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High-Frequency Vibrations of Systems with Concentrated Masses Along Planes

  • D. Gömez
  • M. Lobo
  • M. E. Pérez
Chapter

Abstract

Let Ω be an open bounded domain of ℝ3 with a smooth boundary \(\partial\Omega\). Weassume that Ω is divided into two parts Ω+ and Ω- by the plane \(\gamma: \Omega = \Omega_+ \cup \Omega_- \cup \gamma\) .For simplicity, we assume that the plane { x 3 = 0} cuts Ω and \(\gamma = \Omega \cap \{x_3 = 0\}\). Let ε be a small positive parameter that tends to zero. We denote by ωε the ε-neighborhood of γ, i.e., \(\omega_\varepsilon = \Omega \cup \{|x_3| < \varepsilon\}\); for ε sufficiently small, we assume that \(\omega_\varepsilon = \gamma \times (-\varepsilon, \varepsilon)\)) (see Figure 15.1). Note that this conditions the geometry of Ω near γ. Let us denote by \(\bar{x}\) the two first components of any x = (x 1, x 2, x 3) ε ℝ3, that is,\(\bar{x} = (x_1, x_2)\)

Keywords

Manifold 
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Copyright information

© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Universidad de CantabriaCantabriaSpain

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