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Self-oscillations of a vocal apparatus: a
port-Hamiltonian formulation
Thomas Hélie, Fabrice Silva
To cite this version:
Thomas Hélie, Fabrice Silva. Self-oscillations of a vocal apparatus: a port-Hamiltonian formulation.
Frank Nielsen and Frédéric Barbaresco. Geometric Science of Information: Third International
Con-ference, GSI 2017, Paris, France, November 7-9, 2017, Proceedings, Springer International Publishing,
pp.375–383, 2017, 3rd conference on Geometric Science of Information (GSI), 978-3-319-68445-1.
�hal-01567464�
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 ful l 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 con guration laryngeal con gurations 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 emphaanaly-sis 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.2is a reminder on the port-Hamiltonian systems, Sec.3is 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 con gurations.
2 Port-Hamiltonian Systems
Port-Hamiltonian systems are open passive systems that ful l 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 ˙x w −y = S(x, w) ∇xH z(w) u , with S = −S T = Jx −K Gx KT Jw Gw −GT x −G T w Jy, (1) where state x(t) ∈ X = RN is associated with energy E = H(x) ≥ 0 and where variables w(t) ∈ W = RQare associated with dissipative constitutive laws z such that Pdis= z(w)Tw ≥ 0stands for a dissipated power. Such a system naturally ful ls the power balance dE/dt + Pdis− Pext = 0, where the external power is Pext = yTu. 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 ATB = ATSA = 0, that is, ∇xH(x)T˙x + z(w)Tw − uTy = 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
Bene ting 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 resonatorAcoustic Psub Qsub Prsub Qsubr Psub l Qsubl Ptot− Q− F p l v l F p r vr P+ Q+ Plsup Qsupl Prsup 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 πiof the mass, and the elongations ξiand ζi of the body and cover springs, respectively, to the velocity vi= ˙ζi+ ˙ξiof the cover imposed by the glottal ow, and to the transverse resultants of the pressure forces on the upstream (Psub
i ) and downstream (P sup
i ) faces of the trapezoid-shaped structures (see Fig. 2, left part) :
˙πi= −kiξi− riξi˙ + κiζi− Psub i S sub i − P sup i S sup i . (3)
Fip = −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 in ates).
Port-Hamiltonian modelling of a vocal fold Fi : xFi= πi ξi ζi , uFi= Psub i Pisup vi , yFi= −Qsub i Qsupi −Fip , HFi = 1 2x T Fi 1/mi ki κi xFi, wFi = ˙ξi, zFi(wFi) = riwFi, JFi w = 0, GFwi = O1×3, JFyi = O3×3, JFi x = 0 −1 1 1 0 0 −1 0 0 , K Fi = 1 0 0 , and G Fi x = −Ssub i −S sup i 0 0 0 0 0 0 1 . ki ri mi κi Pisub P sup i Fip Ssubi S sup i Glottal ow Sr Sl | −` | 0 | ` x yr(t) yl(t) y L h Ω(t) S− S+ Flp Frp Ptot− 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 S
land 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 = ym= (yr+ yl)/2(see Fig.2, right part). The simplest kinematics for the uid velocity v(x, y) obeying the Euler equation
˙v + 1 ρ∇ p + 1 2ρ|v| 2 = 0 (4)
and satisfying the normal velocity continuity on the walls is given by: v =vx vy = v0− x ˙ h h ˙ ym+hh˙ (y − ym) ! ∀(x, y) ∈ Ω = [−`, `] × [yr, yl]. (5) The velocity eld is thus parametrised by four macroscopic quantities: h, its time derivative ˙h, and the mean axial and transverse velocities v0 =< vx >Ω and ˙ym =< vy >Ω, respectively. Choosing these quantities as the state allows the exact reduction of the in nite-dimensional problem to a nite-dimension system. The pressure eld p(x, y, t) can also be obtained from Eq. (4), as well as the total pressure p +1
2ρ|v|
2, but are not expanded here for brevity.
The dynamics for the glottal ow is controlled by the mean total pressures Ptot− and Ptot+ on the open boundaries S−(x = −`) and S+(x = +`), respectively, and the resultant Fp
r and F p
l of the pressure forces on the right and left walls, respectively (see App.Afor the derivation of the equations). The kinetic energy of the uid on the domain writes as
ε(t) = HG(xG(t)) = 1 2
m(h)v20+ m(h) ˙y2m+ m3(h) ˙h2 (6) with the total mass of the uid m(h) = 2ρ`Lh(t) , and the eective mass for the transverse expansion motion m3(h) = m(h) 1 + 4`2/h2 /12. The energy could be written as a function of the momenta to yield a canonical Hamiltonian representation (see Ref. [7] for a similar PHS based on normalised momenta).
