A modified averaging operator with some applications

Anh Tay Nguyen, N. D. Anh
Author affiliations


  • Anh Tay Nguyen SUNY Korea, Incheon, Republic of Korea
  • N. D. Anh Institute of Mechanics, VAST, Hanoi, Vietnam




weighted local averaging, Galerkin method, buckling, Euler column, free vibration, strong nonlinearity


The paper presents a new approach to the conventional averaging in which the role of boundary values is considered in a more detailed way. It results in a new weighted local averaging operator (WLAO) taking into account the particular role of boundary values. A remarkable feature of WLAO is that this operator contains a parameter of boundary regulation p and depends on a local value h of the integration domain. By varying these two parameters one can regulate the obtained approximate solutions in order to get more accurate ones. It has been shown that the combination of WLAO with Galerkin method can lead to an effective approximate tool for the buckling problem of columns and for the frequency analysis of free vibration of strongly nonlinear systems. 


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J. T. Oden and J. N. Reddy. Variational methods in theoretical mechanics. Springer-Verlag, NY, (1976).

J. N. Reddy. Applied functional analysis and variational methods in engineering. Krieger Publishing Company, (1991).

J. N. Reddy. An introduction to continuum mechanics with applications. Cambridge University Press, (2008).

J. N. Reddy. Principles of continuum mechanics: A study of conservation principles with applications. Cambridge University Press, (2010).

J. N. Reddy. Energy principles and variational methods in applied mechanics. John Wiley & Sons, (2017).

J. T. Oden and J. N. Reddy. An introduction to the mathematical theory of finite elements. Wiley, (1976).

J. N. Reddy. An introduction to the finite element method. McGraw-Hill Education, (2005).

C. M.Wang, C. Y. Wang, and J. N. Reddy. Exact solutions for buckling of structural members. CRC Press, (2005).

I. Elishakoff, A. P. Ankitha, and A. Marzani. Rigorous versus na¨ıve implementation of the Galerkin method for stepped beams. Acta Mechanica, 230, (11), (2019), pp. 3861–3873. https://doi.org/10.1007/s00707-019-02393-z.

R. T. Fenner and J. N. Reddy. Mechanics of solids and structures. Second edition, CRC Press, (2012).

L. Euler. Die altitudine colomnarum sub proprio pondere corruentium. Acta Academiae Scientiarum Imperialis Petropolitanae, (1778). (in Latin).

V. P. Agrwal and H. H. Denman.Weighted linearization technique for period approximation in large amplitude non-linear oscillations. Journal of Sound and Vibration, 99, (4), (1985), pp. 463–473. https://doi.org/10.1016/0022-460x(85)90534-6.

Y. K. Cheung, S. H. Chen, and S. L. Lau. A modified Lindstedt-Poincar´e method for certain strongly non-linear oscillators. International Journal of Non-Linear Mechanics, 26, (3-4), (1991), pp. 367–378. https://doi.org/10.1016/0020-7462(91)90066-3.

H. S. Y. Chan, K. W. Chung, and Z. Xu. A perturbation-incremental method for strongly non-linear oscillators. International Journal of Non-Linear Mechanics, 31, (1), (1996), pp. 59–72. https://doi.org/10.1016/0020-7462(95)00043-7.

J. Cai, X.Wu, and Y. P. Li. An equivalent nonlinearization method for strongly nonlinear oscillations. Mechanics Research Communications, 32, (5), (2005), pp. 553–560. https://doi.org/10.1016/j.mechrescom.2004.10.004.

J.-H. He. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178, (3-4), (1999), pp. 257–262. https://doi.org/10.1016/s0045-7825(99)00018-3.

J.-H. He. Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B, 20, (10), (2006), pp. 1141–1199. https://doi.org/10.1142/s0217979206033796.

R. E. Mickens. Truly nonlinear oscillations: harmonic balance, parameter expansions, iteration, and averaging methods. World Scientific, (2010).

N. D. Anh, N. Q. Hai, and D. V. Hieu. The equivalent linearization method with a weighted averaging for analyzing of nonlinear vibrating systems. Latin American Journal of Solids and Structures, 14, (9), (2017), pp. 1723–1740. https://doi.org/10.1590/1679-78253488.

