Influence of Geometric Non-Linearities on the Free Vibrations of Orthotropic Open Cylindrical Shells

Abstract

SUMMARYThis paper presents a general approach to predict the influence of geometric non-linearities on the free vibration of elastic, thin, orthotropic and non-uniform open cylindrical shells. The open shells are assumed to be freely simply supported along their curved edges and to have arbitrary straight edge boundary conditions. The method is a hybrid of finite element and classical thin shell theories. The solution is divided into two parts. In part one, the displacement functions are obtained from Sanders' linear shell theory and the mass and linear stiffness matrices are obtained by the finite element procedure. In part two, the modal coefficients derived from the Sanders-Koiter non-linear theory of thin shells are obtained for these displacement functions. Expressions for the second-and third-order non-linear stiffness matrices are then determined through the finite element method. The non-linear equation of motion is solved by the fourth-order Runge-Kutta numerical method. The linear and non-linear natural frequency variations are determined as a function of shell amplitudes for different cases. The results obtained reveal that the frequencies calculated by this method are in good agreement with those obtained by other authors.

Nonlinear vibrations of clamped-free circular cylindrical shells

“…Each model has a different number of degrees of freedom (dofs). In particular, results for 18,20,22,24,26,28,30,40,43 and 47 dofs are presented. The response curves become less and less hardening increasing the number of degrees of freedom, reaching convergence for the model with 43 dofs, which shows a very mild softening behavior.…”

“…[3][4][5][6][7][8][9][10][11]. Only some researchers used more refined nonlinear shell theories [12][13][14][15][16][17][18][19][20][21][22][23][24][25], as the Novozhilov, the Sanders-Koiter (also referred as Sanders) and the Flügge-Lur'e-Byrne nonlinear shell theory, or included shear deformation and rotary inertia. Numerical differences among the four most popular classical nonlinear shell theories have been numerically investigated in Ref.…”

Free-vibration analysis of laminated shells via refined MITC9 elements

“…The generalized Hooke's law is considered, by employing a linear constitutive model for infinitesimal deformations. In a composite material, these equations are obtained in material coordinates (1,2,3) for each orthotropic layer k and then rotated in the general curvilinear reference system (α, β, z). Therefore, the stress-strain relations after the rotation are:…”

“…In most of the practical problems, the solution demand applications of approximated computational methods. Selmane and Lakis [1], [2] have presented a method for the dynamic and static analysis of thin, elastic, anisotropic, and nonuniform open cylindrical shells. The open shells are assumed to be simply supported along their curved edges and to have arbitrary straight-edge boundary conditions.…”

Non-linear Vibrations of Deep Cylindrical Shells by thep-Version Finite Element Method

“…There are some applications of multi-degree-of-freedom models based on finite elements to study the dynamics of non-shallow shells, but these are generally restrained to short time spans and to transient vibrations. In [18], a finite element method model for geometrically non-linear free vibrations of thin, closed or open, shells was presented. Before carrying out the numerical calculations, the equations of motion were transformed into modal coordinates and the non-diagonal terms of the transformed non-linear stiffness matrices were discarded.…”