Abstract
This paper introduces the nonlinear vibration response of an eccentrically stiffened porous functionally graded sandwich cylindrical shell panel with simply supported boundary conditions by using a new analytical model. The cylindrical shell has three layers: an FGM core layer and two layers made of isotropic homogeneous material. The FGM core properties are considered to be porosity dependent and varied in the thickness direction according to power-law distribution in terms of the volume fractions index of the constituents. Governing equations for the dynamic response of porous ES-FGM sandwich cylindrical shell panels are derived by combining classical shell theory and nonlinearity von Karman strains. Besides that, the airy stress function and Galerkin approaches are proposed to obtain the resulted equations: Fundamental natural frequency, dynamic response, and amplitude–frequency relation. Comparisons are made to estimate the reliability of the received results. Eventually, the effects of stiffeners, porous coefficient, face sheet thickness, material gradient, FGM core thickness, slenderness ratios, and excitation force on the natural frequency and nonlinear dynamic response of porous FGM sandwich cylindrical shell panels are also clarified in this paper.
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Abbreviations
- \({{\varvec{A}}}_{{\varvec{i}}{\varvec{j}}},{{\varvec{B}}}_{{\varvec{i}}{\varvec{j}}},{{\varvec{D}}}_{{\varvec{i}}{\varvec{j}}}\) :
-
Coefficients described in Appendix N/ m2
- \({{\varvec{A}}}_{{\varvec{x}}},{{\varvec{A}}}_{{\varvec{y}}}\) :
-
Cross-sectional areas of the stiffeners m2
- \({{\varvec{A}}}_{1},{{\varvec{A}}}_{2},{{\varvec{A}}}_{3}\) :
-
Coefficients described in Appendix
- \({{\varvec{E}}}_{^\circ }\) :
-
Elastic modulus for stiffeners N/ m2
- \({{\varvec{E}}}_{\mathbf{c}}\) :
-
Elastic modulus for ceramic N/ m2
- \({{\varvec{E}}}_{\mathbf{m}}\) :
-
Elastic modulus for metal N/ m2
- \({{\varvec{I}}}_{{\varvec{x}}},{{\varvec{I}}}_{{\varvec{y}}}\) :
-
Second moments of the cross-sectional areas for the stiffeners m4
- \({{\varvec{M}}}_{{\varvec{x}}},\boldsymbol{ }{{\varvec{M}}}_{{\varvec{y}}}, {{\varvec{M}}}_{{\varvec{x}}{\varvec{y}}}\) :
-
Moment’s resultants N.m
- \({{\varvec{N}}}_{{\varvec{x}}},\boldsymbol{ }{{\varvec{N}}}_{{\varvec{y}}}, {{\varvec{N}}}_{{\varvec{x}}{\varvec{y}}}\) :
-
Forces resultants Newton
- \({{\varvec{S}}}_{{\varvec{x}}}{,\boldsymbol{ }{\varvec{S}}}_{{\varvec{y}}}\) :
-
Distance between the transversal and longitudinal stiffeners m
- \({{\varvec{S}}}_{1},{{\varvec{S}}}_{2},{{\varvec{S}}}_{3}\) :
-
Coefficients
- \({{\varvec{Z}}}_{{\varvec{x}}},{{\varvec{Z}}}_{{\varvec{y}}}\) :
-
Eccentricities stiffened m
- \({{\varvec{h}}}_{{\varvec{F}}{\varvec{G}}}\) :
-
Core Thickness m
- \({{\varvec{h}}}_{{\varvec{L}}},{{\varvec{h}}}_{{\varvec{U}}},\) :
-
Skins Thickness m
- \({{\varvec{h}}}_{{\varvec{x}}},{{\varvec{h}}}_{{\varvec{y}}}\), \({{\varvec{d}}}_{{\varvec{x}}},{{\varvec{d}}}_{{\varvec{y}}}\) :
-
Thickness and width of longitudinal and transversal for the rectangle stiffeners m
- a:
-
Panel length m
- b:
-
Span length m
- c:
-
Ceramic
- E:
-
Young modulus N/ m2
- H:
-
Panel thickness m
- m:
-
Metal
- m:
-
Axial wave number unitless
- N:
-
Power law index unitless
- n:
-
Circumferential wave number unitless
- Pr:
-
Effective material properties
- R:
-
Panel radius m
- u, v:
-
Displacement components along x, y directions m
- Vc:
-
Volume fraction of ceramic
- Vm:
-
Volume fraction of metal
- w:
-
The deflection of the panel m
- x, y, z:
-
Panel coordinates m
- \({\varvec{M}},\boldsymbol{ }{\varvec{H}},\boldsymbol{ }{\varvec{K}}\) :
-
Coefficients
- \({\varvec{Q}}\) :
-
Excitation force N/ m2
- \({\varvec{f}}\) :
-
The stress function
- \({\varvec{q}}\) :
-
Uniformly distributed pressure of intensity Pascal
- \({{\varvec{\gamma}}}_{{\varvec{x}}{\varvec{y}}}\) :
-
The shear strain component unitless
- \({{\varvec{\varepsilon}}}_{{\varvec{x}}},\boldsymbol{ }{{\varvec{\varepsilon}}}_{{\varvec{y}}}\) :
-
The normal strains component unitless
- \({{\varvec{\rho}}}_{0}\) :
-
Mass density for stiffeners Kg/ m3
- \({{\varvec{\rho}}}_{1}\) :
-
Coefficient explained in Appendix Kg/ m3
- \({{\varvec{\omega}}}_{{\varvec{L}}}\) :
-
Linear fundamental frequencies Hertz (HZ)
- \({{\varvec{\omega}}}_{{\varvec{N}}{\varvec{L}}}\) :
-
Nonlinear vibration frequency Hertz (HZ)
- \(\boldsymbol{\Omega }\) :
-
Rotational velocity Rad/s
- \({\varvec{\beta}},\boldsymbol{ }\mathbf{B}\mathbf{e}\mathbf{t}\mathbf{a}\) :
-
The factor of the Porosity
- \({\varvec{\eta}}\) :
-
Amplitude of nonlinear vibration m
- \({\varvec{\nu}}\) :
-
Poisson's ratio unitless
- \({\varvec{\rho}}\) :
-
Mass density Kg/ m3
- \({\varvec{\sigma}}\) :
-
Stress component N/ m2
- \({\varvec{\tau}}\) :
-
Shear stress component N/ m2
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Mouthanna, A., Bakhy, S.H. & Al-Waily, M. Analytical Investigation of Nonlinear Free Vibration of Porous Eccentrically Stiffened Functionally Graded Sandwich Cylindrical Shell Panels. Iran J Sci Technol Trans Mech Eng 47, 1035–1053 (2023). https://doi.org/10.1007/s40997-022-00555-4
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DOI: https://doi.org/10.1007/s40997-022-00555-4