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Research Papers

Barycentric Rational Interpolation Iteration Collocation Method for Solving Nonlinear Vibration Problems

[+] Author and Article Information
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;
Institute of Engineering Mechanics,
Shandong Jianzhu University,
1000 Fengming Road,
Lingang Development Zone,
Jinan, Shandong 250101, China

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

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

Bing-Tao Tang

Institute of Engineering Mechanics,
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 11(2), 021001 (Aug 26, 2015) (13 pages) Paper No: CND-13-1250; doi: 10.1115/1.4030979 History: Received October 18, 2013

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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References

Figures

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Fig. 1

A flow chart for the numerical procedure

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Fig. 3

The displacement of BRICM, LPM, and CPM

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Fig. 4

The velocity of BRICM, LPM, and CPM

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Fig. 5

The displacement of BRICM and LPM

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Fig. 6

The velocity of BRICM and LPM

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Fig. 7

Current-carrying wire in the field of an infinite current-carrying conductor

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Fig. 8

The displacement of BRICM, LPM, and CPM

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Fig. 9

The velocity of BRICM, LPM, and CPM

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Fig. 10

The displacement of BRICM and LPM

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Fig. 11

The velocity of BRICM and LPM

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Fig. 12

Particle on a rotating parabola

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Fig. 13

The displacement of BRICM, LPM, and CPM

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Fig. 14

The velocity of BRICM, LPM, and CPM

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Fig. 15

The displacement of BRICM and LPM

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Fig. 16

The velocity of BRICM and LPM

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Fig. 17

Pendulum attached to rolling wheels

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Fig. 18

The displacement of BRICM and CPM

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Fig. 19

The velocity of BRICM and CPM

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Fig. 20

The displacement of BRICM and RKM

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Fig. 21

The velocity of BRICM and RKM

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Fig. 22

Rigid rod rocks on a circular surface

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Fig. 23

The displacement of BRICM and CPM

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Fig. 24

The velocity of BRICM and CPM

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Fig. 25

The displacement of BRICM and RKM

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Fig. 26

The velocity of BRICM and RKM

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