P. Iii, . Source, . Of, ]. E. Bibliography, and . Blackham, The physics of the piano [2] A. Dolge. Pianos and Their Makers: A comprehensive history of the development of the piano from the monochord to the concert grand player piano. Pianos and Their Makers, Scientific american, vol.2133, issue.16, pp.88-95, 1911.

. Phoenix-carbiano-, Phoenix Piano Systems Ltd -Carbiano -The Carbon Fibre Piano, pp.2017-2023

H. Helmholtz, On the sensations of tone (original work by 1877), Translated. A. J. Ellis, vol.8, pp.390-394, 1954.

H. Suzuki and I. Nakamura, Acoustics of pianos, Lieber. On the possibilities of influencing piano touch. Das Musikinstrument, pp.14758-63, 1985.
DOI : 10.1016/0003-682X(90)90043-T

A. Askenfelt and E. Jansson, From touch to string vibrations???The initial course of the piano tone, The Journal of the Acoustical Society of America, vol.81, issue.S1, pp.61-61, 1987.
DOI : 10.1121/1.2024316

B. Gillespie, The virtual piano action: Design and implementation International Computer Music Association, 1994. [12] B. Gillespie. Haptic display of systems with changing kinematic constraints: The virtual piano action, Proceedings of the international Computer Music Conference, pp.167-167, 1996.

M. Hirschkorn, J. Mcphee, and S. Birkett, Dynamic Modeling and Experimental Testing of a Piano Action Mechanism, Journal of Computational and Nonlinear Dynamics, vol.32, issue.1, pp.47-55, 2006.
DOI : 10.1115/1.3423596

A. Izadbakhsh, J. Mcphee, and S. Birkett, Dynamic Modeling and Experimental Testing of a Piano Action Mechanism With a Flexible Hammer Shank, Journal of Computational and Nonlinear Dynamics, vol.81, issue.3, p.31004, 2008.
DOI : 10.1081/SME-100000002

R. Masoudi, S. Birkett, and J. Mcphee, Dynamic Model of a Vertical Piano Action Mechanism, Volume 4: 7th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B and C, pp.389-398, 2009.
DOI : 10.1115/DETC2009-87680

R. Masoudi and S. Birkett, Experimental Validation of a Mechanistic Multibody Model of a Vertical Piano Action, Journal of Computational and Nonlinear Dynamics, vol.10, issue.6, p.61004, 2015.
DOI : 10.1115/1.4028194

A. Thorin, X. Boutillon, J. Lozada, and X. Merlhiot, Non-smooth dynamics for an efficient simulation of the grand piano action, Meccanica, vol.125, issue.2, pp.1-18, 2017.
DOI : 10.1121/1.3125343

URL : https://hal.archives-ouvertes.fr/hal-01331941

A. Thorin, Non-smooth model of the grand piano action, 2013.
URL : https://hal.archives-ouvertes.fr/pastel-00939493

J. Chabassier and M. Duruflé, Energy based simulation of a Timoshenko beam in non-forced rotation. Influence of the piano hammer shank flexibility on the sound, Journal of Sound and Vibration, vol.333, issue.26, pp.7198-7215, 2014.
DOI : 10.1016/j.jsv.2014.08.017

URL : https://hal.archives-ouvertes.fr/hal-00929938

W. Kaufmann, Uber die bewegungen geschlagener klaviersaiten, Ann. Phys, vol.54, pp.675-712

X. Boutillon, Model for piano hammers: Experimental determination and digital simulation, The Journal of the Acoustical Society of America, vol.83, issue.2, pp.746-754, 1988.
DOI : 10.1121/1.396117

A. Chaigne and A. Askenfelt, Numerical simulations of piano strings. II. Comparisons with measurements and systematic exploration of some hammer???string parameters, The Journal of the Acoustical Society of America, vol.95, issue.3, pp.1631-1640, 1994.
DOI : 10.1121/1.408549

A. Stulov, Experimental and computational studies of piano hammers, Acta Acustica united with Acustica, vol.91, issue.6, pp.1086-1097, 2005.

