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Dispersion analysis of improved time discretization for
simply supported prestressed Timoshenko systems.
Application to the stiff piano string.
Juliette Chabassier, Sébastien Imperiale
To cite this version:
Dispersion analysis of improved time discretization for simply supported prestressed Timoshenko systems. Application to the stiff piano string.
J. Chabassier1,∗, S. Imperiale2
1 Magique3d, Inria Sud Ouest, Talence, France.
2 Department of Applied Physics and Applied Mathematics, Columbia University, New York, USA. ∗Email: juliette.chabassier@inria.fr
Abstract
We study the implicit time discretization of Tim-oshenko prestressed beams. This model features two types of waves: flexural and shear waves, that propa-gate with very different velocities. We present a novel implicit time discretization adapted to the physical phenomena occuring at the continuous level. Af-ter analyzing the continuous system and the two branches of eigenfrequencies associated with the standing modes, the classical θ-scheme is studied. A dispersion analysis recalls that θ = 1/12 re-duces the numerical dispersion, but yields a severely constrained stability condition for our application. Therefore we propose a new θ-like scheme based on two parameters adapted to each wave velocity, which reduces the numerical dispersion while relaxing this stability condition. Numerical experiments success-fully illustrate the theoretical results on the specific cas of a realistic piano string. This motivates the extension of the proposed approach for more chal-lenging physics.
Introduction
Piano strings can be modeled as simply supported Timoshenko prestressed beams. This model in-troduced in [2] accounts for inharmonicity of the transversal displacement, via a coupling with a shear angle resulting in the propagation of flexural and shear waves with very different speeds. Our concern in this work is to develop a new implicit time dis-cretization, which will be associated with finite ele-ment methods in space, in order to reduce the nu-merical dispersion of flexural waves while allowing the use of a large time step in spite of the high shear velocity (compared to the maximal time step allowed with the explicit leap-frog scheme).
1 Continuous system
The prestressed Timoshenko model considers two unknowns (u, ϕ) which stand respectively for the transversal displacement and the shear angle of the cross section of the the string. We assume that
the physical parameters (see [1] for definition) are positive and that ES > T0 (which is true in
prac-tice for piano strings). We consider “simply sup-ported” boundary conditions (zero displacement and zero torque). It reads:
Find (u, ϕ) such that ∀ x ∈]0, L[, ∀ t > 0, ρS∂ 2u ∂t2 − T0 ∂2u ∂x2 + SGκ ∂ ∂x ϕ − ∂u ∂x = σ, ρI∂ 2ϕ ∂t2 − EI ∂2ϕ ∂x2 + SGκ ϕ − ∂u ∂x = 0, (1)
with boundary conditions
∂xu(x = 0, t) = 0, ∂xu(x = L, t) = 0,
∂xϕ(x = 0, t) = 0, ∂xϕ(x = L, t) = 0, (2)
where σ stands for a source term. Standard energy techniques for systems of wave equations can be used to show a priori estimates on this system thanks to the following energy identity:
dE dt = Z L 0 ρS σ · ∂tu, with (3) E(t) = 1 2 Z L 0 ρ S |∂tu|2+ 1 2 Z L 0 ρ I |∂tϕ|2+ 1 2 Z L 0 T0 |∂xu|2 +1 2 Z L 0 EI |∂xϕ|2+ 1 2 Z L 0 SGκ |ϕ − ∂xu|2. (4)
If we seek a solution of the form t(u, ϕ)(x, t) =
V (x)e−2iπf t, then there exists (see [3]) ` such that f = f`±, where f`−= `f0− 1 + `2 + O(`5), (5a) where f0−= 1 2L s T0 ρS, = π2 2L2 EI T0 h 1 − T0 ES i . f`+= f0+ 1 + η `2 + O(`4), (5b) where f0+= 1 2π s SGκ ρI , η = π2 2L2 EI + IGκ SGκ . 2 Discretisation
get the following equation: d2
dt2MhUh+ KhUh = MhΣh (6)
where Mh is symmetric positive definite and Kh is
positive semi-definite.
