Nonlinear damping models for linear conservative mechanical systems with preserved eigenspaces: a port-Hamiltonian formulation

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(14) . Hélie, Thomas and Matignon, Denis Nonlinear damping models for linear conservative mechanical systems with preserved eigenspaces: a port-Hamiltonian formulation. (2015) In: 5th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, 4 July 2015 - 7 July 2015 (Lyon, France).. .

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(24) Nonlinear damping models for linear conservative mechanical systems with preserved eigenspaces: a port-Hamiltonian formulation Thomas H´ elie ∗ Denis Matignon ∗∗ ∗. Laboratory of Sciences and Technologies for Music and Sound (UMR9912), IRCAM-CNRS-UPMC, 1, pl. Igor Stravinsky, 75004 Paris, France (e-mail: thomas.helie@ircam.fr). ∗∗ University of Toulouse, ISAE-Supaéro. 10, av. Edouard Belin. BP 54032. 31055 Toulouse Cedex 4, France (e-mail: denis.matignon@isae.fr). Abstract: This paper introduces linear and nonlinear damping models, which preserve the eigenspaces of conservative linear mechanical problems. After some recalls on the finite dimensional case and on Caughey’s linear dampings, an extension to a nonlinear class is introduced. These results are recast in the port-Hamiltonian framework and generalized to infinite dimensional systems. They are applied to an Euler-Bernoulli beam, excited by a distributed force. Simulations yield sounds of xylophone, glockenspiel (etc) and some interpolations for nonlinear dampings. Keywords: energy storage, port-Hamiltonian systems, eigenfunctions, damping, nonlinear model, partial differential equations, sound synthesis 1. INTRODUCTION For conservative mechanical systems, the so-called Caughey series are known to define the class of damping matrices that preserve eigenspaces (see Caughey (1960); Caughey and O’Kelly (1965)). For finite-dimensional systems, these matrices prove to be a polynomial of one reduced matrix, which depends on the mass and stiffness matrices. In Matignon and Hélie (2013), this class of damping has been extended to a class of infinite dimensional and recast in the port-Hamiltonian framework. This paper proposes a new extension of this work to nonlinear damping models. An application on beams is developed to simulate sounds. Linear damping models can be related to wooden (xylophone) or metallic (glockenspiel) sounds (see Lambourg et al. (2001); Hélie and Matignon (2001); Aramaki et al. (2007)). The formalism proposed here allows nonlinear interpolations of such dampings that leave eigenmodes unchanged: the goal is to simulate a variety of sounds with fixed notes, for materials that can evolve e.g. from wood to metal according to the magnitude. Preserving eigenmodes and notes is is directly related to the geometrical structure of the port-Hamiltonian systems. The paper is organized as follows. Section 2 gives some recalls on finite dimensional mechanical linear systems and the damping class of Caughey type. Then, it introduces an extension to nonlinear damping models. In section 3, these results are recast in the framework of port-Hamiltonian systems. A special care is devoted to the formulation ? This work has been done within the context of the French National Research Agency sponsored project HAMECMOPSYS. Further information is available at http://www.hamecmopsys.ens2m.fr/.. of the damping models and several examples of typical target behaviours are proposed. Section 4 is devoted to a generalization of sections 2 and 3 to a class of mechanical infinite dimensional problems. Finally, section 5 presents an application to an Euler-Bernoulli beam with a nonlinear damping for sound synthesis purposes. 2. INTRODUCTION OF A CLASS OF NONLINEAR DAMPING MODELS 2.1 Recalls on linear mechanical systems. We consider a n-d.o.f. finite dimensional harmonic oscillator, excited by an external force. Following Géradin and Rixen (1996), its dynamic equation is usually written in the form: M q¨ + C q˙ + Kq = f , (1) with q(t) ∈ Rn (positions), f (t) ∈ Rn (forces) and where M = M T > 0 (mass matrix), K = K T ≥ 0 (stiffness matrix) and C = C T ≥ 0 (damping matrix). In this model, C has no skew-symmetric part, meaning that no gyroscopic effect is considered (see Matignon and Hélie (2013) for a discussion on this point). This system is passive since its total energy 1 1 (2) E := q˙T M q˙ + q T Kq ≥ 0 2 2 varies as E˙ = q˙T M q¨ + Kq = −q˙T C q˙ + q˙T f , that is, E˙ = −Pdis + Pext ≤ Pext , (3) T T where Pdis = q˙ C q˙ ≥ 0 and Pext = q˙ f ∈ R are the dissipated power and the power supplied by the external force, respectively..

(25) 2.2 Linear damping models of Caughey type. Caughey (1960) proposed a class of matrices C such that the eigenvectors of problem (1) are identical to those of the conservative case (C = 0). This class is defined by n−1 X C = b0 M + b1 K + bl K(M −1 K)l−1 with bl ≥ 0. l=2. This result can be rewritten, based on the polynomial n−1 X P(X) = bl X l with bl ≥ 0, (4) l=0. as follows 1. 1. C = M 2 C M 2, with C := P(K) and K := M. (5) − 12. KM. − 21. ,. 1. The proof that (4-6) provide a sufficient condition to preserve the eigenvectors is straightfoward, using an appro1 priate change of variables. Indeed, introducing q := M 2 q 1 and f := M − 2 f , (1) rewrites (7) ¨q + C q˙ + Kq = f , and E = 12 q˙ T q˙ + 12 q T Kq ≥ 0, Pdis = q˙ T C q˙ ≥ 0, Pext = q˙ T f ∈ R. Let ek be an eigenvector of K and denote ωk2 with ωk ≥ 0 its associated eigenvalue: ωk is a natural pulsation of the conservative system. Now, C ek = P(K) ek = λk ek Pn−1 2 2l with λk = l=0 bl ωk = P(ωk ) so that ek is also an eigenvector of the problem. Note that ek is associated with the double eigenvalue 0 if ωk = 0 and to a pair of eigenvalues with negative real part q s± = ω (−ξ ± ξk2 − 1), (8) k k k with ξk = P(ωk2 )/ωk2 ≥ 0, (9) otherwise. They correspond to damped oscillating dynamics if 0 < ξk < 1 and to damped non-oscillating ones if ξk ≥ 1. A necessary and sufficient condition is also available in Caughey and O’Kelly (1965). It proves to be [C, K] = 0 where [A, B] := AB − BA denotes the commutator. The previous sufficient condition is then recovered as a special case. 2.3 A new class of nonlinear damping models. We propose to generalize the previous class of damping from the linear to the nonlinear case by replacing (4) by a polynomial whose coefficients are non negative continuous functions of (q, q). ˙ Namely, the constant matrix C is replaced by ˙ K l, bl (q, q). (10). where bl ∈ C (X, R+ ) with X = Rn × Rn .. (11). ˙ := C(q, q). l=0 0. This new class still fulfills the looked-for properties:. dynamics but not the eigenvectors); • this nonlinear model introduces dissipation since (3) is satisfied with Pdis = q˙ T C(q, q) ˙ q˙ ≥ 0. Thus, this class of nonlinear damping models is well-posed in the sense that the energy fulfills (3). More precisely, this energy is bounded by the energy Elin of the passive linear system which is fed by the same excitation and for which functions bl have been replaced by the constant coefficients bmin = inf x∈X bl (x) ≥ 0. This energy bound is that of a l dissipative system if min0≤l≤n−1 bmin > 0 and that of a l conservative system if this minimum reaches zero. Typical target examples are introduced in the next section.. (6). where M 2 denotes the non negative square-root of M and 1 1 M − 2 its inverse. Note that M 2 > 0, K ≥ 0 and C ≥ 0 are symmetric matrices.. n−1 X. ˙ k= • the eigenspace structure is preserved since C(q, q)e Pn−1 2l ˙ ωk ek (the nonlinearity affects the l=0 bl (q, q). 3. PORT-HAMILTONIAN FORMULATION A detailed presentation of Port-Hamiltonian Systems (PHS) can be found in Duindam et al. (2009); van der Schaft and Jeltsema (2014). 3.1 Recalls. The standard differential formulation of finite dimensional Port-Hamiltonian systems with state x, input u and output y is x˙ = J − R ∂x H(x) + Gu, (12) y = GT ∂x H(x) + Du, (13) where the definite non negative function H is an energy storing function (E = H(x)), where matrices J = −J T , D = −DT (skew-symmetric), R = RT ≥ 0 (symmetric non negative) and G can depend on x. Such a system is passive since it fulfills the power balance (3) where the dissipated power is Pdis = xT R(x)x ≥ 0 and the external power is Pext = y T u. An algebro-differential formulation of port-Hamiltonian systems, consistent with the power balance, is (see e.g. Falaize et al. (2014))     x˙ ∂x H(x)  w  = S  z(w)  (14) −y u where matrix S = −S T (skew-symmetric) can depend on x and w and where vector w ∈ Rdw and function z : Rdw → Rdw isolate the pairs of co-power variables (effort, flow) of the dissipative components. Such a system still satisfies (3) with Pdis = wT z(x, w) so that the dissipativity is satisfied if z is such that wT z(w) ≥ 0. Note that the standard formulation can be recovered from (14) by eliminating w. This is possible if the second line of (14) admits a unique solution w = fw ∂x H(x), u . 3.2 Case of n-d.o.f. finite dimensional mechanical systems. The differential and algebro-differential PHS formulations of system (1) are given in table 1. Note that compared to (14), the standard formulation (12-13) yields more concise expressions. However, by outsourcing the damping forces, formulation (14) allows the recovery of a canonical global.

