END CONDITIONS OF PIANO STRINGS Palaiseau Cedex,

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1 END CONDITIONS OF PIANO STRINGS Kerem Ege, Atoie Chaige aboratory for Solid Mehais, Eole Polytehique, UMR7649, 98 Palaiseau Cedex, Uité de Méaique, Eole Natioale Supérieure de Tehiques Avaées, Chemi de la Huière, 976 PAAISEAU Cedex, Abstrat The ed oditios of piao strigs a be approximated by the iput admittae at the bridge. Proper measuremets of this value are therefore required. A method of validatio of admittae measuremets o simple strutures is proposed i this paper. High resolutio sigal aalysis performed o strig s vibratios yields a estimate for the iput admittae. This method is implemeted o a simplified devie omposed of a piao strig oupled to a thi steel beam. INTRODUCTION The researh of a trade-off betwee loudess ad sustai (duratio) is a major issue for piao desigers ad maufaturers. The way the eergy of vibratio is trasferred from the piao strig to the soudboard depeds o the ed oditios of the strigs at the bridge: these oditios a be approximated by the iput admittae at the oetig poit betwee the strig ad the resoator. Therefore, proper measuremets of this value are eeded. Give this, we propose here a method of validatio of admittae measuremets o simple strutures. Parameters suh as frequeies ad dampig fators of the strig partials deped diretly o the ed oditios. The aalysis of the vibratory sigal of the strig, based o high resolutio estimatio methods (ESPRIT algorithm), allows us to evaluate effiietly those parameters ad leads to the alulatio of the iput admittae. This method is implemeted o a simplified devie omposed of a piao strig oupled to a thi steel beam. The omparative study of two experimetal ases (isolated strig vs. oupledstrig) leads us to the iput admittae. This value, derived from vibratory measuremets of the strig is ompared to diret admittae measuremets performed o the beam, ad to theoretial preditios, i order to validate the method. Summary of piao aoustis STRING-SOUNDBOARD COUPING IN PIANOS The vibratios of the piao strig are oupled to the soudboard (whih radiates the soud) via the bridge (Figure ). This ouplig determies the toe duratio ad the soud power. The boudary oditios of the strig must esure a ompromise betwee soud power effiiey (high trasverse veloity for a give strig fore) ad toe duratio (low soudboard veloity).

2 Review of the liear model Figure : Priipal seth of the piao, with the mai ompoets. [Asefelt, 990] et us osider the bridge veloity v(, t) (, t) (where ( x, t ) is the vertial trasverse displaemet of the strig), ad the fore trasmitted to the bridge f (, t) T (, t) (where T is the tesio of the strig). Figure shows these x vetors at the ed of the strig. t T f v Figure : Fore ad veloity at the bridge A proper model of this ouplig leads us to osider the mea mehaial power trasmitted from the strig to the soudboard for a steady state exitatio of the soudboard at agular frequey : P m ( ) Re F (, ) V (, ) The iput admittae (at the bridge side) is the ratio betwee the applied fore ad the veloity of the soudboard at poit x (see figure ) : V (, ) Y ( ) G ( ) jb ( ). () F (, ) We have therefore : P ( ) Re F (, ) F (, ) Y ( ) G ( ) Z V (, ) m () T F (, ) where Z T is the harateristi impedae of the strig (at V (, ) T the strig side), with the elerity of the trasverse waves i the strig ad the liear desity of the strig. Proper measuremets of Y are therefore eeded. Previous measuremets of soudboard impedae Z have bee doe (Figure 3) but eed to be reosidered espeially i the upper frequey rage. x

3 a. b. Figure 3: Previous measuremets of soudboard impedae a. [Wogram,984] b. [Giordao,998] I partiular, the fat that the modulus of the impedae dereases with frequey(above Hz for Wogram s resultsad above 7Hz for Giordao s oes) seems to otradits the theory whih predits a ostat asymptotial value (see for example [Bush-Vishia, 98] ): Z ( ) 8 hd (3) where p p 3 is the desity of material, h the thiess, D Eh [ ( )] the rigidity of a isotropi plate, E the Youg s modulus ad the Poisso s ratio.notie that for a orthotropi plate, a similar result applies where E eeds to be replaed by E E. From a physial poit of view, the fat that the impedae dereases would mea that the mobility of the soudboard ireases i the upper frequey rage, whih is rather questioable. The soudboard impedae is defied as the iverse of the soudboard admittae. For a sigle polarisatio of the strig we have: F (, ) G ( ) jb ( ) Z ( ) V (, ) Y ( ) G ( ) B ( ) Strig model The soudboard is a movig ed for the strig whih modifies its eigefrequeies f ad dampig fators i s (iverse of deay times). A usual trasmissio lie model yields the admittae of the strig at the ed: Y ( ) jy ta (4) where Y Z is the harateristi admittae of the strig ad the wave umber. With a first-order approximatio ta we get the perturbatio of the omplex eigefrequeies: jz Y ( ) j Z G ( ) jb ( ) j f (5) where is the agular frequey for the first trasverse mode of the strig.

