scholarly journals The nonlinear thermomechanical vibration of a functionally graded beam on Winkler-Pasternak foundation

2018 ◽  
Vol 148 ◽  
pp. 13004 ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herisanu
Keyword(s):  
Differential Equation ◽  
Elastic Foundation ◽  
Geometric Nonlinearity ◽  
Beam Theory ◽  
Functionally Graded ◽  
Pasternak Foundation ◽  
Bernoulli Beam ◽  
Functionally Graded Beam ◽  
Pasternak Elastic Foundation ◽  
Euler Bernoulli Beam

By using the Optimal Auxiliary Functions Method (OAFM), nonlinear free thermomechanical vibration of functionally graded beam (FGB) on Winkler-Pasternak elastic foundation is studied. Based on von Karman geometric nonlinearity, on Euler-Bernoulli beam theory and also on Galerkin procedure we obtain a second-order nonlinear differential equation with quadratic and cubic nonlinear terms. The results obtained by means of OAFM are compared and shown to be in an excellent agreement with available solutions known in the literature.

Postbuckling and nonlinear free vibration of size-dependent prestressed FG nanobeams resting on elastic foundation based on nonlocal Euler-Bernoulli beam theory

2015 ◽  
Vol 24 (3-4) ◽  
pp. 91-103 ◽  
Author(s):  
Sima Ziaee
Keyword(s):  
Differential Equation ◽  
Elastic Foundation ◽  
Axial Force ◽  
Beam Theory ◽  
Neutral Axis ◽  
Small Scale ◽  
Bernoulli Beam ◽  
Nonlinear Free Vibration ◽  
Size Dependent ◽  
Euler Bernoulli Beam

AbstractVibrations of micro/nanobeams that are subjected to initial stresses due to mismatch between different materials or thermal stresses are important in some devices. The present study is an attempt to present nonlinear free vibration of simply supported size-dependent functionally graded (FG) nanobeams resting on elastic foundation and under precompressive axial force. It is assumed that the material properties of FG materials are graded in the thickness direction. The partial differential equation of motion, which is simplified into an ordinary differential equation using the Galerkin method, is derived based on Euler-Bernoulli beam theory, von Karman geometric nonlinearity, and Eringen’s nonlocal elasticity theory. The final ordinary differential equation is solved using the variational iteration method. The effects of geometrical parameters, small-scale parameter, elastic coefficient of foundation, precompressive axial force, and neutral axis location on dimensionless nonlinear natural frequencies are investigated. In this study, the buckling and postbuckling behavior of FG nanobeams and the effect of neutral axis location on buckling behavior are investigated as well. Results show that the effects of small scale on FG nanobeam frequencies change with the aspect ratio, the values of radius of gyration, and the values of compressive axial force. It is also found that the influence of neutral axis location on the nonlinear fundamental frequency of prestressed FG nanobeams is more than that of prestressed FG nanobeams resting on elastic foundation.

scholarly journals Multi-moving Loads Induced Vibration of FG Sandwich Beams Resting on Pasternak Elastic Foundation

2021 ◽  
Vol 18 (9) ◽  
Author(s):  
Wachirawit SONGSUWAN ◽  
Monsak PIMSARN ◽  
Nuttawit WATTANASAKULPONG
Keyword(s):  
Dynamic Response ◽  
Elastic Foundation ◽  
Excitation Frequency ◽  
Beam Theory ◽  
Equation Of Motion ◽  
Functionally Graded ◽  
Sandwich Beams ◽  
Moving Loads ◽  
Layer Thickness Ratio ◽  
Pasternak Elastic Foundation

The dynamic behavior of functionally graded (FG) sandwich beams resting on the Pasternak elastic foundation under an arbitrary number of harmonic moving loads is presented by using Timoshenko beam theory, including the significant effects of shear deformation and rotary inertia. The equation of motion governing the dynamic response of the beams is derived from Lagrange’s equations. The Ritz and Newmark methods are implemented to solve the equation of motion for obtaining free and forced vibration results of the beams with different boundary conditions. The influences of several parametric studies such as layer thickness ratio, boundary condition, spring constants, length to height ratio, velocity, excitation frequency, phase angle, etc., on the dynamic response of the beams are examined and discussed in detail. According to the present investigation, it is revealed that with an increase of the velocity of the moving loads, the dynamic deflection initially increases with fluctuations and then drops considerably after reaching the peak value at the critical velocity. Moreover, the distance between the loads is also one of the important parameters that affect the beams’ deflection results under a number of moving loads.

Nonlinear Vibration of Nanobeam With Quadratic Rational Bezier Arc Curvature

2014 ◽  
Author(s):  
Hassan Askari ◽  
Zia Saadatnia ◽  
Ebrahim Esmailzadeh
Keyword(s):  
Nonlinear Vibration ◽  
Natural Frequency ◽  
Governing Equation ◽  
Oscillation Amplitude ◽  
Beam Theory ◽  
Nonlinear Ordinary Differential Equation ◽  
Pasternak Foundation ◽  
Bernoulli Beam ◽  
Linear Frequency ◽  
Euler Bernoulli Beam

Nonlinear vibration of nanobeam with the quadratic rational Bezier arc curvature is investigated. The governing equation of motion of the nanobeam based on the Euler-Bernoulli beam theory is developed. Accordingly, the non-uniform rational B-spline (NURBS) is implemented in order to write the implicit form of the governing equation of the structure. The simply-supported boundary conditions are assumed and the Galerkin procedure is utilized to find the nonlinear ordinary differential equation of the system. The nonlinear natural frequency of the system is found and the effects of different parameters, namely, the waviness amplitude, oscillation amplitude, aspect ratio, curvature shape and the Pasternak foundation coefficient are fully investigated. The hardening and softening responses of the natural frequency of structure are detected for variations of the shape and amplitude of the curvature waviness. It is revealed that the ratio of nonlinear to linear frequency increases with an increase in the oscillation amplitudes. It is found that by increasing the Pasternak foundation coefficient, the ratio of nonlinear to linear frequency decreases.

