Nonlinear Vibration of Truncated Conical Shells: Donnell, Sanders and Nemeth Theories

Mehrdad Bakhtiari; Aouni A. Lakis; Youcef Kerboua

doi:10.1515/ijnsns-2018-0377

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Nonlinear Vibration of Truncated Conical Shells: Donnell, Sanders and Nemeth Theories

Mehrdad Bakhtiari , Aouni A. Lakis and Youcef Kerboua

Published/Copyright: October 15, 2019

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From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 21 Issue 1

Abstract

Nonlinear free vibration of truncated conical shells has been investigated for three different shell theories; Donnell, Sanders and Nemeth to investigate the effect of their simplifying assumptions. The displacement field of a finite element model that was obtained from the exact solution of equilibrium equations of Sander’s improved first-approximation theory is used to define the nonlinear strain energy of conical shells. Employing generalized coordinates method the equations of motion are derived and subsequently the amplitude equation of nonlinear vibration of conical shells was developed. The amplitude equation is solved for multiple cases of isotropic materials. Linear and nonlinear free vibration results are validated against the existing studies in scientific literature and demonstrate good accordance. The validated model is used to investigate effects of different parameters including circumferential mode number, cone-half angle, length to radius ratio, thickness to radius ratio and boundary conditions for the nonlinear vibration of conical shells.

Keywords: nonlinear vibration; finite element; Donnell; Sanders; Nemeth

MSC 2010: 74K25

NOMENCLATURE

A1,A2: surface metrics along x and θ directions
R1,R2: principle radii of curvature along x and θ directions
CCˉ0CCˉ0: symmetric constitutive matrix for conical element
E∘E∘: reference surface total strain vector defined by eq. (7)
e11∘,e22∘,e12∘: linear deformation parameters defined by eq. (3)
AQAQ: characteristic polynomial matrix
K11K11,K˜12K˜12,K˜22K˜22: assembled first, second and third order structural stiffness matrices
MTMT, MSMS: assembled translational and structural mass matrices defined by eqs. (28) and (29)
NN: displacement field matrix of a finite element defined by eq. (23)
U_i(i = 1,2,3): displacements along the longitudinal, lateral and normal to surface directions, off the reference surface
u_i(i = 1,2,3): displacements along the longitudinal (U), lateral (V) and normal (W), on the reference surface
cNL,c1,c2,c3: flag parameters to define different shell theories
L: truncated cone element length
n_c: circumferential mode number
α_c: cone half angle
χ11∘,χ22∘,χ12∘,χ∘χ∘: linear deformation parameters defined by eq.(5)
δ_m: nodal degrees of freedom associated with mth node
ϵ11∘,ϵ22∘,ϵ∘ϵ∘: reference surface strains defined by eq. (4)
φ1,φ2, φ: linear rotation parameters defined by eq. (2)
ρ11,ρ22: geodesic radii of curvature radii of curvature along x and θ directions
δδe: vector of element degrees of freedom
δδ: vector of whole system degrees of freedom

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Received: 2018-12-11

Accepted: 2019-09-19

Published Online: 2019-10-15

Published in Print: 2020-02-25

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https://doi.org/10.1515/ijnsns-2018-0377

Keywords for this article

nonlinear vibration; finite element; Donnell; Sanders; Nemeth