Vibrational behavior of isotropic plate structures in contact with a bounded fluid via unified formulation
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摘要
This paper presents an analytical solution for free vibration analysis of thick rectangular isotropic plates coupled with a bounded fluid for various boundary conditions. In order to consider displacement theories of an arbitrary order, the Carrera Unified Formulation(CUF) is used. The eigenvalue problem is obtained by using the energy functional, considering plate and fluid kinetic energies as well as the potential energy of the plate. The Ritz method is used to evaluate the displacement variables, and the functions used in the Ritz series can be adjusted to consider any of the classical boundary conditions. The convergence of the solution is analyzed, and a validation of results considering open literature and 3D finite element software is performed. Parametric studies are carried out to obtain natural frequencies as a function of the side-to-thickness ratio, plate aspect ratio, fluid domain size, plate boundary conditions, and fluid-solid density ratio. Pressure and velocity in the fluid domain are evaluated in order to establish the consistency of the solution.Accurate results for thick plates are obtained with a much lower computational cost compared to that of 3D finite element solutions.
This paper presents an analytical solution for free vibration analysis of thick rectangular isotropic plates coupled with a bounded fluid for various boundary conditions. In order to consider displacement theories of an arbitrary order, the Carrera Unified Formulation(CUF) is used. The eigenvalue problem is obtained by using the energy functional, considering plate and fluid kinetic energies as well as the potential energy of the plate. The Ritz method is used to evaluate the displacement variables, and the functions used in the Ritz series can be adjusted to consider any of the classical boundary conditions. The convergence of the solution is analyzed, and a validation of results considering open literature and 3D finite element software is performed. Parametric studies are carried out to obtain natural frequencies as a function of the side-to-thickness ratio, plate aspect ratio, fluid domain size, plate boundary conditions, and fluid-solid density ratio. Pressure and velocity in the fluid domain are evaluated in order to establish the consistency of the solution.Accurate results for thick plates are obtained with a much lower computational cost compared to that of 3D finite element solutions.
This paper presents an analytical solution for free vibration analysis of thick rectangular isotropic plates coupled with a bounded fluid for various boundary conditions. In order to consider displacement theories of an arbitrary order, the Carrera Unified Formulation(CUF) is used. The eigenvalue problem is obtained by using the energy functional, considering plate and fluid kinetic energies as well as the potential energy of the plate. The Ritz method is used to evaluate the displacement variables, and the functions used in the Ritz series can be adjusted to consider any of the classical boundary conditions. The convergence of the solution is analyzed, and a validation of results considering open literature and 3D finite element software is performed. Parametric studies are carried out to obtain natural frequencies as a function of the side-to-thickness ratio, plate aspect ratio, fluid domain size, plate boundary conditions, and fluid-solid density ratio. Pressure and velocity in the fluid domain are evaluated in order to establish the consistency of the solution.Accurate results for thick plates are obtained with a much lower computational cost compared to that of 3D finite element solutions.
引文
1.Jeong KH,Kim KJ.Hydroelastic vibration of a circular plate submerged in a bounded compressible fluid.J Sound Vib 2005;283(1-2):153-72.
2.Tariverdilo S,Shahmardani M,Mirzapour J,Shabani R.Asymmetric free vibration of circular plate in contact with incompressible fluid.Appl Math Model 2013;37(1-2):228-39.
3.Kerboua Y,Lakis AA,Thomas M,Marcouiller L.Vibration analysis of rectangular plates coupled with fluid.Appl Math Model2008;32(12):2570-86.
4.Chang TP.On the natural frequency of transversely isotropic magneto-electro-elastic plates in contact with fluid.Appl Math Model 2013;37(4):2503-15.
5.Carra S,Amabili M,Garziera R.Experimental study of large amplitude vibrations of a thin plate in contact with sloshing liquids.J Fluids Struct 2013;42:88-111.
6.Kozlovsky Y.Vibration of plates in contact with viscous fluid:Extension of Lamb’s model.J Sound and Vib 2009;326(1-2):332-9.
