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Barycentric Rational Interpolation Iteration Collocation Method For Solving Nonlinear Vibration Problems

Posted on:2016-11-22Degree:MasterType:Thesis
Country:ChinaCandidate:J JiangFull Text:PDF
GTID:2180330461499459Subject:Solid mechanics
Abstract/Summary:PDF Full Text Request
In the project, most of problems are nonlinear phenomenon. Only when facing weak nonlinear factors and no affect on the final results of problem, the elastic theory can be applied to simplify the problem. But for many nonlinear problems, such unreasonable simplification has great errors and even has fundamentally transformed the problem This paper aims to put forward a valid numerical method, barycentric rational interpolation iteration collocation method, for the solution of nonlinear problems in engineering. The controlling equation of nonlinear problems is governed by nonlinear differential equations. Firstly, the linear iterative format of nonlinear differential equations is derived by using direct linearization method or Newton linearization method.Then the nonlinear differential equation is discretized into linear algebraic equation by using barycentric rational interpolation differential matrix. Given control accuracy, the numerical solution of nonlinear algebraic equations is obtained through the calculation of linear iterative. Abundant examples of nonlinear problem will demonstrate the proposed methodological advantages of effectiveness, simple formulations and high precision according to the comparison between numerical solution and analytical solution.Paper has carried out the following research work:1. Describes and discusses the simple, effective and high precision characteristics of barycentric rational interpolation collocation method. Amount of literature and engineering have respectively demonstrate the proposed methodological advantages.2. Detailedly introduce the application of linearization methods, when iteratively calculate nonlinear problems. And then, respectively state the applying process of direct linearization method and Newton linearization method in analyzing nonlinear problems.3. Wield barycentric rational interpolation iterative collocation method to analysis single degree of freedom nonlinear vibration systems and give the derivation process. Clearly show the structure of the proposed method in analyzing engineering problems.4. This method is applied to analysis multiple degrees of freedom nonlinear vibration problems further. Demonstrate the proposed methodological advantages of effectiveness, simple formulations and high precision by analysis and comparison.5. Expand the research of the proposed method to analysis continuum nonlinear vibration. According to the deeper levels of analysis, demonstrate again the proposed methodological advantages of wider applicability and high precision. The combination of the Gauss integral method also is powerful to exploit the using range of the proposed method.For solving the nonlinear problems in the project, the article put forwards barycentric rational interpolation iteration collocation method by integrating direct linearization method, Newton linearization method and iterative method as well as Gauss integral method into barycentric rational interpolation collocation method.Barycentric rational interpolation function is smoothness with infinite time and not exist pole in the whole calculation area. When facing Chebyshev point and equidistant node, barycentric rational interpolation collocation method shows excellent numerical accuracy and computational efficiency. It is given full play to the simple and effective advantage of barycentric rational interpolation collocation method that nonlinear differential equation is transformed into linear differential equation by applying linearization method and iterative method. This article will provide barycentric rational interpolation iterative collocation method to analysis nonlinear problems.
Keywords/Search Tags:nonlinear vibration problem, barycentric rational interpolation collocation method, linearization method, iterative method, differential matrix
PDF Full Text Request
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