, See Supplementary materials at [URL will be inserted by AIP

, Supplementary materials at [URL will be inserted by AIP

V. Acary, Energy conservation and dissipation properties of time-integration methods for nonsmooth elastodynamics with contact, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift f??r Angewandte Mathematik und Mechanik, vol.15, issue.1, pp.96-585, 2016.
DOI : 10.1007/978-3-540-32609-0

URL : https://hal.archives-ouvertes.fr/hal-01067691

V. Acary, M. Brémond, O. Huber, and F. Pérignon, Siconos. a software for modeling and simulation of nonsmooth dynamical systems, 2016.
URL : https://hal.archives-ouvertes.fr/inria-00162911

V. Acary and B. Brogliato, Numerical methods for nonsmooth dynamical systems. Applications in mechanics and electronics, of Lecture Notes in Applied and Computational Mechanics, 2008.
URL : https://hal.archives-ouvertes.fr/inria-00423530

V. Acary, H. De-jong, and B. Brogliato, Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems, Physica D: Nonlinear Phenomena, vol.269, pp.103-119, 2014.
DOI : 10.1016/j.physd.2013.11.013

URL : https://hal.archives-ouvertes.fr/hal-00778412

J. Ahn, A vibrating string with dynamic frictionless impact, Applied Numerical Mathematics, vol.57, issue.8, pp.861-884, 2007.
DOI : 10.1016/j.apnum.2006.07.033

L. Amerio, Continuous solutions of the problem of a string vibrating against an obstacle, pp.67-96, 1978.

T. Bacon, Electric bass guitar Grove Music Online. Oxford Music Online, 2013.

F. Bertails-descoubes, F. Cadoux, G. Davier, and V. Acary, A nonsmooth Newton solver for capturing exact Coulomb friction in fiber assemblies, ACM Transactions on Graphics, vol.30, issue.1, pp.1-6, 2011.
DOI : 10.1145/1899404.1899410

URL : https://hal.archives-ouvertes.fr/inria-00557706

S. Bilbao, Numerical sound synthesis: finite difference schemes and simulation in musical acoustics, 2009.
DOI : 10.1002/9780470749012

S. Bilbao, Time domain simulation and sound synthesis for the snare drum, The Journal of the Acoustical Society of America, vol.131, issue.1, pp.914-925, 2012.
DOI : 10.1121/1.3651240

S. Bilbao, T. , and A. , Numerical Modeling and Sound Synthesis for Articulated String/Fretboard Interactions, Journal of the Audio Engineering Society, vol.63, issue.5, pp.336-347, 2015.
DOI : 10.17743/jaes.2015.0023

S. Bilbao, A. Torin, and V. Chatziioannou, Numerical Modeling of Collisions in Musical Instruments, Acta Acustica united with Acustica, vol.101, issue.1, pp.155-173, 2015.
DOI : 10.3813/AAA.918813

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

H. Cabannes, Periodic motions of a string vibrating against a fixed point-mass obstacle: II, Mathematical Methods in the Applied Sciences, vol.20, issue.1, pp.55-67, 1984.
DOI : 10.1016/0362-546X(83)90076-7

URL : https://hal.archives-ouvertes.fr/hal-01527093

H. Cabannes, Motion of a string vibrating against a rigid fixed obstacle, Mathematical Modelling, vol.8, pp.364-365, 1987.
DOI : 10.1016/0270-0255(87)90606-3

URL : https://hal.archives-ouvertes.fr/hal-01527086

V. Chatziioannou and M. Van-walstijn, Energy conserving schemes for the simulation of musical instrument contact dynamics, Journal of Sound and Vibration, vol.339, pp.262-279, 2015.
DOI : 10.1016/j.jsv.2014.11.017

Q. Chen, V. Acary, G. Virlez, and O. Brüls, scheme for flexible multibody systems with unilateral constraints, International Journal for Numerical Methods in Engineering, vol.23, issue.2, pp.487-511, 2013.
DOI : 10.1007/s11044-009-9178-y

URL : https://hal.archives-ouvertes.fr/hal-00858066

C. Citrini, Vibrating strings with obstacles: the analytic study Advances in kinetic theory and continuum mechanics, pp.249-259, 1991.

