0
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,
Lingang Development Zone,
Jinan, Shandong 250101, China

Zhao-Qing Wang

Institute of Engineering Mechanics,
Shandong Jianzhu University,
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,
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

## Abstract

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.

<>

## References

Hemeda, A. A. , 2012, “Homotopy Perturbation Method for Solving Systems of Nonlinear Coupled Equations,” Appl. Math. Sci., 6(96), pp. 4787–4800.
Kumar, D. , Singh, J. , and Rathore, S. , 2012, “Sumudu Decomposition Method for Nonlinear Equations,” Int. Math. Forum, 7(11), pp. 515–521.
Emad Az-Zo'bi , 2012, “Modified Laplace Decomposition Method,” World Appl. Sci. J., 18(11), pp. 1481–1486.
Abbas, Y. Al-B. , Ann, J. Al-S. , and Merna, A. S. , 2009, “A Multistage Adomian Decomposition Method for Solving the Autonomous Van der Pol System,” Aust. J. Basic Appl. Sci., 3(4), pp. 4397–4407.
Abassy, T. A. , El-Tawil, M. A. , and El-Zoheiry, H. , 2007, “Modified Variational Iteration Method for Boussinesq Equation,” Comput. Math. Appl., 54(7–8), pp. 955–965.
Simo, J. C. , and Armero, F. , 1992, “Geometrically Non-Linear Enhanced Strain Mixed Methods and the Method of Incompatible Modes,” Int. J. Numer. Methods Eng., 33(7), pp. 1413–1449.
Wriggers, P. , and Reese, S. , 1996, “A Note on Enhanced Strain Methods for Large Deformations,” Comput. Methods Appl. Mech. Eng., 135(3–4), pp. 201–209.
Cockburn, B. , 2001, “Devising Discontinuous Galerkin Methods for Non-Linear Hyperbolic Conservation Laws,” J. Comput. Appl. Math., 128(1–2), pp. 187–204.
Cockburn, B. , 2003, “Discontinuous Galerkin Methods,” Z. Angew. Math. Mech., 83(11), pp. 731–754.
Noels, L. , and Radovitzky, R. , 2006, “A New Discontinuous Galerkin Method for Non-Linear Mechanics,” AIAA Paper No. 2006-2122.
Izadian, J. , and Mohammadzade Attar, M. , 2012, “Numerical Solution of Deformation Equations in Homotopy Analysis Method,” Appl. Math. Sci., 6(8), pp. 357–367.
Han, Z. D. , Rajendran, A. M. , and Atluri, S. N. , 2005, “Meshless Local Petrov–Galerkin (MLPG) Approaches for Solving Nonlinear Problems With Large Deformations and Rotations,” Comput. Model. Eng. Sci., 10(1), pp. 1–12.
Zhao, T. , Wu, Y. , and Ma, H. , 2010, “Chebyshev–Legendre Pseudo-Spectral Methods for Nonclassical Parabolic Equations,” J. Inf. Comput. Sci., 7(8), pp. 1809–1817.
Lee, S. Y. , Lu, S. Y. , and Liu, Y. T. , 2008, “Exact Large Deflection Solutions for Timoshenko Beams With Nonlinear Boundary Conditions,” Comput. Model. Eng. Sci., 33(3), pp. 293–312.
Peng, J.-S. , Liu, Y. , and Yang, J. , 2010, “A Semianalytical Method for Nonlinear Vibration of Euler-Bernoulli Beams With General Boundary Conditions,” Math. Probl. Eng., 2010, p. 591786.
Hu, H. Y. , and Chen, J. S. , 2008, “Radial Basis Collocation Method and Quasi-Newton Iteration for Nonlinear Elliptic Problems,” Numer. Methods Partial Differ. Equations, 24(3), pp. 991–1017.
Yan, J.-P. , and Guo, B.-Y. , 2011, “A Collocation Method for Initial Value Problems of Second-Order ODEs by Using Laguerre Functions,” Numer. Math.: Theory Methods Appl., 4(2), pp. 283–295.
Rashidinia, J. , Ghasemi, M. , and Jalilian, R. , 2010, “A Collocation Method for the Solution of Nonlinear One-Dimensional Parabolic Equations,” Math. Sci., 4(1), pp. 87–104.
Shu, C. , Ding, H. , and Yeo, K. S. , 2004, “Solution of Partial Differential Equations by a Global Radial Basis Function-Based Differential Quadrature Method,” Eng. Anal. Boundary Elem., 28(10), pp. 1217–1226.
Ma, H. , and Qin, Q.-H. , 2008, “An Interpolation-Based Local Differential Quadrature Method to Solve Partial Differential Equations Using Irregularly Distributed Nodes,” Commun. Numer. Methods Eng., 24(7), pp. 573–584.
Liu, G. R. , and Wu, T. Y. , 2000, “Numerical Solution for Differential Equations of Duffing-Type Non-Linearity Using the Generalized Differential Quadrature Rule,” J. Sound Vib., 237(5), pp. 805–817.
Tomasiello, S. , 2003, “Simulating Non-Linear Coupled Oscillators by an Iterative Differential Quadrature Method,” J. Sound Vib., 265(3), pp. 507–525.
Pu, J.-P. , and Zheng, J.-J. , 2006, “Structural Dynamic Responses Analysis Applying Differential Quadrature Method,” J. Zhejiang Univ., Sci., A, 7(11), pp. 1831–1838.
Ordokhani, Y. , 2007, “A Collocation Method for Solving Nonlinear Differential Equations Via Hybrid of Rationalized Haar Functions,” J. Sci. Tarbiat Moallem Univ., 7(3), pp. 223–232.
Tsai, C.-C. , Liu, C.-S. , and Yeih, W.-C. , 2010, “Fictitious Time Integration Method of Fundamental Solutions With Chebyshev Polynomials for Solving Poisson-Type Nonlinear PDEs,” Comput. Model. Eng. Sci., 56(2), pp. 131–151.
Chen, L. , 2010, “An Integral Approach for Large Deflection Cantilever Beams,” Int. J. Non-Linear Mech., 45(3), pp. 301–305.
Zhu, T. , Zhang, J. , and Atluri, S. N. , 1998, “A Meshless Local Boundary Integral Equation (LBIE) Method for Solving Nonlinear Problems,” Comput. Mech., 22(2), pp. 174–186.
Li, S.-C. , and Wang, Z.-Q. , 2012, High Precision and Meshless Barycentric Interpolation Collocation Method: Algorithm, Program and Engineering Application, Beijing Science Press, Beijing.
Koçak, H. , and Yıldırım, A. , 2011, “An Efficient New Iterative Method for Finding Exact Solutions of Nonlinear Time-Fractional Partial Differential Equations,” Nonlinear Anal.: Modell. Control, 16(4), pp. 403–414.
Amirfakhrian, M. , and Keighobadi, S. , 2012, “Solution for Partial Differential Equations Involving Logarithmic Nonlinearities,” Aust. J. Basic Appl. Sci., 5(4), pp. 60–66.
Duangpithak, S. , 2012, “Variational Iteration Method for Special Nonlinear Partial Differential Equation,” Int. J. Math. Anal., 6(22), pp. 1071–1077.
Zhong, H.-Z. , and Lan, M.-Y. , 2006, “Solution of Nonlinear Initial-Value Problems by the Spline-Based Differential Quadrature Method,” J. Sound Vib., 296(4–5), pp. 908–918.
Gachpazan, M. , and Kamyad, A. V. , 2004, “Solving of Second Order Nonlinear PDE Problems by Using Artificial Controls With Controlled Error,” J. Appl. Math. Comput., 15(1–2), pp. 173–184.
Weideman, J. A. C. , and Reddy, S. C. , 2000, “A matlab Differentiation Matrix Suite,” ACM Trans. Math. Software, 26(4), pp. 465–519.
Floater, M. S. , and Hormann, K. , 2007, “Barycentric Rational Interpolation With No Poles and High Rates of Approximation,” Numer. Math., 107(2), pp. 315–331.
Nayfeh, A. H. , and Mook, D. T. , 1979, Nonlinear Oscillations, Wiley, New York.

