Active control of the axisymmetric vibration modes of a tom-tom drum

Authors

Marc Wijnand, Brigitte d’Andrea-Novel, Benoit Fabre, Thomas Helie, Lionel Rosier, David Roze

Abstract

This paper deals with an application of active control of percussion instruments. Our setup consists of a tom-tom drum with a circular membrane, a cylindrical cavity and a circular rigid wall on which a loudspeaker is mounted. The current applied to the loudspeaker is controlled in order to modify the frequencies of the drum membrane modes. First, a PDE model of the axisymmetric transverse vibration of the tom-tom membrane is developed. Subsequently, the equation is recast as an infinite-dimensional port-Hamiltonian system. The port-Hamiltonian framework enables us to develop a numerical scheme that preserves the power balance and guarantees a stable simulation. Finally, a control law for the loudspeaker current is designed to modify the frequency of the first axisymmetric vibration mode of the drum membrane, using finite-time and passivity-based methods.

Citation

• Journal: 2019 IEEE 58th Conference on Decision and Control (CDC)
• Year: 2019
• Volume:
• Issue:
• Pages: 6887–6892
• Publisher: IEEE
• DOI: 10.1109/cdc40024.2019.9029960

BibTeX

@inproceedings{Wijnand_2019,
  title={{Active control of the axisymmetric vibration modes of a tom-tom drum}},
  DOI={10.1109/cdc40024.2019.9029960},
  booktitle={{2019 IEEE 58th Conference on Decision and Control (CDC)}},
  publisher={IEEE},
  author={Wijnand, Marc and d’Andrea-Novel, Brigitte and Fabre, Benoit and Helie, Thomas and Rosier, Lionel and Roze, David},
  year={2019},
  pages={6887--6892}
}

Download the bib file

References

• hélie, Systèmes Hamiltoniens à Ports avec approche par composants pour la simulation à passivité garantie de problèmes conservatifs et dissipatifs. 11e Colloque National en Calcul des Structures (2015)
• Theory of Robot Control. Communications and Control Engineering (Springer London, 1996). doi:10.1007/978-1-4471-1501-4 – 10.1007/978-1-4471-1501-4
• Tiwari, S. & Gupta, A. Effects of air loading on the acoustics of an Indian musical drum. The Journal of the Acoustical Society of America 141, 2611–2621 (2017) – 10.1121/1.4979782
• Christian, R. S. et al. Effects of air loading on timpani membrane vibrations. The Journal of the Acoustical Society of America 76, 1336–1345 (1984) – 10.1121/1.391449
• Rhaouti, L., Chaigne, A. & Joly, P. Time-domain modeling and numerical simulation of a kettledrum. The Journal of the Acoustical Society of America 105, 3545–3562 (1999) – 10.1121/1.424679
• Krstic, M. & Smyshlyaev, A. Boundary Control of PDEs. (2008) doi:10.1137/1.9780898718607 – 10.1137/1.9780898718607
• Haimo, V. T. Finite Time Controllers. SIAM J. Control Optim. 24, 760–770 (1986) – 10.1137/0324047
• Bhat, S. P. & Bernstein, D. S. Finite-Time Stability of Continuous Autonomous Systems. SIAM J. Control Optim. 38, 751–766 (2000) – 10.1137/s0363012997321358
• Bernuau, E., Perruquetti, W., Efimov, D. & Moulay, E. Robust finite-time output feedback stabilisation of the double integrator. International Journal of Control 88, 451–460 (2014) – 10.1080/00207179.2014.956340
• kinsler, Fundamentals of acoustics. Fundamentals of Acoustics (1999)
• graff, Wave Motion in Elastic Solids (2012)
• morse, Vibration and Sound (1995)
• wijnand, Contrôle des vibrations d’un oscillateur passif : stabilisation en temps fini et par remodelage d’énergie. Proc Congrès Français d’Acoustique (2018)
• boutin, Méthodes de contrôle actif d’instruments de musique. cas de la lame de xylophone et du violon. Ph D Dissertation (2011)
• d’Andréa-Novel, B. & De Lara, M. Control Theory for Engineers. (Springer Berlin Heidelberg, 2013). doi:10.1007/978-3-642-34324-7 – 10.1007/978-3-642-34324-7
• d’Andréa-Novel, B., Fabre, B. & Coron, J.-M. An Acoustic Model for Automatic Control of a Slide Flute. Acta Acustica united with Acustica 96, 713–721 (2010) – 10.3813/aaa.918325
• meurisse, Contrôle actif appliqué aux instruments de musique à vent. Ph D Dissertation (2014)
• lebrun, Electroacoustic absorbers based on finite-time control of loudspeakers: a numerical investigation. Int Conf on Nonlinear Dynamics (2019)
• benacchio, Contrôle actif modal appliqué aux instruments de musique à cordes. Ph D Dissertation (2014)
• Worland, R. Normal modes of a musical drumhead under non-uniform tension. The Journal of the Acoustical Society of America 127, 525–533 (2010) – 10.1121/1.3268605
• Jossic, M. et al. Modal active control of Chinese gongs. The Journal of the Acoustical Society of America 141, 4567–4578 (2017) – 10.1121/1.4985108
• Fuller, C. R., Elliott, S. J. & Nelson, P. A. Active Control of Vibration in Structures. Active Control of Vibration 153–183 (1996) doi:10.1016/b978-012269440-0/50006-6 – 10.1016/b978-012269440-0/50006-6
• Sathej, G. & Adhikari, R. The eigenspectra of Indian musical drums. The Journal of the Acoustical Society of America 125, 831–838 (2009) – 10.1121/1.3058632
• Elliott, S. J. & Nelson, P. A. Active noise control. IEEE Signal Process. Mag. 10, 12–35 (1993) – 10.1109/79.248551
• Maschke, B. M. & van der Schaft, A. J. Port-Controlled Hamiltonian Systems: Modelling Origins and Systemtheoretic Properties. IFAC Proceedings Volumes 25, 359–365 (1992) – 10.1016/s1474-6670(17)52308-3
• Falaize, A. & Hélie, T. Passive Guaranteed Simulation of Analog Audio Circuits: A Port-Hamiltonian Approach. Applied Sciences 6, 273 (2016) – 10.3390/app6100273
• van der schaft, Port-Hamiltonian systems: an introductory survey. Proceedings of the International Congress of Mathematicians (2006)
• Curtain, R. F. & Zwart, H. An Introduction to Infinite-Dimensional Linear Systems Theory. Texts in Applied Mathematics (Springer New York, 1995). doi:10.1007/978-1-4612-4224-6 – 10.1007/978-1-4612-4224-6
• Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica 38, 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
• villegas, A port-Hamiltonian approach to distributed parameter systems. Ph D Dissertation (2007)
• Hélie, T. & Matignon, D. Nonlinear damping models for linear conservative mechanical systems with preserved eigenspaces: a port-Hamiltonian formulation. IFAC-PapersOnLine 48, 200–205 (2015) – 10.1016/j.ifacol.2015.10.239