C. Vyasarayani, S. Birkett, and J. Mcphee, Modeling the dynamics of a compliant piano action mechanism impacting an elastic stiff string, The Journal of the Acoustical Society of America, vol.125, issue.6, pp.4034-4042, 2009.
DOI : 10.1121/1.3125343

S. Birkett, Experimental investigation of the piano hammer-string interaction, The Journal of the Acoustical Society of America, vol.133, issue.4, pp.2467-2478, 2013.
DOI : 10.1121/1.4792357

A. Chaigne, Reconstruction of piano hammer force from string velocity, The Journal of the Acoustical Society of America, vol.140, issue.5, pp.3504-3517, 2016.
DOI : 10.1121/1.4965965

G. Weinreich, Coupled piano strings, The Journal of Acoustical Society of America, vol.7, p.926, 2004.
DOI : 10.1121/1.2016722

URL : http://asa.scitation.org/doi/pdf/10.1121/1.2016722

N. J. Giordano, Physics of the Piano, 2016.

I. Nakamura, Fundamental theory and computer simulation of the decay characteristics of piano sound., Journal of the Acoustical Society of Japan (E), vol.10, issue.5, pp.289-297, 1989.
DOI : 10.1250/ast.10.289

O. Thomas, D. Rousseau, R. Caussé, and E. Marandas, Comparison of the effect of and mistuning on the double decay of piano tones, ISMA: International Symposium of Music Acoustics, pp.253-258, 1998.
URL : https://hal.archives-ouvertes.fr/hal-01105499

M. Aramaki, J. Bensa, L. Daudet, P. Guillemain, and R. Kronland-martinet, Resynthesis of Coupled Piano String Vibrations Based on Physical Modeling, Journal of New Music Research, vol.30, issue.3, pp.213-226, 2001.
DOI : 10.1076/jnmr.30.3.213.7472

J. Chabassier and S. Imperiale, Stability and dispersion analysis of improved time discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string, Wave Motion, vol.50, issue.3, pp.456-480, 2013.
DOI : 10.1016/j.wavemoti.2012.11.002

URL : https://hal.archives-ouvertes.fr/hal-00738233

M. Ducceschi and S. Bilbao, Linear stiff string vibrations in musical acoustics: Assessment and comparison of models, The Journal of the Acoustical Society of America, vol.140, issue.4, pp.2445-2454, 2016.
DOI : 10.1121/1.4962553

H. and C. Jr, Generation of partials due to nonlinear mixing in a stringed instrument, The Journal of the Acoustical Society of America, vol.105, issue.1, pp.536-545, 1999.

N. Giordano and A. Korty, Motion of a piano string: Longitudinal vibrations and the role of the bridge, The Journal of the Acoustical Society of America, vol.100, issue.6, pp.3899-3908, 1996.
DOI : 10.1121/1.417219

B. Bank and L. Sujbert, Generation of longitudinal vibrations in piano strings: From physics to sound synthesis, The Journal of the Acoustical Society of America, vol.117, issue.4, pp.2268-2278, 2005.
DOI : 10.1121/1.1868212

J. Chabassier and P. Joly, Energy preserving schemes for nonlinear Hamiltonian systems of wave equations: Application to the vibrating piano string, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.45-48, pp.2779-2795
DOI : 10.1016/j.cma.2010.04.013

URL : https://hal.archives-ouvertes.fr/inria-00534473

E. Kurmyshev, Transverse and longitudinal mode coupling in a free vibrating soft string, Physics Letters A, vol.310, issue.2-3, pp.148-160, 2003.
DOI : 10.1016/S0375-9601(03)00264-0

A. Mamou-mani, J. Frelat, and C. Besnainou, Numerical simulation of a piano soundboard under downbearing, The Journal of the Acoustical Society of America, vol.123, issue.4, pp.2401-2406, 2008.
DOI : 10.1121/1.2836787

K. Ege, The piano soundboard -Modal studies in the low-and the midfrequency range, 2009.
URL : https://hal.archives-ouvertes.fr/tel-00460783

A. Askenfelt, Sound radiation and timbre. Mechanics of Playing and Making Musical Instruments, 2006.

J. Berthaut, M. Ichchou, and L. Jézéquel, Piano soundboard: structural behavior, numerical and experimental study in the modal range, Applied Acoustics, vol.64, issue.11, pp.1113-1136, 2003.
DOI : 10.1016/S0003-682X(03)00065-3

H. Suzuki, Vibration and sound radiation of a piano soundboard, The Journal of the Acoustical Society of America, vol.80, issue.6, pp.1573-1582, 1986.
DOI : 10.1121/1.394321

K. Ege, X. Boutillon, and M. Rébillat, Vibroacoustics of the piano soundboard: (Non)linearity and modal properties in the low- and mid-frequency ranges, Journal of Sound and Vibration, vol.332, issue.5, pp.1288-1305, 2013.
DOI : 10.1016/j.jsv.2012.10.012

