Influence of soundboard modelling approaches on piano string vibration

Abstract

This work explores the influence of the dynamics of the piano soundboard on string vibration and on the force acting between the vibrating string and the bridge. Four different soundboard representations of different complexities are considered: (i) a finite element model that considers the complete dynamic behavior of the soundboard at the connection point with the string within the frequency range of interest, (ii) a reduced modal model containing only five modes, (iii) a Kelvin–Voigt system characterized by an equivalent stiffness and damping, and (iv) a rigid soundboard represented by a simply supported boundary condition. The connection between the string and the soundboard is modelled by coupling a simply supported stiff string model with the different representations of the soundboard through a contact stiffness. As well as directly accounting for the string-soundboard coupling, this approach also includes the duplex scaling segment. The latter can be left to vibrate freely or muted with a continuous distribution of dampers. Although the simplest soundboard representation is not dissimilar from the other more complex models, the dynamics of the soundboard affect the decay time of the note, the force transmitted to it, and the vibration of the radiating surface of the soundboard.

Full text

Influence of soundboard modelling approaches on piano string
vibration
Pablo Miranda Valiente,
a)
Giacomo Squicciarini, and David J. Thompson
Institute of Sound and Vibration Research, University of Southampton, Southampton, SO17 1BJ, United Kingdom
ABSTRACT:
This work explores the influence of the dynamics of the piano soundboard on string vibration and on the force acting
between the vibrating string and the bridge. Four different soundboard representations of different complexities are
considered: (i) a finite element model that considers the complete dynamic behavior of the soundboard at the
connection point with the string within the frequency range of interest, (ii) a reduced modal model containing only
five modes, (iii) a Kelvin–Voigt system characterized by an equivalent stiffness and damping, and (iv) a rigid
soundboard represented by a simply supported boundary condition. The connection between the string and the
soundboard is modelled by coupling a simply supported stiff string model with the different representations of the
soundboard through a contact stiffness. As well as directly accounting for the string-soundboard coupling, this
approach also includes the duplex scaling segment. The latter can be left to vibrate freely or muted with a continuous
distribution of dampers. Although the simplest soundboard representation is not dissimilar from the other more com-
plex models, the dynamics of the soundboard affect the decay time of the note, the force transmitted to it, and the
vibration of the radiating surface of the soundboard. V
C2024 Acoustical Society of America.
https://doi.org/10.1121/10.0025925
(Received 25 August 2023; revised 12 March 2024; accepted 17 April 2024; published online 14 May 2024)
[Editor: Andrew Morrison] Pages: 3213–3232
I. INTRODUCTION
The strings of musical instruments, including the piano,
are generally coupled through a bridge to a soundboard,
which radiates sound more efficiently. The vibration of the
strings is affected by this coupling to some extent. The
bridge and soundboard provide a quasi-rigid termination to
the string at its speaking length, allowing it to vibrate at its
fundamental frequency and associated harmonics. Although
most of the vibration energy is reflected back into the string,
parts are also transmitted into the soundboard and past the
bridge into the duplex scaling segment of the string.
1
The
connection with the soundboard can also produce the double
polarization of the strings
2
and can provide coupling
between vertical and longitudinal directions.
In the literature, the connection between the strings and
the soundboard has been modelled either by studying the
dynamics of two separate systems or, less frequently, by
accounting for a full coupling between them. When the
string and the soundboard are modelled as decoupled sys-
tems, the component of the string tension perpendicular to
the soundboard serves as an input to the vibration of the
soundboard.
3–6
In these cases, the length of the string corre-
sponds to the distance between the agraffe and the bridge
(the speaking length, i.e., neglecting the duplex scaling seg-
ment), and its ends are simply supported. String vibration
excited by the hammer can then be studied with numerical
approaches such as finite differences.
7–11
In a fully coupled
approach, however, the dynamics of the soundboard at the
bridge provides a non-rigid boundary for the string. In this
case, the total length of the string between the agraffe and
the hitch pin is included. Models of this type are more com-
plex but can better explain the effect of the soundboard on
the string vibration and hence on the force transmitted to the
soundboard. Fully coupled string-soundboard models for
piano acoustics have been proposed by the authors,
12,13
and
also exist for other instruments.
14–17
Finite difference
approaches
14
or modal models in the time domain
15–17
have
been successfully implemented to obtain the string vibration
and contact forces of the coupled system.
Whether they are seen as part of a coupled system or as
independent vibrating components, the strings and sound-
board can be represented using different modelling techni-
ques. For string vibration, the equation of a stiff string
solved with finite differences is probably the most popular
method in musical acoustics, but finite elements (FE)
4
or
modal models have also been presented.
16,17
The dynamics
of the soundboard at the bridge have been addressed in dif-
ferent ways, including a frequency-dependent boundary
condition for string-only models,
18
or plate models such as
thin plates,
3
Reissner–Mindlin plates
4
(including shear
deformations), linear filters,
6
or FE.
19–21
The soundboard itself was characterized experimentally
at the bridge by Wogram,
22
using different modal analysis
techniques. The coupled vibration of the string soundboard
system was analyzed, and it was found that the decay of
string tones was larger due to the reduced energy transfer
caused by the mismatch in the impedance. Ege et al.
23
a)
Email: pmmv1g14@soton.ac.uk
J. Acoust. Soc. Am. 155 (5), May 2024 V
C2024 Acoustical Society of America 3213
ARTICLE
...................................

characterized the soundboard both numerically and experi-
mentally. Among other things, the authors studied the non-
linear behavior of the soundboard, which was quantified to
be orders of magnitude smaller than the linear one. Their
study also gives an insight into the damping of the sound-
board. Values of damping ratio between 0:005 and 0:015
were obtained. Similar values for the soundboard damping
were obtained by Corradi et al.
24
These authors measured
the vibration of the soundboard at different manufacturing
stages and noted that, at advanced manufacturing stages, the
soundboard presents damping ratios varying between 0:008
and 0:03. In Squicciarini’s thesis,
20
the finished soundboard
fitted into the piano exhibited even higher damping ratios,
ranging between 0:007 and 0:047. Suzuki
25
obtained values
of 0:032 for the first mode and between approximately 0:01
and 0:015 for higher modes, while Berthaut et al.
26
obtained
smaller values of 0.003–0.0065.
Trevisan et al.
27
developed analytical soundboard mod-
els, in which the soundboard was modelled as a
Love–Kirchhoff plate. One of the interesting conclusions
reached by the authors is that geometric and manufacturing
details of the soundboard can have an influence on the first
natural frequencies when compared with experimental
results. Closely related to the present study, reduced sound-
board modelling has been introduced by various authors
(see, for instance, Boutillon and Ege
28
and Corradi et al.
24
).
In these studies, the frequency response of the soundboard is
approximated in an average sense by combining the driving
point impedances of an infinite beam and an infinite plate.
These approaches were formulated directly in the frequency
domain and cannot be directly adopted in a time-domain
solution.
The main aim of this work is to develop a model of a
coupled system, in which a piano string and soundboard are
connected at the bridge, and to evaluate the degree of
complexity required to describe the dynamic behavior of the
soundboard. For this purpose, different dynamic models of a
soundboard, of increasing complexity, are coupled to a
string in the direction perpendicular to the soundboard, as
described in Sec. II. The authors have previously developed
models using one of the simpler soundboard representations
to study the dynamics of the coupled system in two
12
and
three
13
directions. Differently from this, in the current work,
comparisons are made between different soundboard models
that focus on the transverse direction perpendicular to the
soundboard. To calculate the response of the coupled sys-
tem, a time-domain model in a state-space formulation is
implemented in Sec. III. This approach requires the struc-
tural damping of the string alone, for which an experimental
setup is designed, and measurements are performed in Sec.
IV to determine the damping of strings disconnected from
the bridge. The implications of using the different sound-
board representations are discussed in Sec. V, with conclu-
sions given thereafter. The modelling approach considers
only the transverse direction perpendicular to the sound-
board, and no nonlinear phenomena are considered, such as
the generation of phantom partials due to the coupling
between transverse and longitudinal directions on the
strings.
6
Instead, the main novelty of this work lies in the
exploration of the details needed to model the soundboard to
account for the interaction with the string. Furthermore, the
inclusion of a duplex scaling segment in the model, which
can be either be attenuated with a continuous distribution of
dampers and springs or left free to vibrate (see below), is an
aspect that, to the best of the authors’ knowledge, has not
been extensively addressed before.