(26) Energy: H(x) := . state: x=. h. q := q p := M q˙. i. with. damping variable: w := q˙. S :=. i. 0 In Matrices: J := , h−Iin 0 0 G := , In Damping force:. " global matrix:. . K 0 . 0 M −1. W :=. h. input : u := f output : y := q˙. J −G G GT 0 0 −GT 0 0. #. Formulation of (w, z) and (w, z) isolating monomials of degree l. 1 T x W x, 2. h. i. 0 0 R := , 0 C. z(x, w) := diag b0 (x)In , b1 (x)K, . . .. damping var.:   q˙  . w :=  ..  n  q˙. z(x, w) := diag b0 (x)M, b1 (x)K, . . . . . . , bn−1 (x)K(M −1 K)n−2 w.. . . . . , bl (x)K l , . . . , bn−1 (x)K n−1 w.. D := 0.. z(w) := C w.. . damping var.: w :=   q˙   ..  n. .  q˙. 0 0 In  −In 0 −In = 0 In 0 0 −In 0. 0 In 0 0.  1. 1.  .. 0  −In " #  0 J −Gn G  0 0 =  . S := GT n  .. −GT 0 0  0 with Gn = [G, . . . , G]. 0 | {z }. matrix S that isolates and describes the exchanges of power between elementary components.. Note that the dependency of z with respect to x does not alter the power balance. Moreover, formulation (14) also allows the expansion of the polynomial structure (4) to isolate in z each contribution associated with coefficient bl . This specific formulation is described in table 2.. 0 In 0 .. . 0 0.       . n. Damping coefficient range and energy bounds. A possible target is to make a damping coefficient bl lie in a fixed range [γl− , γl+ ] (0 < γl− < γl+ < +∞). This can be achieved by using an interpolation function βl such that βl (ε) = γl− + (γl+ − γl− )f (ε/εl ), (increasing case)(18) βl (ε) = γl+ − (γl+ − γl− )f (ε/εl ), (decreasing case)(19) where function f behaves like, for instance, 4 1 2 f (0) = 0 and f (ε) = 1 / 2 x ( 1+x ) if ε > 0, (20) (see figure 1) and where εl accounts for an energy threshold. 1. 0.8. 0.6 f(ξ). l=0. In 0 ... 0 0 −In . . . −In In .. . 0 In −In 0 . . . 0. Table 2. Class of damping models that preserve the eigenvectors of the conservative system.. The formulations associated with (7) are obtained by replacing all quantities by their underlined versions and with M := In and y = w := q. ˙. In formulations (12-13) and (14), the linear and nonlinear damping classes of Caughey type are obtained using the ˙ defined in (4-6) particular expressions of C, C and C(q, q) and (10-11). In the nonlinear case, it makes z become a function of x and w. For system (7), definition (10-11) can be rewritten as n−1 X bl (x) K l w. z(x, w) = (15). . global matrix (S = S):. Table 1. Linear finite-dimensional mechanical system with linear damping.. 3.3 Linear and nonlinear damping models of Caughey type.. . change of function: bl (x) = bl x = diag(M 2 , M 2 )x. 0.4. 3.4 Examples of nonlinear damping. 0.2. Damping as a function of energies. A typical target behaviour can be to increase(/decrease) the damping according to the total energy. In this cases, function bl will take the form bl (x) = β(xT W x) where W = diag(K, M −1 ), (16) and where βl : R+ → R+ is a non negative increasing (/decreasing) function. This case is presented in the application to generate beam sounds with typical metallic or wooden behaviours according to the total energy. Other behaviours can be obtained by choosing diag(0, M −1 ) (kinetic energy only) or diag(K, 0) (potential energy only) in place of W . Another typical case is to make function bl only depend on the norm associated with its own damping force. In 1 1 this case, function bl (x) = bl x = diag(M 2 , M 2 )x will be such that bl x = [q T , pT ]T = βl (pT K l p). (17). 