URL : https://hal.archives-ouvertes.fr/hal-00743291

T. Moore and S. Zietlow, Interferometric studies of a piano soundboard, The Journal of the Acoustical Society of America, vol.119, issue.3, pp.1783-1793, 2006.
DOI : 10.1121/1.2164989

A. Chaigne, B. Cotté, and R. Viggiano, Dynamical properties of piano soundboards, The Journal of the Acoustical Society of America, vol.133, issue.4, pp.2456-2466, 2013.
DOI : 10.1121/1.4794387

URL : https://hal.archives-ouvertes.fr/hal-01137948

K. Wogram, Acoustical research on pianos: vibrational characteristics of the soundboard, Das Musikinstrument, vol.24, pp.694-702, 1980.

I. Nakamura, The vibrational character of the piano soundboard, Proceedings of the 11th ICA, pp.385-388, 1983.

H. Conklin, Design and tone in the mechanoacoustic piano. Part II. Piano structure, The Journal of the Acoustical Society of America, vol.100, issue.2, pp.695-708, 1996.
DOI : 10.1121/1.416233

N. Giordano, Mechanical impedance of a piano soundboard, The Journal of the Acoustical Society of America, vol.103, issue.4, pp.2128-2133, 1998.
DOI : 10.1121/1.421358

X. Boutillon and G. Weinreich, Three-dimensional mechanical admittance: Theory and new measurement method applied to the violin bridge, The Journal of the Acoustical Society of America, vol.105, issue.6, pp.3524-3533, 1999.
DOI : 10.1121/1.424677

X. Boutillon and K. Ege, Vibroacoustics of the piano soundboard: Reduced models, mobility synthesis, and acoustical radiation regime, Journal of Sound and Vibration, vol.332, issue.18, pp.4261-4279, 2013.
DOI : 10.1016/j.jsv.2013.03.015

URL : https://hal.archives-ouvertes.fr/hal-00817271

B. Trévisan, K. Ege, and B. Laulagnet, A modal approach to piano soundboard vibroacoustic behavior, The Journal of the Acoustical Society of America, vol.141, issue.2, pp.690-709, 2017.
DOI : 10.1121/1.4974860

V. Debut, J. Antunes, M. Marques, and M. Carvalho, Physics-based modeling techniques of a twelve-string Portuguese guitar: A non-linear time-domain computational approach for the multiple-strings/bridge/soundboard coupled dynamics, Applied Acoustics, vol.108, pp.3-18, 2016.
DOI : 10.1016/j.apacoust.2015.10.029

J. Smith, Physical Modeling Using Digital Waveguides, Computer Music Journal, vol.16, issue.4, pp.74-91, 1992.
DOI : 10.2307/3680470

B. Bank, Physics-based sound synthesis of the piano, 2000.

J. Bensa, S. Bilbao, R. Kronland-martinet, J. Smith, and I. , The simulation of piano string vibration: From physical models to finite difference schemes and digital waveguides, The Journal of the Acoustical Society of America, vol.114, issue.2, pp.1095-1107, 2003.
DOI : 10.1121/1.1587146

URL : https://hal.archives-ouvertes.fr/hal-00088329

J. Rauhala, V. Välimäki, and H. Lehtonen, Multi-ripple loss filter for waveguide piano synthesis, Proceeding of International Computer Music Conference, 2005.

J. Bensa, S. Bilbao, R. Kronland-martinet, J. Smith, and T. Voinier, Computational modeling of stiff piano strings using digital waveguides and finite differences, Acta Acustica united with Acustica, vol.91, issue.2, pp.289-298, 2005.

S. Bilbao, Numerical Sound Synthesis: Finite Difference Scheme and Simulation in Musical Acoustics, 2009.
DOI : 10.1002/9780470749012

A. Chaigne and A. Askenfelt, Numerical simulations of piano strings. I. A physical model for a struck string using finite difference methods, The Journal of the Acoustical Society of America, vol.95, issue.2, pp.1112-1118, 1994.
DOI : 10.1121/1.408459

N. Giordano and M. Jiang, Physical modeling of the piano, EURASIP J. Appl. Signal Process, pp.926-933, 2004.

J. Chabassier, A. Chaigne, and P. Joly, Modeling and simulation of a grand piano, The Journal of the Acoustical Society of America, vol.134, issue.1, pp.648-6652577, 1995.
DOI : 10.1121/1.4809649

URL : https://hal.archives-ouvertes.fr/hal-00873089

N. Giordano, Simple model of a piano soundboard, The Journal of the Acoustical Society of America, vol.102, issue.2, pp.1159-1168, 1997.
DOI : 10.1121/1.419868

A. Chaigne, M. Hennet, J. Chabassier, and M. Duruflé, Comparison between three different Viennese pianos of the nineteenth century, Proceedings of the 22nd International Congress on Acoustics, pp.2016-2041, 2016.

