contrary, rbs and board are n drect contact: before bondng a frctonless hard contact whch allows for mutual slp s assumed; after bondng stck condtons are mposed by not allowng further slp and separaton (rough blateral contact or te). The mould was modelled as a rgd surface; contact wth the board s frctonless. The analyss was conducted wth Abaqus/Standard, usng 3D contnuum elements (C3D8R) for the rbs, 3D contnuum shell elements (SC8R and SC6R) for the board, and dscrete rgd elements (R3D3) for the mould. The choce of the contnuum shell elements s motvated by the need of reproducng bendng and tapered geometry of the board and of an accurate descrpton of normal contact on both faces. In Fgure 14 the ntal poston of the varous parts s depcted. An external pressure s exerted on the rbs n order to push the board on the mould and establsh full contact. In ths frst phase mutual slp s allowed between all parts. Afterwards rbs and board are ted together and the external pressure s removed to smulate the elastc sprng-back. At the end of ths step one has a curved geometry (Fgure 15) due to self-stresses (Fgure 16). Fgure 14. The three components of the FE model: the rbs, the board and the mould. A comparson between the shape of the real soundboard after the glung process and the shape obtaned va fnte element modellng s gven n Fgure 17, where the dstances between the control ponts (marked by red dots n the pcture) and a common reference plane are plotted. Consderng that the measures performed on the real board are easly affected by errors due to the hgh flexblty of the structure, the results showed n Fgure 17 are satsfactory and the computed statc confguraton can be safely used as the startng pont for the subsequent egenvalue analyss.

9 U, U3 [m] 6.2e-3 4.9e-3 3.7e-3 2.4e-3 1.1e-3-1.4e-4-2.6e-3-3.9e-3-5.2e-3-6.5e-3-7.8e-3-9.1e-3-1.e-3-1.1e-2-1.1e-3-1.3e-2 U, U3 [m] 3.1e-2 2.9e-2 2.6e-2 2.4e-2 2.3e-2 1.9e-2 1.7e-2 1.5e-2 1.3e-2 1.1e-2 8.8e-3 6.6e-3 4.5e-3 2.3e-3 5.3e-5-2.1e-3 (a) (b) Fgure 15. The smulaton of the glung process: (a) the board s pressed on the mould, (b) the soundboard after sprng-back. Colour maps represent the dsplacement normal to the board. Dstance from least square plane [mm] Fgure 16. Maxmum prncpal stress map (whte corresponds to tracton stress, everywhere else compresson occurs); lower surface (on the left) and upper surface (on the rght). 3 Measured Numercal Nodes 6 8 Fgure 17. Comparson between measured and calculated soundboard confguraton after elastc sprng-back (left). Locaton of the control ponts used for the comparson (rght).

10 5 MODEL VALIDATION Egenvalue analyss was performed on the FEM model descrbed n the prevous secton, at the deformed state attaned after the elastc sprng-back: the results are compared to the correspondng dentfed natural modes. As an example, Fgure 18 and Fgure 19 show the comparson between computed and expermental mode shapes, for the 5 th and the 14 th soundboard natural modes. Numercal and expermental results show very good agreement n terms of both mode shapes and natural frequences (for both modes, the frequency error s lower than 3%). In order to summarze the results obtaned for all the vbraton modes n the -3Hz frequency range, MAC (Modal Assurance Crteron) analyss was performed. The MAC ndex expresses the cross-correlaton between the mode mode shape φ of the -th expermental mode: e, MAC 2 φ φ = ( φ φ ) ( φ φ ) e, n, j j T T T e, e, j n, n, j φ of the j-th numercal mode and the n, j,j=1,, Nmod (3) The subscrpts e and n n equaton (3) respectvely stand for expermental and numercal data and Nmod s the number of natural modes n the consdered frequency range. The result s a square matrx whose elements may vary from (no correlaton) to 1% (full correlaton). In the deal case of a perfect numercal-expermental matchng, the MAC matrx s equal to the dentty matrx. shows the calculated MAC matrx correspondng to the step 1 soundboard vbraton modes n the -3Hz frequency range. Lookng at Fgure 21, t can be notced that the mode pars wth MAC>5% are defntely concentrated on the man dagonal. Moreover, the number of mode pars wth MAC>5% s 24 over a total of 36. Wth reference to the natural frequency matchng, Fgure 21 shows a summary of the percentage errors between numercal and expermental natural frequences, for those mode pars showng hgh correlaton (MAC>6%). The results reported n Fgure 21 demonstrate that the frequency error never exceeds 1%. It s very mportant to pont out that the results shown n Fgures were obtaned wthout any model tunng, thus showng that the accuracy of the mplemented model s really promsng for future model development. In order to further mprove the numercal results, the fnte element parameters wll be optmzed by mnmzng the mean error between computed and measured natural frequences. We suppose n fact that ths error s due to an ncorrect estmate of wood materal propertes whch prmarly affect the soundboard numercal egenvalues and whch typcally show sgnfcant dsperson. (a) (b) Fgure 18. Step 1 soundboard: (a) expermental (3.1Hz) and (b) numercal (3.4Hz) modes.