0 0. 0.5. 1. 1.5. 2. 2.5. 3. 3.5. 4. 4.5. 5. ξ. Fig. 1. Function f : f (ε) is close to 0 if ε 1, it equals 1/2 if ε = 1, it is greater than 1/2 close to 1 and its limit is 1 as ε → +∞. Note that in these cases, denoting x± the trajectories that would result from the linear system with constant Caughey coefficients bl = γl± , the trajectory x of the nonlinear system satisfies H x− (t) ≤ H x(t) ≤ H x+ (t) . (21) 4. INFINITE-DIMENSIONAL SYSTEMS We consider a generalization to a class of infinitedimensional mechanical systems in one space dimension..

(27) For sake of conciseness, the description is presented for the underline-type quantities (see (7)), that is, once the mass has been reduced to identity (M = I). 4.1 Linear mechanical model and settings. In this paper, the generalization of systems (7) is ∂t2 q + C∂t q + Kq = f (22) for all t ∈ T = R (or T = R+ with zero initial conditions q = ∂t q = 0 at t = 0), with the following technical assumptions (see e.g. Curtain and Zwart (1995)). (A1) the configuration space of positions q(t) is a Hilbert space H, (A2) operator K is an unbounded operator on the Hilbert space H := L2 (Ω) with domain D(K), which is supposed to be closed, densely defined, self-adjoint and positive on H. 1 (A3) operator K 2 is the (uniquely defined) square-root 1 1 1 of K with domain D(K 2 ), where H 2 := D(K 2 ) 1 1 equipped with the K 2 -norm (k · k 21 = kK 2 · kH ) H defines a Hilbert space. (A4) operator C is chosen as a polynomial of K: in the case of order 0, C is bounded on H; in the case of order 1, C is unbounded on H with domain D(C) = D(K); and for higher degrees D(C) = D(K p ), where p is the degree of the polynomial. In order to avoid technical difficulties, the initial data of our dynamical system will then be taken in D(K p ), that is, as regular as needed. (A5) the distributed force is a bounded trajectory on H, namely, t 7→ f (t) belongs to L∞ (T, H). 4.2 Port-Hamiltonian formulation and well-posedness. For (22), the PHS formulation (12-13) becomes ∂t x = J − R δx H(x) + Gu, y = G∗ δx H(x),. (23) (24). with state x := [q := q, p := ∂t q]T , input u := f output y := ∂t q and where • the storing-energy function is related to the Hilbert space X as follows 1 (25) H(x) := kxk2X , 2 1 1 2 X := H 2 × H with kxkX = kqk2 1 + kpk2H (26) . H2. • operator δx denotes the variational derivative, • operators Jand R defined by 0 I 0 0 1 and R := (27) J := 0C −K 2 0 are formally skew-symmetric and non negative symmetric, respectively, w. r. t. the scalar product on X, • operators G and G∗ are simply defined by 0 G := and G∗ := [0 I] . (28) I The PHS formulation (14) becomes     " # ∂t x δx H(x) J −G G ∗  w  = S  z(w)  with S := G 0 0 , (29) −G∗ 0 0 −y u. and where w := x2 (= p = ∂t q) and z(w) = Cw. Damping models of Caughey-type. For short, it is a good idea to write C := f (K), where function f is well defined in the cone of symmetric positive operators, which readily amounts to diagonalize the transformation in a Hilbert basis, and apply ci := f (ki ) on each coordinate, provided that ki ≥ 0. Now, a condition for damping is that f (R+ ) ⊂ R+ , so as to ensure C := f (K) ≥ 0, hence C ≥ 0. The useful sub-class considered here is