Global and local synthetic descriptions of the piano soundboard, Proceedings of Forum Acusticum,...

8/10/2019 Global and local synthetic descriptions of the piano soundboard, Proceedings of Forum Acusticum, Aalborg (DK), 27.06.-01.07.2011, EGE, K. + X. BOUTILLO

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Synthetic descriptions of the piano soundboardFORUM ACUSTICUM 2011

27. June - 1. July, Aalborg

A0

B2

x

Woodgraindirection y

33o

p

13cm

A#2

C8

Figure 1. Rear view of the upright piano studied with the woodgrain direction added in green and the two bridges (on the otherside of the soundboard) added in red.

Based on the model corresponding to the findings of themodal analysis, the mobility (or mechanical admittance)at any point of the soundboard can be synthesised (sec-

tion 3), in the spirit given in the introduction. It is com-pared to published measurementsfar fromthe bridge andat the bridge, and also to global approaches of the mo-bility of mechanical structures by Skudrzyk and Langley(section 4).

2. Two vibratory regimes

2.1. Modal density

The modal density n(f)is a global descriptor of the vi-bratory behaviour of the soundboard in the mid-frequencydomain. Based on measured modal frequencies ([3]), themodal density has been estimated as the moving averageof the modal spacing (six successive modes retained foreach estimation). The frequency evolution ofn(f)(Fig. 2)at four points of measurements (see Fig. 1 for the exact lo-cations) reveals two distinct vibratory regimes of the struc-ture.1. Below1.1kHz, the four experimental curves are almost

similar. The modal density increases slowly and tendstowards a constant value of 0.06modes Hz1 inde-pendently of the zones of the board where the measure-ment is done. This means that the corresponding modesextend over the whole soundboard: the ribbed board

behaves as a homogeneous plate. The rise in n in thelowest frequency range is characteristic ofconstrainedboundary conditions. The theoretical asymptotic modaldensity of a isotropic homogeneous plate is:

n(f) =S

2

h

D (1)

where Sand hare respectively the plate area and thick-

ness, is the material density, and D = Eh3

12(1 2)withEand , the Youngs modulus and Poisson ratio.The ribbed zone was replaced by an isotropic thin plate

with surface and surfacic area equal to those of the cor-responding part of the soundboard. The rigidity of thehomogenised plate D was somewhat arbitrarily takenso that itsdynamical rigidityD/(h)equals that of theorthotropic spruce plate in the direction of the grain:

Dhom = Dx,wood = Exh

3

12(1 xyyx) . The mechanicalcharacteristics of the strips of wood are derived frommeasurements made by Berthaut [4] on spruce speciesselected for piano soundboards.

2. For frequencies above1.1kHz,n(f)decreases signifi-cantly and is not exactly the same at each location of thesoundboard. The interpretation is that ribs confine wavepropagation: the soundboard behaves as a set of struc-tural wave-guides. The lowest frequency for which thisphenomenon occurs is such that the inter-rib spacing pcorresponds to a half-wavelength:kx= /p (see Fig. 1for the directions of the x- andy-axes), correspondingroughly to 1.1 kHz in the soundboard case. The modaldensity of such a waveguide can be easily calculated.

The red continuous curve in Fig. 2 corresponds to threetimes the theoretical modal density of one waveguideand corresponds closely to the measured modal densi-ties. The interpretation is that the motion is not strictlyconfined to the inter-rib region where it has been gener-ated but extends over the two adjacent regionswith thesame wavenumberkx= /p. Thus, a given point of thesoundboard "sees" three times more modes than thereare in a single wave-guide and the modal density is alsoto be multiplied by 3. Modes can be organised in fam-ilies, corresponding to the successive transverse modes(like in pipes) whose wavenumbers are kx,m = m/pwith m N. The asymptotic modal density of anyfamily (i.e.restricted to one given transverse mode) isthat of a beam of lengthLy(1/2 dependency):

n(f) f+

Ly2f

h

Dy

1/4(2)

As frequency increases, successive families enter intothe modal population of the guide, multiplying step bystep the modal density given by Eq. 2 by 2, 3, etc., asit will be seen further. The mechanical characteristicsof the waveguide are the one of the interrib orthotropicspruce plate.

