Operational transfer path analysis of a piano
Citation for published version (APA):Tan, J. J., Chaigne, A., & Acri, A. (2018). Operational transfer path analysis of a piano. Applied Acoustics, 140, 39-47. https://doi.org/10.1016/j.apacoust.2018.05.008
DOI:
10.1016/j.apacoust.2018.05.008
Document status and date: Published: 01/11/2018
Document Version:
Accepted manuscript including changes made at the peer-review stage
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page numbers.
Link to publication
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:
www.tue.nl/taverne
Take down policy
If you believe that this document breaches copyright please contact us at:
openaccess@tue.nl
Operational transfer path analysis of a piano
1
Jin Jack Tana,b,1, Antoine Chaignea,2, Antonio Acric,d,3
2
aIWK, University of Music and Performing Arts Vienna, Austria
3
bIMSIA, ENSTA-ParisTech-CNRS-EDF-CEA, France
4
cPolitecnico di Milano, Italy
5
dVirtual Vehicle, Austria
6
Abstract 7
The piano sound is made audible by the vibration of its soundboard. A pianist pushes the key to release a hammer that strikes the strings, which transfer the en-ergy to the soundboard, set it into vibration and the piano sound is heard due to the compression of air surrounding the soundboard. However, as piano is being played, other components such as the rims, cast-iron frame and the lid are also vibrating. This raises a question of how much of their vibrations are contributing to the sound as compared to the soundboard. To answer this question, operational transfer path analysis, a noise source identification technique used widely in automotive acoustics, is carried out on a B¨osendorfer 280VC-9 grand piano. The ”noise” in a piano system would be the piano sound while the ”sources” are soundboard and the aforemen-tioned components. For this particular piano, it is found out that the soundboard is the dominant contributor. However, at high frequencies, the lid contributes the most to the piano sound.
Keywords: piano acoustics, structural vibration, source identification, operational 8
transfer path analysis 9
1. Introduction 10
The study of piano acoustics has traditionally been focused on its piano action 11
[1, 2], interaction between a piano hammer and string [3, 4], string vibration [5, 6, 7], 12
soundboard vibration [8, 9, 10] and its radiation [11, 12, 13]. As computational 13
power becomes cheaper, it is possible to model the piano as a coupled system that 14
involves the string, soundboard and surrounding air [14, 15]. However, this raises 15
a question whether the considered system is complete enough to have a realistic 16
reproduction of the piano sound. The importance and role of a soundboard in 17
1Present affiliation: Faculty of Engineering and the Environment, University of Southampton
Malaysia Campus, Iskandar Puteri, Malaysia
2Present address: 13 all´ee Fran¸cois Jacob, 78530 BUC, France
piano sound production have been extensively studied [16, 17, 18, 19, 20, 21] but 18
not for other components. Is there any other components that are contributing to 19
the piano sound production that has not been accounted for? Anecdotally, when a 20
piano is played, vibration can be felt not only on the soundboard but also on the 21
rim, the frame, the lid etc. In a B¨osendorfer piano, spruce, a wood usually used for 22
the soundboard by other manufacturers, is used extensively in building the case of 23
the piano. B¨osendorfer claims that the use of spruce, especially on the rim of the 24
piano, allows the whole instrument to vibrate and is the reason that gives the unique 25
B¨osendorfer sound [22]. Based on the fact that vibration is felt on other parts of the 26
piano and how B¨osendorfer uses spruce extensively, it necessitates an investigation 27
if the vibrations of these parts contribute to the production of the sound. 28
Current work takes inspiration from noise source identification techniques used 29
commonly in automotive acoustics [23, 24, 25]. However, in the case of piano, 30
the ”noise” is the resulting piano sound and the ”sources” to be identified are the 31
piano components to be investigated. These ”sources” may emit different ”noise” 32
contributions that characterise the resulting ”noise”, i.e. the piano sound. One 33
technique that can be used to identify the contribution of the piano components 34
to the final sound is the operational transfer path analysis (OTPA) [26]. OTPA 35
computes a transfer function matrix to relate a set of input/source(s) measurements 36
to output/response(s) measurements. In this case, the inputs are the vibration of the 37
components of piano and the output is the resulting piano sound. Initial result of the 38
work was first presented at the International Congress of Acoustics [27] but more 39
thorough analysis has since been conducted with more convincing and confident 40
results obtained. These results are being presented in this paper. In Section 2, the 41
theory of OTPA is presented. The experiment designed for OTPA is then detailed 42
