Application of HDQM for the Analysis of Two-Dimensional Nonlinear Dynamics of Axially Accelerating Beam

2014 ◽  
Author(s):  
Dong-Mei Wang ◽  
Wei Zhang ◽  
Mu-Rong Li ◽  
Qian Wang
Keyword(s):  
Nonlinear Dynamics ◽  
Nonlinear Vibrations ◽  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Nonlinear Partial Differential Equations ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Chaotic Motions ◽  
Nonlinear Dynamical ◽  
Out Of Plane

In this paper, the two-dimensional nonlinear dynamics are investigated for the nonplanar nonlinear vibrations of an axially accelerating moving viscoelastic beam with in-plane and out-of-plane vibrations by the harmonic differential quadrature method (HDQM). The coupled nonlinear partial differential equations for the two-dimensional nonplanar nonlinear vibrations are discretized in space and time domains using HDQM and Runge-Kutta-Fehlberg methods respectively. Based on the numerical solutions, the nonlinear dynamical behaviors such as bifurcations and chaotic motions of the nonlinear system are investigated by using of the phase portrait and the bifurcation diagrams. The bifurcation diagrams for the in-plane and out-of-plane displacements via the mean axial velocity and the amplitude of velocity fluctuation are respectively presented while other parameters are fixed.

The Influence of Excitation Frequency on Dynamical Behaviors of Atomic Force Microscope

2012 ◽  
Vol 518-523 ◽  
pp. 3891-3895
Author(s):  
Ran Hui Liu ◽  
Qing Quan Hu
Keyword(s):  
Atomic Force Microscope ◽  
Numerical Solutions ◽  
Excitation Frequency ◽  
Excitation Amplitude ◽  
Bifurcation Diagrams ◽  
Governing Equations ◽  
Chaotic Motions ◽  
Atomic Force ◽  
Dynamical Behaviors ◽  
Amplitude Decrease

This paper deals with dynamical behaviors of Atomic Force Microscope in the different excitation frequency. By using Poincare maps, phase trajectory, Lyapunov exponent, bifurcation diagram, the dynamical behaviors are identified based on the numerical solutions of the governing equations. Bifurcation diagrams are presented in the case that the excitation amplitude increases while other parameters are fixed. Numerical simulations indicate that periodic and chaotic motions occur in the system. At the same, when chaotic motions occur, the excitation amplitude decrease as the excitation frequency increases.

scholarly journals Analytical Solution of Two-Dimensional Sine-Gordon Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Alemayehu Tamirie Deresse ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Test Problems ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Science And Engineering ◽  
Transform Method ◽  
Differential Transform ◽  
Good Agreement

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

scholarly journals A Composite Algorithm for Numerical Solutions of Two-Dimensional Coupled Burgers’ Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Vikas Kumar ◽  
Sukhveer Singh ◽  
Mehmet Emir Koksal
Keyword(s):  
Finite Difference ◽  
Exact Solutions ◽  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Dirichlet Boundary Conditions ◽  
Two Dimensional ◽  
Quadrature Method ◽  
B Spline ◽  
Burgers Equations

In this study, a new composite algorithm with the help of the finite difference and the modified cubic trigonometric B-spline differential quadrature method is developed. The developed method was applied to two-dimensional coupled Burgers’ equation with initial and Dirichlet boundary conditions for computational modeling. The established algorithm is better than the traditional differential quadrature algorithm proposed in literature due to more smoothness of cubic trigonometric B-spline functions. In the development of the algorithm, the first step is semidiscretization in time with the forward finite difference method. Furthermore, the obtained system is fully discretized by the modified cubic trigonometric B-spline differential quadrature method. Finally, we obtain coupled Lyapunov systems of linear equations, which are analyzed by the MATLAB solver for the system. Moreover, comparative study of these solutions with the numerical and exact solutions which are appeared in the literature is also discussed. Finally, it is found that there is good suitability between exact solutions and numerical solutions obtained by the developed composite algorithm. The technique can be extended for various multidimensional Burgers’ equations after some modifications.

On Global Transversality and Chaos in Two-Dimensional Nonlinear Dynamical Systems

2007 ◽  
Vol 2 (3) ◽  
pp. 242-248 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Approximate Expression ◽  
Melnikov Function ◽  
Two Dimensional ◽  
Chaotic Motions ◽  
Nonlinear Dynamical ◽  
Energy Increment ◽  
Exact Energy

In this paper, the global transversality and tangency in two-dimensional nonlinear dynamical systems are discussed, and the exact energy increment function (L-function) for such nonlinear dynamical systems is presented. The Melnikov function is an approximate expression of the exact energy increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are derived. Numerical simulations are carried out for illustrations of the analytical conditions. From analytical and numerical results, the simple zero of the energy increment (or the Melnikov function) may not imply that chaos exists. The conditions for the global transversality and tangency to the separatrix may be independent of the Melnikov function. Therefore, the analytical criteria for chaotic motions in nonlinear dynamical systems need to be further developed. The methodology presented in this paper is applicable to nonlinear dynamical systems without any separatrix.

