An assessment of fluid compressibility influence on the natural frequencies of a submerged plate via unified formulation

Abstract Fluid compressibility of liquids is often neglected in engineering design. However, the error incurred due to this simplification is not well identified. This paper examines the influence of compressibility on the hydroelastic vibration of plates in contact with fluid. An analytical solution for the free vibration of thick rectangular isotropic plates coupled with a bounded compressible inviscid fluid domain is developed. Plate displacement theories with arbitrary order are considered using the 2D Carrera Unified Formulation, which can obtain results very similar to 3D solutions. The eigenvalue problem is obtained by considering the kinetic and potential energy of both the fluid and the plate. The displacement variables are evaluated using the Ritz method. A comparison of the results with open literature and 3D finite element software is performed. Parametric studies are carried out in order to assess the error due to neglecting fluid compressibility as a function of plate geometry, material properties and boundary conditions. The influence of fluid domain size, density and sonic velocity is also assessed. The results indicate that the error due to neglecting fluid compressibility is high when thick, square plates made of light, stiff materials and with rigid boundary conditions are considered.

[1]  Y. Kerboua,et al.  Vibration analysis of truncated conical shells subjected to flowing fluid , 2010 .

[2]  E. Carrera,et al.  Variational Statements and Computational Models for MultiField Problems and Multilayered Structures , 2008 .

[3]  Lorenzo Dozio,et al.  A variable kinematic Ritz formulation for vibration study of quadrilateral plates with arbitrary thickness , 2011 .

[4]  E. Carrera.  TRANSVERSE NORMAL STRAIN EFFECTS ON THERMAL STRESS ANALYSIS OF HOMOGENEOUS AND LAYERED PLATES , 2005 .

[5]  Shahrokh Hosseini-Hashemi,et al.  Natural frequencies of rectangular Mindlin plates coupled with stationary fluid , 2012 .

[6]  C. Chinosi,et al.  MITC9 Shell Elements Based on Refined Theories for the Analysis of Isotropic Cylindrical Structures , 2013 .

[7]  E. Carrera,et al.  Advanced variable kinematics Ritz and Galerkin formulations for accurate buckling and vibration analysis of anisotropic laminated composite plates , 2011 .

[8]  Jakob Kuttenkeuler,et al.  Experimental eigenfrequency study of dry and fully wetted rectangular composite and metallic plates by forced vibrations , 2016 .

[9]  Lorenzo Dozio,et al.  On the use of the Trigonometric Ritz method for general vibration analysis of rectangular Kirchhoff plates , 2011 .

[10]  M. Amabili,et al.  Experimental study of large amplitude vibrations of a thin plate in contact with sloshing liquids , 2013 .

[11]  Lawrence E. Kinsler,et al.  Fundamentals of acoustics , 1950 .

[12]  Ivo Senjanović,et al.  Modified Mindlin plate theory and shear locking-free finite element formulation , 2014 .

[13]  S. Hosseini-Hashemi,et al.  Hydroelastic vibration and buckling of rectangular Mindlin plates on Pasternak foundations under linearly varying in-plane loads , 2010 .

[14]  Moon K. Kwak,et al.  Hydroelastic vibration of rectangular plates , 1996 .

[15]  Catherine N. Phan,et al.  Finite amplitude vibrations of cantilevers of rectangular cross sections in viscous fluids , 2013 .

[16]  Colin Atkinson,et al.  The frequency response of a rectangular cantilever plate vibrating in a viscous fluid , 2007 .

[17]  Chien-Ching Ma,et al.  Vibration characteristics of rectangular plate in compressible inviscid fluid , 2016 .

[18]  J. R. Banerjee,et al.  Axiomatic/asymptotic PVD/RMVT-based shell theories for free vibrations of anisotropic shells using an advanced Ritz formulation and accurate curvature descriptions , 2014 .