in Section 3 before the results are shown and discussed in Section 4. 43
2. The Operational Transfer Path Analysis (OTPA) 44
Operational transfer path analysis (OTPA) is a signal processing technique that 45
studies the noise source propagation pathways of a system based on its operational 46
data [28]. This is in contrast to classical transfer path analysis (TPA) where the 47
source propagation pathways are established by means of experimental investiga-48
tions with specific inputs. Indeed, OTPA was developed in wake of the need of 49
a fast and robust alternative to the classical TPA. In both methods, the source 50
propagation pathways are determined by studying the transfer functions between 51
the sources (inputs) and the responses (outputs). In TPA, the relationship between 52
sources (excitation signals) and responses are estimated by frequency response func-53
tions and these are meticulously determined by series of experimental excitation of 54
known forces (e.g. by shaker or impact hammer). Where necessary, part of the 55
system is also removed or isolated. In this way, TPA is able to trace the flow of 56
vibro-acoustic energy from a source through a set of known structure- and air-borne 57
pathways, to a given receiver location by studying the frequency response functions. 58
On the other hand, OTPA measures directly the source and response signals when 59
the system is operated and establishes the source propagation pathways based on 60
the experimentally determined transfer functions. OTPA is a response-response 61
model where measurement data are collected and analysed while the system, when 62
operated, provides the excitation. Detailed comparisons between the classical TPA 63
and OTPA can be found in [29, 30, 31]. 64
While OTPA may appear to be simpler to use, it is also prone to error if it is 65
not designed and analysed properly. OTPA requires prior knowledge of the system 66
as neglected pathways could not be easily detected. In a multi-component system, 67
cross-coupling between the components could affect the accuracy of an OTPA model. 68
Several techniques, which are detailed in the following Section 2.2, can be employed 69
to mitigate the effects of cross-coupling [28]. 70
2.1. Theory of OTPA 71
In a linear system, the input X and output Y can be related by: 72
Y(jω) = X(jω)H(jω), (1)
where: 73
Y(jω) is the output vector/matrix at the receivers; 74
X(jω) is the input vector/matrix at the sources; 75
H(jω) is the operational transfer function matrix, also known as the transmissibility 76
matrix. The dependency of frequencies for all three matrix is as denoted by (jω) 77
[28]. 78
The inputs and outputs signals can be the forces, displacements, velocities or 79
pressures of the components in the system. Given that there are m inputs and n 80
outputs with p set of measurements, Equation (1) can be written in the expanded 81 form: 82 y11 · · · y1n .. . . .. ... yp1 · · · ypn = x11 · · · x1m .. . . .. ... xp1 · · · xpm H11 · · · H1n .. . . .. ... Hm1 · · · Hmn , (2)
where for clarity purposes, the frequency dependency jω is dropped. In order to 83
quantify the contributions of the inputs to the outputs, the transfer function matrix 84
needs to be solved. If the input matrix X is square and invertible, this can be solved 85
by simply multiplying the inverse of X on both sides: 86
H = X−1Y. (3)
However, in most cases, p 6= m. Thus, for the system to be solvable, it is required 87
that the number of measurement sets is larger than or equal to the number of inputs, 88
i.e. 89
thereby forming an overdetermined system with residual vector µ: 90
XH + µ = Y. (5)
The transmissibility matrix H can then be obtained via the following equation [28]: 91
H = (XTX)−1XTY = X+Y, (6) where the + superscript denotes the Moore-Penrose pseudo-inverse [32].
92
2.2. Enhanced OTPA with singular value decomposition and principal component 93
analysis 94
In essence, the basics of OTPA is analogous to the input multiple-95
output (MIMO) technique in experimental modal analysis [33]. However, solving the 96
transfer function H directly is prone to error if the input signals are highly coherent 97
between each other. High coherence is caused by unavoidable cross-talks between 98
the measurement channels as they are sampled simultaneously. To mitigate this 99
error, an enhanced version of OTPA can be employed. Singular value decomposition 100
(SVD) and principal component analysis (PCA) can be carried out [28, 26]. There 101
are two main reasons in using SVD. Firstly, it can be used to solve for X+, even
102
though it is not the only way to solve for Moore-Penrose pseudo-inverse. Secondly, 103
the singular value matrix can later be repurposed to carry out PCA to reduce the 104
measurement noise. 105
The input matrix X, as decomposed by economy size SVD, can be written as 106
X = UΣVT, (7)
where 107
U is a unitary column-orthogonal matrix; 108
Σ is a square diagonal matrix with the singular values; 109
VT is the transpose of a unitary column-orthogonal matrix, V.