7.Cheung YK,Zhou D.Coupled vibratory characteristics of a rectangular container bottom plate.J Fluids and Struct 2000;14(3):339-57.
8.Cho DS,Kim BH,Vladimir N,Choi TM.Natural vibration analysis of rectangular bottom plate structures in contact with fluid.Ocean Eng 2015;103:171-9.
9.Liao CY,Ma CC.Vibration characteristics of rectangular plate in compressible inviscid fluid.J Sound and Vib 2016;362:228-51.
10.Hashemi SH,Karimi M,Taher HRD.Vibration analysis of rectangular Mindlin plates on elastic foundations and vertically in contact with stationary fluid by the Ritz method.Ocean Eng2010;37:174-85.
11.Shahbaztabar A,Ranji AR.Effects of in-plane loads on free vibration of symmetrically cross-ply laminated plates resting on Pasternak foundation and coupled with fluid.Ocean Eng2016;115:196-209.
12.Cho DS,Kim BH,Kim JH,Vladimir N,Choi TM.Frequency response of rectangular plate structures in contact with fluid subjected to harmonic point excitation force.Thin-Wall Struct2015;95:276-86.
13.Khorshidi K,Farhadi S.Free vibration analysis of a laminated composite rectangular plate in contact with a bounded fluid.Compos Struct 2013;104:176-86.
14.Jeong KH,Kang HS.Free vibration of multiple rectangular plates coupled with a liquid.Int J of Mech Sci 2013;74:161-72.
15.Khorsidi K,Akbari F,Ghadirian H.Experimental and analytical modal studies of vibrating rectangular plates in contact with a bounded fluid.Ocean Eng 2017;140:146-54.
16.Khorsidi K.Effect of hydrostatic pressure and depth of fluid on the vibrating rectangular plates partially in contact with a fluid.Appl Mech Mat 2012;110:927-35.
17.Khorsidi K.Effect of hydrostatic pressure on vibrating rectangular plates coupled with fluid.Sci Iran Trans A:Civ Eng2010;17:415-29.
18.Khorsidi K,Bakhsheshy A.Free vibration analysis of a functionally graded rectangular plate in contact with a bounded fluid.Acta Mech 2015;226:3401-23.
19.Carrera E.Theories and finite elements for multilayered plates and shells:a unified compact formulation with numerical assessment and benchmarking.Arch Comput Meth Eng 2003;10:215-96.
20.Carrera E.Transverse normal strain effects on thermal stress analysis of homogeneous and layered plates.AIAA J 2005;43(10):2232-42.
21.Robaldo A,Carrera E,Benjeddou A.Unified formulation for finite element thermoelastic analysis of multilayered anisotropic composite plates.J Therm Stresses 2005;28:1031-65.
22.Carrera E.Temperature profile influence on layered plates response considering classical and advanced theories.AIAA J2002;40:1885-96.
23.Carrera E,Boscolo M,Robaldo A.Hierarchic multilayered plate elements for coupled multifield problems of piezoelectric adaptive structures:Formulation and numerical assessment.Arch Comput Meth Eng 2007;14:383-430.
24.Carrera E,Brischetto S,Nali P.Variational statements and computational models for multifield problems and multilayered structures.Mech Adv Mater Struct 2008;15:182-98.
25.Carrera E,Brischetto S,Fagiano C,Nali P.Mixed multilayered plate elements for coupled magneto-electro-elastic analysis.Multidiscipline Model Mater Struct 2009;5:251-6.
26.Robaldo A,Carrera E,Benjeddou A.A unified formulation for finite element analysis of piezoelectric adaptive plates.Comput Struct 2006;84:1494-505.
27.Cinefra M,Carrera E,Della Croce L,Chinosi C.Refined shell elements for the analysis of functionally graded structures.Compos Struct 2012;94:415-22.
28.Cinefra M,Chinosi C,Della Croce L.MITC9 shell elements based on refined theories for the analysis of isotropic cylindrical structures.Mech Adv Mater Struct 2013;20(2):91-100.
