Generic passive-guaranteed nonlinear interaction model and structure-preserving spatial discretization procedure with applications in musical acoustics

Authors

Antoine Falaize, David Roze

Abstract

In musical acoustics, the production of sound is usually described by the nonlinear interaction of the musician with a resonator (the instrument). For example a string (resonator) can be bowed or hit by a piano hammer (nonlinear interactions). The aim of this paper is to provide a stable (passive-guaranteed) simulation of such interaction systems. Our approach consists in first defining a generic passive-guaranteed structure for the interaction (finite dimensional) and for the resonator (infinite dimensional) and second constructing a generic procedure for the discretization of the resonator. This is achieved in the Port-Hamiltonian systems framework that decomposes a physical model into a network of energy-storing components, dissipative components and inputs-outputs, thus guaranteeing the passivity of the proposed models. Finally, a well established structure preserving time discretization method is used to provide numerical models which prove to fulfill a discrete power balance, hence the numerical stability. This generic procedure is applied to the sound synthesis of a bowed string and of a string hit by a piano hammer.

Keywords

Port Hamiltonian system; Order reduction; Friction; Collision

Citation

• Journal: Nonlinear Dynamics
• Year: 2025
• Volume: 113
• Issue: 4
• Pages: 3249–3275
• Publisher: Springer Science and Business Media LLC
• DOI: 10.1007/s11071-024-10438-9

BibTeX

@article{Falaize_2024,
  title={{Generic passive-guaranteed nonlinear interaction model and structure-preserving spatial discretization procedure with applications in musical acoustics}},
  volume={113},
  ISSN={1573-269X},
  DOI={10.1007/s11071-024-10438-9},
  number={4},
  journal={Nonlinear Dynamics},
  publisher={Springer Science and Business Media LLC},
  author={Falaize, Antoine and Roze, David},
  year={2024},
  pages={3249--3275}
}