110 111
The singular values obtained along the diagonal of Σ are also the principal 112
components (PC). The PC are defined such that the one with the largest variance 113
within the data is the first PC (the first singular values), the next most varying is the 114
second PC and so on. The smallest singular value thus corresponds to the weakest 115
PC that has little to no variation. A matrix of PC scores can then be constructed 116
as: 117
Z = XV = UΣ. (8)
The contribution of each PC can be evaluated by dividing Z with the sum of all the 118
PC scores. For each PC, this yields a value between 0 to 1. The larger the number, 119
the more significant the PC is. In other words, a weakly contributing PC can be 120
identified and thus be removed by setting it to zero as they are mainly caused by 121
noise influences or external disturbances [28]. Then, the inverse of the singular value 122
matrix Σ−1 can be recalculated by: 123
Σ−1e = (
1/σq if ¯σq ≥ θ,
0 otherwise, (9)
where σq indicates the q-th singular value along the diagonal Σ matrix and the
124
overbar indicates normalised singular value against sum of all singular values [26]. 125
On the other hand, θ represents a threshold value (where 0 ≤ θ ≤ 1) while the 126
subscript e indicates that the matrix has been enhanced by SVD and PCA. 127
Subsequently, the modified pseudo-inverse of X can be written as [34]: 128
X+e = VΣ−1e UT. (10)
Introducing Equation (10) into Equation (6), the treated transmissibility matrix He
129
can then be written as: 130
He= X+eY. (11)
Once the treated transmissibility matrix Heis computed, it is possible to derive the
131
synthesised responses Ys such that [28, 25]:
132
Ys = XHe. (12)
However, to quantify the contribution of each input to the outputs, one could con-133
sider the following summations: 134
cijk = xikhkj, (13)
where i = 1, . . . , p, j = 1, . . . , n and k = 1, . . . , m while xik and hkj represent the
135
element in the matrix X and He respectively. In other words, Equation (13)
quan-136
tifies the contributions c from a transmission path (i.e. input) k to a measurement 137
point (i.e. output) j for a given measurement i. Summing up all measurement sets, 138
one can thus obtain the overall contributions, Cjk, from an input j to an output k:
139 Cjk = p X i=1 cijk, (14)
where the individual k-th synthesised response could be recovered if one sums Cjk
140
overs all the j inputs. 141
3. Experimental setup 142
The soundboard, inner rim, outer rim, the cast-iron frame and the lid are all 143
considered to be potential transmission paths of the piano sound (see Figure 1a). 144
The vibrations of these components are sampled by sets of accelerometers and can 145
be treated to be the inputs of OTPA, X. To fully capture all the modes of interest, 146
the accelerations are sampled extensively over the surfaces in grids of 20cm x 20cm. 147
For the cast-iron frame, only the part highlighted in the green area in Figure 1b are 148
sampled. The other part has a comparatively small surface area, and it is assumed 149
that its contribution does not play a significant role in the final sound. In the same 150
figure, the surface sampled for inner rim and outer rim are also highlighted (albeit 151
partially) in blue and purple respectively. Of course, the outer rim measurement 152
is also extended to the straight part of the rim which is not seen in the figure. 153
The soundboard and lid measurement are sampled on the surface that is visible in 154
Figure 1b (i.e. the soundboard surface that faces up and the lid surface that faces 155
down) but are not highlighted so as to not obscure the presentation. The total 156
number of measurements for each components is outlined in Table 1. Meanwhile, 157
the output Y is the sound pressure at a point away from the piano, recorded by a 158
microphone, marked M in Figure 2. The experiment is conducted in the anechoic 159
chamber of the University of Music and Performing Arts Vienna so as to eliminate 160
wall and floor reflections, and conveniently ignore radiation from the bottom surface 161
of the soundboard. The piano used is a B¨osendorfer 280VC-9 equipped with a CEUS 162
Reproducing System. The CEUS Reproducing System is a computerised system that 163
is able to record and reproduce the playing of the piano by means of optical sensors 164
and solenoids. 165
(a) Illustrated view of piano, showing the components investigated. Modified from [35].
(b) A B¨osendorfer 280VC-9. Highlighted green, blue and purple areas are the frame, inner rim and outer rim respectively.
Figure 1: Anatomy of a piano.
One of the advantages of OTPA is its flexibility to use operational data rather 166
than controlled excitation from a shaker or impact hammer. Thus, measurement 167
y x SB L IR,OR F M
Figure 2: Location of the microphone (marked M) with respect to the outline of the piano sound-board. Assuming that the origin of an (x,y) coordinate system as shown in the figure, the mi-crophone is at x=2.2m, y=1.0m and is the same height with the soundboard. The midpoints on the x-y plane of each summed input are also shown where SB, L, F, IR and OR represent the soundboard, lid, frame, inner and outer rim respectively. The image is to scale.