II. STRING AND SOUNDBOARD
A string of length Lconnected to a soundboard system
is shown in Fig. 1together with the main variables adopted
FIG. 1. (Color online) Schematic representation of a string coupled with different representations of the soundboard. These are a full FE model, a reduced
modal model, and an equivalent Kelvin–Voigt system.
3214 J. Acoust. Soc. Am. 155 (5), May 2024 Miranda Valiente et al.
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for the analysis. These are the displacements yand forces F
at the hammer striking point efrom a distance Lefrom the
agraffe termination and at the connection points between the
string sand soundboard b. Only vibration perpendicular to
the soundboard is considered; this direction will be referred
to as vertical. The connection with the soundboard divides
the string of total length Linto two parts: the speaking
length, Ls, and the remaining vibrating duplex scaling seg-
ment, Ld. With the aim of analyzing the effect of sound-
board dynamics on the string vibration and transmitted
force, three different representations of the soundboard are
developed and compared in Sec. II B. These are also sum-
marized schematically in Fig. 1and are a full modal FE
model, a reduced modal model, and a Kelvin–Voigt (K-V)
system. A contact stiffness, kc, is introduced to represent the
normal contact stiffness associated with local deflection at
the contact point. Although the contact stiffness could be
accounted for implicitly by higher order modes in the modal
summation of the soundboard mobilities presented in Sec.
II B, it is not considered as the modal summations are trun-
cated. In addition a simply supported end at the bridge loca-
tion (i.e., rigid soundboard) is considered.
A. Stiff string model
A model is defined for the vertical motion of the string
uncoupled from the soundboard and simply supported at its
ends defined by the agraffe and hitch pin (see Fig. 1). It is
represented as a stiff string with an equation of motion
given as
29
l@2y
@t2¼T0
@2y
@x2ESK2@4y
@x4;(1)
where yis the vertical motion of the string in a position, x,
across the string at a time, t,lis the mass per unit length, T0
is the tension, Eis the Young’s modulus, Sis the cross-
sectional area, and Kis the radius of gyration. For a pinned
string with length L, the n-th mode shape at a position, x,
from the agraffe and the corresponding natural angular fre-
quency are
29
/nx
ðÞ¼sin npx=L
ðÞ
;xn¼n2pf01þBn2
ðÞ
1=2;(2)
where f0¼T0=l
ðÞ
1
2=2Lis the fundamental frequency of the
string in the absence of bending stiffness and the inharmonic
coefficient B¼p2ESK2=T0L2. Damping is omitted from Eq.
(1), but this will be included in terms of damping ratios (see
Sec. III) in the state-space formulation and also indirectly by
coupling with the soundboard.
In a coupled string-soundboard system, the force at the
bridge is calculated from the interaction between the two
components, as outlined below. In the absence of a sound-
board model (i.e., simply supported string at the bridge), the
input force can be written considering the vertical compo-
nent of the tension and a third order derivative related to the
bending stiffness, evaluated at the string termination x¼Ls,
as
Fss ¼T0
@y
@xx¼LsþESK2@3y
@x3x¼Ls
:(3)
B. Soundboard models
The different soundboard models adopted in this work
are introduced in this subsection. These are an FE model, a
reduced modal soundboard, the response of which is fitted
to the FE model, and a K-V soundboard consisting of a
spring-damper system.
1. FE model
The geometry of the soundboard adopted in this work is
based on a grand piano that was made available to the
authors. An FE model that can represent the complete
soundboard dynamics has been developed in COMSOL
MULTIPHYSICS
V
R
. The thickness of the soundboard varies
between 7 and 9 mm from the edge to the center; its edges
are clamped, and the bridges and the wooden stiffener
beams—often referred to as ribs—are modelled as isotropic
and the soundboard as orthotropic. The assumption of iso-
tropic ribs and bridges is justified from previous studies by
one of the coauthors,
20
in which a design sensitivity analysis
was performed. It was found that the most important param-
eter of the ribs is the Young’s modulus in their longitudinal
direction, with their other directions being negligible. This
matches the purposes of the ribs, which are to stiffen the
soundboard in its weakest direction. The properties of the
wood correspond to Sitka spruce and were obtained from lit-
erature,
20
but the stiffest direction of the Young’s modulus
and the density have been modified to provide a better
agreement with driving point mobility measurements. The
direction convention used for the material properties is illus-
trated in Fig. 2, along with the points at which the mobilities
are obtained along the two bridges. Additionally, a point, k,
is identified in the middle of the soundboard that is used to
compute the vibrational response, as described in Sec. VC.
Regarding the mesh, approximately 49 000 quadratic
FIG. 2. Overview of soundboard geometry, with locations of connection
points for each note and response point k. Conventions used for directions
of orthotropic properties of wood are also shown.
J. Acoust. Soc. Am. 155 (5), May 2024 Miranda Valiente et al. 3215
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tetrahedral elements were used with a minimum size of
kpl=9 in the ribs and the bridge and kpl =10 in the sound-
board, with kpl being the shortest wavelength of a simple
Kirchhoff plate representing the soundboard. The ribs/
bridge and soundboard are considered as elastic bodies that
share common element nodes, corresponding to a perfect
connection between the domains. The main parameters of
the model are listed in Table I. The first three modes calcu-
lated using the FE model are shown in Fig. 3.
To calibrate the FE model, measurements of the mobility
were performed. Impact testing was conducted using an impact
hammer PCB model 086C03 (PCB Piezotronics, Depew, NY)
with sensitivity (615%) 2.25 mV/N, measurement range
62224 N pk, and hammer mass 0.16 kg. The response was
obtained using a miniature uniaxial accelerometer PCB model
352C23, with sensitivity (615%) 1.0 mV/(m/s
2
). Both devices
were connected to a Data Physics Quattro (Data Physics,
Riverside, CA), a portable data acquisition system.
A coherence above 0.9 was achieved for frequencies
between 50 and 4000 Hz. Damping ratios of the first six
modes were estimated from the measured mobilities, using
the circle fitting procedure.
30
These are shown in Table II.
For higher frequencies, damping was chosen to represent the
different behavior that the piano structure may have on the
mobility response.
To cover a frequency range between 50 and 4000 Hz,
1000 modes of the soundboard are included in the modal
summation. The measured driving point mobility at the
bridge location of note D4 is compared with the results of
the FE model in Fig. 4. The FE model gives a similar trend
and level of the mobility magnitude and phase and can be
considered as a realistic representation of a piano sound-
board. It is therefore used in this work as a reference result
for the other simplified representations of the soundboard
dynamics.
2. Reduced modal soundboard
A simpler model of the soundboard is developed using
an equivalent modal system. Five modes are considered suf-
ficient to represent the main low frequency resonances as
well as the high frequency trends. The modal parameters of
this equivalent system are determined by fitting the mobility
to that from the FE model such that the first four modes cap-
ture the first four resonances, while the fifth mode is more
highly damped to represent the average level of the sound-
board mobility at higher frequencies. Although it would be
possible to extract the modal parameters directly from the
FE model to yield correct natural frequencies and mode
shapes, this would not produce the highly damped mode at
higher frequencies that represents the overall trend of the
mobility in this region. A modal fitting approach is therefore
applied by adopting the non-linear least-squares curve fitting
routine in MATLAB to refine the modal parameters of the
reduced model and hence minimize the error between the
FE mobility and the modal summation based on the five
modes. The mass normalized mode shapes, damping ratios,
and natural frequencies are then obtained. The soundboard
mobilities obtained at the bridge positions corresponding to
notes A1, D4, and D5 are shown in Fig. 5, together with
those from the full FE model and the simplified approach
discussed below. The reduced modal model can replicate
both the dynamic behavior of the main modes and the aver-
age flat response at higher frequency. This can represent the
mobilities quite well. However, at higher frequencies, the
reduced modal model can only represent the average
TABLE I. Parameters of FE soundboard model.
Description Value(s) Unit Description Value Unit
Soundboard width 1:39 m Poisson’s ratio, 12 0:37 —
Soundboard length 1:66 m Poisson’s ratio, 13 0:47 —
Young’s modulus, E117:1 GPa Poisson’s ratio, 23 0:43 —
Young’s modulus, E21:04 GPa Shear modulus, G12 1.0 GPa
Young’s modulus, E30:48 GPa Shear modulus, G13 0:96 GPa
Thickness 0:007–0:009 m Shear modulus, G23 0:04 GPa
Density 600 kg=m3——
FIG. 3. (Color online) First three modes of the soundboard calculated using FE.
