A New Analytical Approach for Free Vibration, Buckling and Forced Vibration of Rectangular Nanoplates Based on Nonlocal Elasticity Theory

2018 ◽  
Vol 18 (04) ◽  
pp. 1850055 ◽  
Author(s):  
Dalun Rong ◽  
Junhai Fan ◽  
C. W. Lim ◽  
Xinsheng Xu ◽  
Zhenhuan Zhou
Keyword(s):  
Steady State ◽  
Free Vibration ◽  
Forced Vibration ◽  
Plate Theory ◽  
Nonlocal Elasticity Theory ◽  
Nonlocal Theory ◽  
Symplectic Space ◽  
Bending Moments ◽  
State Response ◽  
Unknown Vector

In this paper, an analytical Hamiltonian-based model for the dynamic analysis of rectangular nanoplates is proposed using the Kirchhoff plate theory and Eringen’s nonlocal theory. In a symplectic space, the dynamic problem is reduced to solving a unified Hamiltonian dual equation formed by a total unknown vector consisting of displacements, rotation angles, bending moments and generalized shear forces. The exact solutions for free vibration, buckling and steady state forced vibration are established by the eigenvalue analysis and expansion of eigenfunction without any trial functions. In addition, the explicit expressions of the characteristic equations, mode functions and steady state response of the nanoplate with two opposite edges that are simply supported or guided supported are obtained. To verify the accuracy and reliability of the present method, numerical results are compared with published solutions and excellent agreement is obtained. Comprehensive benchmark results that consider the nonlocal effect on the dynamic behaviors of rectangular nanoplates are also presented in dimensionless tabular and graphical forms.

NONLOCAL THEORY FOR BUCKLING OF NANOPLATES

2011 ◽  
Vol 11 (03) ◽  
pp. 411-429 ◽  
Author(s):  
S. C. PRADHAN ◽  
J. K. PHADIKAR
Keyword(s):  
Elasticity Theory ◽  
Scale Effect ◽  
Nonlocal Elasticity ◽  
Plate Theory ◽  
Nonlocal Elasticity Theory ◽  
Nonlocal Theory ◽  
Theoretical Development ◽  
Small Scale ◽  
Classical Plate Theory ◽  
Small Scale Effect

Classical plate theory (CLPT) and first-order shear deformation plate theory (FSDT) of plates are reformulated using the nonlocal elasticity theory. Developed nonlocal plate theories have been applied to study buckling behavior of nanoplates. Nonlocal elasticity theory, unlike traditional elasticity theory introduces a length scale parameter into the formulation to take into account the discrete structure of the material to some extent. Both single-layered and multilayered nanoplates have been included in the analysis. Navier's approach has been used to obtain exact solutions for buckling loads for simply supported boundary conditions. Dependence of the small scale effect on various geometrical and material parameters has been investigated. Present study reveals the presence of significant small scale effect on the buckling response of nanoplates. The theoretical development and the numerical results presented in the present work are expected to promote the use of nonlocal theories for more accurate prediction of stability behavior of nanoplates and nanoshells.

Determination of the Steady-State Response of Viscoelastically Supported Rectangular Orthotropic Mass Loaded Plates by an Energy-Based Finite Difference Method

2005 ◽  
Vol 11 (12) ◽  
pp. 1535-1552 ◽  
Author(s):  
Gökhan Altintaş ◽  
Muhiddin Bağci
Keyword(s):  
Steady State ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Plate Theory ◽  
Steady State Response ◽  
Algebraic Equations ◽  
State Response ◽  
Elastic Rectangular Plate ◽  
Spring Coefficient

A method based on a variational procedure in conjunction with a finite difference method is used to examine the free vibration characteristics and steady-state response to a sinusoidally varying force applied orthotropic elastic rectangular plate carrying masses. Using the energy-based finite difference method, the problem reduced to the solution of a system of algebraic equations. Due to the significance of the fundamental natural frequency of the plate, its variation is investigated with respect to the mechanical properties of the plate material, the translational spring coefficient of the supports, the mass distribution, the mass locations and the quantity of mass. The steady-state response of the viscoelastically supported plates was also investigated numerically for the damping coefficient of the supports and the force distribution in addition to the characteristics of the plate system. Many new results are presented and the validity of the present approach is demonstrated by comparing the results with other solutions based on the Kirchhoff-Love plate theory.