11 (a) Fgure 19. Step 1 soundboard: (a) expermental (92.1Hz) and (b) numercal (91.6Hz) modes. (b) 35 MAC 1% 2% Expermental modes % 6% 4% 2% Frequency error 1% -1% Numercal modes -2% Mode no. Fgure 2. MAC matrx for the frst 36 modes of the step 1 soundboard. Fgure 21. Frequency comparson. Postve error stands for a model overestmaton of the natural frequency. 6 CONCLUSIONS The results obtaned n ths prelmnary study are encouragng and show that the effort of creatng a valdated numercal model of the soundboard-frame assembly can be useful to gan a better understandng of the nfluence of the constructon process on ts vbroacoustc behavour. It s the authors opnon that the promsng results obtaned are a consequence of the hgh modellng detal and of the weak smplfyng assumptons (lnear elastc wood behavour; based on the accurate materal selecton, the board s assumed to be a monolthc plate made of homogeneous orthotropc materal wth constant orentaton; same assumpton for the rbs; deal glue wth neglgble thckness). The research actvty s gong on and the numercal-expermental comparson wll be extended to a hgher frequency range. In vew of ths comparson, t s mportant to pont out that the materal propertes have been taken from lterature: the actual propertes of the soundboard materal under testng can be expected to be substantally dfferent. Therefore, a model optmzaton procedure wll be set up, whch wll allow to dentfy the most approprate materal propertes. REFERENCES [1] N. H. Fletcher, T. D. Rossng, The physcs of muscal nstruments, Sprnger, U.S.A, 1998.

12 [2] Harold A. Conkln Jr., A. Askenfelt, E. Janson, D. E. Hall, G. Wenrech, K. Wogram, Fve Lectures on the Acoustc of Pano, [3] Harold A. Conkln Jr., Desgn and tones n the mechanoacoustc pano. Part I. Pano hammers and tonal effects, Journal of Acoust. Soc. Of Amerca, 99 (6), , [4] Harold A. Conkln Jr., Desgn and tones n the mechanoacoustc pano. Part II. Pano structure, Journal of Acoust. Soc. Of Amerca, 1 (2), , [5] Harold A. Conkln Jr., Desgn and tones n the mechanoacoustc pano. Part III. Pano strngs and scale desgn, Journal of Acoust. Soc. Of Amerca, 1 (3), , [6] H. Suzuk, Vbraton and sound radaton of a pano soundboard,, Journal of Acoust. Soc. Of Amerca, 8, , [7] N. Gordano, Mechancal Impedance of a Pano Soundboard, Journal of Acoustc Socety of Amerca, 13 (4), Aprl 1998, (a). [8] N. Gordano, Sound Producton by a Vbratng Pano Soundboard, Journal of Acoustc Socety of Amerca, 14 (3), September 1998, (b). [9] N. Gordano, Smple Model of a Pano Soundboard, Journal of Acoustc Socety of Amerca, 12 (2), Pt.1, August [1] J. Berthaut, M.N. Ichchou, and L. Jézéquel, Pano Soundboard: Structural Behavor, Numercal and Expermental Study n the Modal Range, Appled Acoustcs, 64, , 23. [11] J. Tppner, Konas P., and Dànel V. Influence Factors on Dynamcal Behavor of Pano Soundboard, [12] Mamou-Man, J. Frelat, C. Besnanou, Numercal smulaton of pano soundboard under downbearng, Journal of Acoustc Socety of Amerca, 123 (4), 28. [13] Paul H. Blhuber, C. A. Jhonson, The nfluence of the Soundboard on Pano Tone Qualty, Journal of Acoust. Soc. Of Amerca, 11, , 194. [14] K. Ege, X. Boutllon, B. Davd, Hgh-resoluton modal analyss, Journal of Sound and Vbraton, 325 (4-5), , 29 [15] D. J. Ewns, Modal Testng: Theory and Practce, Research Studes Press LTD, Great Brtan, 1984 [16] V. Bucur, Acoustcs of Wood, Sprnger, 26. [17] D. Keunecke, S. Herng, P. Nemz, Three-dmensonal elastc behavour of common yew and Norway spruce, Wood Sc. Technol., 42: , 28.