3216 J. Acoust. Soc. Am. 155 (5), May 2024 Miranda Valiente et al.
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behavior of the mobility. Alternative methods have been
adopted in the literature
24,28,31
based on the mean value
method proposed by Skudrzyk:
32
although these can approx-
imate the dynamic response of a complex structure in the
frequency domain, they cannot be directly used in a time
domain, whereas the reduced modal model adopted in this
study is well suited to provide a direct structural coupling
between the string and the soundboard.
3. K-V model for the soundboard
Since the soundboard is significantly stiffer than the
strings,
33
it may be sufficient to use a model that repre-
sents the generic trends and average values of the sound-
board mobility without including the full modal
characteristics. The local dynamic behavior of the sound-
board is therefore modelled using a spring and a damper
connected in parallel, which are tuned to fit to the mobil-
ity of the FE model. The stiffness is obtained from the
low frequency asymptote, where the model is stiffness-
controlled. The spring constant kis hence obtained by
considering the reference FE mobility YFE evaluated at a
frequency, x1, that needs to be 10 times smaller than the
first natural frequency of the soundboard. In this case,
k¼jx1=YFE x1
ðÞ
. The damper coefficient is obtained
through the logarithmic average of the mobility at fre-
quencies above 2500 Hz. The soundboard mobility
obtained using this K-V approach is also shown in Fig. 5.
The spring-damper system can approximate the main
trends but does not replicate the modal behavior.
To connect the K-V soundboard model to the string via
the contact stiffness, a small mass is added at the interface
to avoid numerical problems in the time-domain calculation.
This mass should be small enough to ensure that the added
mode is outside the frequency range of interest, since the
trend of the mobility is chosen to be determined by the
spring and damper only in this case.
4. Input force to the soundboard
The connection between the string and the soundboard
is modelled by means of a contact stiffness, which repre-
sents the local stiffness behavior at the contact point.
15,16
This method of connecting dynamic components of a system
is also used in other areas of research, such as railway noise
and vibration research.
34
In the present case, the force is
proportional to the relative displacement between string and
soundboard as
Fb¼kcysyb
ðÞ
;(4)
where ysand ybare the displacement of the string and
soundboard at the bridge, which can be obtained using the
output state-space matrix Cdefined in Sec. III B, and kcrep-
resents the stiffness of the contact zone. An expression for
the contact stiffness kcis derived starting from Hertzian con-
tact theory.
35
For contact between cylindrical bodies, the
normal load per unit length Pacting over a contact patch of
width acan be written as
35
P¼pEa2
4R;(5)
where Ris the equivalent radius of curvature for the two sur-
faces. For a cylinder in contact with a flat surface,
TABLE II. Measured damping ratios.
Natural frequency, Hz Damping ratio
75:00:04
118:80:034
145:30:019
182:80:024
242:20:025
260:90:018
FIG. 4. (Color online) (Left) Location
of measurement. The impact point is
circled in red. (Right) Point mobilities
of the soundboard at the bridge loca-
tion corresponding to note D4. (b)
Magnitude. (c) Phase.
J. Acoust. Soc. Am. 155 (5), May 2024 Miranda Valiente et al. 3217
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representing the case of a string in contact with the bridge’s
surface, this is equivalent to the radius of the string alone.
The equivalent Young’s modulus, E, is given by
E¼EsEw
Es12
w

þEw12
s

;(6)
where Es,sand Ew,ware the Young’s moduli and
Poisson’s ratios of the steel string and the wooden bridge,
the latter corresponding to E3in Table I. Such an approxi-
mation considers the rigidity of the material in the direction
normal to the surface.
Using a contact width of affiffiffiffiffiffi
Rd
p, where dis the
indentation, and multiplying Eq. (5) by the length of the
contact surface Lcgives the contact force, as also derived by
Popov,
36
Fc¼pLcE
4d;(7)
from which the contact stiffness (Fc=d)is
kc¼pLcE
4;(8)
FIG. 5. Soundboard mobilities obtained using different models at different locations. (a) Magnitude. (b) Phase.
3218 J. Acoust. Soc. Am. 155 (5), May 2024 Miranda Valiente et al.
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where Lcis the length of the contact zone. The value of kc
is on the order of LcEw, and for a small contact length,
Lc0:01 m is evaluated as 4:8106N=m.
The mobility of the soundboard obtained from the FE
model at D4 is compared with that of the corresponding
string (full length L) and contact stiffness in Fig. 6.The
soundboard mobility is on average 5 to 6 orders of magnitude
smaller than that of the string at its resonances. The low
structural damping of the string gives pronounced peaks and
dips, unlike in the soundboard, where the damping of the
material is much higher. Consequently, at some antiresonan-
ces of the string, its mobility is comparable or even lower
than the soundboard and contact mobilities. The contact
spring mobility exceeds that of the soundboard for frequen-
cies above about 600Hz. Results for the other notes exhibit
similar trends; however, for A1, the spring mobility from
300 Hz exceeds that of the corresponding soundboard.
III. TIME-DOMAIN MODEL
This section describes the state-space formulation
adopted to represent and solve the dynamics of the cou-
pled system in the time domain. The model adopted for
excitation by the hammer is described first, and this is fol-
lowed by the state-space formulation of the coupled
string-soundboard-hammer system. A brief discussion on
the numerical scheme used for this study is given at the
end of the section.
A. Hammer excitation
Since Ghosh in 1932,
37
different authors have modelled
and showed experimentally that the hammer felt compres-
sion force is nonlinear and can be represented as a power
law
7,38,39
given by
Fe¼KHnp;(9)
while the equation of motion of the hammer can be written
as
Fe¼mH€
yH:(10)
The parameters KH,mH, and pcorrespond to the nonlinear
stiffness, mass, and power law coefficients obtained experi-
mentally for piano hammers.
40
The term €
yHis the hammer
acceleration, while nis the compression of the hammer
which can be expressed as
n¼yHyeif yH>ye;
0 otherwise;
((11)
where yHand yeare the displacement of the hammer and the
string at the excitation point, respectively.
Other approaches have been developed for the hammer-
string interaction, which can include, for example, the
effects of hysteresis.
41
However, as the main focus of this
analysis is the string-soundboard interaction, the simple
power law formulation is adopted.
B. String coupled with soundboard
When the string is connected with any soundboard rep-
resentation, the equations of motion of the system in state-
space form can be expressed as
42
FIG. 6. Mobility of the soundboard (FE model) Yb, the string Ys, and con-
tact stiffness Ycfor A1, D4, and D5.
J. Acoust. Soc. Am. 155 (5), May 2024 Miranda Valiente et al. 3219
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_
x¼Ax þBu þB2Fb:(12)
On the left-hand side of Eq. (12), the state vector xcontains
the modal velocities _
qand modal displacements qof the
string and the soundboard, the velocity _
yH, and displacement
of the hammer yHand can be written as x¼_
q;ð
_
qb;q;qb;_
yH;yHÞT. This approach helps to produce a single
state-space formulation that can be used for all the models
that include the connection with a soundboard. For the
reduced soundboard, there are 5 modal coordinates, and for
the FE soundboard, 1000. For the simplest soundboard rep-
resentation, the K-V model with small added mass, only one
modal coordinate is considered. The state-space matrix A
can be defined as
A¼
diag 2fnxn
ðÞ
Cd0diag x2
n

Kd0j00
0diag 2fnbxnb
 0diag x2
nb
j00
Inn0nn0nn0nnj00
0nbnbInbnb0nbnb0nbnbj00

0000j00
0000j10
2
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
:(13)
For the different soundboard models, matrix Acontains
information about the damping ratio and natural frequencies
of each string and soundboard mode within the correspon-
dent modal damping and stiffness matrices. The damping
ratio fnshould be that of the string disconnected from the
soundboard; this is characterized below in Sec. IV. To atten-
uate the vibration occurring in the damped segment of the
string, a method consisting in coupling several dashpots has
been used previously by Jiolat et al.