2.2. Nonuniformity of the radiation

The structural wave-guide phenomenon deeply modifiesthe acoustical coincidence phenomenon in comparison towhat occurs with a homogeneous plate. A plate radiates ef-ficiently when the structural wavelength is larger than thatin air (supersonic structural waves). For a thin isotropicplate, this occurs above a so-called coincidence frequencygiven by the intersection of the dispersion curves in theplate (blue plain and dotted line in Fig. 3) and in air (dah-dot line in the same figure). Musical consequence: for anote with a fundamental below 1.1 kHz, the lowest par-tials radiate less efficiently (and thus, decrease in time lessrapidly) that the upper ones.


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0 500 1000 1500 2000 2500 3000 3500 4000

102

101

Modaldensity[modesHz

1]

Frequency [Hz]

Figure 2. Modal densities of the upright piano sondboard studied.(), , ), () : measurements at four points of the soundboard(different zones). : theoretical modal density of the homo-geneous equivalent clamped plate (see 2.1). : three timesthe theoretical modal density of one waveguide for the first trans-verse mode (1,n).

The dispersion curve of a structural wave-guide is dif-ferent: it always starts with supersonic structural waves(efficient acoustical radiation) and may present two, one,or even no coincidence frequencies, depending on thevalue of the wave widthp. Accordingly, there will be two,one, or no change in the radiation efficiency. This createsa nonuniformity in the radiation of the soundboard in thetreble range of the instrument compared to the low-rangewhich may explain the well-known difference in timbre.The timbre is influenced by the relative level of the par-tials and also by their relative time-decays. For example,for the key D6 having a fundamental frequency around1245 Hz, the sound level andthe damping factor of thefundamental may be higher (due to the acoustical radia-tion: supersonic structural waves) than those of the nexttwo partials (subsonic structural waves).

It may be possible to establish a connection betweenthe frequency range where the radiation pattern of the pi-ano is ruled by the wave-guide phenomenon (around 1.1kHz) and the so-called killeroctave mentionned by somemanufacturers. The transition between the two vibratoryregimes of the soundboard and the induced nonuniformityof the acoustical radiation may explain why the sustain isso difficult to obtain around the fifth to sixth octave (see

for example comments of the Fandrich Piano Companyspiano maker [5]).

3. Synthesised mechanical mobility of the

soundboard

The purpose here is to give an expression of the pianosoundboard mechanical mobility (in the direction normalto the soundboard) depending on a small number of pa-rameters and valid up to several kHz.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

20

40

60

80

100

120

140

160

Frequency (Hz)

Wavenumber(radm

1)

A6

The

"killer"

octave

A5

= 2p/3

= p

= 2p

Figure 3. Relations of dispersion for flexural waves in the ho-mogeneous isotropic plate (), in the air ( ) and for thetwo first modes ( ) of the waveguide between the second andthird ribs. and : discrete values corresponding to the (1, n)-and(2, n)-modes of the wave-guide, respectively. : partials oftheD6strings. See also the range of the so-called killeroctaveA5-A6([880-1760] Hz).

3.1. Analytical expression: sum of the modal contri-

butionsThe driving point mobility YA at a point (xA, yA) of aweakly dissipative vibrating system can be expressed asthe sum of the mobility of single-degree of freedom lineardamped oscillators:

YA() =VA()

FA()= i

+=1

2(xA, yA)

m(2+ i 2)(3)

wheremis the modal mass, is the modal loss factor,the modal angular frequency and the modal shapeof the mode.

