A Study on the Analytical Solution using Homotopy Perturbation Method of Shell-like Shallow Trusses

Title & Authors
A Study on the Analytical Solution using Homotopy Perturbation Method of Shell-like Shallow Trusses
Shon, Su-Deok; Lee, Seung-Jae;

Abstract
In this study, an analytical solution using the homotopy perturbation method of shell-like shallow trusses was investigated. The dynamic unstable phenomenon that appears in shallow space trusses is sensitive to initial conditions, and its characteristics are analyzed and its instability is examined by obtaining the exact solution or through numerical analysis. In this process, obtaining a more precise solution plays a very important role, and obtaining an analytical solution expressed in infinite terms is one way to get a more accurate result. A homotopy equation, in this study, was derived by formulating a governing equation for a shell-like shallow truss, and a semi-analytical solution was obtained by homotopy perturbation method. Besides, sensitive unstable phenomena were examined and their solutions were compared according to various dynamic behaviors and initial conditions, and the shapes of attractors in the phase space were observed. In conclusion, the analytical solution of the simple nonlinear model using the homotopy method proposed in this study was excellent when compared with the numerical analysis result, and reflected the nonlinear unstable phenomenon of the shell-like shallow truss which was the target model of analysis.
Keywords
Homotopy perturbation method;Analytical solution;Shell-like shallow truss;Dynamic instability;Runge-Kutta method;
Language
Korean
Cited by
References
1.
Adomian, G., & Rach, R. (1992). Generalization of adomian polynomials to functions of several variables. Computers & mathematics with Applications, 24, 11-24.

2.
Alnasr, M., & Erjaee, G. (2011). Application of the multistage homotopy perturbation method to some dynamical systems. International Journal of Science & Technology, A1, 33-38.

3.
Ario, I. (2004). Homoclinic bifurcation and chaos attractor in elastic two-bar truss. International Journal of Non-Linear Mechanics, 39(4), 605-617.

4.
Barrio, R., Blesa, F., & Lara, M. (2005). VSVO formulation of the Taylor method for the numerical solution of ODEs. Computers & mathematics with Applications, 50, 93-111.

5.
Bi, Q., & Dai, H. (2000). Analysis of non-linear dynamics and bifurcations of a shallow arch subjected to periodic excitation with internal resonance, Journal of Sound and Vibration, 233(4), 557-571.

6.
Blair, K., Krousgrill, C., & Farris, T. (1996). Non-linear dynamic response of shallow arches to harmonic forcing. Journal of Sound and Vibration, 194(3), 353-367.

7.
Blendez, A., & Hernandez, T. (2007). Application of He's homotopy perturbation method to the doffing-harmonic oscillator. International Journal of Nonlinear Science and Numerical Simulation, 8(1), 79-88.

8.
Budiansky, B., & Roth, R. (1962). Axisymmetric dynamic buckling of clamped shallow spherical shells; Collected papers on instability of shells structures. NASA TN D-1510, Washington DC, 597-606.

9.
Chen, J., & Li, Y. (2006). Effects of elastic foundation on the snap-through buckling of a shallow arch under a moving point load. International Journal of Solids and Structures, 43, 4220-4237.

10.
Chowdhury, M., & Hashim, I. (2008). Analytical solutions to heat transfer equations by homotopy perturbation method revisited. Physics Letters A, 372, 1240-1243.

11.
Chowdhury, M., Hashim, I., & Abdulaziz, O. (2007). Application of homotopy perturbation method to nonlinear population dynamics models. Physics Letters A, 368, 251-258.

12.
Chowdhury, M., Hashim, I., & Momani, S. (2009). The multistage homotopy-perturbation method: A powerful scheme for handling the Lorenz system. Chaos Solitons & Fractal, 40, 1929-1937.

13.
Chowdhury, S. (2011). A Comparison between the modified homotopy perturbation method and adomain decomposition method for solving nonlinear heat transfer equations. Journal of Applied Sciences, 11(8), 1416-1420.

14.
Compean, F., Olvera, D., Campa, F., Lopez, L., Elias-Zuniga, A., & Rodriguez C. (2012). Characterization and stability analysis of a multivariable milling tool by the enhanced multistage homotopy perturbation method. International Journal of Machine Tools & Manufacture, 57, 27-33.

15.
De Rosa, M., & Franciosi, C. (2000). Exact and approximate dynamic analysis of circular arches using DQM. International Journal of Solids and Structures, 37, 1103-1117.

16.
Ganji, D. (2006). The application of He's homotopy perturbation method to nonlinear equations arising in heat transfer. Physics Letters A, 355, 337-341.

17.
Ha, J., Gutman, S., Shon, S., & Lee S. (2014). Stability of shallow arches under constant load. International Journal of Non-Linear Mechanics, 58, 120-127.

18.
He, J. (1999). Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178, 257-262.

19.
He, J. (2004). The homotopy perturbation method for nonlinear oscillators with discontinuities. Applied Mathematics and Computation, 151, 287-292.

20.
He, J. (2005). Application of homotopy perturbation method to nonlinear wave equations. Chaos Solitons & Fractals, 26, 695-700.

21.
Jianmin, W., & Zhengcai, C. (2007). Sub-harminic resonances of nonlinear oscillations with parametric excitation by means of the homotopy analysis method. Physics Letters A, 371, 427-431.

22.
Kim, S., Kang, M., Kwun, T., & Hangai, Y. (1997). Dynamic instability of shell-like shallow trusses considering damping. Computers and Structures, 64, 481-489.

23.
Lacarbonara, W., & Rega, G. (2003). Resonant nonlinear normal modes-part2: activation/ orthogonality conditions for shallow structural systems. International Journal of Non-linear Mechanics, 38, 873-887.

24.
Lin, J., & Chen, J. (2003). Dynamic snap-through of a laterally loaded arch under prescribed end motion. International Journal of Solids and Structures, 40, 4769-4787.

25.
Rashidi, M., Shooshtari, A., & Anwar Beg, O. (2012). Homotopy perturbation study of nonlinear vibration of Von Karman rectangular plates. Computers and Structures, 106-107, 46-55.

26.
Sadighi, A., Ganji, D., & Ganjavi, B. (2007). Travelling wave solutions of the sine-gordon and the coupled sine-gordon equations using the homotopy perturbation method. Scientia Iranica Transaction B: Mechanical Engineering, 16(2), 189-195.

27.
Shon, S., Ha, J., & Lee, S. (2012). Nonlinear dynamic analysis of space truss by using multistage homotopy perturbation method. Journal of Korean Society for Noise and Vibration Engineering, 22(9), 879-888.

28.
Shon, S., Lee, S., & Lee, K. (2013). Characteristics of bifurcation and buckling load of space truss in consideration of initial imperfection and load mode. Journal of Zhejiang University-SCIENCE A, 14(3), 206-218.

29.
Shon, S., & Lee, S. (2015). Semi-analytical solution of shallow sinusoidal arches by using multistage homotopy perturbation method. Journal of Architectural Institute of Korea Structure & construction, 31(4), 21-28.

30.
Shon, S., Lee, S., Ha, J., & Cho, G. (2015). Semi-analytic solution and stability of a space truss using a multi-step Taylor series method. Materials, 8(5), 2400-2414.

31.
Wang, S., & Yu, Y. (2012). Application of multistage homotopy perturbation Method for the solutions of the chaotic fractional order systems. International Journal of Nonlinear Science, 13(1), 3-14.

32.
Yu, Y., & Li, H. (2009). Application of the multistage homotopy perturbation method to solve a class of hyperchaotic systems. Chaos, Solitons and Fractals, 42, 2330-2337.