data are collected as the piano is being played. As vibrational data need to be taken 168
in batches due to limited number of accelerometers, the CEUS Reproducing System 169
is used where the piano playing can be reproduced with the same force and timing. 170
The CEUS system can be controlled via pre-produced .boe files, which contains 171
information on the keys to play, their pushing profiles and the hammer velocities 172
[36]. For this experiment, two .boe files are prepared via in-house MATLAB/GNU 173
Octave script, each corresponding to different excitation patterns. 174
For an OTPA, it is desired that the inputs are as incoherent as possible to 175
minimise cross-couplings between them. The two excitations are designed with that 176
in mind. The first is simply playing each of the 88 notes sequentially, from the 177
lowest A0 (27.5Hz) to the highest C8 (4186 Hz). Each note is played for 3 seconds 178
with a rest interval of 0.6 seconds in between. Each 3-second sequence is a unique 179
measurement and thus p = 88 as is defined in Equation (2). The second sets of 180
measurement contains 12 playing sequences with each of them playing all the same 181
notes, starting with all A (i.e. A0, A1, A2 ... A7) and followed by all of the A#s, 182
all of the Bs, all of the Cs until finally all of the G#s as is illustrated in Figure 3. 183
Each sequence is played for 5 seconds with a 1 second interval in between and 184
yields p = 12. Both sequences are played at a moderate dynamic level that gives a 185
subjective loudness of mezzoforte. 186
As outlined in Table 1, there are a total of 154 vibration signals that can be 187
treated as inputs. This would violate Equation (4) if m = 154. It is thus necessary 188
to either increase the number of measurements or reduce the number of inputs. For 189
this study, the latter is performed where the vibrational signals are summed via 190
Figure 3: Illustration of the 12-sequence excitation. Vertical axis indicates the note number played. The dips indicate the keys being pressed.
Rayleigh integral so as to obtain an input variable with the dimension of a sound 191
pressure. The net pressure p at point R as summed by Rayleigh integral can be 192 defined as: 193 p(R, ω) = ρ 2π X i ai(ω)Si ri e−jkri, (15)
where ρ, ai, ri, k and Si represent density of the air, the acceleration, distance to the
194
summed point, wavenumber and area covered by source point i respectively and the 195
term ω indicates dependence on frequency. The acceleration data of each component 196
are summed to the midpoint of the accelerometer locations of each component (see 197
Figure 2), with an averaged uniform area assumed (i.e. total calculated area divided 198
by number of measurements on each component). As a result, this yields a total of 199
only 5 sources, i.e. m = 5, making them suitable to be used for OTPA. 200
Table 1: Number of accelerometers for each components.
Components Number of accelerometers
Soundboard 35
Inner rim 24
Outer rim 24
Frame 25
Lid 46
The use of Rayleigh integral is an approximation as it is defined for an infinitely 201
large flat baffle [37]. A rapid calculation allows to estimate the frequency range for 202
which the Rayleigh approximation may lead to an overestimation of the sound pres-203
sure. The soundboard and lid have a nominal length Ln of about 1.47m. With this
204
order of magnitude for the acoustic wavelength, an acoustical short circuit might 205
appear for wave modes below 230Hz in the reality, thus leading to a sound pressure 206
lower than the one predicted by the Rayleigh integral. However, the overestimation 207
may not be so prominent due to the complex shape of the piano and the presence 208
of the semi-opened cavity defined by the rim and the lid. In practice, this overesti-209
mation should not have appreciable consequences in the analysis, since the Rayleigh 210
integral is used here for the purpose of reducing the number of inputs, only, and 211
also it is a comparative study. The only consequence is that the contributions of 212
the smaller components (such as the frame) might be slightly overestimated in the 213
low-frequency range, compared to the large components (soundboard, lid) through 214
the use of the Rayleigh integral. 215
4. Results and discussion 216
A quality check of OTPA can be performed by comparing the experimentally 217
measured output against the synthesised output as obtained from Equation (12). 218
The error between the two outputs can be defined as: 219
ε = |Y − XHe|. (16)
Using different threshold values θ as defined in Equation (9) will yield different 220
synthesised output and as a result, different ε. To minimise the overall ε, different θ 221
is used for each frequency point and is chosen for when ε is lowest at that particular 222
frequency point. This is possible since the system is linear, and its linearity is further 223
verified by varying the number of measurement blocks, i.e. p with no significant 224
difference in the transmissibility matrix, He [28].
225
The first 300Hz of the outputs of the first sequence of both 88-sequence and 226
12-sequence excitations are as shown in Figure 4. For the 88-sequence excitation, 227
the averaged threshold value θ is 1.7% and varies between 1% and 4% while for the 228
12-sequence excitation, the averaged θ value is 1.2% and varies between 1% and 6%. 229
The distribution of number of principal component (PC) kept after the principal 230
component analysis is performed is also presented in Table 2. Comparing between 231
the measured and synthesised data in Figure 4, there are very good agreements 232
except at the frequency range between 10 Hz and 40Hz for the 88-sequence excita-233
tion. Additionally, the synthesised responses produce few extra peaks that are not 234
observed in the measured responses, such as at 124Hz for both excitations. These 235
peaks appear to persist even when a high value of θ (e.g. 25%) is used, or when p is 236
varied, ruling them out being noises or effect due to nonlinearity. It must be high-237
lighted that the variable θ approach across the whole frequency range is important 238
as it manages to remove several other peaks that would otherwise be present if a 239
common θ value is used instead. 240
The presences of these peaks might be due to the contribution of other input 241
signals that are not recorded in the experiment. Indeed, there has been study 242
that suggest that the presence of finger-key impact and the key-bottom impact 243
are perceivable to listeners [36]. The former does not exist in the experiment as 244
all keys are played by the CEUS system while the role of the latter might not be 245
significant as the key is not played with excessive force and to a mezzoforte level 246
only. Nonetheless, it remains a point to revisit in future study.