scholarly journals Numerical solutions of nonlinear fractional model arising in the appearance of the strip patterns in two-dimensional systems

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Sunil Kumar ◽  
Amit Kumar ◽  
Shaher Momani ◽  
Mujahed Aldhaifallah ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Power Series ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Analytical Techniques ◽  
Approximate Solutions ◽  
Two Dimensional ◽  
Power Series Method ◽  
Analytical Technique ◽  
Residual Power Series ◽  
Residual Power

Abstract The main aim of this paper is to present a comparative study of modified analytical technique based on auxiliary parameters and residual power series method (RPSM) for Newell–Whitehead–Segel (NWS) equations of arbitrary order. The NWS equation is well defined and a famous nonlinear physical model, which is characterized by the presence of the strip patterns in two-dimensional systems and application in many areas such as mechanics, chemistry, and bioengineering. In this paper, we implement a modified analytical method based on auxiliary parameters and residual power series techniques to obtain quick and accurate solutions of the time-fractional NWS equations. Comparison of the obtained solutions with the present solutions reveal that both powerful analytical techniques are productive, fruitful, and adequate in solving any kind of nonlinear partial differential equations arising in several physical phenomena. We addressed $L_{2}$ L 2 and $L_{\infty }$ L ∞ norms in both cases. Through error analysis and numerical simulation, we have compared approximate solutions obtained by two present aforesaid methods and noted excellent agreement. In this study, we use the fractional operators in Caputo sense.

scholarly journals Different Approximations to the Solution of Upper-Convected Maxwell Fluid over a Porous Stretching Plate

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Vasile Marinca ◽  
Remus-Daniel Ene ◽  
Bogdan Marinca ◽  
Romeo Negrea
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Maxwell Fluid ◽  
Approximate Solutions ◽  
Two Dimensional ◽  
Rapid Convergence ◽  
Similarity Transformations ◽  
Optimal Homotopy Asymptotic Method ◽  
Stretching Plate ◽  
Good Agreement

In the present paper, we consider an incompressible magnetohydrodynamic flow of two-dimensional upper-convected Maxwell fluid over a porous stretching plate with suction and injection. The nonlinear partial differential equations are reduced to an ordinary differential equation by the similarity transformations and taking into account the boundary layer approximations. This equation is solved approximately by means of the optimal homotopy asymptotic method (OHAM). This approach is highly efficient and it controls the convergence of the approximate solutions. Different approximations to the solution are given, showing the exceptionally good agreement between the analytical and numerical solutions of the nonlinear problem. OHAM is very efficient in practice, ensuring a very rapid convergence of the solutions after only one iteration even though it does not need small or large parameters in the governing equation.

scholarly journals Modeling Transient Flows in Heterogeneous Layered Porous Media Using the Space–Time Trefftz Method

2021 ◽  
Vol 11 (8) ◽  
pp. 3421
Author(s):  
Cheng-Yu Ku ◽  
Li-Dan Hong ◽  
Chih-Yu Liu ◽  
Jing-En Xiao ◽  
Wei-Po Huang
Keyword(s):  
Porous Media ◽  
Time Domain ◽  
Numerical Solutions ◽  
Space Time ◽  
Two Dimensional ◽  
Transient Flows ◽  
Layered Porous Media ◽  
Time Marching ◽  
Time Basis ◽  
Marching Scheme

In this study, we developed a novel boundary-type meshless approach for dealing with two-dimensional transient flows in heterogeneous layered porous media. The novelty of the proposed method is that we derived the Trefftz space–time basis function for the two-dimensional diffusion equation in layered porous media in the space–time domain. The continuity conditions at the interface of the subdomains were satisfied in terms of the domain decomposition method. Numerical solutions were approximated based on the superposition principle utilizing the space–time basis functions of the governing equation. Using the space–time collocation scheme, the numerical solutions of the problem were solved with boundary and initial data assigned on the space–time boundaries, which combined spatial and temporal discretizations in the space–time manifold. Accordingly, the transient flows through the heterogeneous layered porous media in the space–time domain could be solved without using a time-marching scheme. Numerical examples and a convergence analysis were carried out to validate the accuracy and the stability of the method. The results illustrate that an excellent agreement with the analytical solution was obtained. Additionally, the proposed method was relatively simple because we only needed to deal with the boundary data, even for the problems in the heterogeneous layered porous media. Finally, when compared with the conventional time-marching scheme, highly accurate solutions were obtained and the error accumulation from the time-marching scheme was avoided.

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