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References

• Arnold, D. N. Finite Element Exterior Calculus. (2018) doi:10.1137/1.9781611975543 – 10.1137/1.9781611975543
• Ayoub, R., Hamdouni, A. & Razafindralandy, D. A new Hodge operator in discrete exterior calculus. Application to fluid mechanics. Communications on Pure and Applied Analysis vol. 20 2155–2185 (2021) – 10.3934/cpaa.2021062
• Bensa, J., Bilbao, S., Kronland-Martinet, R., Smith III, J.O.: A power normalized non-linear lossy piano hammer. In: Proceedings of the stockholm music acoustics conference (SMAC03) (2003). URL https://www.speech.kth.se/music/smac03/programme.html
• Bensoam, J., Misdariis, N., Vergez, C., Causse, R.: Integral formalism and finite element method applied to sound synthesis by physical modeling. In: ICA : International congress of acoustics (2001)
• Bilbao, S. Wave and Scattering Methods for Numerical Simulation. (2004) doi:10.1002/0470870192 – 10.1002/0470870192
• Bilbao, S. Numerical Sound Synthesis. (2009) doi:10.1002/9780470749012 – 10.1002/9780470749012
• Bilbao, S., Torin, A. & Chatziioannou, V. Numerical Modeling of Collisions in Musical Instruments. Acta Acustica united with Acustica vol. 101 155–173 (2015) – 10.3813/aaa.918813
• Boutillon, X. Model for piano hammers: Experimental determination and digital simulation. The Journal of the Acoustical Society of America vol. 83 746–754 (1988) – 10.1121/1.396117
• Cardoso-Ribeiro, F. L., Matignon, D. & Lefèvre, L. A structure-preserving Partitioned Finite Element Method for the 2D wave equation. IFAC-PapersOnLine vol. 51 119–124 (2018) – 10.1016/j.ifacol.2018.06.033
• Chabassier, J., Chaigne, A. & Joly, P. Modeling and simulation of a grand piano. The Journal of the Acoustical Society of America vol. 134 648–665 (2013) – 10.1121/1.4809649
• Chaigne, A. & Kergomard, J. Acoustics of Musical Instruments. Modern Acoustics and Signal Processing (Springer New York, 2016). doi:10.1007/978-1-4939-3679-3 – 10.1007/978-1-4939-3679-3
• Chatziioannou, V., van Walstijn, M.: An energy conserving finite difference scheme for simulation of collisions. In: Proceedings of the stockholm music acoustics conference (SMAC13) (2013). URL https://www.kth.se/is/tmh/publications/archive-1.859266
• Chatziioannou, V. & van Walstijn, M. Energy conserving schemes for the simulation of musical instrument contact dynamics. Journal of Sound and Vibration vol. 339 262–279 (2015) – 10.1016/j.jsv.2014.11.017
• Ciarlet, P. G. The Finite Element Method for Elliptic Problems. (2002) doi:10.1137/1.9780898719208 – 10.1137/1.9780898719208
• Cohen, D. & Hairer, E. Linear energy-preserving integrators for Poisson systems. BIT Numerical Mathematics vol. 51 91–101 (2011) – 10.1007/s10543-011-0310-z
• Dahl, P. R. Solid Friction Damping of Mechanical Vibrations. AIAA Journal vol. 14 1675–1682 (1976) – 10.2514/3.61511
• Canudas de Wit, C., Olsson, H., Astrom, K. J. & Lischinsky, P. A new model for control of systems with friction. IEEE Transactions on Automatic Control vol. 40 419–425 (1995) – 10.1109/9.376053
• Desvages, C.: Physical modelling of the bowed string and applications to sound synthesis. Ph.D. thesis, University of Edinburgh (2018)
• Desvages, C. & Bilbao, S. Two-Polarisation Physical Model of Bowed Strings with Nonlinear Contact and Friction Forces, and Application to Gesture-Based Sound Synthesis. Applied Sciences vol. 6 135 (2016) – 10.3390/app6050135
• Ducceschi, M. & Bilbao, S. Non-Iterative Simulation Methods for Virtual Analog Modelling. IEEE/ACM Transactions on Audio, Speech, and Language Processing vol. 30 3189–3198 (2022) – 10.1109/taslp.2022.3209934
• Ducceschi, M., Bilbao, S., Desvages, C.: Modelling collisions of nonlinear strings against rigid barriers: conservative finite difference schemes with application to sound synthesis. In: Proceedings of the 22nd international congress on acoustics (2016)
