scholarly journals Effect of Material Nonlinearity on Large Deflection of Variable-Arc-Length Beams Subjected to Uniform Self-Weight

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Chainarong Athisakul ◽  
Boonchai Phungpaingam ◽  
Gissanachai Juntarakong ◽  
Somchai Chucheepsakul
Keyword(s):  
Differential Equations ◽  
Large Deflection ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Optimization Technique ◽  
Material Constant ◽  
Constitutive Law ◽  
Arc Length ◽  
Stress Strain Relation ◽  
Equilibrium Configurations

This paper presents a large deflection of variable-arc-length beams, which are made from nonlinear elastic materials, subjected to its uniform self-weight. The stress-strain relation of materials obeys the Ludwick constitutive law. The governing equations of this problem, which are the nonlinear differential equations, are derived by considering the equilibrium of a differential beam element and geometric relations of a beam segment. The model formulation presented herein can be applied to several types of nonlinear elastica problems. With presence of geometric and material nonlinearities, the system of nonlinear differential equations becomes complicated. Consequently, the numerical method plays an important role in finding solutions of the presented problem. In this study, the shooting optimization technique is employed to compute the numerical solutions. From the results, it is found that there is a critical self-weight of the beam for each value of a material constantn. Two possible equilibrium configurations (i.e., stable and unstable configurations) can be found when the uniform self-weight is less than its critical value. The relationship between the material constantnand the critical self-weight of the beam is also presented.

Finite Deflections of a Nonlinearly Elastic Bar

1970 ◽  
Vol 37 (1) ◽  
pp. 48-52 ◽  
Author(s):  
J. T. Oden ◽  
S. B. Childs
Keyword(s):  
Differential Equations ◽  
Axial Force ◽  
Classical Theory ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Finite Deflections ◽  
Hyperbolic Tangent ◽  
Elastic Bar ◽  
Elastoplastic Materials ◽  
Nonlinearly Elastic

The problem of finite deflections of a nonlinearly elastic bar is investigated as an extension of the classical theory of the elastica to include material nonlinearities. A moment-curvature relation in the form of a hyperbolic tangent law is introduced to simulate that of a class of elastoplastic materials. The problem of finite deflections of a clamped-end bar subjected to an axial force is given special attention, and numerical solutions to the resulting system of nonlinear differential equations are obtained. Tables of results for various values of the parameters defining the material are provided and solutions are compared with those of the classical theory of the elastica.

scholarly journals Homotopy Perturbation Method

2017 ◽  
pp. 24-61
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Perturbation Method ◽  
Exact Solutions ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Homotopy Perturbation ◽  
Wide Range ◽  
Transfer Problems

The homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy perturbation method in comparison with the previous ones in solving heat transfer problems. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.

scholarly journals Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-14 ◽  
Author(s):  
Hijaz Ahmad ◽  
Tufail A. Khan ◽  
Predrag S. Stanimirović ◽  
Yu-Ming Chu ◽  
Imtiaz Ahmad
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Nonlinear Models ◽  
Nonlinear Differential Equations ◽  
Variational Iteration Method ◽  
Iteration Algorithm ◽  
Compact Finite Difference Method ◽  
Variational Iteration ◽  
Linear And Nonlinear ◽  
The Stability

Variational iteration method has been extensively employed to deal with linear and nonlinear differential equations of integer and fractional order. The key property of the technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The current study presents an improved algorithm to the variational iteration algorithm-II (VIA-II) for the numerical treatment of diffusion as well as convection-diffusion equations. This newly introduced modification is termed as the modified variational iteration algorithm-II (MVIA-II). The convergence of the MVIA-II is studied in the case of solving nonlinear equations. The main advantage of the MVIA-II improvement is an auxiliary parameter which makes sure a fast convergence of the standard VIA-II iteration algorithm. In order to verify the stability, accuracy, and computational speed of the method, the obtained solutions are compared numerically and graphically with the exact ones as well as with the results obtained by the previously proposed compact finite difference method and second kind Chebyshev wavelets. The comparison revealed that the modified version yields accurate results, converges rapidly, and offers better robustness in comparison with other methods used in the literature. Moreover, the basic idea depicted in this study is relied upon the possibility of the MVIA-II being utilized to handle nonlinear differential equations that arise in different fields of physical and biological sciences. A strong motivation for such applications is the fact that any discretization, transformation, or any assumptions are not required for this proposed algorithm in finding appropriate numerical solutions.

