• Ocean Systems Engineering
• /
• v.11 no.1
• /
• pp.59-81
• /
• 2021

The nonlinear formulation using the principle of virtual work-energy for free vibration of a large-sag extensible catenary riser in two dimensions is presented in this paper. A support at one end is hinged and the other is a free-sliding roller in the horizontal direction. The catenary riser has a large-sag configuration in the static equilibrium state and is assumed to displace with large amplitude to the motion state. The total virtual work of the catenary riser system involves the virtual strain energy due to bending, the virtual strain energy due to axial deformation, the virtual work done by the effective weight, and the inertia forces. The nonlinear equations of motion for two-dimensional free vibration in the Cartesian coordinate system is developed based on the difference between the Euler's equations in the static state and the displaced state. The linear and nonlinear stiffness matrices of the catenary riser are obtained and the eigenvalue problem is solved using the Galerkin finite element procedure. The natural frequencies and mode shapes are obtained. The results are validated with regard to the reference research addressing the accuracy and efficiency of the proposed nonlinear formulation. The numerical results for free vibration and the effect of the nonlinear behavior for catenary riser are presented.

• Steel and Composite Structures
• /
• v.41 no.6
• /
• pp.787-803
• /
• 2021

The size-dependent nonlinear thermomechanical vibration analysis of pre- and post-buckled tapered two-directional functionally graded (2D-FG) microbeams is presented in this study. In the context of the modified couple stress theory, the formulations are derived based on the parabolic shear deformation beam theory and von Karman nonlinear strains. Different thermomechanical material properties are assumed to be temperature-dependent and smoothly vary in both length and thickness directions using the power law and the physical neutral axis concept is employed. The nonlinear governing equations are derived using the Hamilton principle and the resulting variable coefficient equations of motion are solved using the differential quadrature method (DQM) and iterative Newton's method for clamped-clamped and simply supported boundary conditions. Comparison studies are presented to validate the derived model and solution procedure. The impacts of induced thermal moments, temperature power index, two gradient indices, nonuniform cross-section, and microstructure length scale parameter on the frequency-temperature configurations are explored for both clamped and simply supported microbeams.

• Steel and Composite Structures
• /
• v.45 no.5
• /
• pp.641-652
• /
• 2022

Fluid-conveying tubes are widely used to transport oil and natural gas in industries. As an advanced composite material, functionally graded carbon nanotube-reinforced composites (FG-CNTRC) have great potential to empower the industry. However, nonlinear free vibration of the FG-CNTRC fluid-conveying pipe has not been attempted in thermal environment. In this paper, the nonlinear free vibration characteristic of functionally graded nanocomposite fluid-conveying pipe reinforced by single-walled carbon nanotubes (SWNTs) in thermal environment is investigated. The SWCNTs gradient distributed in the thickness direction of the pipe forms different reinforcement patterns. The material properties of the FG-CNTRC are estimated by rule of mixture. A higher-order shear deformation theory and Hamilton's variational principle are employed to derive the motion equations incorporating the thermal and fluid effects. A two-step perturbation method is implemented to obtain the closed-form asymptotic solutions for these nonlinear partial differential equations. The nonlinear frequencies under several reinforcement patterns are presented and discussed. We conduct a series of studies aimed at revealing the effects of the flow velocity, the environment temperature, the inner-outer diameter ratio, and the carbon nanotube volume fraction on the nature frequency.

• Steel and Composite Structures
• /
• v.50 no.2
• /
• pp.149-158
• /
• 2024

Considering that different boundary conditions can have an important impact on structural vibration characteristics. In this paper, the nonlinear forced vibration behavior of functionally graded material (FGM) doubly curved shells with initial geometric imperfections under different boundary conditions is studied. Considering initial geometric imperfections and von Karman geometric nonlinearity, the nonlinear governing equations of FGM doubly curved shells are derived using Reissner's first order shear deformation (FOSD) theory. Three different boundary conditions of four edges simply supported (SSSS), four edges clamped (CCCC), clamped-clamped-simply-simply (CCSS) were studied, and a system of nonlinear ordinary differential equations was obtained with the help of Galerkin principle. The nonlinear forced vibration response of the FGM doubly curved shell is obtained by using the modified Lindstedt Poincare (MLP) method. The accuracy of this method was verified by comparing it with published literature. Finally, the effects of curvature ratio, power law index, void coefficient, prestress, and initial geometric imperfections on the resonance of FGM doubly curved shells under different boundary conditions are fully discussed. The relevant research results can provide certain guidance for the design and application of doubly curved shell.

• Journal of the Chungcheong Mathematical Society
• /
• v.25 no.3
• /
• pp.381-391
• /
• 2012

In this paper, the formal linearization method is used to construct multisoliton solutions for three model of shallow water waves equations. The three models are completely integrable. The formal linearization method is an efficient method for obtaining exact multisoliton solutions of nonlinear partial differential equations. The method can be applied to nonintegrable equations as well as to integrable ones.

• Journal of the Korean Mathematical Society
• /
• v.37 no.6
• /
• pp.1031-1042
• /
• 2000

The equations prescribed in Ω⊂R(sup)n are called loaded, if they contain some operations of the traces of desired solution on manifolds (of dimension which is strongly less than n) from closure Ω. These equations result from approximations of nonlinear equations by linear ones, in the problems of optimal control when the control when the control actions depends on a part of independent variables, in investigations of the inverse problems and so on. In present work we study the nonlocal boundary value problems for first-order loaded differential operator equations. Criterion of unique solvability is established. We illustrate the obtained results by examples.

• Journal of applied mathematics & informatics
• /
• v.30 no.5_6
• /
• pp.981-992
• /
• 2012

Integro-differential equations arise in modeling various physical and engineering problems. Several numerical and analytical methods have been developed to solving such equations. We introduce the NHPM for solving nonlinear integro-differential equations. Several examples for solving integro-differential equations are presented to illustrate the efficiency of the proposed NHPM.

• Communications of the Korean Mathematical Society
• /
• v.12 no.2
• /
• pp.467-478
• /
• 1997

A nonlinear Galerkin method for the Burgers equation is considered. Due to the lack of the divergence free condition, the nonlinear term is treated differently compared to that of the Navier-Stokes equations. Strong convergence results are proved for the nonlinear Galerkin method.

• Journal of the Chungcheong Mathematical Society
• /
• v.7 no.1
• /
• pp.141-152
• /
• 1994

We consider a numerical scheme for solving a nonlinear boundary integral equation (BIE) obtained by reformulation the nonlinear boundary value problem (BVP). We give a simple alternative to the standard collocation method for the nonlinear BIE. This method consists of one conventional linear system and another coupled linear system resulting from an auxiliary BIE which is obtained by differentiating both side of the nonlinear interior integral equations. We obtain an analogue BIE through the perturbation of the fundamental solution of Laplace's equation. We procure the super-convergence of approximate solutions.

• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.4
• /
• pp.883-893
• /
• 2011

In this paper we study h-stability of the solutions of nonlinear difference system via the notion of $n_{\infty}$-summable similarity between its variational systems. Also, we show that two concepts of h-stability and h-stability in variation for nonlinear difference systems are equivalent. Furthermore, we characterize h-stability for nonlinear difference systems by using Lyapunov functions.