2.1 Time discretisation with a classical θ-scheme Using a classical θ-scheme leads to :
Mh Uhn+1− 2Un h + U n−1 h ∆t2
+ Kh θUhn+1+ (1 − 2θ)Uhn+ θUhn−1 = MhΣh (7)
Stability of this numerical scheme and a priori esti-mates can be shown with energy techniques. First, any numerical solution is shown to satisfy an energy identity. If θ ≥ 1/4, this discrete energy is always positive, while if θ < 1/4, the time step ∆t must be lower than a maximal value ∆tθ. Then, the scheme is shown to be stable if the energy is positive. Orignial proofs of stability are proposed in [1].
We also remind that if we seek a solution of the form Un
h = Vh0e2iπ fhn∆t, then there exists ` such that
fh = fh,`, where fh,`= f`+ f`3 2 1 12 − θ ∆t2+ O(∆t4+ h4) (8) where f` = f`± is one of the eigenfrequencies of the
continuous problem given in (5). Choosing θ = 1/12 reaches fourth order of accuracy. This value being lower than 1/4, it leads to a conditionally stable scheme. This condition will be very severe, because of the large velocity of the shear waves of the system of a realistic piano string.
Our goal is to construct a numerical scheme with a small numerical dispersion on the flexural wave, without undergoing the time step restriction coming from the shear waves.
2.2 Time discretisation with a new θ-scheme. The idea is to separate the matrix Kh into the
sum of two matrices Kh and Kh, respectively
in-ducing the energy terms T0|∂xu|2 and EI |∂xϕ|2 +
SGκ |ϕ − ∂xu|2. We consider the following scheme,
with (θ, θ) ∈ [0, 1/2]2: Mh Un+1h − 2Un h + U n−1 h ∆t2 + Kh{Uh}nθ + Kh{Uh}nθ = MhΣnh (9)
where the θ-approximation of Uh(tn) is the weighted
average on three time steps:
{Uh}nθ = θ Un+1h + (1 − 2θ) Unh+ θ Un−1h , (10) Stability of this scheme and a priori estimates can be shown via energy techniques. Sufficient conditions of stability can be given according to the values of (θ, θ) (see [1]).
We show that if we seek a solution of the form Uhn= Vh0e2iπ fhn∆t, then there exists ` such that f
h= fh,`, where ( f− h,` = ` f − 0 1 + ∆t`2 + O(`5+ ∆t4+ h4), (11a) fh,`+ = f0,∆t+ 1 + η∆t`2 + O(`3+ ∆t4+ h4), (11b)
with (f0−, f0+, and η were defined in (5)) ∆t = + 2π2∆t2 1 12− θ f0−2 , (12a) f0,∆t+ = f0+ 1 + (2πf0+)2 1 12 − θ ∆t2 , (12b) η∆t= η + π2(E + Gκ) 2ρL2 θ − 1 12 ∆t2. (12c) We note that the value θ = 1/12 provides fourth order accuracy for the approximation fh,`− of the flex-ural eigenfrequencies given by (5a), for small `.
The main interest of this scheme is to choose, for the slow wave, a value of θ that diminishes numerical dispersion, and for the fast wave, a value of θ that en-sures stability under acceptable conditions, typically (θ, θ) = (1/12, 1/4).
Numerical illustrations that show the interest of this scheme for piano strings will be displayed. References
[1] J. Chabassier and S. Imperiale Stability and dispersion analysis of improved time discretisa-tion for prestressed Timoshenko systems. Appli-cation to the stiff piano string. Wave Motion, to appear, doi : 10.1016/j.wavemoti.2012.11.002.. [2] S Timoshenko. On the correction for shear of
the differential equation for transverse vibra-tions of bars of uniform cross-section. Philosoph-ical Magazine, vol 41, pp 744–746, 1921.