Large Deflections and Buckling Loads of Cantilever Columns with Constant Volume

2017 ◽  
Vol 17 (08) ◽  
pp. 1750091 ◽  
Author(s):  
Joon Kyu Lee ◽  
Byoung Koo Lee
Keyword(s):  
Differential Equations ◽  
Cross Section ◽  
Large Deflection ◽  
Nonlinear Differential Equations ◽  
Constant Volume ◽  
Large Deflections ◽  
Buckling Loads ◽  
Geometrical Nonlinear ◽  
Deflection Theory ◽  
Large Deflection Theory

This paper deals with the large deflections and buckling loads of tapered cantilever columns with a constant volume. The column member has a solid regular polygonal cross-section. The depth of this cross-section is functionally varied along the column axis. Geometrical nonlinear differential equations, which govern the buckled shape of the column, are derived using the large deflection theory, considering the effect of shear deformation. The buckling load of the column is approximately equivalent to the load under which a very small tip deflection occurs. In regard to the numerical results, both the elastica and buckling loads with varying column parameters are discussed. The configurations of the strongest column are also presented.

Snap-Through Phenomenon and Self-Contact of Spatial Elastica Subjected to Mid-Torque

2015 ◽  
Vol 07 (04) ◽  
pp. 1550057
Author(s):  
Boonchai Phungpaingam ◽  
Lawrence N. Virgin ◽  
Somchai Chucheepsakul
Keyword(s):  
Differential Equations ◽  
Shooting Method ◽  
Contact Point ◽  
Point Load ◽  
The Other ◽  
Arc Length ◽  
System Of Differential Equations ◽  
Kinematic Singularity ◽  
Euler Parameters ◽  
Equilibrium Configurations

This paper presents the snap-through phenomenon and effect of self-contact of the spatial elastica subjected to mid-length torque. One end of the elastica is clamped while the other end is placed in a sleeve joint. The total arc-length of the elastica can be varied by sliding the end through the sleeve joint. At a certain value of total arc-length, the sleeve joint is clamped and an external torque is applied at the mid-length of the elastica. The system of governing differential equations is derived from the equilibrium of an elastica segment and geometric relations of the inextensible elastica. The transformation matrix formulated in terms of Euler parameters is utilized to avoid the kinematic singularity. To display the behavior of the elastica, the system of differential equations needs to be integrated numerically from one end to the other end. The integration is performed so that the boundary conditions and some constraint conditions of the problem are satisfied, i.e., a shooting method is used. The effect of self-contact is taken into account by considering the contact force as a point load applying at contact point. From the results, the snap-through phenomenon, effect of self-contact and equilibrium configurations are highlighted herein.

A Comparative Assessment of Simplified Models for Simulating Parametric Roll

2017 ◽  
Vol 139 (2) ◽  
Author(s):  
Abhilash S. Somayajula ◽  
Jeffrey Falzarano
Keyword(s):  
Differential Equations ◽  
Analytical Solutions ◽  
Time Domain ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Phenomenon ◽  
Concept Design ◽  
Parametric Roll ◽  
Statistical Confidence ◽  
Restoring Forces