4 This fially yields the perturbatio of the eigefrequeies (real part) ad to the dampig fators (imagiary part): T G ( ) ad B ( ) (6) f T The method ad its validatio Sie these modifiatios of strig s eigefrequeies ad deay times are diretly related to the soudboard admittae, we propose to measure these quatities i order to derive the soudboard admittae. I additio, diret measuremet of the admittae at the bridge will allow to he whether the admittae derived from strig s measuremets is really the oe that is see by the strig. The validatio of the method is divided i four steps: (a) Measuremets of eigefrequeies ad deay times o a isolated strig. (b) Same measuremets with the strig loaded at oe ed by a ow admittae. () Derive the edadmittae from these measuremets. (d) Compare the results with alulated or diretly measured load admittae. The prototype EXPERIMENTA SET-UP The experimetal devie is omposed of a piao strig strethed out betwee two eds srewed to a massive support (a alumium plate). Two ases are osidered: the first, Figure 4a, is a isolated strig (with two fixed eds assumig to have a ifiite admittae) ; the seod, Figure 4b, is a strig oupled to a thi steel beam (with a ow load admittae). b. z y o F ix ed ed A x a. F ixed ed B l = 6:6 m F ix ed ed A E d B : y T hi st eel b eam o z x E m = 8: m E m = 8: m Figure 4: The prototype a. Isolated strig b. Strig loaded at oe ed by a ow admittae (thi beam) Experimetal diffiulties The mai diffiulties result from the fat that we have to ompare two experimetal ases. Fist of all we must eep exatly the same legth ad the same tesio of the strig i both experimetal ases: with ad without the beam. This is

5 ritial as log as a small error of oe of these two parameters diretly affets the eigefrequeies of the strig. Aother diffiulty is that whe attahed to the strig, the beam is prestressed, whih modifies its properties ompared to the isolated beam. A additioal soure of experimetal diffiulty lies i the fat that we must avoid exitig the horizotal polarizatio of the strig, otherwise we get double peas. Fially, beause the frequey shifts due to the load are very small, we have the eessity of usig powerful estimatio methods (see the ext Setio). RESUTS Ifluee of the strig o the load ad ifluee of the load o the strig The system uder study is a strig oupled to the beam at oe ed. The aim of the first series of measuremets osists i ivestigatig the ifluee of oe ompoet of the system oto the other oe. For both the strig ad beam, we ompare measuremets doe for the uoupled ompoets (i blue o the figures below) to measuremets performed o the oupled system (i red). a. b. Figure 5: Ifluee of the ouplig a. Beam Admittae (beam aloe i blue,beam oupled to the strig i red) b. Strig s spetrum (below 3.5 Hz) (isolated strig i blue, strig loaded i red) Figure 5a shows the beam admittae at the bridge poit for the two ases. These diret measuremets were oduted with a impedae head srewed to a mehaial shaer. The ompariso of the admittaes i both ases shows that the ormal flexio modes frequeies have bee slightly ireased after the ouplig. This a be explaied by the fat that the prestressig of the beam due to the tesio of the strig adds stiffess to the system. The ifluee of the load o the strig is showed i Figure 5b. These spetra are obtaied by mea of the Fast Fourier Trasform (FFT) applied to the reorded trasverse veloity of the strig (isolated ad oupled) at a give poit with the help of a laser vibrometer. Two features a be poited out: the strig s eigefrequeies are slightly shifted up after the loadig ad ew spetral ompoets appear. The first three ompoets (at 05 Hz, 390 Hz ad 97 Hz respetively) are the first flexural modes of the beam; the ompoet at 530 Hz is a torsioal mode of the beam.