An Analytical Solution for Free Vibration of Piezoelectric Nanobeams Based on a Nonlocal Elasticity Theory

2015 ◽  
Vol 32 (2) ◽  
pp. 143-151 ◽  
Author(s):  
A. A. Jandaghian ◽  
O. Rahmani
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Nonlocal Parameter ◽  
Free Vibration Analysis ◽  
Nonlocal Elasticity Theory ◽  
Nonlocal Theory ◽  
Beam Model ◽  
Euler Beam ◽  
Charge Condition ◽  
Thin Beam

ABSTRACTIn the present study, an exact solution for free vibration analysis of piezoelectric nanobeams based on the nonlocal theory is obtained. The Euler beam model for a long and thin beam structure is employed, together with the electric potential satisfying the surface free charge condition for free vibration analysis. The governing equations and the boundary conditions are derived using Hamilton's principle. These equations are solved analytically for the vibration frequencies of beams with various end conditions. The model has been verified with the previously published works and found a good agreement with them. A detailed parametric study is conducted to discuss the influences of the nonlocal parameter, on the vibration characteristics of piezoelectric nanobeams. The exact vibration solutions should serve as benchmark results for verifying numerically obtained solutions based on other beam models and solution techniques.

FORCED VIBRATION OF TIP-MASSED CANTILEVER WITH NONLINEAR MAGNETIC INTERACTIONS

2014 ◽  
Vol 06 (02) ◽  
pp. 1450015 ◽  
Author(s):  
GUO-CE ZHANG ◽  
LI-QUN CHEN ◽  
HU DING
Keyword(s):  
Steady State ◽  
Forced Vibration ◽  
Viscoelastic Model ◽  
Magnetic Interactions ◽  
Stable Steady State ◽  
Steady State Response ◽  
Response Curves ◽  
Approximate Methods ◽  
State Response ◽  
Tip Mass

This paper predicts the nonlinear relative motion of a cantilever with a tip mass and magnets based on a distributed-parameter model. The Kelvin viscoelastic model is used to account for the damping in the cantilever. Under the harmonic base excitation, the governing equation is deduced via a coordinate transformation linked to its static equilibrium. To qualitatively validate those results captured from the approximate methods, the finite difference method and the multi-scale method are respectively employed to determine the natural frequency and the steady-state response of forced vibration. It is analytically demonstrated that those modes uninvolved in a certain resonance actually have no effect on the response amplitude of the stable steady-state motion. The effects of forcing amplitude, viscoelastic damping, and tip mass on the steady-state response are detailed via the amplitude–frequency response curves. For the first time, the current works theoretically illustrated the conversion from the hardening-type behavior to the softening-type versus the augmenting magnetic force, as well as the opposing effect of different tip masses on the first mode and the higher modes.

Nonlinear vibration and stability analysis of a size-dependent viscoelastic cantilever nanobeam with axial excitation

2020 ◽  
pp. 095440622095910
Author(s):  
Mohammad Noroozi ◽  
Majid Ghadiri
Keyword(s):  
Steady State ◽  
Bifurcation Point ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Multiple Scale ◽  
Harmonic Excitation ◽  
Steady State Response ◽  
Size Dependent ◽  
State Response ◽  
Point Types

In the present paper, nonlinear forced vibrations of an axial moving nanobeam which is vertically influenced by an external harmonic excitation and gravity is analyzed by considering the effects of linear damping. Considering certain assumptions, a nonlinear Euler-Bernoulli beam theory is developed. With the implementation of the nonlocal elasticity theory, the governing integro-partial-differential equation is obtained by using the Hamilton principle. The multiple scale method is employed to obtain a steady-state response for the size-dependent viscoelastic nanobeam with fixed-free boundary conditions. Subsequently, the trivial and non-trivial steady-state response and the bifurcation point types are examined. Finally, the effects of damping coefficient and nonlocal parameter on stability and bifurcation of trivial and non-trivial solutions are studied. It is found that the effect of nonlocal parameter on the steady-state response and the bifurcation point types is quite important.

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