R. W. Cottle, J. Pang, and R. E. Stone, The Linear Complementarity Problem, 1992.

C. Desvages and S. Bilbao, Two-polarisation finite difference model of bowed strings with nonlinear contact and friction forces, International Conference on Digital Audio Effects, p.15, 2015.
DOI : 10.3390/app6050135

URL : http://www.mdpi.com/2076-3417/6/5/135/pdf

D. Doyen, A. Ern, and S. Piperno, Time-Integration Schemes for the Finite Element Dynamic Signorini Problem, SIAM Journal on Scientific Computing, vol.33, issue.1, pp.223-249, 2011.
DOI : 10.1137/100791440

URL : https://hal.archives-ouvertes.fr/hal-00440128

M. Ducceschi, S. Bilbao, and C. Desvages, Modelling collisions of nonlinear strings against rigid barriers: conservative finite difference schemes with application to sound synthesis, International Congress on Acoustics, 2016.

N. H. Fletcher, R. , and T. D. , The physics of musical instruments, 1998.
DOI : 10.1119/1.16685

W. Goldsmith, , 2001.

O. Inácio, L. Henrique, and J. Antunes, The dynamics of tibetan singing bowls, Acta Acustica united with Acustica, vol.92, issue.4, pp.637-653, 2006.

C. Issanchou, S. Bilbao, L. Carrou, J. Touzé, C. Doaré et al., 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

C. Issanchou, L. Carrou, J. Touzé, C. Fabre, B. Doaré et al., String/frets contacts in the electric bass sound: Simulations and experiments, Applied Acoustics, vol.129, pp.217-228, 2018.
DOI : 10.1016/j.apacoust.2017.07.021

URL : https://hal.archives-ouvertes.fr/hal-01582578

O. Janin and C. Lamarque, Comparison of several numerical methods for mechanical systems with impacts, International Journal for Numerical Methods in Engineering, vol.51, issue.9, pp.1101-1132, 2001.
DOI : 10.1002/nme.206

URL : https://hal.archives-ouvertes.fr/hal-00814950

M. Jean, Frictional contact in collections of rigid or deformable bodies: numerical simulation of geomaterial motions, Mechanics of Geomaterial Interfaces, pp.463-486, 1995.
DOI : 10.1016/S0922-5382(06)80022-X

URL : https://hal.archives-ouvertes.fr/hal-01826980

M. Jean, The non-smooth contact dynamics method, Computer Methods in Applied Mechanics and Engineering, vol.177, issue.3-4, pp.235-257, 1999.
DOI : 10.1016/S0045-7825(98)00383-1

URL : https://hal.archives-ouvertes.fr/hal-01390459

M. Jean and J. Moreau, Dynamics in the Presence of Unilateral Contacts and Dry Friction: A Numerical Approach, pp.151-196, 1987.
DOI : 10.1007/978-3-7091-2967-8_10

URL : https://hal.archives-ouvertes.fr/hal-01864213

A. K. Mandal and P. Wahi, Natural frequencies, modeshapes and modal interactions for strings vibrating against an obstacle: Relevance to Sitar and Veena, Journal of Sound and Vibration, vol.338, pp.42-59, 2015.
DOI : 10.1016/j.jsv.2014.06.010

M. B. Meingast, M. Legrand, P. , and C. , A linear complementarity problem formulation for periodic solutions to unilateral contact problems, International Journal of Non-Linear Mechanics, vol.66, pp.18-27, 2014.
DOI : 10.1016/j.ijnonlinmec.2014.01.007

URL : https://hal.archives-ouvertes.fr/hal-00955647

J. Moreau, Bounded variation in time, Topics in Nonsmooth Mechanics, pp.1-74, 1988.
URL : https://hal.archives-ouvertes.fr/hal-01363799

J. Moreau, Numerical aspects of the sweeping process, Computer Methods in Applied Mechanics and Engineering, vol.177, issue.3-4, pp.329-349, 1999.
DOI : 10.1016/S0045-7825(98)00387-9

URL : https://hal.archives-ouvertes.fr/hal-01349847

N. S. Nguyen and B. Brogliato, Multiple Impacts in Dissipative Granular Chains, of Lecture Notes in Applied and Computational Mechanics, 2014.
DOI : 10.1007/978-3-642-39298-6

