scholarly journals Free Vibration Analysis of Triclinic Nanobeams Based on the Differential Quadrature Method

2019 ◽  
Vol 9 (17) ◽  
pp. 3517 ◽  
Author(s):  
Behrouz Karami ◽  
Maziar Janghorban ◽  
Rossana Dimitri ◽  
Francesco Tornabene
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Strain Gradient ◽  
Differential Quadrature Method ◽  
Free Vibration Analysis ◽  
Differential Quadrature ◽  
Vibration Response ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Beam Model

In this work, the nonlocal strain gradient theory is applied to study the free vibration response of a Timoshenko beam made of triclinic material. The governing equations of the problem and the associated boundary conditions are obtained by means of the Hamiltonian principle, whereby the generalized differential quadrature (GDQ) method is implemented as numerical tool to solve the eigenvalue problem in a discrete form. Different combinations of boundary conditions are also considered, which include simply-supports, clamped supports and free edges. Starting with some pioneering works from the literature about isotropic nanobeams, a convergence analysis is first performed, and the accuracy of the proposed size-dependent anisotropic beam model is checked. A large parametric investigation studies the effect of the nonlocal, geometry, and strain gradient parameters, together with the boundary conditions, on the vibration response of the anisotropic nanobeams, as useful for practical engineering applications.

Free nonlinear vibration analysis of nano-truncated conical shells based on modified strain gradient theory

2021 ◽  
pp. 146442072110419
Author(s):  
Alireza Sheykhi ◽  
Shahrokh Hosseini-Hashemi ◽  
Adel Maghsoudpour ◽  
Shahram E Haghighi
Keyword(s):  
Boundary Conditions ◽  
Strain Gradient ◽  
Couple Stress ◽  
Equations Of Motion ◽  
Differential Quadrature ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Modified Strain Gradient Theory ◽  
Conical Shells ◽  
Modified Couple Stress

In this study, the nonlinear free vibrations behaviour of nano-truncated conical shells was analysed, using the first-order shear deformable shell model. The analysis took into account the structure size through modified strain gradient theory, and differential quadrature and Fréchet derivative methods in von Kármán-Donnell-type approach to kinematic nonlinearity. The governing equations were obtained, utilizing Hamilton's principle. Partial differential equations plus the non-classical and classical boundary conditions were used to obtain the shells’ equations of motion. Discretizing the boundary conditions and equations of motion were performed based on a generalized differential quadrature analogy. The eigenvalue system was considered based on the harmonic balance technique. The Galerkin and Fréchet derivative approaches were used to determine the nonlinear free vibration behaviour of the carbon nano-cone, which was modelled in the simply- and clamped-supported boundary conditions. Comparisons were made between the findings from the new model versus the couple and classical stress theories, indicating that the classical and modified couple stress theories are distinct representations of modified strain gradient theory. The results also revealed that the degree of hardening of nano-truncated conical shells in the modified strain gradient theory is less than that of modified couple stress and classical theories. This led to a rise in the non-dimensional amplitude and frequency ratios. This study investigated the effect of size on free nonlinear vibrations of nano-truncated conical shells for various apex angles and lengths. Finally, we evaluated and compared our findings versus those reported by previous studies, which confirmed the precision and accuracy of our results.

NONLINEAR STRAIN GRADIENT THEORY BASED VIBRATION AND INSTABILITY OF BORON NITRIDE MICRO-TUBES CONVEYING FERROFLUID

2014 ◽  
Vol 06 (05) ◽  
pp. 1450060 ◽  
Author(s):  
ALI GHORBANPOUR ARANI ◽  
ABDOLREZA JALILVAND ◽  
REZA KOLAHCHI
Keyword(s):  
Magnetic Field ◽  
Boron Nitride ◽  
Boundary Conditions ◽  
Electric Fields ◽  
Strain Gradient ◽  
Differential Quadrature Method ◽  
Fluid Velocity ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Nonlinear Frequency

Nonlinear vibration and instability of a boron nitride micro-tube (BNMT) conveying ferrofluid under the combined magnetic and electric fields are investigated. Based on Euler–Bernoulli beam (EBB), piezoelasticity strain gradient theory and Hamilton's principle, high order equations of motion are derived for three boundary conditions namely as clamped–clamped (C–C), simply–simply (S–S) and clamped–simply (C–S). The differential quadrature method (DQM) is applied to discretize the motion equations in order to obtain the nonlinear frequency and critical fluid velocity using a direct iterative method. A detailed parametric study is conducted to elucidate the influences of the various boundary conditions, size diameter and magnetic field on vibrational characteristic of BNMT. Numerical results indicate that the effect of magnetic field appears in higher speed of ferrofluid and increases the critical velocity or enlarges the stability region. The results are in good agreement with the previous researches. The results of this study can be used to manufacture smart micro/nano electromechanical systems in advanced biomechanics applications with magnetic and electric fields as parametric controllers.

Free Vibration Analysis of Functionally Graded Moderately Thick Annular Sector Plates Using Differential Quadrature Method

2011 ◽  
Vol 110-116 ◽  
pp. 2990-2998 ◽  
Author(s):  
S.H. Mirtalaie ◽  
M.A. Hajabasi ◽  
F. Hejripour
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Volume Fraction ◽  
Differential Quadrature Method ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Differential Quadrature ◽  
Radius Ratio ◽  
Quadrature Method ◽  
Annular Sector

In this paper, the free vibration of moderately thick annular sector plates made of functionally graded materials is studied using the Differential Quadrature Method (DQM). The material properties of the functionally graded plate are assumed to vary continuously through the thickness, according to a power-law distribution. The governing differential equations of motion are derived based on the First order Shear Deformation plate Theory (FSDT) and then solved numerically using DQM under different boundary conditions. The results for the isotropic plates which are derivable with this approach are presented and compared with the literature and they are in good agreement. The natural frequencies of the functionally graded moderately thick annular sector plates under various combinations of clamped, simple supported and free boundary conditions are presented for the first time. The effects of boundary conditions, sector angle, radius ratio, thickness to outer radius ratio, volume fraction exponent and variation of the Poisson’s ratio on the free vibration behavior of the plate are studied

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