17
In the present work, a
distributed damper and spring are used to modify the modal
damping and stiffness matrix, respectively, embedded in the
state-space matrix Ain Eq. (12). They take the form
Cd¼ðLsþLd
Ls
cd/nx
ðÞ
/T
nx
ðÞ
dx;
Kd¼ðLsþLd
Ls
kd/nx
ðÞ
/T
nx
ðÞ
dx;(14)
where Ldis the length of the duplex scaling segment and cd
and kdare the damping and stiffness coefficients that are
defined (see below Sec. VE) to represent the strip of felt
that usually mutes the duplex scaling segment in the mid-
low register. The row vector /nx
ðÞcorresponds to the mode
shape function evaluated across Ld.
The matrix Bin Eq. (12) can be written as
B¼
/T
e
0

1=mH
0
2
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
5
:(15)
Matrix Bis a column vector containing string mode shapes
at the excitation point e,/T
e, as well as the inverse of the
hammer mass. It is used to transform the force due to the
hammer Feinto modal forces. The external force vector uis
simply composed of Fe. The remaining modal force term,
B2Fb,is
B2Fb¼
/T
S0
0/T
b
00
––
00
00
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
Fb
Fb
"#
;(16)
which introduces the modal contact force between the string
and the soundboard. This term couples the soundboard and
the string at their connection point using the string mode
shapes /sand the soundboard mode shapes /bat the con-
nection point b.
The physical displacements and velocities of the differ-
ent parts of the system are calculated as
y¼Cx;(17)
where the matrix Chas the function of converting the modal
coordinates to physical coordinates, and the output ycon-
tains the physical velocities and displacements of the string,
the soundboard at the connection point and hammer; these
are required to calculate the forces in Eq. (4) and Eq. (9) at
each time step of the integration. Expanding, this equation
takes the form
3220 J. Acoust. Soc. Am. 155 (5), May 2024 Miranda Valiente et al.
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_
ys
_
yb
ys
yb
_
yH
yH
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
¼
/s00000
0/b0000
00/s000
000/b00
000010
000001
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
_
q
_
qb
q
qb
_
yH
yH
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
:(18)
The mode shapes /are obtained at the excitation point and
at the connection point to obtain the relevant physical quan-
tities. Evaluating the mode shapes at different positions and
modifying Caccordingly can give the vibration response of
the string at any arbitrary position.
C. Simply supported string
In the case where the connection between the string and
the soundboard is not included in the model, the string is
considered as simply supported at its ends, with a total
length equal to its speaking length, Ls. Equation (12)
becomes
_
x¼Ax þBu;(19)
where now the state space vector includes only the modal
coordinates of the string and the physical velocities and dis-
placements of the hammer x¼_
q;q;_
yH;yH
ðÞ
T. The state-
space matrix Ais now reduced to include only the modal
damping and modal stiffness matrix of the string:
A¼
diag 2fnxn
ðÞdiag x2
n
j00
Inn0nnj00
  
00j00
00j10
2
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
5
:(20)
The modal forcing term Bu remains unaltered.
D. Numerical integration
Numerical time integration is performed to obtain the
time-dependent output vector from the state-space model,
containing the velocities and displacements of the coupled
system at the connection point. The time resolution of the
response dt is defined as dt ¼1=fsin terms of a required
sample frequency, fs. This, in turn, is defined in terms of the
maximum natural frequency analyzed and set to fs¼10fmax ,
where f
max
is the highest natural frequency considered for
the string. The initial conditions of the state-space vector x
are
_
q;_
qb;q;qb¼0;
_
yH¼2:5m=s;yH¼0:05 m;(21)
where initial conditions for the modal coordinates are appli-
cable for both string and soundboard. The numerical time
integration of the input state-space equation for the different
cases [i.e., in Eq. (12) or Eq. (19)] is performed in MATLAB
using ode45, which is based on the fourth-order
Runge–Kutta method.
IV. DAMPING OF A PIANO STRING
Although most of the string parameters are available in
the literature, the model developed in this work requires the
structural damping of the string disconnected from the
bridge and tensioned between the agraffe and the hitch pin.
This is generally not available and was therefore measured
experimentally. To uncouple the string from the sound-
board, two example strings were lifted using a metal cylin-
der placed near the hitch pin and fitted between the string
and the cast-iron frame, as shown in Fig. 7. The string was
therefore lifted by 2–3 mm to avoid contact with the bridge.
Measurements were also performed with the string coupled
to the soundboard for comparison.
Free vibration of the string was excited by means of a
piano hammer, and the vibration decay was analyzed to
evaluate the damping. The transverse velocity response at a
point close to the hammer striking position was recorded
using a laser Doppler velocimeter.
Prior to the calculation of the energy decay, the veloci-
ties were filtered in frequency bands of bandwidth Df¼f0,
where f0is the fundamental frequency; the bands were cen-
tered at the partials of the recorded tone. The energy decay
in each band was calculated using the Schroeder integral
as
43
Et
ðÞ¼ð1
t
v2
fs
ðÞ
ds;(22)
where vfrepresents the filtered measured velocity of the
string in the direction perpendicular to the soundboard.
Negligible double polarization was observed with this setup,
whereas in the case of the string coupled to the soundboard,
the energy associated with the decay will be influenced by
the different polarizations of the string. The reverberation
time T60 can be calculated by fitting a straight line to a
FIG. 7. (Color online) Cylinder used to lift the string to decouple it from
the soundboard.
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manually chosen range of the energy decay curve in decibels
to obtain the slope. Then the damping ratio f, corresponding
to a frequency band with central frequency fc, was found
from
44
f¼1:1
T60 fc
:(23)
Estimated values ffor two uncoupled strings, a C2 (copper-
wound) string and F4 (steel) string, are shown as diamonds
in Fig. 8and Fig. 9. The estimated damping for the case
where the strings are connected with the soundboard are
also shown (circles). In the uncoupled strings, the results for
F4 are more disperse than for C2, due to limitations of the
measurement technique used to try to avoid the hammer hit-
ting other unison strings, which in reality are not indepen-
dent strings: they consist of a single string that is chorded
through the hitch pin of the piano, as shown in the photo of
the piano string configuration in Fig. 10.
Thus, hitting other strings will result in undesired vibra-
tion in the measured string segment. These string segments
were therefore lifted, which produced a change in tension,
causing variable string frequencies. However, the resultant
damping ratios did not vary significantly.
The measured damping ratios are compared for refer-
ence with the viscous damping model (SVM), as proposed
by Chabassier
4,45,46
and also used more recently by Tan.
47
There is generally a good agreement between the measured
values and the SVM, which is here implemented using the
values available in the literature for this note and not by
curve fitting it with this set of measurements.
Although the results are slightly frequency dependent, a
constant value for fof all the modes could be also used as a
reasonable approximation for the decoupled string. This was
set to 5 105for the copper-wound string and to
7:5105for the steel string (continuous lines in Figs.
8and 9). In comparison to the structural damping of the
string alone, the damping ratios of the string coupled to the
soundboard are higher at low frequency by about a factor of
5–10, while they tend towards the damping ratios of the
string alone at higher frequencies. It is to be expected that
the model of the coupled string (see below) would be able to
replicate this effect.
V. TIME-DOMAIN MODEL RESULTS
Results from the time-domain model are shown here to
illustrate the various physical phenomena occurring in the
piano string vibration. The hammer-string contact force is
shown as well as the subsequent string vibration at different
locations. The interaction force between string and sound-
board is also described, including the vibration response at a
point on the soundboard. Comparisons are made using the
different models described in Sec. II B and the simply sup-
ported string (Sec. III C) used as a reference to emphasize
the presence of additional damping added by the connection
with the soundboard. Following this, a minimal model is
introduced, composed of a simply supported string with the
damping adjusted to include that introduced by the connec-
tion with the soundboard. Finally, the influence of including
a distributed damping and stiffness along the duplex scaling
segment of a string is shown.
FIG. 8. (Color online) Estimated
damping ratios ffor C2 string. Circles
(red) represent string coupled to the
soundboard. Diamonds (black) repre-
sent decoupled string.
FIG. 9. (Color online) Estimated damp-
ing ratios ffor F4 string. Circles (red)
represent string coupled to the sound-
board. Diamonds (black) represent
decoupled string.
3222 J. Acoust. Soc. Am. 155 (5), May 2024 Miranda Valiente et al.
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Three different notes across the piano range are consid-
ered for the analysis: A1, D4, and D5. For the different
strings analyzed, the soundboard dynamics is that of the cor-
responding connection point. The parameters used for the
strings are summarized in Table III. Some of these were
obtained from measurements on the available piano, while
others were obtained from literature, as indicated.