that of polynomials, defined explicitly by C := P (K), such as Rayleigh damping when deg(P ) = 1. For the nonlinear damping models, functions bl become functionals bl : X → R+ that can be built as in § 3.4, based on functions βl and (weighted) norms of x. 5. APPLICATION TO THE EULER-BERNOULLI BEAM. This section addresses the model and the simulation of a pinned damped beam excited by a distributed force. The damping is composed of the two first terms of (4), which allow the sound synthesis of metallic and wooden beams (see Hélie and Matignon (2001)). First, the linear problem is described (§ 5.1) and its wellposedness is analyzed (§ 5.2). Second, a finite dimensional approximation is provided, based on a modal projection (§ 5.3), from which simulations can be derived . Third, nonlinear damping models are introduced (§ 5.4) and numerical results are illustrated and examined (§ 5.5). 5.1 Linear model. We consider a pinned beam model excited by a distributed force, based on: (H1) the assumption of Euler-Bernoulli kinematics (Géradin and Rixen (1996); Graff (1991)): any cross-section before deformation remains straight after deformation; (H2) the linear approximation for the conservative problem. (H3) some viscous and structural dampings based on the first degree polynomial (4), invloving linear (H3a) or nonlinear (H3b) phenomena. A dimensionless version of the model governing the deflection waves for (H1-H3a) is given by, for all z ∈ Ω = [0, 1] and t ∈ T = R+ , ∂t2 q + (b0 + b1 ∂z4 ) ∂t q + ∂z4 q = f, (30) with zero initial conditions q(z, 0) = 0 and ∂t q(z, 0) = 0, (31) where b0 > 0 and b1 > 0 are fluid and structural damping parameters. The excitation is a distributed force f which belongs to U = L∞ (T, U) with U = L2 (0, 1). The boundary conditions are q(z, t) = 0 (fixed extremities) and 2 ∂z q(z, t) = 0 (no momentum) at z ∈ {0; 1}.. (32). 5.2 Well-posedness and PHS fomulation. The beam model can be rewritten as (22) where K is the unbounded operator on H = L2 (0, 1) with domain.

(28) D(K) = q ∈ H 4 (0, 1) s.t. q(0) = q(1) = 0,. q 00 (0) = q 00 (1) = 0 ,. First, the time-step T for the simulation is chosen such that the first mode would correspond to a frequency f1 = 220 Hz with no damping, when the signal is heard using such that K q = q (4) , for all q ∈ D(K), and where a sound card with sampling frequency fs = 48 kHz. This C := b0 I+b1 K. Equation (12) can be rewritten ∂t x = Ax+ time-step is given by T = κ fs , where κ = 2πf1 /=mλ1 is Gu where A : x 7→ [J − R]δx H(x) is derived by using (27), the adimensionalization parameter and where (λ1 , λ1 ) are 1 the eigenvalues associated with the first mode (see (34)). δH/δq = K 2 q, and δH/δp = p. This operator The number of modes is N = 9 so that, with no damping, A : D(A) −→ X (33) the higher frequency N 2 f1 = 17820 kHz is just lower q 0 I q than the maximal perceptible audio frequency (≈ 20 kHz). 