URL : https://hal.archives-ouvertes.fr/hal-00908612

L. Paoli and M. Schatzman, A Numerical Scheme for Impact Problems I: The One-Dimensional Case, SIAM Journal on Numerical Analysis, vol.40, issue.2, pp.702-733, 2002.
DOI : 10.1137/S0036142900378728

URL : https://hal.archives-ouvertes.fr/hal-01885729

A. Paté, L. Carrou, J. Fabre, and B. , Predicting the decay time of solid body electric guitar tones, The Journal of the Acoustical Society of America, vol.135, issue.5, pp.3045-3055, 2014.
DOI : 10.1121/1.4871360

C. Raman, On some indian stringed instruments Indian Association for the Cultivation of, Science, issue.7, pp.29-33, 1921.

M. Renouf, F. Dubois and P. Allart, A parallel version of the non smooth contact dynamics algorithm applied to the simulation of granular media, Journal of Computational and Applied Mathematics, vol.168, issue.1-2, pp.375-382, 2004.
DOI : 10.1016/j.cam.2003.05.019

URL : https://hal.archives-ouvertes.fr/hal-00450571

L. Rhaouti, A. Chaigne, J. , and P. , Time-domain modeling and numerical simulation of a kettledrum, The Journal of the Acoustical Society of America, vol.105, issue.6, pp.3545-3562, 1999.
DOI : 10.1121/1.424679

M. Schatzman, A hyperbolic problem of second order with unilateral constraints: The vibrating string with a concave obstacle, Journal of Mathematical Analysis and Applications, vol.73, issue.1, pp.138-191, 1980.
DOI : 10.1016/0022-247X(80)90026-8

URL : https://hal.archives-ouvertes.fr/hal-01620859

S. Siddiq, A Physical Model of the Nonlinear Sitar String, Archives of Acoustics, vol.125, issue.6, pp.73-79, 2012.
DOI : 10.1121/1.3123403

URL : http://www.degruyter.com/downloadpdf/j/aoa.2012.37.issue-1/v10168-012-0010-y/v10168-012-0010-y.xml

A. Signorini, Sopra alcune questioni di statica dei sistemi continui, Annali della Scuola Normale Superiore di Pisa, pp.231-251, 1933.

C. Studer, Numerics of unilateral contacts and friction, 47 of Lecture Notes in Applied and Computational Mechanics, 2009.

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.11-12, 2017.
DOI : 10.1121/1.3125343

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

L. Trautmann and R. Rabenstein, Multirate Simulations of String Vibrations Including Nonlinear Fret-String Interactions Using the Functional Transformation Method, EURASIP Journal on Advances in Signal Processing, vol.2004, issue.7, pp.949-963, 2004.
DOI : 10.1155/S1110865704312059

URL : https://doi.org/10.1155/s1110865704312059

C. Valette, C. , and C. , Mécanique de la corde vibrante, 1993.

C. Valette, C. Cuesta, C. Besnainou, and M. Castellengo, The tampura bridge as a precursive wave generator, Acustica, vol.74, pp.201-208, 1991.

M. Van-walstijn and J. Bridges, Simulation of distributed contact in string instruments: A modal expansion approach, 2016 24th European Signal Processing Conference (EUSIPCO), pp.1023-1027, 2016.
DOI : 10.1109/EUSIPCO.2016.7760403

M. Van-walstijn, J. Bridges, and S. Mehes, A real-time synthesis oriented tanpura model, International Conference on Digital Audio Effects, p.16, 2016.

C. P. Vyasarayani, S. Birkett, and J. Mcphee, Modeling the dynamics of a vibrating string with a finite distributed unilateral constraint: Application to the sitar, The Journal of the Acoustical Society of America, vol.125, issue.6, 2009.
DOI : 10.1121/1.3123403

URL : https://hal.archives-ouvertes.fr/hal-01525495

C. Yoong, A. Thorin and M. LegrandNonsmooth modal analysis of an elastic bar subject to a unilateral contact constraintNonlinear Dynamics, vol.91, issue.4, pp.2453-2476, 2018.
DOI : 10.1007/s11071-017-4025-9

URL : https://hal.archives-ouvertes.fr/hal-01471341

J. J. MoreauBounded Variation in TimeTopics in Nonsmooth Mechanics, 1987.
URL : https://hal.archives-ouvertes.fr/hal-01363799