29.Carrera E,Brischetto S,Robaldo A.Variable kinematic model for the analysis of functionally graded material plates.AIAA J2008;46:194-203.
30.Carrera E,Giunta G,Petrolo M.Beam structures,classical and advanced theories.Chichester:Wiley;2011.
31.Carrera E,Brischetto S,Nali P.Plates and shells for smart structures:Classical and advanced theories for modeling and analysis.Chichester:Wiley;2011.
32.Carrera E,Cinefra M,Petrolo M,Zappino E.Finite element analysis of structures through unified formulation.Chichester:Wiley;2014.
33.Leissa AW.The historical bases of the Rayleigh and Ritz methods.J Sound Vib 2005;287(4-5):961-78.
34.Gander MJ,Wanner G.From Euler,Ritz,and Galerkin to modern computing.SIAM Rev 2012;54:627-66.
35.Ilanko S,Monterrubio L,Mochida Y.The Rayleigh-Ritz method for structural analysis.London:Wiley;2014.
36.Fazzolari FA,Carrera E.Accurate free vibration analysis of thermo-mechanically pre/post-buckled anisotropic multilayered plates based on a refined hierarchical trigonometric Ritz formulation.Compos Struct 2013;95:381-402.
37.Fazzolari FA,Carrera E.Advanced variable kinematics Ritz and Galerkin formulations for accurate buckling and vibration analysis of anisotropic laminated composite plates.Compos Struct2011;94:50-67.
38.Fazzolari FA,Carrera E.Coupled thermoelastic effect in free vibration analysis of anisotropic multilayered plates and FGMplates by using a variable-kinematics Ritz formulation.Eur JMech A Solids 2014;44:157-74.
39.Fazzolari FA,Carrera E.Free vibration analysis of sandwich plates with anisotropic face sheets in thermal environment by using the hierarchical trigonometric Ritz formulation.Compos B2013;50:67-81.
40.Fazzolari FA.Reissner’s mixed variational theorem and variable kinematics in the modelling of laminated composite and FGMdoubly-curved shells.Compos B 2016;89:408-23.
41.Fazzolari FA,Banerjee JR.Axiomatic/asymptotic PVD/RMVT-based shell theories for free vibrations of anisotropic shells using an advanced Ritz formulation and accurate curvature descriptions.Compos Struct 2014;108:91-110.
42.Fazzolari FA,Carrera E.Advances in the Ritz formulation for free vibration response of doubly-curved anisotropic laminated composite shallow and deep shells.Compos Struct2013;101:111-28.
43.Dozio L.Natural frequencies of sandwich plates with FGM core via variable-kinematic 2-D Ritz models.Compos Struct2013;96:561-8.
44.Dozio L.On the use of the trigonometric Ritz method for general vibration analysis of rectangular Kirchhoff plates.Thin-Wall Struct 2011;49:129-44.
45.Dozio L.In-plane free vibrations of single-layer and symmetrically laminated rectangular composite plates.Compos Struct2011;93:1787-800.
46.Dozio L.Free in-plane vibration analysis of rectangular plates with arbitrary elastic boundaries.Mech Res Commun 2010;37(7):627-35.
47.Vescovini R,Dozio L.A variable-kinematic model for variable stiffness plates:vibration and buckling analysis.Compos Struct2016;142:15-26.
48.Dozio L,Carrera E.A variable kinematic Ritz formulation for vibration study of quadrilateral plates with arbitrary thickness.JSound Vib 2011;330(18-19):4611-32.
49.Kondo H.Axisymmetric free vibration analysis of a circular cylindrical tank.Bull Japan Soc Mech Eng 1981;24:215-21.
2.Tariverdilo S,Shahmardani M,Mirzapour J,Shabani R.Asymmetric free vibration of circular plate in contact with incompressible fluid.Appl Math Model 2013;37(1-2):228-39.
3.Kerboua Y,Lakis AA,Thomas M,Marcouiller L.Vibration analysis of rectangular plates coupled with fluid.Appl Math Model2008;32(12):2570-86.
4.Chang TP.On the natural frequency of transversely isotropic magneto-electro-elastic plates in contact with fluid.Appl Math Model 2013;37(4):2503-15.
5.Carra S,Amabili M,Garziera R.Experimental study of large amplitude vibrations of a thin plate in contact with sloshing liquids.J Fluids Struct 2013;42:88-111.
6.Kozlovsky Y.Vibration of plates in contact with viscous fluid:Extension of Lamb’s model.J Sound and Vib 2009;326(1-2):332-9.
7.Cheung YK,Zhou D.Coupled vibratory characteristics of a rectangular container bottom plate.J Fluids and Struct 2000;14(3):339-57.
8.Cho DS,Kim BH,Vladimir N,Choi TM.Natural vibration analysis of rectangular bottom plate structures in contact with fluid.Ocean Eng 2015;103:171-9.
9.Liao CY,Ma CC.Vibration characteristics of rectangular plate in compressible inviscid fluid.J Sound and Vib 2016;362:228-51.
10.Hashemi SH,Karimi M,Taher HRD.Vibration analysis of rectangular Mindlin plates on elastic foundations and vertically in contact with stationary fluid by the Ritz method.Ocean Eng2010;37:174-85.
11.Shahbaztabar A,Ranji AR.Effects of in-plane loads on free vibration of symmetrically cross-ply laminated plates resting on Pasternak foundation and coupled with fluid.Ocean Eng2016;115:196-209.
12.Cho DS,Kim BH,Kim JH,Vladimir N,Choi TM.Frequency response of rectangular plate structures in contact with fluid subjected to harmonic point excitation force.Thin-Wall Struct2015;95:276-86.
13.Khorshidi K,Farhadi S.Free vibration analysis of a laminated composite rectangular plate in contact with a bounded fluid.Compos Struct 2013;104:176-86.
14.Jeong KH,Kang HS.Free vibration of multiple rectangular plates coupled with a liquid.Int J of Mech Sci 2013;74:161-72.
15.Khorsidi K,Akbari F,Ghadirian H.Experimental and analytical modal studies of vibrating rectangular plates in contact with a bounded fluid.Ocean Eng 2017;140:146-54.
16.Khorsidi K.Effect of hydrostatic pressure and depth of fluid on the vibrating rectangular plates partially in contact with a fluid.Appl Mech Mat 2012;110:927-35.
17.Khorsidi K.Effect of hydrostatic pressure on vibrating rectangular plates coupled with fluid.Sci Iran Trans A:Civ Eng2010;17:415-29.
18.Khorsidi K,Bakhsheshy A.Free vibration analysis of a functionally graded rectangular plate in contact with a bounded fluid.Acta Mech 2015;226:3401-23.
19.Carrera E.Theories and finite elements for multilayered plates and shells:a unified compact formulation with numerical assessment and benchmarking.Arch Comput Meth Eng 2003;10:215-96.
20.Carrera E.Transverse normal strain effects on thermal stress analysis of homogeneous and layered plates.AIAA J 2005;43(10):2232-42.
21.Robaldo A,Carrera E,Benjeddou A.Unified formulation for finite element thermoelastic analysis of multilayered anisotropic composite plates.J Therm Stresses 2005;28:1031-65.
22.Carrera E.Temperature profile influence on layered plates response considering classical and advanced theories.AIAA J2002;40:1885-96.
23.Carrera E,Boscolo M,Robaldo A.Hierarchic multilayered plate elements for coupled multifield problems of piezoelectric adaptive structures:Formulation and numerical assessment.Arch Comput Meth Eng 2007;14:383-430.
24.Carrera E,Brischetto S,Nali P.Variational statements and computational models for multifield problems and multilayered structures.Mech Adv Mater Struct 2008;15:182-98.
25.Carrera E,Brischetto S,Fagiano C,Nali P.Mixed multilayered plate elements for coupled magneto-electro-elastic analysis.Multidiscipline Model Mater Struct 2009;5:251-6.
26.Robaldo A,Carrera E,Benjeddou A.A unified formulation for finite element analysis of piezoelectric adaptive plates.Comput Struct 2006;84:1494-505.
27.Cinefra M,Carrera E,Della Croce L,Chinosi C.Refined shell elements for the analysis of functionally graded structures.Compos Struct 2012;94:415-22.
28.Cinefra M,Chinosi C,Della Croce L.MITC9 shell elements based on refined theories for the analysis of isotropic cylindrical structures.Mech Adv Mater Struct 2013;20(2):91-100.
29.Carrera E,Brischetto S,Robaldo A.Variable kinematic model for the analysis of functionally graded material plates.AIAA J2008;46:194-203.
30.Carrera E,Giunta G,Petrolo M.Beam structures,classical and advanced theories.Chichester:Wiley;2011.
31.Carrera E,Brischetto S,Nali P.Plates and shells for smart structures:Classical and advanced theories for modeling and analysis.Chichester:Wiley;2011.
32.Carrera E,Cinefra M,Petrolo M,Zappino E.Finite element analysis of structures through unified formulation.Chichester:Wiley;2014.
33.Leissa AW.The historical bases of the Rayleigh and Ritz methods.J Sound Vib 2005;287(4-5):961-78.
34.Gander MJ,Wanner G.From Euler,Ritz,and Galerkin to modern computing.SIAM Rev 2012;54:627-66.
35.Ilanko S,Monterrubio L,Mochida Y.The Rayleigh-Ritz method for structural analysis.London:Wiley;2014.
36.Fazzolari FA,Carrera E.Accurate free vibration analysis of thermo-mechanically pre/post-buckled anisotropic multilayered plates based on a refined hierarchical trigonometric Ritz formulation.Compos Struct 2013;95:381-402.
37.Fazzolari FA,Carrera E.Advanced variable kinematics Ritz and Galerkin formulations for accurate buckling and vibration analysis of anisotropic laminated composite plates.Compos Struct2011;94:50-67.
38.Fazzolari FA,Carrera E.Coupled thermoelastic effect in free vibration analysis of anisotropic multilayered plates and FGMplates by using a variable-kinematics Ritz formulation.Eur JMech A Solids 2014;44:157-74.
39.Fazzolari FA,Carrera E.Free vibration analysis of sandwich plates with anisotropic face sheets in thermal environment by using the hierarchical trigonometric Ritz formulation.Compos B2013;50:67-81.
40.Fazzolari FA.Reissner’s mixed variational theorem and variable kinematics in the modelling of laminated composite and FGMdoubly-curved shells.Compos B 2016;89:408-23.
41.Fazzolari FA,Banerjee JR.Axiomatic/asymptotic PVD/RMVT-based shell theories for free vibrations of anisotropic shells using an advanced Ritz formulation and accurate curvature descriptions.Compos Struct 2014;108:91-110.
42.Fazzolari FA,Carrera E.Advances in the Ritz formulation for free vibration response of doubly-curved anisotropic laminated composite shallow and deep shells.Compos Struct2013;101:111-28.
43.Dozio L.Natural frequencies of sandwich plates with FGM core via variable-kinematic 2-D Ritz models.Compos Struct2013;96:561-8.
44.Dozio L.On the use of the trigonometric Ritz method for general vibration analysis of rectangular Kirchhoff plates.Thin-Wall Struct 2011;49:129-44.
45.Dozio L.In-plane free vibrations of single-layer and symmetrically laminated rectangular composite plates.Compos Struct2011;93:1787-800.
46.Dozio L.Free in-plane vibration analysis of rectangular plates with arbitrary elastic boundaries.Mech Res Commun 2010;37(7):627-35.
47.Vescovini R,Dozio L.A variable-kinematic model for variable stiffness plates:vibration and buckling analysis.Compos Struct2016;142:15-26.
48.Dozio L,Carrera E.A variable kinematic Ritz formulation for vibration study of quadrilateral plates with arbitrary thickness.JSound Vib 2011;330(18-19):4611-32.
49.Kondo H.Axisymmetric free vibration analysis of a circular cylindrical tank.Bull Japan Soc Mech Eng 1981;24:215-21.