7−→ p −K −(b0 I + b1 K) p This frequency is also lower than the Shannon frequency 1 1 4 4 with D(A) = {(q, p) ∈ H 2 × H 2 , (b0 + b1 ∂z )p + ∂z q ∈ H}, fs /2 = 24 kHz in order to avoid aliasing effects. the minimal damping coefficients are chosen as generates a C0 contraction semigroup S on X (Jacob et al., Second, γ0− = 0.02 and γ1− = 10−6 . They are typical of a “metallic 2008, (A1-A2), p. 6). beam”. The maximal damping coefficients are chosen as Note also that operator G : u 7→ [0, u]T , which belongs to γ − = 0.04 and γ − = 10−4 . They are typical of a “wooden 0 1 L(U, X) with U = H, is such that kGk = 1. beam”. The nonlinearity is built such that the dampings increase (case 1, see (18)) or decrease (case 2, see (19)), 5.3 Finite dimensional approximation. with respect to the total energy (thresholds in (20) are ε0 = ε1 = 1). Following (Jacob et al., 2008, corollary 5.2), A is a Riesz spectral operator on X which generates an analytic semi- Third, the excitation is chosen such that f1 = · · · = fN := group S, provided that −2/b0 is not in the point spectrum F (approximation of the Dirac space distribution near the (k = 0, . . . , 4) of A. Moreover, the spectrum of A lies in the closed left boundary z = 0) where F (t) is a sequence−4 of pulses with constant force during t = 10 s every halfαt e half-plane and S is such that, for all t ∈ T, kS(t)kX ≤ e the magnitude Ae of impacts with α = sup <e(Spec(A)). Its point spectrum is composed second. In this sequence, increases as Ak = k A∗ with A∗ = 0.1κ/te in case 1 and 4 2 4 of the roots λ± m of Pn (λ) = λ + (b0 + b1 kn )λ + kn for all A∗ = 0.25κ/te in case 2. n ≥ 1, where kn = nπ. The associated eigenfunctions are √ The results are presented in figures 2-4 for the case 1: the 1/λ± n e e± bn (z) with ebn (z) = 2 sin(kn z), (34) beam behaves as a metallic beam when the total energy n (z) = 1 is low and as wooden beam when it is high (compared to where {b en }n∈N∗ is a Hilbert basis of H. the unit). As expected (see fig. 3), the energy stored the We consider the finite dimensional linear model which is higher modes decreases much faster than the lower modes composed of the collection of modes en for 1 ≤ n ≤ N . It (wooden beam) for a high total energy (last impacts). is described by (12-14) and tables 1-2 with Energy. 6. K := K = π 4 diag(1, . . . , N )4 , (35) C := b0 In + b1 K, z(w) := diag b0 I, b1 K w. (36). M := M = In ,. 0. 0.5. 1. 1.5. 2. 0. 0.5. 1. 1.5. 2. 3. 3.5. 4. 4.5. 5. 2.5. 3. 3.5. 4. 4.5. 5. 3. 3.5. 4. 4.5. 5. y. 0. -5 t (in s). 5 x 10 1. 0. -1 0. Nonlinear case. For the nonlinear case (see § 2.3), coefficients bl are replaced by (H3b) functions of the total energy, according to equations (16), and (18-20). Coefficients γl± are chosen so that (H4-H5) are satisfied.. 2.5. 5. dy/dt. Linear case and hypotheses. Additionally, it is assumed that b0 > 0 and b1 > 0 satisfy b0 (H4): 2 + b1 π 2 < 2, and (H5): b1 (b0 − b1 π 4 ) ≤ 2, π so that the first mode (n = 1) has a damped oscillating dynamics (H4) and that higher modes are more damped (H5).. 2 0. and where q = [q1 , . . . , qN ]T , p = w = y = [q˙1 , . . . , q˙N ]T and u = [f1 , . . . , fN ]T denote the modal projections of the corresponding quantities q, p = w = y and f . 5.4 Damping.. 4. 0.5. 1. 1.5. 2. 2.5 t (in s). Fig. 2. (Case 1) Top: total energy. Middle: output signal y (velocity, co-power variable of the input). Bottom: signal dy/dt (proportionnal to a “sound pressure”).. 5.5 Numerical results. We choose parameters that are appropriate to the sound synthesis of e.g. vibraphones, xylophones, marimbas, etc.. For the case 2 (see fig. 5), the beam behaves as a wooden beam when the total energy is low and as metallic beam when it is high (compared to 1)..