• Ducceschi, M., Bilbao, S., Willemsen, S. & Serafin, S. Linearly-implicit schemes for collisions in musical acoustics based on energy quadratisation. The Journal of the Acoustical Society of America vol. 149 3502–3516 (2021) – 10.1121/10.0005008
• Duindam, V., Macchelli, A., Stramigioli, S. & Bruyninckx, H. Modeling and Control of Complex Physical Systems. (Springer Berlin Heidelberg, 2009). doi:10.1007/978-3-642-03196-0 – 10.1007/978-3-642-03196-0
• Dupont, P., Hayward, V., Armstrong, B. & Altpeter, F. Single state elastoplastic friction models. IEEE Transactions on Automatic Control vol. 47 787–792 (2002) – 10.1109/tac.2002.1000274
• Falaize, A. & Hélie, T. Passive Guaranteed Simulation of Analog Audio Circuits: A Port-Hamiltonian Approach. Applied Sciences vol. 6 273 (2016) – 10.3390/app6100273
• Falaize, A. & Hélie, T. Passive simulation of the nonlinear port-Hamiltonian modeling of a Rhodes Piano. Journal of Sound and Vibration vol. 390 289–309 (2017) – 10.1016/j.jsv.2016.11.008
• Ghosh, M.: Experimental study of the duration of contat of an elastic hammer striking a damped pianoforte string (1932)
• Giordano, N. & Winans, J. P., II. Piano hammers and their force compression characteristics: Does a power law make sense? The Journal of the Acoustical Society of America vol. 107 2248–2255 (2000) – 10.1121/1.428505
• Gong, Y., Wang, Q. & Wang, Z. Structure-preserving Galerkin POD reduced-order modeling of Hamiltonian systems. Computer Methods in Applied Mechanics and Engineering vol. 315 780–798 (2017) – 10.1016/j.cma.2016.11.016
• Gonzalez, O. & Simo, J. C. On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry. Computer Methods in Applied Mechanics and Engineering vol. 134 197–222 (1996) – 10.1016/0045-7825(96)01009-2
• Geometric Numerical Integration. Springer Series in Computational Mathematics (Springer-Verlag, 2006). doi:10.1007/3-540-30666-8 – 10.1007/3-540-30666-8
• Ishikawa, A., Michels, D. L. & Yaguchi, T. Geometric-integration tools for the simulation of musical sounds. Japan Journal of Industrial and Applied Mathematics vol. 35 511–540 (2018) – 10.1007/s13160-017-0292-6
• Issanchou, C.: Vibrations non linéaires de cordes avec contact unilatéral. application aux instruments de musique. Ph.D. thesis, Université Pierre et Marie Curie—Paris VI (2017). URL https://theses.hal.science/tel-01631495
• Issanchou, C., Bilbao, S., Le Carrou, J.-L., Touzé, C. & Doaré, O. 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. 393 229–251 (2017) – 10.1016/j.jsv.2016.12.025
• Itoh, T. & Abe, K. Hamiltonian-conserving discrete canonical equations based on variational difference quotients. Journal of Computational Physics vol. 76 85–102 (1988) – 10.1016/0021-9991(88)90132-5
• Jacob, B. & Zwart, H. J. Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces. (Springer Basel, 2012). doi:10.1007/978-3-0348-0399-1 – 10.1007/978-3-0348-0399-1
• Kaselouris, E., Bakarezos, M., Tatarakis, M., Papadogiannis, N. A. & Dimitriou, V. A Review of Finite Element Studies in String Musical Instruments. Acoustics vol. 4 183–202 (2022) – 10.3390/acoustics4010012
• Kotyczka, P., Maschke, B. & Lefèvre, L. Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems. Journal of Computational Physics vol. 361 442–476 (2018) – 10.1016/j.jcp.2018.02.006
• Leimkuhler, B. & Reich, S. Simulating Hamiltonian Dynamics. (2005) doi:10.1017/cbo9780511614118 – 10.1017/cbo9780511614118
• Marx, D., Bailliet, H. & Valière, J.-C. Analysis of the Acoustic Flow at an Abrupt Change in Section of an Acoustic Waveguide Using Particle Image Velocimetry and Proper Orthogonal Decomposition. Acta Acustica united with Acustica vol. 94 54–65 (2008) – 10.3813/aaa.918008
• Maschke, B. M., Van Der Schaft, A. J. & Breedveld, P. C. An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators. Journal of the Franklin Institute vol. 329 923–966 (1992) – 10.1016/s0016-0032(92)90049-m
• ME McIntyre. McIntyre, M.E., Woodhouse, J.: On the fundamentals of bowed-string dynamics. Acta Acust. United Acust. 43(2), 93–108 (1979) (1979)