The motion of a ship/offshore platform at sea is governed by a coupled set of nonlinear differential equations. In general, analytical solutions for such systems do not exist and recourse is taken to time-domain simulations to obtain numerical solutions. Each simulation is not only time consuming but also captures only a single realization of the many possible responses. In a design spiral when the concept design of a ship/platform is being iteratively changed, simulating multiple realizations for each interim design is impractical. An analytical approach is preferable as it provides the answer almost instantaneously and does not suffer from the drawback of requiring multiple realizations for statistical confidence. Analytical solutions only exist for simple systems, and hence, there is a need to simplify the nonlinear coupled differential equations into a simplified one degree-of-freedom (DOF) system. While simplified methods make the problem tenable, it is important to check that the system still reflects the dynamics of the complicated system. This paper systematically describes two of the popular simplified parametric roll models in the literature: Volterra GM and improved Grim effective wave (IGEW) roll models. A correction to the existing Volterra GM model described in current literature is proposed to more accurately capture the restoring forces. The simulated roll motion from each model is compared against a corresponding simulation from a nonlinear coupled time-domain simulation tool to check its veracity. Finally, the extent to which each of the models captures the nonlinear phenomenon accurately is discussed in detail.

An implicit differential equation governing lumped capacitance, radiation dominated, unsteady, heat transfer

2012 ◽  
Vol 22 (7) ◽  
pp. 896-906
Author(s):  
Lawrence J. De Chant
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Simple Model ◽  
Differential Equations ◽  
Large Scale ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Implicit Differential Equation ◽  
Content Type ◽  
Integration Schemes

PurposeAlthough most physical problems in fluid mechanics and heat transfer are governed by nonlinear differential equations, it is less common to be confronted with a “so – called” implicit differential equation, i.e. a differential equation where the highest order derivative cannot be isolated. The purpose of this paper is to derive and analyze an implicit differential equation that arises from a simple model for radiation dominated heat transfer based upon an unsteady lumped capacitance approach.Design/methodology/approachHere we discuss an implicit differential equation that arises from a simple model for radiation dominated heat transfer based upon an unsteady lumped capacitance approach. Due to the implicit nature of this problem, standard integration schemes, e.g. Runge‐Kutta, are not conveniently applied to this problem. Moreover, numerical solutions do not provide the insight afforded by an analytical solution.FindingsA predictor predictor‐corrector scheme with secant iteration is presented which readily integrates differential equations where the derivative cannot be explicitly obtained. These solutions are compared to numerical integration of the equations and show good agreement.Originality/valueThe paper emphasizes that although large‐scale, multi‐dimensional time‐dependent heat transfer simulation tools are routinely available, there are instances where unsteady, engineering models such as the one discussed here are both adequate and appropriate.

scholarly journals Finite-Energy Dressed String-Inspired Dirac-Like Monopoles

Universe ◽  
2018 ◽  
Vol 5 (1) ◽  
pp. 8 ◽  
Author(s):  
Nikolaos E. Mavromatos ◽  
Sarben Sarkar
Keyword(s):  
Differential Equations ◽  
Shooting Method ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Finite Energy ◽  
Spherically Symmetric ◽  
New Class ◽  
The Standard Model ◽  
Monopole Solution ◽  
Asymptotic Analyses

On extending the Standard Model (SM) Lagrangian, through a non-linear Born–Infeld (BI) hypercharge term with a parameter β (of dimensions of [mass] 2 ), a finite energy monopole solution was claimed by Arunasalam and Kobakhidze. We report on a new class of solutions within this framework that was missed in the earlier analysis. This new class was discovered on performing consistent analytic asymptotic analyses of the nonlinear differential equations describing the model; the shooting method used in numerical solutions to boundary value problems for ordinary differential equations is replaced in our approach by a method that uses diagonal Padé approximants. Our work uses the ansatz proposed by Cho and Maison to generate a static and spherically-symmetric monopole with finite energy and differs from that used in the solution of Arunasalam and Kobakhidze. Estimates of the total energy of the monopole are given, and detection prospects at colliders are briefly discussed.

Sign in / Sign up

Export Citation Format

Share Document