Table 2: Distribution of number of PC kept after principal component analysis is performed.
Number of PC 1 2 3 4 5 12-sequence 740 2276 2400 3728 56392 88-sequence 1141 2823 2211 1832 57529 247 0 50 100 150 200 250 300 40 60 80 100 120 140 Frequency (Hz) Sound pressure (d B) measured synthesised
(a) 88-sequence excitation, playing only A0 note.
0 50 100 150 200 250 300 40 60 80 100 120 140 Frequency (Hz) Sound pressure (d B) measured synthesised
(b) 12-sequence excitation, playing all A note.
Figure 4: Comparison of the synthesised and measured output (sound pressure at M as shown in Figure 2). Good agreements between the data indicate that the models are representative of the measurements.
The next part of the analysis is to study the contributions of each source in the 248
output. The contribution level of each input can be obtained via Equation (14) over 249
the whole frequency range. To represent the result in a clear and readable manner, 250
five frequency groups, as divided based on the partition made by the frame of the 251
B¨osendorfer piano (see Figure 1b on how the stress bars terminate on top of the 252
keyboard) are identified and all results obtained within each frequency group are 253
averaged and presented in bar charts as shown in Figure 5. The frequency group is 254
summarised in Table 3.
Table 3: Key number of the frequency groups and their corresponding frequency ranges.
Key number Frequency range (Hz) 1 to 20 27.5 to 82.4 21 to 38 82.4 to 233.1 39 to 54 233.1 to 587.3 55 to 72 587.3 to 1661.2 73 to 88 1661.2 to 4186.0 255
Figure 5 shows the contribution level of each input (i.e. soundboard, inner rim, 256
outer rim, frame and lid) and the corresponding output (labeled ”total”) in two sets 257
of bar charts: the top graph shows the results from the 88-sequence excitation, while 258
the bottom shows results from the 12-sequence excitation. On top of each individual 259
bar, the sound pressure in dB (relative to 20µPa) is displayed. From both results, 260
some main observations can be made: 261
• From key 1 to 72, the soundboard is the main contributor to the sound. 262
• From key 73 to 88, the lid is the main contributor. 263
• From key 73 to 88, the rims and the frame have similar level of contributions 264
compared to the soundboard. 265
However, there are also some discrepancies between the two sets of results: 266
• The differences in sound pressure between soundboard and the other compo-267
nents are noticeably smaller in key 1 to 20 and key 55 to 72 in the 88-sequence 268
excitation (about 4dB) compared to the 12-sequence excitation (8 to 10 dB). 269
• The frame is a clear secondary contributor from key 1 to 38 in the 12-sequence 270
excitation, but less obvious in the 88-sequence excitation. 271
• Overall, 12-sequence excitation has higher sound pressure level. 272
The consistent results between the two types of excitation confirm the importance 273
of soundboard in piano sound production, which is not surprising. However, the 274
results also reveal some new findings, particularly at higher frequencies (key 73 and 275
above), that the lid plays an important role in sound production. The result can be 276
best illustrated by plotting the contribution of input and the output sound pressure 277
as is shown in Figure 6 and 7. 278
In Figure 6, the synthesised output of the 28th sequence in the 88-sequence 279
excitation is shown together with the contribution of each component. The thick 280
1-20 21-38 39-54 55-72 73-88 40 60 80 63 67 64 55 45 58 50 56 47 46 56 50 55 50 46 57 58 53 50 46 59 57 53 51 50 66 68 66 58 54 Sound pressure (dB) 1-20 21-38 39-54 55-72 73-88 40 60 80 70 75 73 63 48 56 56 60 52 49 53 54 63 53 49 62 66 58 53 48 59 61 55 52 51 71 76 74 65 56 Key number Sound pressure (dB)
soundboard innner rim outer rim frame lid total
Figure 5: Average contribution level from the analysis for (on the top) 88-sequence excitation and (on the bottom) 12-sequence excitation. The frequency range is defined based on the piano key number.