In Table III, the steel core of the string and its copper
winding, noted with the subscripts “core” and “wound,” respec-
tively, are used to obtain the corrected physical properties of
the copper-wound string—in this case A1, as in Conklin.
50
The
agraffe segment corresponds to the distance from the agraffe
termination of the string to the point at which the hammer
strikes. It is calculated using the relative striking position a
given in Chaigne and Askenfelt,
40
Le¼aLs,whereacan
take the values 0:12 for A1 and D4 and 0:11 for D5.
For ease of comparison, the time-domain results are
shown for the first 5 s even when the decay lasts longer than
this. Four different approaches are compared: (i) FE sound-
board, (ii) reduced modal soundboard, (iii) K-V soundboard,
and (iv) simply supported string (no soundboard, noted as
SS). The influence of using different representations of the
soundboard dynamics is discussed.
The results presented below initially include a duplex scal-
ing segment left to vibrate, which highlights the capability of
this modelling approach to include this feature that is adopted
in most grand pianos in the mid-high register. At the same
time, as discussed in Sec. VA, the inclusion of distributed
damping and stiffness along the duplex scaling length is shown
to be an effective modelling approach to mute the vibration of
this segment of the strings in the mid-low register.
1
A. Hammer-string contact force
The hammer-string forces obtained with the four differ-
ent models are generally not affected by the soundboard
model. For note A1, the results in Fig. 11 show that the
forces do not vary when using different soundboard repre-
sentations. The results of the contact profile are similar to
the ones produced by other modelling approaches taken in
the literature for C2.
40
The variations in the contact force
for D5 are also negligible, while for D4, the simply sup-
ported string results in a force that is different from the other
models, whereas for the three models that include the sound-
board, the forces are identical (results not shown here).
B. String vibration and decay
To verify the effects of the different modelling
approaches on the vibration velocity of the string in the time
TABLE III. String and hammer parameters.
Description Variable Unit A1 D4 D5
Speaking length Lsm1:30:59 0:308
Agraffe segment Lem1:56 1017:11023:4103
Duplex length Ldm0:11 0:15 0:05
Tension TN 1821 637 1420
Young’s modulus (Ref. 40)EGPa 200 200 200
Density of string steel core (Ref. 48)qckg=m37860 7860 7860
Density of copper winding (Ref. 49)qwkg=m38960 — —
Diameter of steel string core dcore m110311031103
Diameter of copper winding dwound m1:5103——
Fundamental frequency of string f0Hz 55 294 584
Piano hammer mass (Ref. 40)mHkg 10:41038:61037:8103
Piano hammer stiffness coefficient (Ref. 40)KHN=mp2:15 1087:45 1093:23 1010
Piano hammer nonlinear coefficient (Ref. 40)p—2:28 2:41 2:58
FIG. 11. Hammer-string contact force for string A1.
FIG. 10. (Color online) Piano string configuration in the middle register of
the piano.
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domain, the envelopes of the time histories are analyzed in
this section and the damping ratios of the string-soundboard
systems are also extracted.
The envelopes of the vibration velocity obtained at the
excitation point are shown in Fig. 12 for the three notes con-
sidered and the four different models. The vibration decays
obtained with the simply supported string are longer than all
the other models. This is due to the absence of the damping
provided by the connection with the soundboard, the only
damping mechanism being the structural damping estimated
in Sec. IV. The other three models provide similar results.
For A1 and D5, some differences occur at the beginning of
the response, where the reduced soundboard model is closer
to the full FE representation than the K-V model.
To quantify the differences seen in Fig. 12 and evaluate
the modelling approaches, it is of interest to calculate the
damping introduced by the connection with the soundboard
numerically and compare it with the experimental results
shown in Figs. 8and 9. The comparisons are presented in
Fig. 13 for the A1 string, where the damping estimations
from both measurements and numerical models are obtained
using the same procedure, described in Sec. IV.
The damping of the simply supported case corresponds to
the constant structural damping ratio estimated from measure-
ments, while the other models follow the trend of the measure-
ments performed with the string connected to the soundboard.
The reduced modal representation gives results closest to the
FE soundboard model, particularly at lower frequencies below
600 Hz. At high frequencies, the estimated damping of the
coupled models tends to be that of the simply supported string.
Similar conclusions hold for the other strings analyzed, but the
results are not presented here for brevity.
It can be seen that the damping ratios exhibit a squence
of dips. Although they are not clearly identifiable in the
measurements, the dips in the modal soundboard results
align with the duplex scaling resonances. This is because
damping estimates for frequencies at and close to the reso-
nance of the duplex scaling segment are affected by vibra-
tion transmission beyond the bridge and into this segment.
This is a typical characteristic of piano string vibration since
it was patented by Steinway
51,52
and is a feature of the
model, and its effect on the estimated damping ratio is there-
fore considered.
Although the general trend of the damping ratios is sat-
isfactory in comparison with the measurements, some differ-
ences occur. These can be explained by the limitations and
assumptions of the current models. The differences below
100 Hz, where the damping from the measurements is higher
than that of the models, could be associated with differences
between experimental and model mobilities. These differ-
ences have less influence at higher frequencies, but here
double polarization may have a significant effect. Double
polarization can produce a two-stage decay, hence smaller
apparent damping. An experimental method for estimating
the damping of the two transverse directions of vibration of
a uncoupled string was shown by Tan.
47
Nonetheless, it
assumes that strings have a non-linear behaviour and imper-
fections in the excitation that can excite the direction paral-
lel to the soundboard. Herein, the model is assumed to be
linear, the hammer excitation is perfectly transverse, and the
response in the second polarization is caused only by the
connection with soundboard. Tan’s experimental results
were processed using high resolution modal analysis techni-
ques consisting of the separation of the signal into sinuo-
soids with a corresponding decay, phase, and natural
frequency.
53,54
FIG. 12. Envelopes of time histories of piano string vibration velocity at the
excitation point.
3224 J. Acoust. Soc. Am. 155 (5), May 2024 Miranda Valiente et al.
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To conclude the assessment of string vibration, the
spectra of vibration velocity at the excitation point and the
connection point with the bridge are shown in Fig. 14 for
the A1 string by way of example. The simply supported
model is included for reference only and is fixed at the con-
nection point, and hence this model is not present in Fig.
14(b). The main feature of the spectra is the familiar series
of partials for this note. At the excitation point in Fig. 14(a),
there is no identifiable presence of the soundboard resonan-
ces, but the frequencies associated with the duplex scaling
segment can be identified by small peaks at 660 Hz and its
higher harmonics, indicated in the figure. At the bridge con-
nection point shown in Fig. 14(b), two low-frequency
soundboard resonances are present for the two models that
include them. Other resonances occur close to the funda-
mental frequency of this note and are therefore not sepa-
rately identifiable. The duplex scaling frequencies are also
present and more distinct than at the hammer striking point.
Considering the three models that include a connection
with the soundboard, the string partials have similar levels.
However, more similiarities exist between the two modal
soundboard models. An example is highlighted in Fig.
14(b), where at the first string partial, the K-V model has a
lower level than the other two models. A higher resonance
is also highlighted, showing that the differences are smaller
than at lower frequencies. This is caused by differences in
the mobility of the soundboard (see Fig. 6) at lower frequen-
cies, where it is comparable to that of the contact stiffness,
whereas at higher frequencies, the contact spring mobility is
higher than that of the soundboard and the soundboard has
negligible influence on the string vibration. This shows that
the higher frequency range, in which the simpler soundboard
representations are less accurate (see Fig. 5), is dominated
by the interaction between the string and the contact stiff-
ness, whereas the soundboard representation has a dimin-
ished role. For D4 or D5, where the string partials are higher
FIG. 13. (Color online) Damping ratios
for A1 string connected to the sound-
board, estimated from modelling and
experiments. Vertical dashed lines indi-
cate resonances of the duplex scaling
segment.
FIG. 14. (Color online) Spectra of
vibration velocity for note A1. Duplex
scaling resonances are indicated at the
excitation point by the arrows in panel
(a). Spectra are shown at the connec-
tion point in panel (b).
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in frequency, the interaction is dominated by the contact
stiffness as its mobility is comparable to or higher than that
of the soundboard.
