Self-oscillations of a vocal apparatus: a port-Hamiltonian formulation

07/11/2017
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Port Hamiltonian systems (PHS) are open passive systems that full a power balance: they correspond to dynamical systems composed of energy-storing elements, energy-dissipating elements and external ports, endowed with a geometric structure (called Dirac structure) that encodes conservative interconnections. This paper presents a minimal PHS model of the full vocal apparatus. Elementary components are: (a) an ideal subglottal pressure supply, (b) a glottal ow in a mobile channel, (c) vocal-folds, (d) an acoustic resonator reduced to a single mode. Particular attention is paid to the energetic consistency of each component, to passivity and to the conservative interconnection. Simulations are presented. They show the ability of the model to produce a variety of regimes, including self-sustained oscillations. Typical healthy or pathological conguration laryngeal congurations are explored.

Self-oscillations of a vocal apparatus: a port-Hamiltonian formulation

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application/pdf Self-oscillations of a vocal apparatus: a port-Hamiltonian formulation Thomas Hélie, Fabrice Silva
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Self-oscillations of a vocal apparatus: a port-Hamiltonian formulation
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Self-oscillations of a vocal apparatus: a port-Hamiltonian formulation Thomas Hélie1 and Fabrice Silva2 1 S3AM Team, UMR STMS 9912, IRCAM-CNRS-UPMC, Paris, France 2 Aix Marseille Univ., CNRS, Centrale Marseille, LMA, Marseille, France Abstract. Port Hamiltonian systems (PHS) are open passive systems that full a power balance: they correspond to dynamical systems com- posed of energy-storing elements, energy-dissipating elements and exter- nal ports, endowed with a geometric structure (called Dirac structure) that encodes conservative interconnections. This paper presents a mini- mal PHS model of the full vocal apparatus. Elementary components are: (a) an ideal subglottal pressure supply, (b) a glottal ow in a mobile channel, (c) vocal-folds, (d) an acoustic resonator reduced to a single mode. Particular attention is paid to the energetic consistency of each component, to passivity and to the conservative interconnection. Simu- lations are presented. They show the ability of the model to produce a variety of regimes, including self-sustained oscillations. Typical healthy or pathological conguration laryngeal congurations are explored. 1 Motivations Many physics-based models of the human vocal apparatus were proposed to help understanding the phonation and its pathologies, with a compromise between the complexity introduced in the modelling and the vocal features that can be reproduced by analytical or numerical calculations. Except recent works based on nite elements methods applied to the glottal ow dynamics, most of the models rely on the description of the aerodynamics provided by van den Berg [1] for a glottal ow in static geometries, i.e., that ignores the motion of the vocal folds. Even if enhancements appeared accounting for various eects, they failed to represent correctly the energy exchanges between the ow and the surface of the vocal folds that bounds the glottis. The port-Hamiltonian approach oers a framework for the modelling, analy- sis and control of complex system with emphasis on passivity and power bal- ance [2]. A PHS for the classical body-cover model has been recently pro- posed [3] without connection to a glottal ow nor to a vocal tract, so that no self-oscillations can be produced. The current paper proposes a minimal PHS model of the full vocal apparatus. This power-balanced numerical tool enables the investigation of the various regimes that can be produced by time-domain simulations. Sec. 2 is a reminder on the port-Hamiltonian systems, Sec. 3 is dedi- cated to the description of the elementary components of the full vocal apparatus and their interconnection. Sec. 4 presents simulation and numerical results for typical healthy and pathological laryngeal congurations. 2 Port-Hamiltonian Systems Port-Hamiltonian systems are open passive systems that full a power bal- ance [2,4]. A large class of such nite dimensional systems with input u(t) ∈ U = RP , output y(t) ∈ Y = U, can be described by a dierential algebraic equation   ẋ w −y   = S(x, w)   ∇xH z(w) u   , with S = −ST =   Jx −K Gx KT Jw Gw −GT x −GT w Jy,   (1) where state x(t) ∈ X = RN is associated with energy E = H(x) ≥ 0 and where variables w(t) ∈ W = RQ are associated with dissipative constitutive laws z such that Pdis = z(w)T w ≥ 0 stands for a dissipated power. Such a system naturally fulls the power balance dE/dt + Pdis − Pext = 0, where the external power is Pext = yT u. This is a straightforward consequence of the skew-symmetry of matrix S, which encodes this geometric structure (Dirac structure, see [2]). Indeed, rewriting Eq. (1) as B = SA, it follows that AT B = AT SA = 0, that is, ∇xH(x)T ẋ + z(w)T w − uT y = 0 (2) Moreover, connecting several PHS through external ports yields a PHS. This modularity is used in practice, by working on elementary components, separately. 3 Vocal apparatus Beneting from this modularity, the full vocal apparatus is built as the intercon- nection of the following elementary components: a subglottal pressure supply, two vocal folds, a glottal ow, and an acoustic resonator (see Fig. 1). Pressure supply 0 Glottal ow Left fold Right fold 0 Acoustic resonator Psub Qsub Psub r Qsub r Psub l Qsub l P− tot Q− F p l v l F p r v r P+ Q+ Psup l Qsup l Psup r Qsup r Pac Qac Fig. 1. Components of the vocal apparatus. The interconnection takes place via pairs of eort (P) and ux (Q) variables. The 0 connection expresses the equality of eorts and the division of ux. See Ref. [4] for an introduction to bond graphs. 3.1 The one-mass model of vocal folds The left and right vocal folds (Fi = L or R with i = l or r, respectively), are modelled as classical single-d.o.f. oscillators (as in Ref. [5], mass mi, spring ki and damping ri) with a purely elastic cover (as in Ref. [6], spring κi). Their dynamics relates the momentum πi of the mass, and the elongations ξi and ζi of the body and cover springs, respectively, to the velocity vi = ζ̇i + ˙ ξi of the cover imposed by the glottal ow, and to the transverse resultants of the pressure forces on the upstream (Psub i ) and downstream (Psup i ) faces of the trapezoid-shaped structures (see Fig. 2, left part) : π̇i = −kiξi − ri ˙ ξi + κiζi − Psub i Ssub i − Psup i Ssup i . (3) Fp i = −κiζi is the transverse feedback force opposed by the fold to the ow. The motion of the fold produces the additional owrates Qsub i (pumping from the subglottal space, i.e., positive when the fold compresses) and Qsup i (pulsated into the supraglottal cavity, i.e., positive when the fold inates). Port-Hamiltonian modelling of a vocal fold Fi : xFi =   πi ξi ζi   , uFi =   Psub i Psup i vi   , yFi =   −Qsub i Qsup i −Fp i   , HFi = 1 2 xT Fi   1/mi ki κi   xFi , wFi = ˙ ξi, zFi (wFi ) = riwFi , JFi w = 0, GFi w = O1×3, JFi y = O3×3, JFi x =   0 −1 1 1 0 0 −1 0 0   , KFi =   1 0 0   , and GFi x =   −Ssub i −Ssup i 0 0 0 0 0 0 1   . ki ri mi κi Psub i Psup i Fp i Ssub i Ssup i Glottal ow Sr Sl | −` | 0 | ` x yr(t) yl(t) y L h Ω(t) S− S+ Fp l Fp r P− tot P+ tot Fig. 2. Left: Schematic of a vocal fold. Right: Schematics of the glottal ow with open boundaries S− and S+ and mobile walls Sl and Sr. 3.2 Glottal ow We consider a potential incompressible ow of an inviscid uid of density ρ between two parallel mobile walls located at y = yl(t) and y = yr(t), respectively. The glottis G has width L, length 2` and height h = yl − yr, its mid-line being located at y