In this article, a powerful computational methodology, named as barycentric rational interpolation iteration collocation method (BRICM), for obtaining the numerical solutions of nonlinear vibration problems is presented. The nonlinear vibration problems are governed by initial-value problems of nonlinear differential equations. Given an initial guess value of the unknown function, the nonlinear differential equations can be transformed into linear differential equations. By applying barycentric rational interpolation and differential matrix, the linearized differential equation is discretized into algebraic equations in the matrix form. The latest solution of nonlinear differential equation is obtained by solving the algebraic equations. The numerical solution of nonlinear vibration problem can be calculated by iteration method under given control precision. Then, the velocity and acceleration can be obtained by differential matrix of barycentric rational interpolation, and the period of nonlinear vibration is also computed by BRICM. Some examples of nonlinear vibration demonstrate the proposed methodological advantages of effectiveness, simple formulations, and high precision.

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Institute of Engineering Mechanics,

Shandong Jianzhu University,

1000 Fengming Road,

Lingang Development Zone,

Jinan, Shandong 250101, China
Institute of Engineering Mechanics,

Shandong Jianzhu University,

1000 Fengming Road,

Lingang Development Zone,

Jinan, Shandong 250101, China

e-mail: sdjzujiang@gmail.com
Center for Marine Geotechnical

Engineering Research,

Department of Civil Engineering,

State Key Laboratory of Ocean Engineering,

Shanghai Jiao Tong University,

Shanghai 200240, China
Institute of Engineering Mechanics,

Shandong Jianzhu University,

1000 Fengming Road,

Lingang Development Zone,

Jinan, Shandong 250101, China

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March 2016

Research-Article

# Barycentric Rational Interpolation Iteration Collocation Method for Solving Nonlinear Vibration Problems

Jian Jiang,

Jian Jiang

Center for Marine Geotechnical

Engineering Research,

Department of Civil Engineering,

State Key Laboratory of Ocean Engineering,

Shanghai Jiao Tong University,

Shanghai 200240, China;

Engineering Research,

Department of Civil Engineering,

State Key Laboratory of Ocean Engineering,

Shanghai Jiao Tong University,

Shanghai 200240, China;

Institute of Engineering Mechanics,

Shandong Jianzhu University,

1000 Fengming Road,

Lingang Development Zone,

Jinan, Shandong 250101, China

Shandong Jianzhu University,

1000 Fengming Road,

Lingang Development Zone,

Jinan, Shandong 250101, China

Search for other works by this author on:

Zhao-Qing Wang,

Zhao-Qing Wang

Institute of Engineering Mechanics,

Shandong Jianzhu University,

1000 Fengming Road,

Lingang Development Zone,

Jinan, Shandong 250101, China

e-mail: sdjzujiang@gmail.com

Shandong Jianzhu University,

1000 Fengming Road,

Lingang Development Zone,

Jinan, Shandong 250101, China

e-mail: sdjzujiang@gmail.com

Search for other works by this author on:

Jian-Hua Wang,

Jian-Hua Wang

Center for Marine Geotechnical

Engineering Research,

Department of Civil Engineering,

State Key Laboratory of Ocean Engineering,

Shanghai Jiao Tong University,

Shanghai 200240, China

Engineering Research,

Department of Civil Engineering,

State Key Laboratory of Ocean Engineering,

Shanghai Jiao Tong University,

Shanghai 200240, China

Search for other works by this author on:

Bing-Tao Tang

Bing-Tao Tang

Institute of Engineering Mechanics,

Shandong Jianzhu University,

1000 Fengming Road,

Lingang Development Zone,

Jinan, Shandong 250101, China

Shandong Jianzhu University,

1000 Fengming Road,

Lingang Development Zone,

Jinan, Shandong 250101, China

Search for other works by this author on:

Jian Jiang

Center for Marine Geotechnical

Engineering Research,

Department of Civil Engineering,

State Key Laboratory of Ocean Engineering,

Shanghai Jiao Tong University,

Shanghai 200240, China;

Engineering Research,

Department of Civil Engineering,

State Key Laboratory of Ocean Engineering,

Shanghai Jiao Tong University,

Shanghai 200240, China;

Shandong Jianzhu University,

1000 Fengming Road,

Lingang Development Zone,

Jinan, Shandong 250101, China

Zhao-Qing Wang

Shandong Jianzhu University,

1000 Fengming Road,

Lingang Development Zone,

Jinan, Shandong 250101, China

e-mail: sdjzujiang@gmail.com

Jian-Hua Wang

Engineering Research,

Department of Civil Engineering,

State Key Laboratory of Ocean Engineering,

Shanghai Jiao Tong University,

Shanghai 200240, China

Bing-Tao Tang

Shandong Jianzhu University,

1000 Fengming Road,

Lingang Development Zone,

Jinan, Shandong 250101, China

1Corresponding author.

Manuscript received October 18, 2013; final manuscript received June 27, 2015; published online August 26, 2015. Assoc. Editor: Carmen M. Lilley.

*J. Comput. Nonlinear Dynam*. Mar 2016, 11(2): 021001 (13 pages)

**Published Online:**August 26, 2015

Article history

Received:

October 18, 2013

Revision Received:

June 27, 2015

Citation

Jiang, J., Wang, Z., Wang, J., and Tang, B. (August 26, 2015). "Barycentric Rational Interpolation Iteration Collocation Method for Solving Nonlinear Vibration Problems." ASME. *J. Comput. Nonlinear Dynam*. March 2016; 11(2): 021001. https://doi.org/10.1115/1.4030979

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