Although the soundboard models produce similar
results, the computational times required differ significantly.
For the D4 case, on a standard desktop computer and with
the ode45 MATLAB routine, the computational times required
to calculate 10 s of time history with a sampling frequency
of fS63 kHz are as follows. The reduced representation
requires about 37 s, the K-V model takes 54 s, and the FE
representation requires approximately 840 s. The K-V model
requires more computational time than the more complex
reduced model because the former includes high damping.
This hypothesis was tested by running the K-V model under
different damping coefficients, showing that computational
time can be reduced as much as to 25 s when reducing
damping by orders of magnitude. The computational time
are also related to the ode45 solver adopted in study and can
vary with the numerical implementation.
C. Force at the bridge and soundboard response
In the interaction between the string and the soundboard,
it is of interest to compute the interaction force between the
two systems. This force is the responsible for the coupling,
and it can be used as an input for obtaining the vibratory
response of the soundboard. In the case of the simply sup-
ported string, in the absence of the soundboard, the force is
givenbyEq.(3). Once computed, the forces from the different
models can be used for sound generation in future work.
The envelopes of the time histories of the forces at the
bridge are shown in Fig. 15. The general trend of these
results confirms that lower tones last longer and result in
higher values of the force. Comparing the different sound-
board models, the simply supported approach produces
decays that are much longer than the others, as found for the
string vibration. Although the reduced soundboard model is
closer to the FE representation, the differences in terms of
transmitted force are more subtle than for the string vibra-
tion. The largest difference is for D5, where the response is
larger in the FE model. The time histories of the force decay
do not seem to favour a particular representation of the
soundboard. The readers can make their own subjective
judgement from the audio files Mm. 1, Mm. 2, Mm. 3, and
Mm. 4 for the A1 model and Mm. 5, Mm. 6, Mm. 7, and
Mm. 8 for the D4 model.
Mm. 1. Transmitted force to the soundboard Fb, for A1,
simply supported string.
Mm. 2. Transmitted force to the soundboard Fb, for A1, FE
soundboard model.
Mm. 3. Transmitted force to the soundboard Fb, for A1,
reduced soundboard model.
Mm. 4. Transmitted force to the soundboard Fb, for A1, K-
V soundboard model.
Mm. 5. Transmitted force to the soundboard Fb, for D4,
simply supported string.
Mm. 6. Transmitted force to the soundboard Fb, for D4, FE
soundboard model.
Mm. 7. Transmitted force to the soundboard Fb, for D4,
reduced soundboard model.
FIG. 15. Envelopes of time histories of force transmitted to the soundboard.
3226 J. Acoust. Soc. Am. 155 (5), May 2024 Miranda Valiente et al.
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Mm. 8. Transmitted force to the soundboard Fb, for D4, K-
V soundboard model.
The force spectra of the different representations are
shown in Fig. 16. The simply supported case is omitted for
ease of comparisons between the other models. There is no
sign of soundboard resonances in the force spectra, in con-
trast to what is seen in the string vibration at the connection
point [see Fig. 14(b)]. The duplex scaling resonances are
present in the transmitted force at 660 Hz for the A1 string
and 1110 Hz for D4, as well as their higher harmonics. For
D5, only one resonance is present at 3.8 kHz.
Generally, the models that include the dynamics of the
soundboard have similar peaks in their force spectra, both in
level and in frequency, while the K-V model has peaks that
are slightly different in magnitude and frequency: see, for
instance, the highlighted first peak for A1. The case of D4
shows that the differences are consistent through the fre-
quency range. At this location, there exists a greater imped-
ance mismatch between the string and soundboard than in
the other locations on the bridge. Consequently, the sound-
board model does not influence the results significantly and
the interaction with the contact stiffness is predominant.
Although the differences are generally small, the reduced
model is closer to the FE model than the K-V model.
The forces at the bridge can then be used to evaluate the
vibration response at any point on the soundboard by com-
bining them with the impulse response between the bridge,
point b(see Fig. 1), and a generic point, k, on the sound-
board shown in Fig. 3. The transfer impulse responses are
FIG. 16. (Color online) Spectra of force transmitted to the soundboard.
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obtained by calculating the inverse Fourier transform of the
transfer mobilities Ykb evaluated with the full FE model
(irrespective of which soundboard model is used in calculat-
ing the string vibration). Mathematically, the vibration
velocity at k, caused by a bridge force, Fb, is the convolution
between the bridge force and the impulse response as
vkt
ðÞ¼IFFT Ykb
ðÞFbt
ðÞ
:(24)
The envelopes of the time histories of the soundboard vibra-
tion are shown in Fig. 17 for excitation of strings A1 and
D4. These allow two cases to be compared in which the
soundboard representation has a different importance.
Whereas for the transmitted force in Fig. 15, the soundboard
models did not exhibit notable differences, the response of
the soundboard using the K-V model has more noticeable
differences, particularly in the case of A1, where the sound-
board model plays a more significant role. Compared with
the forces shown in Fig. 15, the profiles of the resultant
velocity vkare more regular, and, moreover, the soundboard
appears to enhance the differences between the K-V model
and the modal soundboard models in the case where its
mobility plays a more important role.
The spectra of the vibration of the soundboard are
shown in Fig. 18. The modes of the soundboard are now pre-
sent in the vibration response for all the models due to the
use of the same transfer mobility Ykb in each case. Two
example string partials are highlighted, showing that, as in
the transmitted force, the models with soundboard dynamics
produce similar resonances, while those of the K-V model
are slightly different. As was stated in other comparisons,
the differences are more significant at lower frequencies,
where the soundboard has a more significant influence.
The sound samples for the vibration velocity obtained
via the reduced model for A1 and D4 are attached as Mm.
9 and Mm. 10. The listener should notice the influence of
the soundboard. As the spectra are now richer in lower fre-
quencies than the force sound samples, the sound is now
“fuller.”
Mm. 9. Vibration velocity at the soundboard vk, for A1,
reduced soundboard model.
Mm. 10. Vibration velocity at the soundboard vk, for D4,
reduced soundboard model.
The simply supported model, not shown, gives a spec-
trum that reproduces adequately the main features obtained
with the other approaches, but its decay is larger and unreal-
istic. It cannot include the duplex scaling resonances, but
the damping value could be adjusted to represent the effect
of the soundboard in an equivalent way. This is presented in
Sec. VD.
D. A minimal model
The simply supported string model is modified to
include the damping arising from the connection with the
FIG. 17. Vibration velocity envelope at a position on the soundboard.
FIG. 18. (Color online) Spectra of
vibration velocity at the soundboard.
3228 J. Acoust. Soc. Am. 155 (5), May 2024 Miranda Valiente et al.
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soundboard, which would yield shorter computational times
while maintaining the characteristics of the piano tone due
to the coupling between the string and the soundboard. The
formulation of the minimal model is the one already pre-
sented in Sec. III C, but in this case, the modal damping
matrix is modified to include the estimated damping rations
of the string coupled to the FE soundboard. The real part of
the mobility is not used in this case. To test this, the esti-
mated damping ratios of the string coupled to the FE sound-
board model, shown in Fig. 13, are used as an input to the
modal damping matrix of a simply supported A1 string. The
envelope of the string vibration velocity at the excitation
point is compared in Fig. 19 with that obtained from the
string coupled with the FE soundboard model. The
agreement between the two models is satisfactory, and some
differences in the envelopes can be attributed to uncertainty
in estimating low values of damping ratios.
The comparisons are also made in the frequency
domain: the spectra of the vibration velocity and the force
transmitted to the soundboard are shown in Fig. 20. The
spectra have good agreement, but this minimal model fails
to represent the vibration of the duplex scaling segment,
which can be seen in the force spectrum in Fig. 20(b) at
approximately 660 Hz and higher harmonics.
E. Attenuating the duplex scaling segment
It has been shown in Secs. VB and VC that one of the
features of the models that include the soundboard is the
presence of resonances of the duplex scaling segment.