black line (labeled ”Total”) represents the final sound and the other lines correspond 281
to the inputs. The 28th sequence corresponds to playing the C3 note and its equal 282
temperament frequency is shown as a dotted vertical line at 130.8Hz. The measured 283
frequency of C3 of the piano investigated is 131.1Hz, but the difference of +4 cents 284
is essentially negligible. From the figure, the spectral content of the played note, 285
as represented as the highest peak, is made up almost entirely by the soundboard. 286
At other peaks (about 10dB less than the C3 peak), soundboard, frame and lid are 287
seen to be contributing as well. Similar spectra can be observed at some other notes 288
of lower ranges, which explain the high contribution of soundboard. 289
Next, a higher key belonging to the highest frequency group is investigated. The 290
spectra in which the 80th key (E7) is played is as shown in Figure 7. The equal 291
temperament frequency of E7 is 2,637Hz (shown as dotted line) but for this piano, 292
it is tuned at 2,672Hz (+23 cents), which is in agreement with the Railsback curve 293
[38]. One can see that while soundboard still contributes at the frequency of the 294
played note (i.e. at 2,672Hz), for other spectral contents, the lid plays a dominant 295
contribution, such as at around 2,549Hz and 2,651Hz. These additional components 296
116
118
120
122
124
126
128
130
132
134
0
20
40
60
Frequency (Hz)
Sound
pressure
(dB)
C3
Soundboard
Outer rim
Lid
Total
Inner rim
Frame
Figure 6: Contribution of the five inspected components for the synthesised output (labeled ”To-tal”) for the 28th sequence of the 88-sequence excitation. The dotted vertical line corresponds to the equal temperament frequency of the played note (C3).
might be due to eigenmodes of the lid or other elements of the piano structure which 297
are transmitted to the lid through prop and hinges. However, such an assumption 298
could only be confirmed after a thorough theoretical and experimental analysis of 299
the transmission of vibrations through the various components of the piano. 300
Attempt was made to determine whether the contribution of the lid to the total 301
sound field is only due to transmission of vibrations through prop and hinges (i.e. 302
structure-borne transmission), or reflection of the pressure radiated by the sound-303
board (i.e. air-borne transmission), especially in the treble range. For this purpose, 304
Figure 8 shows the measured acceleration map of the lid for the two played notes 305
A6 (Nr 73, 1771 Hz) and A7 (Nr 85, 3587 Hz). Both maps show higher acceleration 306
level in the vicinity of the prop (red circle), and of some hinges (2nd hinge from top 307
for A6, first hinge from bottom for A7). In contrast, both pressure maps computed 308
in the lid plane (in the absence of the lid) due to the radiation of the soundboard 309
for these two notes show a much smoother pattern, with a regular decrease of the 310
pressure from the left bottom to the outer edge (see Figure 9). These results sug-311
gest that the acceleration of the lid is not entirely due to the pressure field radiated 312
by the soundboard, and that transmission of vibrations from other piano parts to 313
the lid also exists. More experimental work is needed for clearly separating the 314
transmission-induced from the radiation-induced vibrations on the lid. 315
B¨osendorfer claims that the rims play a role in the sound production as part 316
of their ”resonance case principle”. The main idea is to construct the rims using 317
spruce, the wood used widely in the construction of soundboard. This will result in 318
a complete resonating body consisting of the soundboard and rim that is responsible 319
for the sound production [22]. However, from the analysis performed in this study, 320
their contributions appear to only be influential from 1661Hz (key 73) onwards. 321
2,540
2,560
2,580
2,600
2,620
2,640
2,660
2,680
10
20
30
40
Frequency (Hz)
Sound
pressure
(dB)
E7
Soundboard
Outer rim
Lid
Total
Inner rim
Frame
Figure 7: Contribution of the five inspected components for the synthesised output (labeled ”To-tal”) for the 80th sequence of the 88-sequence excitation. The dotted vertical line corresponds to the equal temperament frequency of the played note (E7).
Finally, the discrepancies between the two types of excitation need to be ad-322
dressed. Physically, the excitations differ by the number of key pushed simultane-323
ously and the frequency content of each excitation and the corresponding output. 324
Pushing more keys at the same time introduce more energy to the piano, which 325
explains the higher sound pressure level in the 12-sequence excitation. Since the 326
frequency contents of the signals of two excitation actually differ, it is thus probable 327
that the same processing would yield slightly different results on the sound pressure 328
level of secondary contributors. 329
5. Conclusion 330
An operational transfer path analysis (OTPA) has been conducted for a B¨osendorfer 331
piano to identify any additional vibrating components (other than the soundboard) 332
that could contribute to the sound production. A variable threshold approach is 333
implemented during the principal component analysis to enhance the computation. 334
Across the frequency range inspected, the soundboard has been the major contrib-335
utor except for the highest frequency range (≥ 1661.2Hz). In that range, the lid 336
contributes the most to the piano sound. 337
Current study represents a rare interdisciplinary example of technique originally 338
developed for automotive analysis being applied to musical instruments. However, 339
it is indeed possible to apply other source identification techniques to investigate 340
current problem and obtain a better understanding on the transfer pathways from 341
the strings to the listener. For recommendation, one can conduct a classical transfer 342
path analysis. Components like lid can be removed and boundary conditions between 343
the soundboard and the rims can be artificially modified to isolate the components. 344
0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x (m) y (m) −40 −30 −20 (a) Note A6 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x (m) y (m) −40 −30 −20 (b) Note A7
Figure 8: Lid acceleration map in dB (re 1 m/s2) for the two notes A6 (1771 Hz) and A7 (3587 Hz). Red circle: position of the prop. Red rectangles: positions of the hinges. The circle and rectangle are not to scale with actual prop and hinges.