J. Chabassier, Modeling the influence of the piano hammer shank flexibility on the sound, The Journal of the Acoustical Society of America, vol.136, issue.4, pp.2133-2133, 2014.
DOI : 10.1121/1.4899693

J. Chabassier, A. Chaigne, and P. Joly, Time domain simulation of a piano. Part 1: model description, ESAIM: Mathematical Modelling and Numerical Analysis, vol.48, issue.5, pp.1241-1278, 2014.
DOI : 10.1121/1.381677

URL : https://hal.archives-ouvertes.fr/hal-01085477

G. Weinreich, The Coupled Motions of Piano Strings, Scientific American, vol.240, issue.1, pp.118-127, 1979.
DOI : 10.1038/scientificamerican0179-118

R. Hanson, J. Anderson, and H. Macomber, Measurements of nonlinear effects in a driven vibrating wire, The Journal of the Acoustical Society of America, vol.96, issue.3, pp.1549-1556, 1994.
DOI : 10.1121/1.410233

R. Hanson, H. Macomber, A. Morrison, and M. Boucher, Primarily nonlinear effects observed in a driven asymmetrical vibrating wire, The Journal of the Acoustical Society of America, vol.117, issue.1, pp.400-412, 2005.
DOI : 10.1121/1.1828511

P. Morse and K. Ingard, Theoretical Acoustics, 1968.

J. Chabassier, Modélisation et simulation numérique d'un piano par modèles physiques, 2012.

S. Bilbao, Conservative numerical methods for nonlinear strings, The Journal of the Acoustical Society of America, vol.118, issue.5, pp.3316-3327, 2005.
DOI : 10.1121/1.2046787

C. Touzé, O. Thomas, and A. Chaigne, ASYMMETRIC NON-LINEAR FORCED VIBRATIONS OF FREE-EDGE CIRCULAR PLATES. PART 1: THEORY, Journal of Sound and Vibration, vol.258, issue.4, p.649, 2002.
DOI : 10.1006/jsvi.2002.5143

O. Thomas, A. Lazarus, and C. Touzé, A Harmonic-Based Method for Computing the Stability of Periodic Oscillations of Non-Linear Structural Systems, Volume 5: 22nd International Conference on Design Theory and Methodology; Special Conference on Mechanical Vibration and Noise, 2010.
DOI : 10.1115/DETC2010-28407

URL : https://hal.archives-ouvertes.fr/hal-01148758

A. Manevitch and L. Manevitch, Free oscillations in conservative and dissipative symmetric cubic two-degree-of-freedom systems with closed natural frequencies, Meccanica, 2003.

C. Valette and C. Cuesta, Evolution temporelle de la vibration des cordes de clavecin, Acta Acustica united with Acustica, vol.66, issue.1, pp.37-45, 1988.

C. Desvages, S. Bilbao, and M. Ducceschi, Improved frequency-dependent damping for time domain modelling of linear string vibration, Proceedings of the 22nd International Congress on Acoustics, pp.2016-821, 2016.

C. Issanchou, S. Bilbao, J. Carrou, C. Touzé, and O. Doaré, A modal-based approach to the nonlinear vibration of strings against a unilateral obstacle: Simulations and experiments in the pointwise case, Journal of Sound and Vibration, vol.393, pp.229-251, 2017.
DOI : 10.1016/j.jsv.2016.12.025

URL : https://hal.archives-ouvertes.fr/hal-01461730

B. Chabassier and M. Durufle, Physical parameters for piano modeling, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00688679

J. J. Thomsen, Vibrations and Stability: Advanced Theory, Analysis, and Tools, 2003.
DOI : 10.1007/978-3-662-10793-5

J. J. Tan, C. Touzé, and B. Cotté, Double polarisation in nonlinear vibrating piano strings, Vienna talk 2015 on music Acoustics, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01194580