3.2. Synthesis in the low-frequency range

In the low-frequency range, the vibration extends over thewhole soundboard and the modal analysis has shown thatthe soundboard was similar to a homogeneous plate in thisfrequency range. Whatever the wavenumbers are (depend-ing on the boundary conditions, for example), the modalshapes can be approximated locally by the product of twosines:

(x, y) = sin(2kxx + x) sin(2kyy+ y)(4)

Since we do not want to take into account the detailsof the geometry, location, etc., we may as well replace the

local dependency of any modal shape by a random distri-bution:

(xA, yA) = sin(2 + ) sin(2+ ) (5)

where the random quantities and are uniformly dis-tributed in[0,1].

For this type of modal shape and in order to obtain theorthogonalityin terms of the mass matrix, we consider thatthe modal mass of any mode is m =M/4. For this low-frequency range,Mis the mass of the whole soundboard(including ribs, bridges and the two fir bars) and almostequal to 9 kg for our upright piano.


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Figure 4. Dispersion law of the flexural waves in the bridge.

The spacing between successive modal frequenciesf+1 f is considered as the reciprocal of the modaldensity, as modelled in 2.1. The first mode is slightly dif-ferent from one piano to another; a typical value of 70 Hzwas chosen here. In order to account for the non-regularity

of the soundboard geometry, a random proportion of theinter-modal interval is added to each modal frequency:

f+1 = f+ 1

n(f)[1 + ] (6)

where is a random quantity uniformly distributed in[1/2,+1/2].

Accordingto the experimentalresults of modal analysis,we took 2% as a uniform value for modal loss factors .

3.3. Limit of the low-frequency range

It was shown in 2.1 that the vibration regime of thesoundboard changes at around 1.1 kHz, that is whenkx=/p(p: inter-rib spacing). In general, 1.1 kHz should thusbe considered as the limit of the low-frequency regime.However, this global description must be considered morecarefully when waves are generated locally (by a string,for example) and that a local singularity of the structure,namely the bridge, alters significantly the wave-guide dy-namics. The impedance of the bridge a slightly curvedbar, almost orthogonal to the direction of the ribs ismuch larger than that of the table (or the ribs) for fre-quencies higher than the kHz (almost five times greaterat 4kHz). The dispersion curve of transverse waves in the

bridge only, considered as a beam, is shown in Fig. 4. Itappears that bending in the x-direction that would corre-spond tokx = /p cannot occur below 4.2 kHz1. Belowthis frequency, the bridge prevents the vibration from be-ing localised in a single waveguide (or in three of them,with the same kx = /p); in effect, it makes the sys-tem plate+bridge+ribs more or less homogeneous up to4.2kHz. As a first approximation, we consider that thelow-frequency regime for the mobility at the bridge ex-tends up to 4.3 kHz.

1 Surprisingly (or not. . . ) this frequency is the fundamental frequency ofC8, the highest piano note.

102

103

104

106

105

104

103

102

101

Frequency [Hz]

Re

(Y)

Figure 5. Real part of the mobilityfar from the bridge. : syn-thesised mobility. : mean value (see 4.1).: envelopes(see 4.2).

3.4. Mobility above 1.1 kHz far from the bridge

The approach is the same as in the low-frequency range.As explained above, modes above 1.1 kHz are localised

in a region extending over approximately three inter-ribbands, with kx= /p. The reasoning on the modal shapescan thus be repeatedverbatim. The massMis consideredto be three times the mass located between two ribs, i.e.0.7 kg overall. The modal masses are here also m =M/4. The modal frequencies are taken as explained in 2.1 for the case of the wave-guide. Although the acousti-cal radiation regime changes in that frequency region, weconsider that the modal loss factors are the same as in thelow-frequency region. This approximation needs to be re-considered in a future research, when a global approach,similar to that performed on the vibrations, will have beendone on the acoustical radiation of the soundboard.

We present in figure 5 the real part of the synthetisedmobility, according to Eq. 3, far from the bridge. Themagnitude of the synthesised impedance is presented inFig. 6 and compares very well with published measure-ments by Giordano [6]. The fall of impedance measuredaround 1 kHz is well predicted by our model.