However, in actual pianos, these are muted in the lower reg-
ister. The same models can still be adopted, provided a suit-
able value for the distributed dampers and springs is applied
in the duplex scaling segment. Using Eq. (14), different val-
ues of damping coefficients cdin a range between 1 and
4Ns=m2are used to determine what value can be used to
obtain the desired effect without significantly affecting the
other string resonances. The value of the distributed spring
was fixed. The results are shown in Fig. 21 for the spectra of
the acceleration at the connection point of the soundboard b
for the note D4 using a reduced soundboard model. Again,
the readers can compare the different outcomes by listening
to the audio files provided in Mm. 11, Mm. 12, Mm. 13, and
Mm. 14 for increasing values of cd. The stiffness by itself
can provide a sufficient attenuation of the first resonance of
the duplex scaling. Within the range of values tested, a value
of cd¼1Ns=m2reduces the higher duplex scaling resonan-
ces in the numerical model adequately: the effect on the
adjacent string resonances is smaller than when using higher
FIG. 19. Time histories of piano string vibration velocity at excitation point
for the A1 string. The black line represents string coupled with the FE
soundboard model. The gray line represents simply supported string with
the adjusted damping ratio.
FIG. 20. Comparisons of spectra
between FE and minimal model for the
A1 case. (a) Spectrum of vibration
velocity at the excitation point. (b)
Spectrum of force transmitted to the
soundboard.
J. Acoust. Soc. Am. 155 (5), May 2024 Miranda Valiente et al. 3229
https://doi.org/10.1121/10.0025925

values of cd. This is shown in the highlighted resonance in
Fig. 21(a). For comparison, the measurement of acceleration
at the bridge pin at the D4 location is shown, evidencing the
lack of duplex scaling vibration in this piano. The modelled
result has a satisfactory agreement with the measurement
and confirms that the model approach taken and its coupling
with the string is correct.
Mm. 11. Acceleration at the connection point of the
soundboard ab, for D4, using cd¼0Ns=m2.
Mm. 12. Acceleration at the connection point of the
soundboard ab, for D4, using cd¼1Ns=m2.
Mm. 13. Acceleration at the connection point of the
soundboard ab, for D4, using cd¼2Ns=m2.
Mm. 14. Acceleration at the connection point of the
soundboard ab, for D4, using cd¼4Ns=m2.
VI. CONCLUSIONS
Soundboard models of differing complexities, a detailed
FE model, a reduced modal model, and a simplified K-V
model, have been coupled to a string in a state-space
approach. The scope is to address the effects of the assump-
tions in using different soundboard models differing in com-
plexity in their modal representation. Nonetheless, it is not
intended to produce complex soundboard model accounting
for nonlinear effects due to prestresses or other effects of its
manufacturing. Moreover, the modelling approaches pre-
sented in this study assume a linear behaviour of the differ-
ent systems associated with piano sound generation, with
the exception of the hammer excitation.
The assumptions related to the soundboard have only a
small effect on the results for the string response, affecting
the lower frequency range more than higher frequencies.
This is explained by the differences between mobilities of
the string, the soundboard, and the contact spring used for
the coupling between them. At lower frequencies, the mobil-
ity levels of the soundboard are similar to those of the con-
tact spring. Consequently, the string and soundboard can
interact, and the soundboard resonances have an influence on
the resultant string vibration in this frequency range. On the
other hand, at higher frequencies, the contact spring mobility
is much greater than that of the soundboard. Hence, the
response of the system is dependent on the interaction
between the string and the contact spring, and the sound-
board model does not significantly influence the results in
FIG. 21. (a) Spectrum of acceleration at the connection point of the soundboard for D4 string, using different damping coefficients for the duplex scaling
segment. (b) Spectrum of comparison between measured acceleration and modelled using highly damped duplex scaling.
3230 J. Acoust. Soc. Am. 155 (5), May 2024 Miranda Valiente et al.
https://doi.org/10.1121/10.0025925

this part of the spectrum. These observations are applicable
at widely different locations in the piano range, for string
vibration, interaction force between string and soundboard,
and for the resultant vibration of the soundboard.
In contrast, the damping estimation is affected by the
soundboard representation to some extent. The trend of the
damping ratios shows irregularities that can be attributable
to the presence of the duplex scaling vibrations in the mod-
els. Differences between the experimental results and the
models at low frequencies may be caused by deviations
between the experimental and modelled soundboard mobili-
ties at the connection point. Another cause of differences
could be the presence of double polarization in the experi-
mental measurements, producing a longer decay and hence
smaller damping. Double polarization is not yet included in
the numerical models.
To provide a minimal model, a simply supported string
model was modified to include the damping corresponding
to that provided by the connection with the soundboard,
yielding similar levels and spectra as a string connected
with an FE soundboard model. However, this simply sup-
ported model does not include the duplex scaling segment
resonances. Such a model could therefore be useful in low-
mid range of the piano where this segment is muted.
Finally, a brief study on the attenuation of the duplex
scaling segment shows that a small damping coefficient is
needed to accomplish the desired effect, so as not to affect
greatly other adjacent resonances. This was achieved by
using a distributed damping and stiffness approach along the
duplex scaling segment. Results were compared with mea-
surements showing a satisfactory agreement. However, it is
yet to be confirmed experimentally what is the damping
introduced by the felt in the duplex scaling.
Overall, it is clear that detailed FE soundboard models
are not needed to represent the main characteristics of
string-soundboard interaction. Simpler soundboard repre-
sentations can achieve the expected results and can model
the trend of the damping with less computational effort. The
reduced model is more similar to the FE soundboard model
than the K-V model, particularly at low frequencies, where
the soundboard can influence vibration. Nonetheless, the K-
V soundboard model is not far from representing the FE
soundboard behavior. For a full three-dimensional (3D) rep-
resentation of strings and soundboard, a model considering a
reduced modal soundboard would need to be extended to
include different directions and to consider other coupling
mechanisms, such as the interaction between unison strings.
The force generated by these string-soundboard models
could then be used to predict the vibration response across
the soundboard, and hence, the radiated sound, by applying
the interaction force to a suitably complete model of the
soundboard. To give an initial estimate of soundboard vibra-
tion, the velocity was obtained at a point on the soundboard,
from which it can be heard from the corresponding audio
files and seen how important the soundboard resonances are
in the generated tone, producing a response that is enriched
at lower frequencies.
ACKNOWLEDGMENTS
The authors are grateful to Professor David O. Norris
for facilitating access to the piano and Cesar Hernandez for
the information about the soundboard. The first author is
funded by the National Agency for Research and
Development (ANID)/Scholarship Program/DOCTORADO
BECAS CHILE/2020—72210046.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
DATA AVAILABILITY
The data that support the findings of this study are
available from the corresponding author upon reasonable
request.
1
F. Oberg and A. Askenfelt, “Acoustical and perceptual influence of
duplex stringing in grand pianos,” J. Acoust. Soc. Am. 131, 856–871
(2012).
2
G. Weinreich, “Coupled piano strings,” J. Acoust. Soc. Am. 62,
1474–1484 (1977).
3
N. Giordano and M. Jiang, “Physical modeling of the piano,” EURASIP
J. Appl. Signal Process. 7, 926–933 (2004).
4
J. Chabassier, A. Chaigne, and P. Joly, “Modeling and simulation of a
grand piano,” J. Acoust. Soc. Am. 134, 648–665 (2013).
5
B. Elie, B. Cotte, and X. Boutillon, “Physically-based sound synthesis
software for computer-aided-design of piano soundboards,” Acta Acust.
6, 30 (2022).
6
B. Bank and L. Sujbert, “Generation of longitudinal vibrations in piano
strings: From physics to sound synthesis,” J. Acoust. Soc. Am. 117,
2268–2278 (2005).
7
A. Chaigne and A. Askenfelt, “Numerical simulations of piano strings. I.
A physical model for a struck string using finite difference methods,”
J. Acoust. Soc. Am. 95, 1112–1118 (1994).
8
N. Giordano, “The physics of vibrating strings,” Comput. Phys. 12,
138–145 (1998).
9
J. Bensa, S. Bilbao, R. Kronland-Martinet, and J. O. Smith III, “The simu-
lation of piano string vibration: From physical models to finite difference
schemes and digital waveguides,” J. Acoust. Soc. Am. 114, 1095–1107
(2003).
10
F. Pfeifle and R. Bader, “Real-time finite difference physical models of
musical instruments on a Field-Programmable Gate Array (FPGA),” in
Proceedings of the 15th International Conference on Digital Audio
Effects, York, UK (September 17–21, 2012).
