On the solution of nonlinear algebraic equations following periodic forced response analysis of nonlinear structures using different nonlinear solvers

Date

2021-01-01

Author

Kizilay, H. Sefa
Ciğeroğlu, Ender

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

233
views

0
downloads

In periodic forced response analysis of nonlinear structures, most of the time analytical solutions cannot be obtained due to the complex behavior of the nonlinearity and/or due to the number of nonlinear equations to be solved. Therefore, numerical methods are widely used. For periodic forced response analysis of nonlinear systems, generally Harmonic Balance Method (HBM) or Describing Function Method (DFM), which transform the nonlinear differential equations into a set of nonlinear algebraic equations, are used. In the literature, there exist several nonlinear algebraic equation solvers based on Newton’s method which have different convergence properties and computational expense. In this paper, comparison of computational performance of different nonlinear algebraic equation solvers are studied where, solvers with different convergence order are selected based on the number of Jacobian matrix and vector function evaluations. In order to compare the performance of these selected nonlinear solvers, a lumped parameter model with cubic stiffness nonlinearity is considered. Several case studies are performed and nonlinear solvers are compared to each other in terms of solution time based on error tolerance used.

Subject Keywords

Newton’s method, Newton-like solvers, Nonlinear algebraic equations, Numerical solution, Nonlinear vibrations, Harmonic balance method

URI

https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85091564359&origin=inward
https://hdl.handle.net/11511/90909

DOI

https://doi.org/10.1007/978-3-030-47626-7_20

Conference Name

38th IMAC, A Conference and Exposition on Structural Dynamics, 2020

Collections

Department of Mechanical Engineering, Conference / Seminar

Suggestions

OpenMETU
Core

On general form of the Tanh method and its application to nonlinear partial differential equations Hamidoğlu, Ali (American Institute of Mathematical Sciences (AIMS), 2016-6) The tanh method is used to compute travelling waves solutions of one-dimensional nonlinear wave and evolution equations. The technique is based on seeking travelling wave solutions in the form of a finite series in tanh. In this article, we introduce a new general form of tanh transformation and solve well-known nonlinear partial differential equations in which tanh method becomes weaker in the sense of obtaining general form of solutions.
On General Form of Tanh Method and Its Application to Medical Problems Ali, Hamidoglu (2016-09-02) The tanh method is used to compute travelling waves solutions of one-dimensional non-linear wave and evolution equations. The technique is based on seeking travelling wave solutions in the form of a finite series in tanh. However, the mentioned method is not always efficient method to solve some types of one dimensional non-linear partial differential equations in more general sense. In this article, we construct new general transformation of tanh function which is more effective in the sense of getting gen...
A Novel Computational Method to Calculate Nonlinear Normal Modes of Complex Structures Samandarı, Hamed; Ciğeroğlu, Ender (2019-01-31) In this study, a simple and efficient computational approach to obtain nonlinear normal modes (NNMs) of nonlinear structures is presented. Describing function method (DFM) is used to capture the nonlinear internal forces under periodic motion. DFM has the advantage of expressing the nonlinear internal force as a nonlinear stiffness matrix multiplied by a displacement vector, where the off-diagonal terms of the nonlinear stiffness matrix can provide a comprehensive knowledge about the coupling between the mo...
On the consistency of the solutions of the space fractional Schroumldinger equation (vol 53, 042105, 2012) Bayin, Selcuk S. (2012-08-01) Recently we have reanalyzed the consistency of the solutions of the space fractional Schroumldinger equation found in a piecewise manner, and showed that an exact and a proper treatment of the relevant integrals prove that they are consistent. In this comment, for clarity, we present additional information about the critical integrals and describe how their analytic continuation is accomplished.
Numerical solution of nonlinear reaction-diffusion and wave equations Meral, Gülnihal; Tezer, Münevver; Department of Mathematics (2009) In this thesis, the two-dimensional initial and boundary value problems (IBVPs) and the one-dimensional Cauchy problems defined by the nonlinear reaction- diffusion and wave equations are numerically solved. The dual reciprocity boundary element method (DRBEM) is used to discretize the IBVPs defined by single and system of nonlinear reaction-diffusion equations and nonlinear wave equation, spatially. The advantage of DRBEM for the exterior regions is made use of for the latter problem. The differential quad...

Citation Formats

H. S. Kizilay and E. Ciğeroğlu, “On the solution of nonlinear algebraic equations following periodic forced response analysis of nonlinear structures using different nonlinear solvers,” presented at the 38th IMAC, A Conference and Exposition on Structural Dynamics, 2020, Texas, Amerika Birleşik Devletleri, 2021, Accessed: 00, 2021. [Online]. Available: https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85091564359&origin=inward.