[19]  Lorenzo Dozio,et al.  Free in-plane vibration analysis of rectangular plates with arbitrary elastic boundaries , 2010 .

[20]  Y. Kozlovsky,et al.  Vibration of plates in contact with viscous fluid: Extension of Lamb's model , 2009 .

[21]  T. Hughes.  Generalization of selective integration procedures to anisotropic and nonlinear media , 1980 .

[22]  Kyeong-Hoon Jeong,et al.  Hydroelastic vibration of two annular plates coupled with a bounded compressible fluid , 2006 .

[23]  L. Dozio.  In-plane free vibrations of single-layer and symmetrically laminated rectangular composite plates , 2011 .

[24]  Y. K. Cheung,et al.  Coupled vibratory characteristics of a rectangular container bottom plate , 2000 .

[25]  Kyeong-Hoon Jeong,et al.  Hydroelastic vibration of a cantilever cylindrical shell partially submerged in a liquid , 2010 .

[26]  L. Dozio.  Natural frequencies of sandwich plates with FGM core via variable-kinematic 2-D Ritz models , 2013 .

[27]  M. Karimi,et al.  Vibration analysis of rectangular Mindlin plates on elastic foundations and vertically in contact with stationary fluid by the Ritz method , 2010 .

[28]  Arthur W. Leissa,et al.  The historical bases of the Rayleigh and Ritz methods , 2005 .

[29]  Erasmo Carrera,et al.  Finite Element Analysis of Structures through Unified Formulation , 2014 .

[30]  Joshua H. Gordis,et al.  Investigation of vibrational characteristics of composite beams with fluid–structure interaction , 2013 .

[31]  Alfredo Bermúdez,et al.  Finite element computation of the vibrations of a plate-fluid system with interface damping , 2001 .

[32]  Medhat A. Haroun,et al.  Reduced and selective integration techniques in the finite element analysis of plates , 1978 .

[33]  Erasmo Carrera,et al.  Laminated beam analysis by polynomial, trigonometric, exponential and zig-zag theories , 2013 .

[34]  Kyeong-Hoon Jeong,et al.  Free vibration of multiple rectangular plates coupled with a liquid , 2013 .

[35]  E. Carrera,et al.  Refined shell elements for the analysis of functionally graded structures , 2012 .

[36]  Erasmo Carrera,et al.  Plates and Shells for Smart Structures: Classical and Advanced Theories for Modeling and Analysis , 2011 .

[37]  Tran Ich Thinh,et al.  Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid , 2016 .

[38]  Nikola Vladimir,et al.  Natural vibration analysis of rectangular bottom plate structures in contact with fluid , 2015 .

[39]  Jaromír Horáček,et al.  Analysis of the free vibration of a coupled plate/fluid interacting system and interpretation using sub-system modal energy , 2007 .

[40]  Marco Amabili,et al.  EIGENVALUE PROBLEMS FOR VIBRATING STRUCTURES COUPLED WITH QUIESCENT FLUIDS WITH FREE SURFACE , 2000 .

[41]  Erasmo Carrera,et al.  Temperature Profile Influence on Layered Plates Response Considering Classical and Advanced Theories , 2002 .

[42]  C. Fagiano,et al.  REFINED MULTILAYERED PLATE ELEMENTS FOR COUPLED MAGNETO-ELECTRO-ELASTIC ANALYSIS , 2009 .

[43]  R. Firouz-Abadi,et al.  Free vibrations of moderately thick truncated conical shells filled with quiescent fluid , 2016 .

[44]  E. Carrera,et al.  Coupled thermoelastic effect in free vibration analysis of anisotropic multilayered plates and FGM plates by using a variable-kinematics Ritz formulation , 2014 .

[45]  O. C. Zienkiewicz,et al.  Reduced integration technique in general analysis of plates and shells , 1971 .

[46]  Erasmo Carrera,et al.  Unified Formulation for Finite Element Thermoelastic Analysis of Multilayered Anisotropic Composite Plates , 2005 .

