Applied Mathematics and Mechanics

, Volume 33, Issue 9, pp 1167–1178 | Cite as

Free vibration of membrane/bounded incompressible fluid

  • S. Tariverdilo
  • J. Mirzapour
  • M. Shahmardani
  • G. Rezazadeh
Article
First Online:
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Revised:

Abstract

Vibration of a circular membrane in contact with a fluid has extensive applications in industry. The natural vibration frequencies for the asymmetric free vibration of a circular membrane in contact with a bounded incompressible fluid are derived in this paper. Considering small oscillations induced by the membrane vibration in an incompressible and inviscid fluid, the velocity potential function is used to describe the fluid field. Two approaches are used to derive the free vibration frequencies of the system, which include a variational formulation and an approximate solution employing the Rayleigh quotient method. A good correlation is found between free vibration frequencies evaluated by these methods. Finally, the effects of the fluid depth, the mass density, and the radial tension on the free vibration frequencies of the coupled system are investigated.

Key words

Rayleigh quotient asymmetric free vibration membrane variational formulation 

Nomenclature

a

radius of the bounded fluid

h

depth of the bounded fluid

w

membrane deflection

Aij

membrane modal amplitude corresponding to the ith circular and the jth diametrical nodes

Bij

fluid modal amplitude corresponding to the ith circular and the jth diametrical nodes

F

fluid pressure

Ji

Bessel function of the first kind of order i

L

Lagrangian of the coupled system

T

radial tension per unit length of the membrane

Tm

kinetic energy of the membrane

Tf

kinetic energy of the fluid

Vm

potential energy of the membrane

βij

eigen parameter of the fluid

ρ

membrane mass density

λij

frequency parameter of the membrane

ω

eigen-frequency of the coupled system

ψ

fluid velocity potential function

ρf

fluid mass density

δim

Kronecker delta

Chinese Library Classification

O353.1 

2010 Mathematics Subject Classification

76A10 76D10 74S25 76M22 
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Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • S. Tariverdilo
    • 1
  • J. Mirzapour
    • 1
  • M. Shahmardani
    • 1
  • G. Rezazadeh
    • 2
  1. 1.Department of Civil Engineering, Faculty of EngineeringUrmia UniversityUrmiaIran
  2. 2.Department of Mechanical Engineering, Faculty of EngineeringUrmia UniversityUrmiaIran

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