(29) From the practical point of view, the introduced damping models allow to handle a variety of behaviours with a very few parameters and to make them evolve according to the dynamics. As an example, the application shows how these results can impact sound synthesis: it provides original morphed sounds based on physical grounds that are controlled by the dynamics (here, the total energy), capable to evolve between metallic and wooden beams. These very first results fully deserve a more theoretical setting, as the original one proposed in this paper.. Magnitude (dB scale) 18000 16000 14000. f (in Hz). 12000 10000 8000 6000 4000 2000. Some perspectives are still left open:. 0 0.5. 1. 1.5. 2. 2.5. 3. 3.5. 4. 4.5. t (in s). Fig. 3. (Case 1) Spectrogram of dy/dt. 1 Energies. Energies. 0.4 0.2. 0.5. 0. 0 1.246 1.248 1.25 1.252 1.254. 1.746 1.748 1.75 1.752 1.754 t. 2. 2. 0. 0. y. y. t. -2. -2 1.246 1.248 1.25 1.252 1.254. 1.746 1.748 1.75 1.752 1.754. t (in s). t (in s). 5000 dy/dt. dy/dt. 5000 0. 0. -5000. -5000 1.246 1.248 1.25 1.252 1.254. 1.746 1.748 1.75 1.752 1.754. t (in s). t (in s). Fig. 4. (Case 1) Zoom on signals of figure 2 with additionnal signals in the top figure: total (black), kinetic (red) and potential (blue) energies. 20. Energy. 15 10 5 0 0. 0.5. 1. 1.5. 2. 2.5. 3. 3.5. 4. Magnitude (dB scale) 18000 16000 14000. f (in Hz). 12000 10000 8000 6000 4000 2000 0 0.5. 1. 1.5. 2. 2.5. 3. 3.5. t (in s). Fig. 5. (Case 2) total energy and spectrogram of dy/dt. 6. CONCLUSION AND PERSPECTIVES. The main contribution of this paper is to propose a class parametrized nonlinear damping models for linear conservative mechanical problems that preserve the original eigenstructure, including for some infinite dimensional systems. The port-Hamiltonian formulation and passivity analysis help the examination of well-posedness conditions and provide energy bounds for the trajectories (see (21).. (1) apply this methodology to other 1D-models of musical interest (waves, strings) and analyze the nonlinear damping effects produced on the synthesized sounds; (2) try to extend the proposed framework to useful 2Dcases, as far as realistic musical synthesis is concerned; (3) in the infinite-dimensional case, where non-standard boundary conditions are to be found, extend the functional framework to a Riesz-spectral operator K 1 with no eigenvalues on R− , so that K 2 can still be uniquely defined; (4) for the domain of the general C operator: either C is a polynomial in K and its domain is well defined, or C is a rational function of K and the characterization of its domain is not so easy (even in the classical case when K is positive and self-adjoint); thus, an adequate functional calculus must be used, in order for these equations to make sense; (5) enlarge the class of application to the modelling of damping in structural dynamics, with possibly interesting applications in aeronautics. REFERENCES Aramaki, M., Baill´ eres, H., Brancheriau, L., Kronland-Martinet, R., and Ystad, S. (2007). Sound quality assessment of wood for xylophone bars. J. Acoust. Soc. Amer., 121(4), 2407–2420. Caughey, T.K. (1960). Classical normal modes in damped linear dynamic systems. ASME, J. Applied Mechanics, 27, 269–271. Caughey, T.K. and O’Kelly, M.E.J. (1965). Classical normal modes in damped linear dynamic systems. ASME, J. Applied Mechanics, 583–588. Curtain, R.F. and Zwart, H.J. (1995). An introduction to Infinitedimensional Linear Systems Theory. Springer. Duindam, V., Macchelli, A., Stramigioli, S., and Bruyninckx, H. (eds.) (2009). Modeling and Control of Complex Physical Systems: The Port-Hamiltonian Approach. Springer. Falaize, A., Lopes, N., H´ elie, T., Matignon, D., and Maschke, B. (2014). Energy-balanced models for acoustic and audio systems: a port-hamiltonian approach. In Unfold Mechanics for Sounds and Music, 1–8. Paris, France. G´ eradin, M. and Rixen, D. (1996). Th´ eorie des vibrations. Masson. Graff, K.F. (1991). Wave motion in elastic solids. Dover, New York. H´ elie, T. and Matignon, D. (2001). Damping models for the sound synthesis of bar-like instruments. In the 5th Conf. on Systemics, Cybernetics and Informatics, (10) 541–546. Orlando, USA. Jacob, B., Trunk, C., and Winklmeier, M. (2008). Analyticity and Riesz basis property of semigroups associated to damped vibrations. J. Evol. Eq., 8(2), 263–281. Lambourg, C., Chaigne, A., and Matignon, D. (2001). Time-domain simulation of damped impacted plates. part ii. numerical model and results. J. Acoust. Soc. Amer., 109, 1433–1447. Matignon, D. and H´ elie, T. (2013). A class of damping models preserving eigenspaces for linear conservative port-Hamiltonian systems. European Journal of Control, 19-6, 486–494. van der Schaft, A. and Jeltsema, D. (2014). Port-Hamiltonian Systems Theory: An Introductory Overview. Now Publishers Inc..

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