Estimation of vibration frequencies of linear elastic membranes

Luca Sabatini

Applications of Mathematics (2018)

  • Volume: 63, Issue: 1, page 37-53
  • ISSN: 0862-7940

Abstract

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The free motion of a thin elastic linear membrane is described, in a simplyfied model, by a second order linear homogeneous hyperbolic system of partial differential equations whose spatial part is the Laplace Beltrami operator acting on a Riemannian 2-dimensional manifold with boundary. We adapt the estimates of the spectrum of the Laplacian obtained in the last years by several authors for compact closed Riemannian manifolds. To make so, we use the standard technique of the doubled manifold to transform a Riemannian manifold with nonempty boundary ( M , ∂ M , g ) to a compact Riemannian manifold ( M ♯ M , g ˜ ) without boundary. An easy numerical investigation on a concrete semi-ellipsoidic membrane with clamped boundary tests the sharpness of the method.

How to cite

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Sabatini, Luca. "Estimation of vibration frequencies of linear elastic membranes." Applications of Mathematics 63.1 (2018): 37-53. <http://eudml.org/doc/294656>.

@article{Sabatini2018,
abstract = {The free motion of a thin elastic linear membrane is described, in a simplyfied model, by a second order linear homogeneous hyperbolic system of partial differential equations whose spatial part is the Laplace Beltrami operator acting on a Riemannian 2-dimensional manifold with boundary. We adapt the estimates of the spectrum of the Laplacian obtained in the last years by several authors for compact closed Riemannian manifolds. To make so, we use the standard technique of the doubled manifold to transform a Riemannian manifold with nonempty boundary $(M, \partial M, g)$ to a compact Riemannian manifold $(M\sharp M, \widetilde\{g\})$ without boundary. An easy numerical investigation on a concrete semi-ellipsoidic membrane with clamped boundary tests the sharpness of the method.},
author = {Sabatini, Luca},
journal = {Applications of Mathematics},
keywords = {membrane; Laplacian; estimation of frequencies},
language = {eng},
number = {1},
pages = {37-53},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Estimation of vibration frequencies of linear elastic membranes},
url = {http://eudml.org/doc/294656},
volume = {63},
year = {2018},
}

TY - JOUR
AU - Sabatini, Luca
TI - Estimation of vibration frequencies of linear elastic membranes
JO - Applications of Mathematics
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 1
SP - 37
EP - 53
AB - The free motion of a thin elastic linear membrane is described, in a simplyfied model, by a second order linear homogeneous hyperbolic system of partial differential equations whose spatial part is the Laplace Beltrami operator acting on a Riemannian 2-dimensional manifold with boundary. We adapt the estimates of the spectrum of the Laplacian obtained in the last years by several authors for compact closed Riemannian manifolds. To make so, we use the standard technique of the doubled manifold to transform a Riemannian manifold with nonempty boundary $(M, \partial M, g)$ to a compact Riemannian manifold $(M\sharp M, \widetilde{g})$ without boundary. An easy numerical investigation on a concrete semi-ellipsoidic membrane with clamped boundary tests the sharpness of the method.
LA - eng
KW - membrane; Laplacian; estimation of frequencies
UR - http://eudml.org/doc/294656
ER -

References

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  9. Sabatini, L., Estimations of the topological invariants and of the spectrum of the Laplacian for Riemannian manifolds with boundary, Submitted to Result. Math. (2017). (2017) MR1362899
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