## Figures

Fig. 1

A flow chart for the numerical procedure

Fig. 2

Simple pendulum

Fig. 3

The displacement of BRICM, LPM, and CPM

Fig. 4

The velocity of BRICM, LPM, and CPM

Fig. 5

The displacement of BRICM and LPM

Fig. 6

The velocity of BRICM and LPM

Fig. 7

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

Fig. 8

The displacement of BRICM, LPM, and CPM

Fig. 9

The velocity of BRICM, LPM, and CPM

Fig. 10

The displacement of BRICM and LPM

Fig. 11

The velocity of BRICM and LPM

Fig. 12

Particle on a rotating parabola

Fig. 13

The displacement of BRICM, LPM, and CPM

Fig. 14

The velocity of BRICM, LPM, and CPM

Fig. 15

The displacement of BRICM and LPM

Fig. 16

The velocity of BRICM and LPM

Fig. 17

Pendulum attached to rolling wheels

Fig. 18

The displacement of BRICM and CPM

Fig. 19

The velocity of BRICM and CPM

Fig. 20

The displacement of BRICM and RKM

Fig. 21

The velocity of BRICM and RKM

Fig. 22

Rigid rod rocks on a circular surface

Fig. 23

The displacement of BRICM and CPM

Fig. 24

The velocity of BRICM and CPM

Fig. 25

The displacement of BRICM and RKM

Fig. 26

The velocity of BRICM and RKM

## Errata

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections