caa a22 4500 445812257 CHVBK 20180317145215.0 cr unu---uuuuu 170323e20110801xx s 000 0 eng 10.1007/s00419-010-0460-5 doi (NATIONALLICENCE)springer-10.1007/s00419-010-0460-5 Effects of exponential volume fraction law on the natural frequencies of FGM cylindrical shells under various boundary conditions [Elektronische Daten] [Shahid Arshad, Muhammad Naeem, Nazra Sultana, Zafar Iqbal, Abdul Shah] In the present work, vibration characteristics of thin functionally graded cylindrical shells are studied under the influence of various boundary conditions. Fabrication of FGM cylindrical shell is carried out by using exponential volume fraction law. Strain- and curvature-displacements relationships are taken from Love's thin shell theory. The frequency equation in the form of eigenvalue problem is obtained by adapting the Rayleigh-Ritz method. Characteristic beam functions are assumed to approximate the axial modal dependence. Effects of exponential volume fraction law on the natural frequencies of the FGM cylindrical shells for various boundary conditions are studied against circumferential wave number, length to radius ratio and thickness to radius ratio for different values of power law exponents. Results evaluated show good agreement with those available in the literature. Springer-Verlag, 2010 Cylindrical shells nationallicence Functionally graded materials nationallicence Exponential volume fraction law nationallicence Love's thin shell theory nationallicence Rayleigh-Ritz method nationallicence Lagrangian functional nationallicence Boundary conditions nationallicence Natural frequencies nationallicence A , B , D : Extensional, coupling and bending stiffness matrices nationallicence A 1, B 1, C 1 : Vibration amplitude coefficients nationallicence A 11, A 12, A 22, A 66 : Extensional stiffnesses nationallicence B 11, B 12, B 22, B 66 : Coupling stiffnesses nationallicence D 11, D 12, D 22, D 66 : Bending stiffnesses nationallicence a 1, a 2, a 3, a 4 : Constant for beam function different for different boundary conditions nationallicence e 1, e 2, γ : Surface strain components nationallicence E 1, E 2 : Young's modulus of the constituents nationallicence E FGM : Resultant Young's modulus of FGM nationallicence F : Lagrangian functional nationallicence { H } : Matrix of the extensional, coupling and bending stiffnesses nationallicence h : Shell thickness nationallicence h / R , L / R : Non-dimensional shell geometrical parameters nationallicence k 1, k 2, 2 τ : Surface curvature components nationallicence K : Kinetic energy nationallicence [ K ] : Stiffness matrix nationallicence L : Length of the shell nationallicence m : Axial wave number nationallicence M1 : Constituent material one nationallicence M2 : Constituent material two nationallicence [ M ] : Mass matrix nationallicence n : Circumferential wave number nationallicence p : Power law exponent nationallicence Q m , n : Reduced stiffnesses nationallicence R : Mean radius of the cylindrical shell nationallicence S : Strain energy nationallicence t : Time variable nationallicence T : Transpose of a matrix nationallicence u , v , w : Axial, circumferential and radial deformation components of the shell mid surface nationallicence u ·, v ·, w · : Derivative of axial, circumferential and radial deformation components of the shell mid surface w.r.t. time nationallicence U , V , W : Axial modal dependence in the axial, circumferential and radial directions nationallicence $${\overline{U},\overline{V},\overline{W}}$$ : Non-dimensional axial modal dependence in the axial, circumferential and radial directions nationallicence V 1 : Volume fraction of 1st constituent material nationallicence V 2 : Volume fraction of 2nd constituent material nationallicence x : Axial coordinate nationallicence x , θ , z : Orthogonal coordinate system nationallicence X : Non-dimensional axial coordinate nationallicence z : Thickness variable of the cylindrical shell nationallicence z / h : Intrinsic thickness variable nationallicence ε : Column matrix of surface strain and surface curvature nationallicence λ : Characteristic value of beam function nationallicence θ : Circumferential coordinate nationallicence ν : Poisson's ratio nationallicence ν FGM : Resultant Poisson's ratio of FGM nationallicence ν 1, ν 2 : Poisson ratio of the constituents nationallicence ρ : Mass density nationallicence ρ FGM : Resultant mass density of FGM nationallicence ρ t : Mass density per unit length nationallicence ρ 1, ρ 2 : Mass density of the constituents nationallicence ω : Angular frequency nationallicence Ω : Frequency parameter nationallicence Ψ( X ) : Beam characteristic function nationallicence Arshad Shahid Department of Mathematics, University of Sargodha, Sargodha, Pakistan aut Naeem Muhammad Department of Mathematics, G. C. University Faisalabad, Faisalabad, Pakistan aut Sultana Nazra Department of Mathematics, University of Sargodha, Sargodha, Pakistan aut Iqbal Zafar Department of Mathematics, University of Sargodha, Sargodha, Pakistan aut Shah Abdul Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, Pakistan aut Archive of Applied Mechanics Springer-Verlag 81/8(2011-08-01), 999-1016 0939-1533 81:8<999 2011 81 419 https://doi.org/10.1007/s00419-010-0460-5 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00419-010-0460-5 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Arshad Shahid Department of Mathematics, University of Sargodha, Sargodha, Pakistan aut NATIONALLICENCE 700 1- Naeem Muhammad Department of Mathematics, G. C. University Faisalabad, Faisalabad, Pakistan aut NATIONALLICENCE 700 1- Sultana Nazra Department of Mathematics, University of Sargodha, Sargodha, Pakistan aut NATIONALLICENCE 700 1- Iqbal Zafar Department of Mathematics, University of Sargodha, Sargodha, Pakistan aut NATIONALLICENCE 700 1- Shah Abdul Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, Pakistan aut NATIONALLICENCE 773 0- Archive of Applied Mechanics Springer-Verlag 81/8(2011-08-01), 999-1016 0939-1533 81:8<999 2011 81 419 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer