Some Intriguing Results Pertaining to Functionally Graded Columns

2013 ◽  
Vol 80 (4) ◽  
Author(s):  
Isaac Elishakoff ◽  
Yohann Miglis
Keyword(s):  
Inverse Method ◽  
Vibration Mode ◽  
Flexural Rigidity ◽  
Axial Loading ◽  
Functionally Graded ◽  
Concentrated Load ◽  
Closed Form Solutions ◽  
Order Polynomial ◽  
Governing Differential Equation ◽  
Fourth Order Polynomial

Some intriguing results are reported in conjunction with closed form solutions obtained for a clamped-free vibrating inhomogeneous column under an axial concentrated load using the semi-inverse method. Fourth order polynomial is postulated for both the vibration mode shape and buckling displacement. Solution is provided for the flexural rigidity and the natural frequency. It is shown that, for each level of axial loading, there may exist up to five flexural rigidities satisfying the governing differential equation and boundary conditions.

SOME CONVENTIONAL AND UNCONVENTIONAL EDUCATIONAL COLUMN STABILITY PROBLEMS

2006 ◽  
Vol 06 (01) ◽  
pp. 139-151 ◽  
Author(s):  
I. ELISHAKOFF ◽  
C. GENTILINI ◽  
R. SANTORO
Keyword(s):  
Differential Equation ◽  
Inverse Method ◽  
Order Differential Equation ◽  
Flexural Rigidity ◽  
Functionally Graded ◽  
The Novel ◽  
Main Approach ◽  
Graded Materials ◽  
Closed Form Solutions ◽  
The Difference

Two interesting problems are considered for enriching the curriculum of the Strength of Materials course, in the light of recently developed functionally graded materials (FGMs), characterized with the smooth variation of the elastic modulus. These are problems associated with buckling of columns with variable flexural rigidity along the axis of the column. A simple semi-inverse method is proposed for determining closed-form solutions of axially inhomogeneous columns. In order for the presentation to be given in one package, the conventional problems are also recapitulated along with the novel ones. The main approach adopted here is the use of the second-order differential equation, instead of the fourth-order one, obtained by integrating twice and having a physical meaning, since it could be derived based on moment equilibrium. This is essentially the same idea as utilized in the textbook by Timoshenko and Gere;1 the difference is that this paper develops a semi-inverse formulation.

Displacement Analysis of the Generalized Clemens Coupling, The R-R-S-R-R Spatial Linkage

1975 ◽  
Vol 97 (2) ◽  
pp. 575-580 ◽  
Author(s):  
D. M. Wallace ◽  
F. Freudenstein
Keyword(s):  
Fourth Order ◽  
Constant Velocity ◽  
Degrees Of Freedom ◽  
Input Output ◽  
Closed Form Solutions ◽  
Input And Output ◽  
Order Polynomial ◽  
Special Case ◽  
Output Equation ◽  
Fourth Order Polynomial

The Clemens Coupling is a constant-velocity, universal-type joint for nonparallel intersecting shafts. This mechanism is a spatial linkage with five links connected by four revolute pairs, R, and one spherical pair (ball-and-socket joint), S, which is located symmetrically with respect to the input and output shafts. The Clemens Coupling is a special case of the R-R-S-R-R spatial linkage with general proportions, which will, therefore, be called the Generalized Clemens Coupling. This paper gives the algebraic derivation of the input-output equation for the general R-R-S-R-R linkage and demonstrates that it is a fourth-order polynomial in the half tangents of the crank angles. The effect of housing-error tolerances on the displacements of the Clemens Coupling has also been considered. The results demonstrate feasibility of closed-form solutions for five-link mechanisms with kinematic pairs having more than two degrees of freedom.

scholarly journals Thermoelastic analysis of FGM hollow cylinder for variable parameters and temperature distributions using FEM

2020 ◽  
Vol 9 (1) ◽  
pp. 256-264
Author(s):  
Dinkar Sharma ◽  
Ramandeep Kaur
Keyword(s):  
Stress Field ◽  
Hollow Cylinder ◽  
Numerical Study ◽  
Axial Stress ◽  
Circumferential Stress ◽  
Functionally Graded ◽  
Gradient Index ◽  
Variable Parameters ◽  
Graded Material ◽  
Governing Differential Equation

AbstractThis paper presents, numerical study of stress field in functionally graded material (FGM) hollow cylinder by using finite element method (FEM). The FGM cylinder is subjected to internal pressure and uniform heat generation. Thermoelastic material properties of FGM cylinder are assumed to vary along radius of cylinder as an exponential function of radius. The governing differential equation is solved numerically by FEM for isotropic and anistropic hollow cylinder. Additionally, the effect of material gradient index (β) on normalized radial stresses, normalized circumferential stress and normalized axial stress are evaluated and shown graphically. The behaviour of stress versus normalized radius of cylinder is plotted for different values of Poisson’s ratio and temperature. The graphical results shown that stress field in FGM cylinder is influenced by some of above mentioned parameters.