3.5. Mobility above 1.1 kHz at the bridge

As explained in 3.3, we considered that the bridge ex-tends the frequency range where the soundboard can beconsidered as homogeneous, namely up to 4.2 kHz. Be-yond that limit, we adopt for the mobility at the bridge

the same approach as for the mobility in a wave-guide, farfrom the bridge, as described in 3.4. There is clearly herea lack of selfconsistency which, hopefully, will be resolvedby future research.

The synthesised mobility far from the bridge is given infigure 7) and the magnitude of the synthesised impedancein Fig. 8. The mean value of the synthesised impedance be-tween 100 and 1000 Hz is approximately 800 kg s1. Thisvalue is consistent with the measurements at the bridgepublished by Wogram [7] or Giordano [6]: these authorsmeasured a mean impedance for typical upright piano ofabout103 kg s1. Moreover, the fluctuations of the mobil-ity for those frequencies are10-15 dB, which is also con-


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102

103

104

101

102

103

104

105

Frequency [Hz]

|

Z|

=

|

Y|

1

Figure 6. Magnitude of the impedance far from the bridge. Top:synthesised values. Bottom: measurements published by Gior-dano [6]. : mean value according to Skudrzyk (see 4.1).

102

103

104

106

105

104

103

102

101

Frequency [Hz]

R

e(Y)

Figure 7. Real part of the mobility at the bridge. : synthe-sised mobility. : mean value (see 4.1).: envelopes(see 4.2).

sistent with measurements published by Conklin [8] forexample. Nevertheless a discrepency on the average valueof the impedance (particularly visible around 1-3kHz, onfigure 8) exists. Our model must be improved in orderto take into account more properly the influence of thebridge.

4. Comparison with global approaches

4.1. Skudrzyk mean-value theorem

Skudrzyks mean-value method(proposed in [9][10] andtheorised in its final form in [11]) predicts the mean valueand the asymptotic value of the driving point admittance

102

103

104

101

102

103

104

105

Frequency [Hz]

|

Z|

=

|

Y|

1

Figure 8. Magnitude of the impedanceat the bridge. Top: synthe-sised values. Bottom: measurements published by Giordano [6]. : mean value according to Skudrzyk (see 4.1).

of a weakly dissipative vibrating structure. This approachis not frequency limited.

The principal results obtained by Skudrzyk are recalledhere. Skudrzyks idea consists in replacing the discretesummation in equation 3 by a continous integral. Aftersimplification of the denominator in the hypothesis of

small damping, the transformation of equation 3 takes theform:

YC =

+0

i2(xA, yA)

m(2 2)d=GC+ i BC(7)

where 2 = 2(1 + i), GC =(YC), BC =(YC),

and = d

d = 2f =

2

n(f)is the average modal

spacing, written here for angular frequencies. Finally, byuse of the residue theorem, the real part of the driving pointadmittance is given by:

(YA())+ GC =

2 M=

n(f)

4M (8)

In this frequency domain, the real part of the admittancedepends only on the modal density and the mass of thestructure. For a thin plate, the imaginary part BCvanishesat high frequency [11]:

YA()+

GC = 1

4h2

3(1 2xy)

E (9)

written here in the isotropic case.GC is equivalent to thedriving point admittance of the infinite plate [12]. It de-pends neither on the frequency, nor on the surface but only


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on the thickness h and on the elastic constants of plate:the Youngs modulus E, the Poissons ratio xy, and thedensity. By extrapolating towards the low frequencies,Skudrzyks theory predicts the mean value of the admit-tance:GC = (YC)is the geometric mean of the values atresonancesGresand antiresonances Gares.

4.2. Langleys envelopes calculationsLangley [13] evaluates analytically the envelopes of theanalytical summation given in equation 3. Bidimensionalstructures, such as plates, can present repeated resonances,degeneracy and thus irregular modal spacing. Langley in-troduces semi-empirical modifications in order to take intoaccount these irregularities. His approach is derived fromthe one of repartition of the resonances in room acoustics.Under the assumption that the modal spacing conforms tothe Poissons law, the amplitudes of resonance frequen-cies of a bi-dimensional rectangular structure are given by([13]):

Gres GC(1 + 12

) coth [(1 + 12

)2

2 ](10)

where the modal overlap factor (f) = n(f)f (de-fined as the ratio between the half-power modal band-width and the average modal spacing) is modified in2 =[1(L1L2)1] in order to take into accountthe repeatedfrequencies.2depends on the natural numbersL1and L2related to the aspect ratio of the rectangular structure byL2/L1 = Ly/Lx. Similarly, Langley gives the amplitudeof antiresonances Gares.