• McLachlan, R. I., Perlmutter, M. & Quispel, G. R. W. On the Nonlinear Stability of Symplectic Integrators. BIT Numerical Mathematics vol. 44 99–117 (2004) – 10.1023/b:bitn.0000025088.13092.7f
• Morrison, J. D. & Adrien, J.-M. MOSAIC: A Framework for Modal Synthesis. Computer Music Journal vol. 17 45 (1993) – 10.2307/3680569
• Onofrei, M. G., Willemsen, S. & Serafin, S. Real-Time Implementation of a Friction Drum Inspired Instrument Using Finite Difference Schemes. 2021 24th International Conference on Digital Audio Effects (DAFx) 168–175 (2021) doi:10.23919/dafx51585.2021.9768291 – 10.23919/dafx51585.2021.9768291
• Raibaud, M.: Modélisation et simulation de systèmes discrétisés par la méthode des éléments finis dans le formalisme des systèmes hamiltoniens à ports : application à la synthèse sonore. Master’s thesis, Sorbonne Université (2018)
• Rath, M., Rocchesso, D., Avanzini, F.: Physically based real-time modeling of contact sounds. In: Proceedings international computer music conference (2002). URL http://hdl.handle.net/2027/spo.bbp2372.2002.046
• Rhaouti, L., Chaigne, A. & Joly, P. Time-domain modeling and numerical simulation of a kettledrum. The Journal of the Acoustical Society of America vol. 105 3545–3562 (1999) – 10.1121/1.424679
• Rodet, X. & Vergez, C. Nonlinear Dynamics in Physical Models: From Basic Models to True Musical-Instrument Models. Computer Music Journal vol. 23 35–49 (1999) – 10.1162/014892699559878
• Russo, R., Ducceschi, M., Bilbao, S.: Efficient simulation of the bowed string in modal form. In: Proceedings of the 25th international conference on digital audio effects (DAFx20in2022) (2022)
• Salisbury, K., Conti, F. & Barbagli, F. Survey - Haptic rendering: introductory concepts. IEEE Computer Graphics and Applications vol. 24 24–32 (2004) – 10.1109/mcg.2004.1274058
• Sato, S., Miyatake, Y. & Butcher, J. C. High-order linearly implicit schemes conserving quadratic invariants. Applied Numerical Mathematics vol. 187 71–88 (2023) – 10.1016/j.apnum.2023.02.005
• van der Schaft, A. Port-Hamiltonian systems: an introductory survey. Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006 1339–1365 (2007) doi:10.4171/022-3/65 – 10.4171/022-3/65
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• Serafin, S., Avanzini, F., Rocchesso, D.: Bowed string simulation using an elasto-plastic friction model. In: Proceedings of the Stockholm music acoustics conference (2003). URL https://hdl.handle.net/2434/656637
• Slotine, J.J.E., Li, W.: Applied Nonlinear Control, vol. 199. Prentice-hall Englewood Cliffs, NJ (1991)
• JM Souriau. Souriau, J.M.: Structure of Dynamical Systems: A Symplectic View of Physics, vol. 149. Springer (1997) (1997)
• Stulov, A. Hysteretic model of the grand piano hammer felt. The Journal of the Acoustical Society of America vol. 97 2577–2585 (1995) – 10.1121/1.411912
• Stulov, A.: Experimental and theoretical studies of piano hammer. In: Proceedings of the Stockholm music acoustics conference, vol. 485 (2003). URL https://www.speech.kth.se/music/smac03/programme.html
• Torin, A.: Percussion instrument modelling in 3d: sound synthesis through time domain numerical simulation. Ph.D. thesis, University of Edinburgh (2015)
• Trenchant, V., Ramirez, H., Le Gorrec, Y. & Kotyczka, P. Finite differences on staggered grids preserving the port-Hamiltonian structure with application to an acoustic duct. Journal of Computational Physics vol. 373 673–697 (2018) – 10.1016/j.jcp.2018.06.051
• Välimäki, V., Pakarinen, J., Erkut, C. & Karjalainen, M. Discrete-time modelling of musical instruments. Reports on Progress in Physics vol. 69 1–78 (2005) – 10.1088/0034-4885/69/1/r01
• Willemsen, S., Bilbao, S., Serafin, S.: Real-time implementation of an elasto-plastic friction model applied to stiff strings using finite-difference schemes. In: 22nd Int. conference on digital audio effects (2019)