# Application of multistage homotopy perturbation method to the nonlinear space truss model

- 92 Downloads
- 1 Citations

## Abstract

The purpose of this study is to investigate the applicability of multistage homotopy perturbation method (MHPM) to space truss structures composed of discrete members in order to obtain an analytical solution. For this research purpose, a nonlinear dynamic governing equation of the space truss structures was formulated in consideration of geometrical nonlinearity, and homotopy equations were derived using a formulated differential equation. The result of carrying out a dynamic analysis on a single-free-node model and a double-free-nodes model were compared with the classical homotopy perturbation method and the 4^{th} order Runge-Kutta method (RK4), and it was found that the dynamic response by MHPM concurred with the numerical results. Additionally, the pattern of the response and attractor in the phase space could delineate the dynamic snapping properties of the space truss structures under excitations, and the attraction of the model in consideration of damping reflected well the convergence and asymptotic stable.

## Keywords

multistage homotopy perturbation method (MHPM) analytical solution space truss nonlinear dynamic system geometric nonlinearity## Preview

Unable to display preview. Download preview PDF.

## References

- Allahverdizadeh, A., Oftadeh, R., Mahjoob, M. J., and Naei, M. H. (2014). “Homotopy perturbation solution and periodicity analysis of nonlinear vibration of thin rectangular functionally graded plates.”
*Acta Mechanica Solida Sinica*, 27(2), pp. 210–220.CrossRefGoogle Scholar - Alnasr, M. H. and Erjaee, G. H. (2011). “Application of the multistage homotopy perturbation method to some dynamical systems.”
*International Journal of Science & Technology*, A1, pp. 33–38.MathSciNetzbMATHGoogle Scholar - Ario, I. (2004). “Homoclinic bifurcation and chaos attractor in elastic two-bar truss.”
*International Journal of Non-Linear Mechanics*, 39(4), pp. 605–617.CrossRefzbMATHGoogle Scholar - Belytschko, T. (1976). “A survey of numerical methods and computer programs for dynamic structural analysis.”
*Nuclear Engineering and Design*, 37, pp. 23–34.CrossRefGoogle Scholar - Blandford, G. E. (1996). “Progressive failure analysis of inelastic space truss structures.”
*Computers & Structures*, 58, pp. 981–990.CrossRefGoogle Scholar - Blendez, A. and 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), pp. 79–88.Google Scholar - Chowdhury, M. S. H. and Hashim, I. (2008). “Analytical solutions to heat transfer equations by homotopy perturbation method revisited.”
*Physics Letters*, A(372), pp. 1240–1243.MathSciNetCrossRefzbMATHGoogle Scholar - Chowdhury, M. S. H. and Hashim, I. (2009). “Application of multistage homotopy-perturbation method for the solutions of the Chen system.”
*Nonlinear Analysis: Real World Applications*, 10, pp. 381–391.MathSciNetCrossRefzbMATHGoogle Scholar - Chowdhury, M. S. H., Hashim, I., and Abdulaziz, O. (2007). “Application of homotopy perturbation method to nonlinear population dynamics models.”
*Physics Letters*, A(368), pp. 251–258.MathSciNetCrossRefGoogle Scholar - Chowdhury, M. S. H., Hashim, I., and Momani, S. (2009). “The multistage homotopy-perturbation method: A powerful scheme for handling the Lorenz system.”
*Chaos, Solitons and Fractals*, 40(4), pp. 1929–1937.CrossRefzbMATHGoogle Scholar - Chowdhury, S. H. (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), pp. 1416–1420.Google Scholar - Coan, C. H. and Plaut, R. H. (1983). “Dynamic stability of a lattice dome.”
*Earthquake Engineering and Structural Dynamics*, 11, pp. 269–274.CrossRefGoogle Scholar - Compean, F. I., Olvera, D., Campa, F. J., Lopez de Lacalle, L. N., Elias-Zuniga, A., and Rodriguez, C. A. (2012). “Characterization and stability analysis of a multivariable milling tool by the enhanced multistage homotopy perturbation method.”
*International Journal of Machine Tools & Manufacture*, 57, pp. 27–33.CrossRefGoogle Scholar - Ganji, D. D. (2006). “The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer.”