J. Carrou, F. Gautier, and R. Badeau, Sympathetic String Modes in the Concert Harp, Acta Acustica united with Acustica, vol.95, issue.4, pp.744-752, 2009.
DOI : 10.3813/AAA.918202

URL : https://hal.archives-ouvertes.fr/hal-00945199

K. Ege, X. Boutillon, and B. David, High-resolution modal analysis, Journal of Sound and Vibration, vol.325, issue.4-5, pp.852-869, 2009.
DOI : 10.1016/j.jsv.2009.04.019

URL : https://hal.archives-ouvertes.fr/hal-00477701

A. Chaigne and J. Kergomard, Acoustics of Musical Instruments, 2016.
DOI : 10.1007/978-1-4939-3679-3

URL : https://hal.archives-ouvertes.fr/hal-01338980

G. Derveaux, Modélisation numérique de la guitare acoustique, 2002.

A. Chaigne, J. Chabassier, and N. Burban, The acoustics of pianos, SMAC 2013-Stockholm Music Acoustics Conference 2013, 2013.
DOI : 10.1121/1.4755009

J. J. Tan, A. Chaigne, and A. Acri, Contribution of the vibration of various piano components in the resulting piano sound, Proceedings of the 22nd International Congress on Acoustics, pp.2016-171, 2016.

A. Stulov and D. Kartofelev, Vibration of strings with nonlinear supports, Applied Acoustics, vol.76, pp.223-229, 2014.
DOI : 10.1016/j.apacoust.2013.08.010

R. Corradi, S. Miccoli, G. Squicciarini, and P. Fazioli, Modal analysis of a grand piano soundboard at successive manufacturing stages, Applied Acoustics, vol.125, pp.113-127, 2017.
DOI : 10.1016/j.apacoust.2017.04.010

N. B. Roozen, Q. Leclere, and C. Sandier, Operational transfer path analysis applied to a small gearbox test set-up, Proceedings of the Acoustics 2012 Nantes Conference, pp.3467-3473, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00811346

D. De-klerk, M. Lohrmann, M. Quickert, and W. Foken, Application of operational transfer path analysis on a classic car, Proceedings of the International Conference on Acoustics NAG, pp.776-779, 2009.

J. Putner, H. Fastl, M. Lohrmann, A. Kaltenhauser, and F. Ullrich, Operational transfer path analysis predicting contributions to the vehicle interior noise for different excitations from the same sound source, Proc. Internoise, pp.2336-2347, 2012.

M. Toome, Operational transfer path analysis: A study of source contribution predictions at low frequency, 2012.

D. De-klerk and A. Ossipov, Operational transfer path analysis: Theory, guidelines and tire noise application, Mechanical Systems and Signal Processing, vol.24, issue.7, pp.1950-1962, 2010.
DOI : 10.1016/j.ymssp.2010.05.009

A. Diez-ibarbia, M. Battarra, J. Palenzuela, G. Cervantes, S. Walsh et al., Comparison between transfer path analysis methods on an electric vehicle, Applied Acoustics, vol.118, pp.83-101, 2017.
DOI : 10.1016/j.apacoust.2016.11.015

P. Gajdatsy, K. Janssens, W. Desmet, and H. Van-der-auweraer, Application of the transmissibility concept in transfer path analysis, Mechanical Systems and Signal Processing, vol.24, issue.7, pp.1963-1976, 2010.
DOI : 10.1016/j.ymssp.2010.05.008

C. Sandier, Q. Leclere, and N. B. Roozen, Operational transfer path analysis: theoretical aspects and experimental validation, Proceedings of the Acoustics 2012 Nantes Conference, pp.3461-3466, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00810966

R. Penrose, A generalized inverse for matrices, Mathematical Proceedings of the Cambridge Philosophical Society, vol.11, issue.03, pp.406-413, 1955.
DOI : 10.1093/qmath/2.1.189

N. Maia, J. Silva, and A. Ribeiro, THE TRANSMISSIBILITY CONCEPT IN MULTI-DEGREE-OF-FREEDOM SYSTEMS, Mechanical Systems and Signal Processing, vol.15, issue.1, pp.129-137, 2001.
DOI : 10.1006/mssp.2000.1356

W. Goebl, R. Bresin, and I. Fujinaga, Perception of touch quality in piano tones, The Journal of the Acoustical Society of America, vol.136, issue.5, pp.2839-2850, 2014.
DOI : 10.1121/1.4896461