H.-L. Zhang. Periodic solutions for some strongly nonlinear oscillations by He’s energy balance method. Computers & Mathematics with Applications, 58, (11-12), (2009), pp. 2480–2485. https://doi.org/10.1016/j.camwa.2009.03.068.

H.-M. Liu. Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt–Poincare method. Chaos, Solitons & Fractals, 23, (2), (2005), pp. 577–579. https://doi.org/10.1016/j.chaos.2004.05.004.

M. Rafei, D. D. Ganji, H. Daniali, and H. Pashaei. The variational iteration method for nonlinear oscillators with discontinuities. Journal of Sound and Vibration, 305, (4-5), (2007), pp. 614–620. https://doi.org/10.1016/j.jsv.2007.04.020.

S.-Q.Wang and J.-H. He. Nonlinear oscillator with discontinuity by parameter-expansion method. Chaos, Solitons & Fractals, 35, (4), (2008), pp. 688–691. https://doi.org/10.1016/j.chaos.2007.07.055.

N. A. Khan, M. Jamil, and A. Ara. Multiple-parameter Hamiltonian approach for higher accurate approximations of a nonlinear oscillator with discontinuity. International Journal of Differential Equations, 2011, (2011), Article ID 649748. https://doi.org/10.1155/2011/649748.

B. S. Wu, W. P. Sun, and C. W. Lim. An analytical approximate technique for a class of strongly non-linear oscillators. International Journal of Non-Linear Mechanics, 41, (6-7), (2006), pp. 766–774. https://doi.org/10.1016/j.ijnonlinmec.2006.01.006.

R. E. Mickens. Mathematical and numerical study of the Duffing-harmonic oscillator. Journal of Sound and Vibration, 244, (3), (2001), pp. 563–567. https://doi.org/10.1006/jsvi.2000.3502.

C. W. Lim and B. S. Wu. A new analytical approach to the Duffing-harmonic oscillator. Physics Letters A, 311, (4-5), (2003), pp. 365–373. https://doi.org/10.1016/s0375-9601(03)00513-9.

S. B. Tiwari, B. Nageswara Rao, N. Shivakumar Swamy, K. S. Sai, and H. R. Nataraja. Analytical study on a Duffing-harmonic oscillator. Journal of Sound and Vibration, 285, (4-5), (2005), pp. 1217–1222. https://doi.org/10.1016/j.jsv.2004.11.001.

C. W. Lim, B. S. Wu, and W. P. Sun. Higher accuracy analytical approximations to the Duffing-harmonic oscillator. Journal of Sound and Vibration, 296, (4-5), (2006), pp. 1039–1045. https://doi.org/10.1016/j.jsv.2006.02.020.

H. Hu. Solutions of the Duffing-harmonic oscillator by an iteration procedure. Journal of Sound and Vibration, 298, (1-2), (2006), pp. 446–452. https://doi.org/10.1016/j.jsv.2006.05.023.

T. Ozis and A. Yildirim. Determination of the frequency-amplitude relation for a Duffing-harmonic oscillator by the energy balance method. Computers and Mathematics with Applications, 54, (7), (2007), pp. 1184–1187. https://doi.org/10.1016/j.camwa.2006.12.064.

A. Beléndez, D. I. Méndez, E. Fernández, S. Marini, and I. Pascual. An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method. Physics Letters A, 373, (32), (2009), pp. 2805–2809. https://doi.org/10.1016/j.physleta.2009.05.074.

M. Bayat, I. Pakar, and G. Domairry. Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: A review. Latin American Journal of Solids and Structures, 9, (2), (2012), pp. 1–93. https://doi.org/10.1590/s1679-78252012000200003.

K.-J. Bathe. Finite element procedures. Second edition, by Klaus-Jurgen Bathe, (2014).

L. Zhang and K.-J. Bathe. Overlapping finite elements for a new paradigm of solution. Computers & Structures, 187, (2017), pp. 64–76. https://doi.org/10.1016/j.compstruc.2017.03.008.

J. N. Reddy. Mechanics of laminated composite plates and shells: theory and analysis. Second edition, CRC Press, (2004).






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