*Physics Letters*, A(355), pp. 337–341.MathSciNetCrossRefzbMATHGoogle Scholar - Hashim, I. and Chowdhury, M. S. H. (2008). “Adaptation of homotopy perturbation method for numeric-analytic solution of system of ODEs.”
*Physics Letter*s, A(372), pp. 470–481.MathSciNetCrossRefzbMATHGoogle Scholar - Hashim, I., Chowdhury, M. S. H., and Mawa, S. (2008). “On multistage homotopy perturbation method applied to nonlinear biochemical reaction model.”
*Chaos, Solitons & Fractals*, 36, pp. 823–827.CrossRefzbMATHGoogle Scholar - He, J. H. (1999). “Homotopy perturbation technique.”
*Computer Methods in Applied Mechanics and Engineering*, 178, pp. 257–262.MathSciNetCrossRefzbMATHGoogle Scholar - He, J. H. (2000). “A coupling method of homotopy technique and perturbation technique for nonlinear problems.”
*International Journal of Non-Linear Mechanics*, 35(1), pp. 37–43.MathSciNetCrossRefzbMATHGoogle Scholar - He, J. H. (2005). “Homotopy perturbation method for bifurcation of nonlinear problems.”
*International Journal of Nonlinear Sciences and Numerical Simulation*, 6(2), pp. 207–208.MathSciNetGoogle Scholar - Hill, C. D., Blandford, G. E., and Wang, S. T. (1989). “Postbucking analysis of steel space trusses.”
*Journal of Structural Engineering*, 115, pp. 900–919.CrossRefGoogle Scholar - Kassimali, A. and Bidhendi, E. (1988). “Stability of trusses under dynamic loads.”
*Computers & Structures*, 29, pp. 381–392.CrossRefzbMATHGoogle Scholar - Kim, S. D., Kang, M. M., Kwun, T. J., and Hangai, Y. (1997). “Dynamic instability of shell-like shallow trusses considering damping.”
*Computers & Structures*, 64(1-4), pp. 481–489.CrossRefzbMATHGoogle Scholar - Liao, S. J. (1992). The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. Thesis, Shanghai Jiaotong University.Google Scholar
- Lopez, A., Puente, I., and Serna, M. A. (2007). “Numerical model and experimental tests on single-layer latticed domes with semi-rigid joints.”
*Computers & Structures*, 85, pp. 360–374.CrossRefGoogle Scholar - Mang, H. A., Schranz, C., and Mackenzie-Helnwein, P. (2006). “Conversion from imperfection-sensitive into imperfectioninsensitive elastic structures I: Theory.”
*Computer Methods in Applied Mechanics and Engineering*, 195, pp. 1422–1457.MathSciNetCrossRefzbMATHGoogle Scholar - Rashidi, M. M., Shooshtari, A., and Anwar Beg, O. (2012). “Homotopy perturbation study of nonlinear vibration of Von Karman rectangular plates.”
*Computers and Structures*, 106-107, pp. 46–55.CrossRefGoogle Scholar - Sadighi, A., Ganji, D. D., and Ganjavi, B. (2007). “Travelling wave solutions of the sine-gordon and the coupled sinegordon equations using the homotopy perturbation method.”
*Scientia Iranica Transaction B: Mechanical Engineering*, 16(2), pp. 189–195.Google Scholar - Saffari, H., Mansouri, I., Bagheripour, M. H., and Dehghani, H. (2012). “Elasto–plastic analysis of steel plane frames using Homotopy Perturbation Method.”
*Journal of Constructional Steel Research*, 70, pp. 350–357.CrossRefGoogle Scholar - Sajid, M., Hayat, T., and Asghar, S. (2007). “Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt.”
*Nonlinear Dynamics*, 50(1-2), pp. 27–35.MathSciNetCrossRefzbMATHGoogle Scholar - Shon, S. D., Lee, S. J., and Lee, K. G. (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), pp. 206–218.CrossRefGoogle Scholar - Tada, M. and Suito, A. (1998). “Static and dynamic postbuckling behavior of truss structures.”
*Engineering Structures*, 20, pp. 384–389.CrossRefGoogle Scholar - Wang, S. and 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), pp. 3–14.MathSciNetzbMATHGoogle Scholar - Wen, J. and Cao, Z. (2008). “Nonlinear oscillations with parametric excitation solved by homotopy analysis method.”
*Acta Mechanica Sinica*, 24(3), pp. 325–329.MathSciNetCrossRefzbMATHGoogle Scholar - Yu, Y. and Li, H. X. (2009). “Application of the multistage homotopy-perturbation method to solve a class of hyperchaotic systems.”
*Chaos, Solitons and Fractals*, 42, pp. 2330–2337.CrossRefzbMATHGoogle Scholar