Damping of Vibrations of Layered Elastic-Viscoelastic Beams

1993 ◽  
Vol 46 (11S) ◽  
pp. S305-S311 ◽  
Author(s):  
Richard B. Hetnarski ◽  
Ray A. West ◽  
Joseph S. Torok
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Beam Theory ◽  
Equation Of Motion ◽  
Bernoulli Beam ◽  
Viscoelastic Effects ◽  
Viscoelastic Layers ◽  
Euler Bernoulli Beam ◽  
Damping Of Vibrations ◽  
Fourth Order Differential Equation

A five-layer cantilever beam consisting of an elastic core, two symmetric viscoelastic layers, and two elastic constraining layers is considered. The viscoelastic effects are incorporated in the Euler-Bernoulli beam theory. If the contraction and extension of the constraining layers is neglecterd a fourth order differential equation of motion is received. Inclusion of contraction and extension of the constraining layers results in a more accurate sixth order differential equation. Appropriate boundary conditions are derived. Laplace transforms are used extensively. Both the analytical solution and the numerical results are presented.

DYNAMIC STABILITY OF A BEAM ON AN ELASTIC FOUNDATION INCLUDING STRESS WAVE EFFECTS

2012 ◽  
Vol 04 (02) ◽  
pp. 1250017 ◽  
Author(s):  
YING LIU ◽  
G. LU
Keyword(s):  
Dynamic Stability ◽  
Elastic Foundation ◽  
Stress Wave ◽  
Axial Load ◽  
Elastic Beam ◽  
Beam Theory ◽  
Stress Wave Propagation ◽  
Bernoulli Beam ◽  
Wave Effects ◽  
Euler Bernoulli Beam

This paper examines the dynamic stability of an elastic beam on the elastic foundation, in which the stress wave effect is taken into account. Based on Euler–Bernoulli beam theory, the dynamic response of the elastic beam on the elastic foundation to a small transverse perturbation is analyzed. By considering the stress wave propagation in the beam and the constraint of the elastic foundation, the critical bifurcation condition of elastic beam is derived, and the critical axial load of the elastic beam is predicted. Furthermore, the effects of the elastic foundation and the beam length on buckling condition are discussed by using numeric examples. Finally, an approximate solution of critical axial load for elastic beam on the elastic foundation is provided, which may be used to investigate elastic beam buckling problem.

Geometrically Nonlinear Free Vibration of Composite Materials: Clamped-Clamped Functionally Graded Beam with an Edge Crack Using Homogenisation Method

2017 ◽  
Vol 730 ◽  
pp. 521-526 ◽  
Author(s):  
Mohcine Chajdi ◽  
El Bekkaye Merrimi ◽  
Khalid El Bikri
Keyword(s):  
Free Vibration ◽  
Edge Crack ◽  
Volume Fraction ◽  
Crack Depth ◽  
Beam Theory ◽  
Functionally Graded ◽  
Geometrically Nonlinear ◽  
Nonlinear Free Vibration ◽  
Functionally Graded Beam ◽  
Euler Bernoulli Beam

The problem of geometrically nonlinear free vibration of a clamped-clamped functionally graded beam containing an open edge crack in its center is studied in this paper. The study is based on Euler-Bernoulli beam theory and Von Karman geometric nonlinearity assumptions. The cracked section is modeled by an elastic spring connecting two intact segments of the beam. It is assumed that material properties of the functionally graded composites are graded in the thickness direction and estimated through the rule of mixture. The homogenisation method is used to reduce the problem to that of isotropic homogeneous cracked beam. Direct iterative method is employed for solving the eigenvalue equation for governing the frequency nonlinear vibration, in order to show the effect of the crack depth and the influences of the volume fraction on the dynamic response.

scholarly journals Nonlinear Analysis of a Functionally Graded Beam Resting on the Elastic Nonlinear Foundation

2014 ◽  
Vol 44 (2) ◽  
pp. 71-82 ◽  
Author(s):  
M. Arefi
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Beam Theory ◽  
Functionally Graded ◽  
Functionally Graded Beam ◽  
Nonlinear Foundation ◽  
Simply Supported ◽  
Nonlinear Responses ◽  
Fg Beam ◽  
Linear And Nonlinear

Abstract This paper evaluates the nonlinear responses of a function- ally graded (FG) beam resting on a nonlinear foundation. After derivation of fundamental nonlinear differential equation using the Euler-Bernouli beam theory, a semi analytical method has been used to study the response of the problem. The responses can be evaluated for both linear and nonlinear isotropic and FG beams individually. Adomians Decomposition and successive approximation methods have been used for solution of nonlinear differential equation. As numerical investigation, the beams with simply supported ends and linear and nonlinear foundations have been analyzed using this method.

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