[47]  Moon K. Kwak,et al.  AXISYMMETRIC VIBRATION OF CIRCULAR PLATES IN CONTACT WITH FLUID , 1991 .

[48]  Y. Kerboua,et al.  Vibration analysis of rectangular plates coupled with fluid , 2008 .

[49]  Kyeong-Hoon Jeong,et al.  Hydroelastic vibration of a circular plate submerged in a bounded compressible fluid , 2005 .

[50]  Arash Shahbaztabar,et al.  Effects of in-plane loads on free vibration of symmetrically cross-ply laminated plates resting on Pasternak foundation and coupled with fluid , 2016 .

[51]  El Mostafa Daya,et al.  Comparison of non-linear eigensolvers for modal analysis of frequency dependent laminated visco-elastic sandwich plates , 2016 .

[52]  F. Alijani,et al.  Nonlinear vibrations and multiple resonances of fluid filled arbitrary laminated circular cylindrical shells , 2014 .

[53]  E. Carrera.  Theories and Finite Elements for Multilayered Plates and Shells:A Unified compact formulation with numerical assessment and benchmarking , 2003 .

[54]  F. Fazzolari.  Reissner's Mixed Variational Theorem and variable kinematics in the modelling of laminated composite and FGM doubly-curved shells , 2016 .

[55]  Gaetano Giunta,et al.  Beam Structures: Classical and Advanced Theories , 2011 .

[56]  Kyeong-Hoon Jeong,et al.  Free vibration analysis of a circular plate partially in contact with a liquid , 2009 .

[57]  T.-P. Chang.  On the natural frequency of transversely isotropic magneto-electro-elastic plates in contact with fluid , 2013 .

[58]  Non-linear vibration analysis of an elastic plate subjected to heavy fluid loading in magnetic field , 2009 .

[59]  Martin J. Gander,et al.  From Euler, Ritz, and Galerkin to Modern Computing , 2012, SIAM Rev..

[60]  E. Carrera,et al.  Variable Kinematic Model for the Analysis of Functionally Graded Material plates , 2008 .

[61]  Nikola Vladimir,et al.  Frequency response of rectangular plate structures in contact with fluid subjected to harmonic point excitation force , 2015 .

[62]  Korosh Khorshid,et al.  Free vibration analysis of a laminated composite rectangular plate in contact with a bounded fluid , 2013 .

[63]  J. Mirzapour,et al.  Asymmetric free vibration of circular plate in contact with incompressible fluid , 2013 .

[64]  E. Carrera,et al.  Accurate free vibration analysis of thermo-mechanically pre/post-buckled anisotropic multilayered plates based on a refined hierarchical trigonometric Ritz formulation , 2013 .

[65]  E. Carrera,et al.  Analysis of laminated composites and sandwich structures by trigonometric, exponential and miscellaneous polynomials and a MITC9 plate element , 2016 .

[66]  Erasmo Carrera,et al.  A unified formulation for finite element analysis of piezoelectric adaptive plates , 2006 .

[67]  Nonlinear vibrations of cantilevered circular cylindrical shells in contact with quiescent fluid , 2014 .

[68]  Lorenzo Dozio,et al.  A variable-kinematic model for variable stiffness plates: Vibration and buckling analysis , 2016 .

[69]  E. Carrera,et al.  Free vibration analysis of sandwich plates with anisotropic face sheets in thermal environment by using the hierarchical trigonometric Ritz formulation , 2013 .

[70]  Erasmo Carrera,et al.  Advances in the Ritz formulation for free vibration response of doubly-curved anisotropic laminated composite shallow and deep shells , 2013 .

[71]  Erasmo Carrera,et al.  Hierarchic Multilayered Plate Elements for Coupled Multifield Problems of Piezoelectric Adaptive Structures: Formulation and Numerical Assessment , 2007 .