6. Acknowledgement 345
The research work presented is funded by the European Commission (EC) within 346
the BATWOMAN Initial Training Network (ITN) of Marie Sk lodowska-Curie action, 347
under the seventh framework program (EC grant agreement no. 605867). The 348
authors thank the Network for the generous funding and support. The research has 349
been further supported by the Lise-Meitner Fellowship M1653-N30 and the stand-350
alone project P29386 of the Austrian Science Fund (FWF) attributed to Antoine 351
Chaigne. 352
The authors also thank Werner Goebl for his guidance on using the CEUS Re-353
producing System and Alexander Mayer for his contribution in setting up the ex-354
periment. 355
References 356
[1] A. Thorin, X. Boutillon, J. Lozada, X. Merlhiot, Non-smooth dynamics for an efficient
simu-357
lation of the grand piano action, Meccanica (2017) 1–18 doi:10.1007/s11012-017-0641-1.
358
[2] R. Masoudi, S. Birkett, Experimental validation of a mechanistic multibody model of a
ver-359
tical piano action, Journal of Computational and Nonlinear Dynamics 10 (6) (2015) 061004–
360
061004–11.
0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x (m) y (m) 50 60 70 (a) Note A6 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x (m) y (m) 50 60 70 (b) Note A7
Figure 9: Pressure map in the lid plane in dB (re 2.0 10−5 Pa) for the two notes A6 (1771 Hz) and A7 (3587 Hz).
[3] A. Chaigne, Reconstruction of piano hammer force from string velocity, The Journal of the
362
Acoustical Society of America 140 (5) (2016) 3504–3517.
363
[4] S. Birkett, Experimental investigation of the piano hammer-string interaction, The Journal
364
of the Acoustical Society of America 133 (4) (2013) 2467–2478.
365
[5] A. Chaigne, A. Askenfelt, Numerical simulations of piano strings. I. a physical model for a
366
struck string using finite difference methods, The Journal of the Acoustical Society of America
367
95 (2) (1994) 1112–1118.
368
[6] J. Chabassier, S. Imperiale, Stability and dispersion analysis of improved time discretization
369
for simply supported prestressed Timoshenko systems. Application to the stiff piano string,
370
Wave Motion 50 (3) (2013) 456–480.
371
[7] A. Stulov, D. Kartofelev, Vibration of strings with nonlinear supports, Applied Acoustics 76
372
(2014) 223–229.
373
[8] A. Chaigne, B. Cott´e, R. Viggiano, Dynamical properties of piano soundboards, The Journal
374
of the Acoustical Society of America 133 (4) (2013) 2456–2466.
375
[9] J. Berthaut, M. Ichchou, L. Jezequel, Piano soundboard: structural behavior, numerical and
376
experimental study in the modal range, Applied Acoustics 64 (11) (2003) 1113–1136.
377
[10] R. Corradi, S. Miccoli, G. Squicciarini, P. Fazioli, Modal analysis of a grand piano soundboard
378
at successive manufacturing stages, Applied Acoustics 125 (2017) 113–127.
379
[11] H. Suzuki, Vibration and sound radiation of a piano soundboard, The Journal of the Acoustical
380
Society of America 80 (6) (1986) 1573–1582.
381
[12] H. Suzuki, I. Nakamura, Acoustics of pianos, Applied Acoustics 30 (1990) 147–205.
382
doi:10.1016/0003-682X(90)90043-T.
[13] P. Derogis, R. Causs´e, Computation and modelisation of the sound radiation of an upright
384
piano using modal formalism and integral equations, in: Proceedings of the 11th ICA, Vol. 3,
385
Trondheim, Norway, 1995, pp. 409–412.
386
[14] J. Chabassier, A. Chaigne, P. Joly, Modeling and simulation of a grand piano, The Journal
387
of the Acoustical Society of America 134 (1) (2013) 648–665.
388
[15] N. Giordano, M. Jiang, Physical modeling of the piano, EURASIP J. Appl. Signal Process.
389
2004 (2004) 926–933. doi:10.1155/S111086570440105X.
390
URL http://dx.doi.org/10.1155/S111086570440105X
391
[16] K. Wogram, Acoustical research on pianos, Vibrational characteristics of the soundboard, Das
392
Musikinstrument 24 (1980) 694–702, 776–782, 872–880.