Downstream the glottis, the ow enters the supraglottal space which has a cross section area much larger than that of the glottis. For positive owrate (Q+ = Lhvx(`) > 0), the ow separates from the walls at the end point of the (straight) channel. The downstream jet then spreads due to the shear-layer vortices until the jet has lost most of its kinetic energy into heat and fully mixed with the quiescent uid. This phenomenon is modelled as a dissipa-tive component with variable wG = Q+ and dissipation function zG(w
G) = (1/2)ρ(wG/Lh)2Θ(wG)where Θ is the Heaviside step function. The pressure in the supraglottal space then writes P+= P+
tot− zG. Port-Hamiltonian modelling of the glottal ow G :
xG = v0 ˙ ym ˙h h , uG = Ptot− P+ Flp Frp , yG = −Q−= −Lhvx(−`) +Q+= Lhvx(`) −vl= + ˙yl −vr= − ˙yr , HG(xG) = m(h) 2 v 2 0+ ˙y 2 m + m3(h) 2 ˙h 2, wG = Q+, zG = ρ 2 wG Lh 2 Θ(wG), JGw= O1×1, GGw= O1×4 and JGy = O4×4,
JGx = 0 0 0 0 0 0 0 0 0 0 0 −m1 3 0 0 m1 3 0 , KG= Lh m 0 −L` m3 0 , and GGx = Lh m − Lh m 0 0 0 0 −1 m 1 m L` m3 L` m3 − 1 2m3 − 1 2m3 0 0 0 0 . 3.3 Vocal tract
We assume a modal representation of the input impedance of the vocal tract as seen from the supraglottal cavity, i.e., the supraglottal pressure Pac is de ned as the sum of pressure components pn (for n = 1, N, denoted Pn in the Fourier domain) related to the input owrate Qac through 2nd order transfer functions:
Zin(ω) = P ac(ω) Qac(ω) = N X n=1 Pn(ω) Qac(ω) = N X n=1 jωan ω2 n+ jqnωnω − ω2 (7) where ω is the angular frequency, ωn are the modal angular frequencies, qn are the modal dampings and anthe modal coe cients. Each mode corresponds to a resonance of the vocal tract, and so to an expected formant in the spectrum of the radiated sound. We follow the convention de ned in Ref. [8] for the internal variables of this subsystem.
Port-Hamiltonian modelling of the acoustic resonator A : xA= p1/a1, . . . , pN/aN,Rt 0p1(t 0)dt0, . . . ,Rt 0pN(t 0)dt0T, HA(xA) = N X n=1 1 2 p2n an +ω 2 n an Z t 0 pn(t0)dt0 2! , wA= (p1, . . . , pN) T , zA= q1ω1 a1 wA1, . . . , qNωN aN wA,N , uA= (Qac) , yA= (−Pac) , JAx = ON ×N −IN ×N IN ×N ON ×N , KA= IN ×N ON ×N , GAx = 1N ON ×1 , GAw= ON ×1, JAw= ON ×N, and JAy = O1×1.
where IN ×N is the identity matrix of dim N × N, and 1N is the column vector N × 1 lled with 1.
3.4 Full system
We assume that the lower airways acts as a source able to impose the pressure Psub in the subglottal space of the larynx. The owrate Qsubcoming from this source splits into the owrate Q− entering the glottis and the owrate Qsub
l and Qsub
r pumped by the lower conus elasticus of the left and right vocal folds, respectively, so that Qsub= Q−+ Qsub
l + Qsubr with Psub= Prsub= Plsub= P − tot.
Conversely, the owrate Q+ sums up with the owrates Qsup l and Q
sup
r pulsated by the left and right vocal folds, respectively. The resulting owrate Qac that enters the acoustic resonator is then Qac= Q++Qsup
l +Q sup
r with Pac= Prsup= Plsup= P+.
The elementary components described above are now put together to assem-bly the full vocal apparatus. In order to simplify the Dirac structure, the ports of the subsystems have been chosen to be complementary: a port with sink con-vention is always connected to a port with source concon-vention. As a result, it is trivial to expand the port Hamiltonian modelling of the full system with the following variables, dissipation functions, ports and energy:
x = xR xL xG xA , w = wR wL wG wA , z = zR zL zG zA , u = Psub , y = −Qsub , H(x) = HR(xR) + HL(xL) + HG(xG) + HA(xA).
The matrices Jx, K, Gx, Gw, Jw and Jy can be obtained using automated generation tools like the PyPHS software [9].
4 Simulations and results
We here brie y present some preliminary results. In the port-Hamiltonian mod-elling of the full system, the dissipation variables w do not explicitly depend on z (i.e., Jw = O), so that they can be eliminated leading to a dierential realisation that can be numerical integrated (e.g., using the Runge-Kutta 4 scheme). The parameters have the following values: mi = 0.2 g, ri = 0.05 kg/s, L = 11 mm, ` = 2 mm, ρ = 1.3 kg/m3. Due to the sparse data available on the input impedances of vocal tract notably in terms of modal amplitudes an, we consider a resonator with a single pole (N = 1) with ωn= 2π×640 rad/s, qn= .4 and an = 1 MΩ (from Ref. [10]). The system is driven by a subglottal pressure Psub that increases from 0 to 800 Pa within 20 ms and is then maintained.
In the rst simulation, the folds are symmetric (kr = kl = 100 N/m, κr = κl = 3kr) and initially separated by a width h = 1 mm. In such conditions, the folds are pushed away from their rest position (until h ∼ 3 mm), but this equilibrium does not become unstable and the system does not vibrate.