4.3. Comparisons

The mean values GCand envelopesGresand Garesare rep-resented in figure 5-8. The synthesised quantities displaysthe right properties in terms of mean value, asymptot, andenvelopes.

5. Conclusion

The approach presented in this communication avoids thedetailed description of the soundboard, based on a verylarge number of parameters. It can be used to predictglobal changes of the driving point mobility, and possibly

of the sound radiation in the treble range, resulting fromstructural modifications. Synthesised impedances matchwell with published measurements. Our model must beimproved in order to take into account more properly theinfluence of the bridge. Lets conclude with a remark byConklin [8] concerning the effect of bridges on the toneproduction.The design of the soundboard bridges affectsprofoundly the tone of a piano. In coupling the strings to

the soundboard, the bridges functions as impedance trans-

formers presenting a higher impedance to the strings than

would exist in the case of direct coupling. [...]If the stringswere terminated directly on the soundboard, the result

would be a louder-than-normal but relatively unpleasant

tone of comparatively short duration. By adjusting the de-

sign of the bridges, the designer of a piano can change the

loudness, the duration, and the quality of the tone, within

a certain range, in order to suit the intended use of the in-

strument.Some more research is clearly needed in orderto put some numbers onto these assertions.

References

[1] K. Ege, X. Boutillon: Synthetic description of the pianosoundboard mechanical mobility. Proc. 20th InternationalSymposium on Music Acoustics, ISMA 2010, Sydney andKatoomba,5.

[2] K. Ege, X. Boutillon, B. David: High-resolution modal anal-ysis. Journal of Sound and Vibration32(2009) 852-869.

[3] K. Ege: The piano soundboard - Modal studies in the low-and the mid-frequency range. PhD Thesis, cole Polytech-nique, Palaiseau, France, 2009.

[4] J. Berthaut, M. N. Ichchou, L. Jezequel: Piano soundboard:structural behavior, numerical and experimental study in the

modal range. Applied Acoustics64(2003) 1113-1136.[5] D. Fandrich: Soundboard Technology and Manufacture. The

Designers Notebook, PianoBuilders/NW, 1995.

[6] N. Giordano: Mechanical impedance of a piano soundboard.Journal of the Acoustical Society of America 103 (1998)2128-2133.

[7] K. Wogram: Acoustical research on pianos. Part I: Vibra-tional characteristics of the Soundboard. Das Musikinstru-ment24(1980) 694-702, 776-782, 872-880.

[8] H. A. Conklin: Design and tone in the mechanoacoustic pi-ano. Part 2. Piano structure. Journal of the Acoustical Soci-ety of America100(1996) 695-708.

[9] E. J. Skudrzyk: Vibrations of a System with a Finite or an

Infinite Number of Resonances. Journal of the AcousticalSociety of America30(1958) 1140-1152.

[10] E. J. Skudrzyk: Simple and complex vibratory systems.Pennsylvania State University Press, 1968.

[11] E. J. Skudrzyk: The Mean-Value Method of Predicting theDynamic-Response of Complex Vibrators. Journal of theAcoustical Society of America67(1980) 1105-1135.

[12] L. Cremer, M. Heckl, B. A. T. Petersson: Structure-BorneSound, Springer-Verlag, 2005.

[13] R. S. Langley: Spatially Averaged Frequency-ResponseEnvelopes for One-Dimensional and 2-Dimensional Struc-tural Components. Journal of Sound and Vibration 178(1994) 483-500.


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Global and local synthetic descriptions of the piano soundboard, Proceedings of Forum Acusticum,...

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