Analysis of Bifurcations for a Functionally Graded Materials Plate

2013 ◽  
Vol 300-301 ◽  
pp. 988-991 ◽  
Author(s):  
Wei Qin Yu
Keyword(s):  
Functionally Graded Materials ◽  
Functionally Graded ◽  
Periodic Motions ◽  
Graded Materials ◽  
Normal Form Theory ◽  
Closed Form Solutions ◽  
Simply Supported ◽  
Form Theory ◽  
Bifurcation And Stability ◽  
Numerical Approaches

Using the analytical and numerical approaches, the nonlinear dynamic behaviors in the vicinity of a compound critical point are studied for a simply supported functionally graded materials (FGMs) rectangular plate. Normal form theory, bifurcation and stability theory are used to find closed form solutions for equilibria and periodic motions. Stability conditions of these solutions are obtained explicitly and critical boundaries are also derived. Finally, numerical results are presented to confirm the analytical predictions

Manufacturing Process for Circular-Arc Curvilinear Cylindrical Gears with Predesigned Fourth Order Transmission Error

2010 ◽  
Vol 37-38 ◽  
pp. 623-627 ◽  
Author(s):  
Jin Zhan Su ◽  
Zong De Fang
Keyword(s):  
Fourth Order ◽  
Gear Tooth ◽  
Transmission Error ◽  
Circular Arc ◽  
Cylindrical Gears ◽  
Polynomial Curve ◽  
Order Polynomial ◽  
Tooth Surfaces ◽  
The Stability ◽  
Fourth Order Polynomial

A fourth order transmission error was employed to improve the stability and tooth strength of circular-arc curvilinear cylindrical gears. The coefficient of fourth order polynomial curve was determined, the imaginary rack cutter which formed by the rotation of a head cutter and the imaginary pinion were introduced to determine the pinion and gear tooth surfaces, respectively. The numerical simulation of meshing shows: 1) the fourth order transmission error can be achieved by the proposed method; 2) the stability transmission can be performed by increasing the angle of the transfer point of the cycle of meshing; 3) the tooth fillet strength can be enhanced.

Effect of Temperature Gradient on the Mechanical Buckling Resistance of FGM Shallow Arches

2014 ◽  
Author(s):  
M. Bateni ◽  
M. R. Eslami
Keyword(s):  
Temperature Gradient ◽  
Thermal Environment ◽  
Mechanical Load ◽  
Functionally Graded ◽  
Effect Of Temperature ◽  
Power Law Distribution ◽  
Concentrated Load ◽  
Shallow Arches ◽  
Buckling Resistance ◽  
Graded Material

This work presents a closed form investigation on the effect of temperature gradient on the buckling resistance of functionally graded material (FGM) shallow arches. The constituents are assumed to vary smoothly through the thickness of the arch according to the power law distribution and they are assumed to be temperature dependent. The arches subjected to the both uniform distributed radial load and central concentrated load and both boundary supports are supposed to be pinned. The temperature field is approximated by one-dimensional linear gradient through the thickness of the arch and the displacement field approximated by classical arches model. Also, Donnell type kinematics is utilized to extract the suitable strain-displacement relations for shallow arches. Adjacent equilibrium criterion is used to buckling analysis, and, critical bifurcation load is obtain in the complete presence of pre-buckling deformations. Results discloses the usefulness of using the FGM shallow arches in thermal environment because the temperature gradient enhances the buckling resistance of these structures when they are subjected to a lateral mechanical load.

Thermo-Mechanical Buckling Analysis of Functionally Graded Skew Laminated Plates with Initial Geometric Imperfections

2018 ◽  
Vol 10 (02) ◽  
pp. 1850014 ◽  
Author(s):  
Sanjay Singh Tomar ◽  
Mohammad Talha
Keyword(s):  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Laminated Plates ◽  
Geometric Imperfections ◽  
Efficiency And Effectiveness ◽  
Governing Differential Equation ◽  
Parametric Studies ◽  
Mechanical Buckling ◽  
Initial Geometric Imperfections ◽  
Continuous Displacement

The aim of the present study is to investigate thermo-mechanical buckling response of skew functionally graded laminated plates (FGLP) with initial geometric imperfections. The formulation has been performed using Reddy’s higher order shear deformation theory (HSDT) with the [Formula: see text] continuous displacement field. A nine-noded isoparametric element has been employed to discretize the domain of the plate. Variational principle has been used to derive the governing differential equation of the problem. Several examples with various comparison and parametric studies have been shown to prove the efficiency and effectiveness of the present formulation. The numerical results have been highlighted with different system parameters and boundary conditions.

Explicit solutions to nonlinear Chen–Lee–Liu equation

2021 ◽  
pp. 2150438
Author(s):  
Lanre Akinyemi ◽  
Najib Ullah ◽  
Yasir Akbar ◽  
Mir Sajjad Hashemi ◽  
Arzu Akbulut ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Nonlinear Phenomena ◽  
Explicit Solutions ◽  
Mathematical Algorithm ◽  
New Exact Solutions ◽  
Order Polynomial ◽  
Fourth Order Polynomial

In this work, a generalized [Formula: see text]-expansion method has been used for solving the nonlinear Chen–Lee–Liu equation. This method is a more common, general, and powerful mathematical algorithm for finding the exact solutions of nonlinear partial differential equations (NPDEs), where [Formula: see text] follows the Jacobi elliptic equation [Formula: see text], and we let [Formula: see text] be a fourth-order polynomial. Many new exact solutions such as the hyperbolic, rational, and trigonometric solutions with different parameters in terms of the Jacobi elliptic functions are obtained. The distinct solutions obtained in this paper clearly explain the importance of some physical structures in the field of nonlinear phenomena. Also, this method deals very well with higher-order nonlinear equations in the field of science. The numerical results described in the plots were obtained by using Maple.

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