393
[17] I. Nakamura, The vibrational character of the piano soundboard, in: Proceedings of the 11th
394
ICA, Vol. 4, 1983, pp. 385–388.
395
[18] A. Mamou-Mani, J. Frelat, C. Besnainou, Numerical simulation of a piano soundboard under
396
downbearing, The Journal of the Acoustical Society of America 123 (4) (2008) 2401–2406.
397
[19] K. Ege, X. Boutillon, M. R´ebillat, Vibroacoustics of the piano soundboard:(non) linearity
398
and modal properties in the low-and mid-frequency ranges, Journal of Sound and Vibration
399
332 (5) (2013) 1288–1305.
400
[20] X. Boutillon, K. Ege, Vibroacoustics of the piano soundboard: Reduced models, mobility
401
synthesis, and acoustical radiation regime, Journal of Sound and Vibration 332 (18) (2013)
402
4261–4279.
403
[21] B. Tr´evisan, K. Ege, B. Laulagnet, A modal approach to piano soundboard vibroacoustic
404
behavior, The Journal of the Acoustical Society of America 141 (2) (2017) 690–709.
405
[22] F. Br¨au, The Touching Sound - Part 1, The Resonance Case Principle, B¨osendorfer, the
406
magazine by B¨osendorfer Austria (2008) 14–15.
407
[23] N. B. Roozen, Q. Leclere, C. Sandier, Operational transfer path analysis applied to a small
408
gearbox test set-up, in: Proceedings of the Acoustics 2012 Nantes Conference, 2012, pp.
409
3467–3473.
410
[24] D. De Klerk, M. Lohrmann, M. Quickert, W. Foken, Application of operational transfer
411
path analysis on a classic car, in: Proceeding of the International Conference on Acoustics
412
NAG/DAGA 2009, 2009, pp. 776–779.
413
[25] J. Putner, H. Fastl, M. Lohrmann, A. Kaltenhauser, F. Ullrich, Operational transfer path
414
analysis predicting contributions to the vehicle interior noise for different excitations from the
415
same sound source, in: Proc. Internoise, 2012, pp. 2336–2347.
416
[26] M. Toome, Operational transfer path analysis: A study of source contribution predictions at
417
low frequency, Master’s thesis, Chalmers University of Technology, Sweden (2012).
418
URL http://publications.lib.chalmers.se/records/fulltext/162546.pdf
419
[27] J. Tan, A. Chaigne, A. Acri, Contribution of the vibration of various piano components in
420
the resulting piano sound, in: Proceedings of the 22nd International Congress on Acoustics,
421
2016, pp. ICA2016–171.
422
[28] D. de Klerk, A. Ossipov, Operational transfer path analysis: Theory, guidelines and tire noise
423
application, Mechanical Systems and Signal Processing 24 (7) (2010) 1950–1962.
424
[29] A. Diez-Ibarbia, M. Battarra, J. Palenzuela, G. Cervantes, S. Walsh, M. De-la Cruz, S.
Theo-425
dossiades, L. Gagliardini, Comparison between transfer path analysis methods on an electric
426
vehicle, Applied Acoustics 118 (2017) 83–101.
427
[30] P. Gajdatsy, K. Janssens, W. Desmet, H. Van Der Auweraer, Application of the
transmis-428
sibility concept in transfer path analysis, Mechanical Systems and Signal Processing 24 (7)
429
(2010) 1963–1976.
430
[31] C. Sandier, Q. Leclere, N. B. Roozen, Operational transfer path analysis: theoretical aspects
431
and experimental validation, in: Proceedings of the Acoustics 2012 Nantes Conference, 2012,
432
pp. 3461–3466.
433
[32] R. Penrose, A generalized inverse for matrices, Mathematical Proceedings of the Cambridge
434
Philosophical Society 51 (3) (1955) 406–413.
[33] N. Maia, J. Silva, A. Ribeiro, The transmissibility concept in multi-degree-of-freedom systems,
436
Mechanical Systems and Signal Processing 15 (1) (2001) 129–137.
437
[34] G. Golub, C. Van Loan, Matrix computations, Vol. 3, JHU Press, 2012.
438
[35] E. Blackham, The physics of the piano, Scientific american 213 (6) (1965) 88–95.
439
[36] W. Goebl, R. Bresin, I. Fujinaga, Perception of touch quality in piano tones, The Journal of
440
the Acoustical Society of America 136 (5) (2014) 2839–2850.
441
[37] A. Chaigne, J. Kergomard, Acoustics of Musical Instruments, Springer-Verlag, New York,
442
2016.
443
[38] D. W. Martin, W. Ward, Subjective evaluation of musical scale temperament in pianos, The
444
Journal of the Acoustical Society of America 33 (5) (1961) 582–585.