11
C. Issanchou, S. Bilbao, J. L. L. Carrou, C. Touze, and O. Doare, “A
modal-based approach to the nonlinear vibration of strings against a uni-
lateral obstacle: Simulations and experiments in the pointwise case,”
J. Sound Vib. 393, 229–251 (2017).
12
P. M. Valiente, G. Squicciarini, and D. J. Thompson, “Piano string vibra-
tion modelling using coupled mobilities and a state-space approach,” in
Proceedings of Internoise 2022, Glasgow, UK (August 21–24, 2022)
(Institute of Noise Control Engineering, Wakefield, MA), pp. 3940–3948.
13
P. M. Valiente, G. Squicciarini, and D. J. Thompson, “Modeling the inter-
action between piano strings and the soundboard,” in Proceedings of the
Fourth Vienna Talk on Music Acoustics, Vienna, Austria (September
11–14, 2022).
14
G. Derveaux, A. Chaigne, P. Joly, and E. Becache, “Time-domain simula-
tion of a guitar: Model and method,” J. Acoust. Soc. Am. 114, 3368–3383
(2003).
15
O. Inacio, J. Antunes, and M. C. M. Wright, “Computational modelling of
string-body interaction for the violin family and simulation of wolf
notes,” J. Sound Vib. 310, 260–286 (2008).
16
V. Debut, J. Antunes, M. Marques, and M. Carvalho, “Physics-based
modeling techniques of a twelve-string Portuguese guitar: A non-linear
J. Acoust. Soc. Am. 155 (5), May 2024 Miranda Valiente et al. 3231
https://doi.org/10.1121/10.0025925

time-domain computational approach for the multiple-strings/bridge/
soundboard coupled dynamics,” Appl. Acoust. 108, 3–18 (2016).
17
J. T. Jiolat, C. d’Alessandro, and J.-L. L. Carrou, “Toward a physical
model of the clavichord,” J. Acoust. Soc. Am. 150, 2350–2363 (2021).
18
L. Trautmann, S. Petrausch, and M. Bauer, “Simulations of string vibra-
tions with boundary conditions of third kind using the functional transfor-
mation method,” J. Acoust. Soc. Am. 118, 1763–1775 (2005).
19
A. M. Mani, J. Frelat, and C. Besnainou, “Numerical simulation of a
piano soundboard under downbearing,” J. Acoust. Soc. Am. 123,
2401–2406 (2008).
20
G. Squicciarini, “Vibroacoustic investigation of a grand piano
soundboard,” Ph.D. dissertation, Politecnico di Milano, Milan, 2012.
21
A. Chaigne, B. Cotte, and R. Viggiano, “Dynamical properties of piano
soundboards,” J. Acoust. Soc. Am. 133, 2456–2466 (2013).
22
K. Wogram, “The strings and the soundboard,” in Five Lectures on the
Acoustics of the Piano, edited by A. Askenfelt (Royal Swedish Academy
of Music, Stockholm, 1990).
23
K. Ege, X. Boutillon, and M. Rebillat, “Vibroacoustics of the piano
soundboard: (Non)linearity and modal properties in the low- and mid-
frequency ranges,” J. Sound Vib. 332, 1288–1305 (2013).
24
R. Corradi, S. Miccoli, G. Squicciarini, and P. Fazioli, “Modal analysis of
a grand piano soundboard at successive manufacturing stages,” Appl.
Acoust. 125, 113–127 (2017).
25
H. Suzuki, “Vibration and sound radiation of a piano soundboard,”
J. Acoust. Soc. Am. 80, 1573–1582 (1986).
26
J. Berthaut, M. N. Ichchou, and L. Jezequel, “Piano soundboard:
Structural behavior, numerical and experimental study in the modal
range,” Appl. Acoust. 64, 1113–1136 (2003).
27
B. Trevisan, K. Ege, and B. Laulagnet, “A modal approach to piano sound-
board vibroacoustic behavior,” J. Acoust. Soc. Am. 141, 690–709 (2017).
28
X. Boutillon and K. Ege, “Vibroacoustics of the piano soundboard:
Reduced models, mobility synthesis, and acoustical radiation regime,”
J. Sound Vib. 332, 4261–4279 (2013).
29
N. Fletcher and T. Rossing, “Vibrations of a stiff string,” in The Physics
of Musical Instruments (Springer, New York, 1998), pp. 64–65.
30
D. J. Ewins, Modal Testing: Theory, Practice and Application (Wiley,
Hoboken, NJ, 2003).
31
B. Elie, F. Gautier, and B. David, “Macro parameters describing the
mechanical behavior of classical guitars,” J. Acoust. Soc. Am. 132,
4013–4024 (2012).
32
E. Skudrzyk, “The mean-value method of predicting the dynamic response
of complex vibrators,” J. Acoust. Soc. Am. 67, 1105–1135 (1980).
33
A. Askenfelt, Five Lectures on the Acoustics of the Piano (Royal Swedish
Academy of Music, Stockholm, 1990).
34
D. Thompson, Railway Noise and Vibration: Mechanisms, Modelling and
Means of Control (Elsevier, Philadelphia, PA, 2009).
35
K. L. Johnson, Contact Mechanics (Cambridge University Press,
Cambridge, UK, 1985).
36
V. Popov, Contact Mechanics and Friction: Physical Principles and
Applications (Springer, New York, 2010).
37
M. Ghosh, “An experimental study of the duration of contact of an elastic
hammer striking a damped pianoforte string,” Ind. J. Phys. 7, 365–382
(1932).
38
D. E. Hall, “Piano string excitation in the case of small hammer mass,”
J. Acoust. Soc. Am. 79, 141–147 (1986).
39
X. Boutillon, “Model for piano hammers: Experimental determination
and digital simulation,” J. Acoust. Soc. Am. 83, 746–754 (1988).
40
A. Chaigne and A. Askenfelt, “Numerical simulations of piano strings.
II. Comparisons with measurements and systematic exploration of
some hammer-string parameters,” J. Acoust. Soc. Am. 95, 1631–1640
(1994).
41
A. Stulov, “Hysteretic model of the grand piano hammer felt,” J. Acoust.
Soc. Am. 97, 2577–2585 (1995).
42
W. Brogan, “State variables and the state space description of dynamic
systems,” in Modern Control Theory, 2nd ed. (Prentice-Hall, Hoboken,
NJ, 1985), pp. 225–255.
43
M. R. Schroeder, “New method of measuring reverberation time,”
J. Acoust. Soc. Am. 37, 409–412 (1965).
44
F. Fahy and J. Walker, “Simple freely vibrating spring-mass system,”
in Fundamentals of Noise and Vibration (Spon Press, New York,
1998).
45
J. Chabassier, “Mod
elisation et simulation num
erique d’un piano
par mode` les physiques” (“Modeling and numerical simulation of a
piano by physical models”), Docteur en Sciences dissertation, Ecole
Polytechnique, Palaiseau, France, 2012.
46
J. Chabassier and M. Durufle, “Physical parameters for piano modeling,”
Technical Report No. RT-0425, INRIA, Le Chesnay-Rocquencourt,
France (2012), p. 24.
47
J. J. Tan, “Piano acoustics: String’s double polarisation and piano source
identification,” Ph.D. dissertation, Universit
e Paris-Saclay, Orsay, France,
2018.
48
A. Stulov, “Physical modelling of the piano string scale,” Appl. Acoust.
69, 977–984 (2008).
49
Information on the copper density available at https://www.rsc.org/
periodic-table/element/29/copper (Last viewed February 2023).
50
H. Conklin, “Design and tone in the mechanoacoustic piano. Part II.
Piano structure,” J. Acoust. Soc. Am. 100, 695–708 (1996).
51
C. F. T. Steinway, “Improvement in duplex agraffe scales for piano-
fortes,” U.S. patent US126848A (May 14, 1872).
52
A. Askenfelt and E. Jansson, “From touch to string vibrations. III: String
motion and spectra,” J. Acoust. Soc. Am. 93, 2181–2196 (1993).
53
R. Roy and T. Kailath, “Esprit—estimation of signal parameters via rota-
tional invariance techniques,” IEEE Trans. Acoust, Speech, Signal
Process. 37, 984–995 (1989).
54
K. Ege, X. Boutillon, and B. David, “High-resolution modal analysis,”
J. Sound Vib. 325, 852–869 (2009).
3232 J. Acoust. Soc. Am. 155 (5), May 2024 Miranda Valiente et al.
https://doi.org/10.1121/10.0025925