6 A alterative to Fourier aalysis: the ESPRIT algorithm As it has bee uderlied earlier (Figure 5b) the frequey shifts due to the load are very small. Therefore, the usual FFT aalysis does ot appear to be the most appropriate tool for estimatig the strig s eigefrequeies ad dampig fators. Here, aother estimatio method has bee used: the ESPRIT algorithm. It is a high-resolutio estimatio method whih presets iterestig alteratives to lassial Fourier trasform. This method is based o the assumptio that the aalyzed sigal x[ ] sampled at frequey F is omposed of damped siusoids (equatio 7). This fially e yields estimates for the followig parameters: frequeies ( f ( F e ), amplitudes A ad phases. F e ), dampig fators i i i x[ ] A e os( ) i i (7) For more details of this method, we ivite the reader to refer to [Roy et al., 986] ad [Roy et al.,989]. Aother method has also bee applied: the Hilbert method. This lassial method is based o a demodulatio tehique. Assumig that eah ompoets of the sigal a be isolated from the others by badpass filterig (equatio 8), a omplex sigal is assoiated to the dampig siusoids by meas of Hilbert trasform (equatio 9). x [ ] A e os( ) (8) ad the assoiated omplex sigal is give by: y i ( [ ] ( x [ ]) A e e ) (9) Amplitude ad phase detetio fially leads to the estimatios of the siusoids ompoets through liear regressio: l y [ ] l A (0) arg y [ ] () Dampig fators Usig the aalysis tools preseted above, we are ow able to determie the omplex eigefrequeies of the strig. Figure 6 shows the dampig fators obtaied for the strig s partials i both ases of our study. These data are give i the Table below.

7 a. b. Figure 6: Dampig fators of the strig partials (obtai via ESPRIT method) a. Frequey rage [0-8 Hz] (isolated strig i blue, strig loaded i red) b. Zoom-i. Frequey rage [0-5.5 Hz] (isolated strig i blue, strig loaded i red) Isolated strig f (Hz) Coupled strig f (Hz) Isolated strig ( s ) Coupled strig ( s ) Isolated strig A Coupled strig A It a be observed i Figure 6b ad i the Table that the first partials of the oupled strig are more damped tha the other oes. Furthermore, the amplitudes are lowered ompared to the isolated strig. It meas that, i this frequey rage, there is a eergy trasfer betwee strig ad beam. For the first partial of the strig (the fudametal), the deay time is oly of 0.3 seods for the oupled strig whereas it is equal to.3 seods for the isolated strig. Fially, the high dampig fators for the upper partials(more tha 0 s above 0Hz) are maily due to itrisi dissipatio i the strig. oad admittae: ompariso betwee diret measuremets ad alulatio From measuremets of dampig fators ad frequey shifts, we a derive the load admittae for eah omplex frequey : Y ( ) i i f () Z Z The results, preseted i Figure 7, are importat. The good agreemet (for frequeies smaller tha 7 Hz) betwee diret measuremets, theory, ad admittae derived from alulatio validates the method. These omparisos also ofirm that the measured isolated load is the oe that is see by the strig. Suh a approah should be of iterest for future measuremets o piao soudboard, sie, due to the omplexity of strig-soudboard ouplig, oe ruial questio is to ow

8 withoutambiguity whether or ot measuremets performed o the soudboard itself are of real sigifiae with regard to strig s behavior ad, i tur, to piao soud. Figure 7: oad ( soudboard ) admittae Compariso betwee three methods Diret measuremets oduted with a mehaial shaer: i red. Calulatio derived from oupled strig s spetrum (equatio ): bla poits Theoretial asymptoti value (derived from equatio 3): i gree CONCUSIONS I this paper a method for validatig admittae measuremets o simple oupled strutures has bee preseted. A prototype omposed of a sigle piao strig, isolated or oupled to a thi steel beam, has bee desiged. High resolutio sigal aalysis performed o strig s vibratios yields a estimate for the iput admittae. This method gives aurate results ad should be ow applied to measuremets o piao strigs mouted o real piaos i order to derive soudboard admittae ad validate diret measuremets. REFERENCES Asefelt, A. (990). Five letures o the aoustis of the piao, Itrodutio, RoyalSwedish Aademy of Musi No64, 990, Stoholm Bush-Vishia, I.J. (98). Drive poit impedae of a ifiite orthotropi plate uder tesio, J. Aoust. So. Am. 7(), pp Giordao, N. (998). Mehaial impedae of a piao soudboard, J. Aoust. So. Am. 03(4), pp Roy, R.; Kailath, T. (989) ESPRIT - Estimatio of Sigal Parameters via Rotatioal Ivariae Tehiques, IEEE Tras. o Aous., Speeh, ad Sig. Pro.37(7), pp Roy, R.; Paulraj, A.; Kailath, T. (986) ESPRIT - A subspae rotatio approah to estimatio of parameters of isoids i oise, IEEE Tras. o Aous., Speeh, ad Sig. Pro.34(5),pp Wogram, K. (980). Aoustial Researh o Piaos: Vibratioal Charateristis of the Soudboard, Das Musiistrumet, Vol. 4, pp , ,