If some adduction is performed bringing the folds closer together (h = 0.1 mm), the glottis rst widens (until h ∼ 2 mm) and the folds then start to vibrate and the acoustic pressure oscillates in the vocal tract (see Fig. 3, top). The sound is stable even if the two folds are slightly mistuned (kr = 100 N/m and kl= 97 N/m).
The right fold is then hardened (kr = 150 N/m). The system still succeeds to vibrate, but, as visible on Fig. 3 (bottom), the oscillation is supported by the soft left fold at rst, and then this latter decays while the hardened right fold starts to vibrate and nally maintains the sound production (even if the oscillations seem intermittent).
Fig. 3. Adducted (top) and asymmetric (bottom) con gurations.
5 Conclusion
To the best knowledge of the authors, this paper proposes the rst port-Hamiltonian model of a full vocal apparatus. This ensures passivity and the power balance. Simulations provide a variety of regimes that can be qualitatively related to aphonia (stable equilibrium), phonation (nearly periodic regimes) and dyspho-nia (irregular oscillations). This preliminary work provides a proof-of-concept for the relevance/interest of the passive and geometric approach.
Further work will be devoted to: (1) analyse regimes and bifurcations of the current model with respect to a few biomechanic parameters, (2) improve the realism of elementary components (separately), (3) account for possible contact between the vocal-folds, and (4) investigate on the synchronisation of coupled asymmetric vocals-folds and explore strategies to treat pathological voices [11]. Acknowledgement. The rst author acknowledges the support of the Collab-orative Research DFG and ANR projectINFIDHEM ANR-16-CE92-0028.
References
1. van den Berg, J.: On the Air Resistance and the Bernoulli Eect of the Human Larynx. J. Acous. Soc. Am. 29(5), 626631 (1957).
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3. Encina, M., et al: Vocal fold modeling through the port-Hamiltonian systems ap-proach. IEEE Conf. on Control App. (CCA), 1558-1563 (2015).
4. Maschke, B., et al: An intrinsic Hamiltonian formulation of network dynamics: non-standard Poisson structures and gyrators, J. Frankl. Inst. 329(5), 923966 (1992). 5. Flanagan, J., Landgraf, L.: Self-Oscillating Source for Vocal-Tract Synthesizers.
IEEE Trans. Audio Electroacous. AU-16(1), 57-64 (1968)
6. Awrejcewicz, J.: Numerical Analysis of the Oscillations of Human Vocal Cords. Nonlin. Dyn. 2, 35-52 (1991)
7. Lopes, N. , Hélie, T.: Energy balanced model of a jet interacting with a brass player's lip. Acta Acust united Ac. 102(1), 141154 (2016).
8. Lopes, N.: Approche passive pour la modélisation, la simulation et l'étude d'un banc de test robotisé pour les instruments de type cuivre. PhD thesis, UPMC, Paris, 2016. 9. Falaize, A.: PyPHS: Passive modeling and simulation in python. Software available
athttps://afalaize.github.io/pyphs/(last viewed on April 21st, 2017). 10. Badin, P., Fant, G.: Notes on vocal tract computation. STL-QPSR 25(2-3), 53108
(1984).
11. Giovanni, A., et al: Nonlinear behavior of vocal fold vibration:The role of coupling between the vocal folds . J. Voice 13(4), 465-476 (1999)
A Dynamics of the glottal ow
The dynamics for the mean velocities can also be derived from the volume inte-gration of the Euler equation (4). Using the gradient theorem, it comes that
m(h) ˙v0= Lh(t) Ptot− − P + tot and m(h)¨ym= Frp− F p l. (8) The energy balance for the glottal ow writes down as:
˙ ε(t) + Z S−∪S+ p + 1 2ρ|v| 2 (v · n) + Z Sl∪Sr p (v · n) = 0 (9) where n is the outgoing normal. As the normal velocity is uniform on the walls, the last term of the energy balance reduces to
Z Sl∪Sr p (v · n) = − ˙yr Z Sr p + ˙yl Z Sl p = ˙ym(Flp− Fp r) + ˙h 2(F p r + F p l) . (10) The same applies on S−∪ S+ where v · n = ±vx(x = ±`)does not depend on y:
Z S−∪S+ ptot(v · n) = vx(`) Z S+ ptot− vx(−`) Z S− ptot = Lv0 Ptot+ − P − tot − L` ˙h h P + tot+ P − tot . (11) Thus, ˙ε = ˙ym(Frp− Flp)−˙h 2(F p r + F p l)+Lh(t)v0 Ptot− − P + tot+L` ˙h P − tot+ P + tot . In the meanwhile, the kinetic energy in Eq. (6) can be derived against time:
˙
ε = m(h) (v0˙v0+ ˙ymym) + m3(h) ˙h¨¨ h + ∂H
∂h ˙h. The identi cation of the contri-bution of the mean axial and transverse velocities (see Eq. (8)) leads to the dynamics of the glottal channel expansion rate :
m3¨h = L` Ptot− + P + tot − Fp r + F p l 2 − ∂H ∂h. (12)
