Homotopy Perturbation–Based Dynamic Analysis of Structural Systems
Publication: Journal of Engineering Mechanics
Volume 146, Issue 12
Abstract
This paper presents the dynamic analysis of structural systems using the homotopy perturbation method (HPM). HPM was applied to the first-order differential equations of motion defined by the state vector. This approach is in the framework of time-history matrix analyses. According to the formulas obtained from HPM, response of the system was achieved by sequential product of the characteristic matrix of the system. This method was theoretically compared with the exponential matrix method (EMM). Based on the obtained formulas, nonlinear structural analysis was performed in addition to the linear method. The applicability and efficiency of the proposed method were tested with the classical and nonclassical damping structures and with different dynamic behaviors. Solutions of this method were compared with solutions of EMM and the time-integration method (TIM). The results of the proposed method were in good agreement with those of the two other methods in linear and nonlinear analyses.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
Some or all data, models, or code generated or used during the study are available from the corresponding author by request. The list of items includes the numerical models of the linear and nonlinear structural analysis using HPM, TIM, and EMM with capability of the torsional behavior and various stiffness irregularity in height, as well as with a base isolation system and tuned mass damping.
Acknowledgments
The authors would like to thank Professor Kamy Sepehrnoori and Dr. Chowdhury K. Mamun for their help in completing this research and preparing the manuscript.
References
Abbasbandy, S. 2006. “Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method.” Appl. Math. Comput. 172 (1): 485–490. https://doi.org/10.1016/j.amc.2005.02.014.
Abdul-Aziz, O., I. Hashim, and S. Momani. 2008. “Application of homotopy perturbation method to fractional IVPs.” J. Comput. Appl. Math. 216 (2): 574–584. https://doi.org/10.1016/j.cam.2007.06.010.
ASCE. 2016. Minimum design loads for buildings and other structures. Reston, VA: ASCE.
Baleanu, D., K. Diethelm, E. Scalas, and J. J. Trujillo. 2012. Fractional calculus models and numerical methods (Series on complexity, nonlinearity and chaos). Boston: World Scientific.
Biazar, J., and H. Ghazvini. 2007. “He’s homotopy perturbation method for solving systems of Volterra integral equations.” Chaos, Solitons Fractals 39 (2): 770–777. https://doi.org/10.1016/j.chaos.2007.01.108.
Brenan, K. E., S. L. Campbell, and L. R. Petzold. 1989. Problems in differential algebraic equations. New York: Elsevier.
Brenan, K. E., and B. E. Engquist. 1985. Backward differentiation approximations of non-linear DAEs. Uppsala, Sweden: Uppsala Univ.
Chang, C. J., and B. Mohraz. 1990. “Modal analysis of nonlinear systems with classical and non-classical damping.” Comput. Struct. 36 (6): 1067–1080. https://doi.org/10.1016/0045-7949(90)90214-M.
Chowdhury, M. S. H., and I. Hashim. 2007. “Solutions of class of singular second-order IVPs by homotopy perturbation method.” Phys. Lett. A 365 (5–6): 439–447. https://doi.org/10.1016/j.physleta.2007.02.002.
Clough, R. W., and J. Penzien. 1975. Dynamics of structures. New York: McGraw-Hill.
Ghasemi, M., M. Tavassoli-Kajani, and E. Babolian. 2007. “Numerical solutions of nonlinear Volterra-Fredholm integral equations by using homotopy perturbation method.” Appl. Math. Comput. 188 (1): 446–449. https://doi.org/10.1016/j.amc.2006.10.015.
Golbabai, A., and B. Keramati. 2008. “Modified homotopy perturbation method for solving Fredholm integral equations.” Chaos, Solitons Fractals 37 (5): 1528–1537. https://doi.org/10.1016/j.chaos.2006.10.037.
Hairer, E., C. Lubich, and M. Roche. 1989. The numerical solution of DAE systems by Runge-Kutta. Berlin: Springer.
Hairer, E., S. P. Norsett, and G. Wanner. 1992. Solving ordinary differential equations II: Stiff and differential algebraic problems. New York: Springer.
Hart, G. C., and K. Wong. 1999. Structural dynamics for structural engineers. New York: Wiley.
He, J. H. 1999. “Homotopy perturbation technique.” Comput. Method Appl. Mech. 178 (3–4): 257–262. https://doi.org/10.1016/S0045-7825(99)00018-3.
He, J. H. 2005. “Application of homotopy perturbation method to nonlinear wave equations.” Chaos, Solitons Fractals 26 (3): 295–700. https://doi.org/10.1016/j.chaos.2005.03.006.
He, J. H. 2006. “Homotopy perturbation method for solving boundary value problems.” Phys. Lett. A 350 (1–2): 87–88. https://doi.org/10.1016/j.physleta.2005.10.005.
Juang, J. N., and M. Q. Phan. 2004. Identification and control of mechanical systems. Cambridge, UK: Cambridge University Press.
Kadem, A., and D. Baleanu. 2011. “Homotopy perturbation method for the coupled fractional Lotka-Volterra equations.” Rom. J. Phys. 56 (3): 332–338.
Li, G. Q., and J. J. Li. 2007. Advanced analysis and design of steel frames. New York: Wiley.
Newmark, N. M. 1959. “A method of computation for structural dynamics.” J. Eng. Mech. 85 (3): 67–94.
Owen, D. R. J., and E. Hinton. 1980. Finite elements in plasticity: Theory and practice. Northampton, UK: Pineridge Press.
Pastia, C., and S. G. Luca. 2013. “Vibration control of a frame structure using semi-active tuned mass damper.” Buletinul Institutului Politehnic din Lasi. Sectia Constructii, Arhitectura 59 (4): 31.
Petzold, L. R. 1995. “Numerical solution of differential algebraic equations.” Adv. Numer. Anal. 4 (Aug): 1017–1029.
Salehi1, F., M. A. Asadi, and M. M. Hosseini. 2012. “Solving system of DAEs by modified homotopy perturbation method.” J. Comput. Sci. Comput. Math. 2 (6): 1–5.
Singh, M. P., and M. Ghafory-Ashtiany. 1986. “Modal time history analysis of non-classically damped structures for seismic motions.” Earthquake Eng. Struct. Dyn. 14 (1): 133–146. https://doi.org/10.1002/eqe.4290140110.
Soong, T. T. 1990. Active structural control: Theory and practice. London: Longman Scientific & Technical.
Wilson, E. L., I. Farhoomand, and K. J. Bathe. 1973. “Nonlinear dynamic of complex structure.” J. Earthquake Eng. Struct. Dyn. 1 (3): 241–252. https://doi.org/10.1002/eqe.4290010305.
Yıldırım, A. 2008a. “Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method.” Comput. Math. Appl. 56 (12): 3175–3180. https://doi.org/10.1016/j.camwa.2008.07.020.
Yıldırım, A. 2008b. “The homotopy perturbation method for approximate solution of the modified KdV equation.” Z. Naturforsch 63a (1): 621–626. https://doi.org/10.1515/zna-2008-10-1102.
Yıldırım, A. 2010. “He’s homotopy perturbation method for nonlinear differential-difference equations.” Int. J. Comput. Math. 65 (1): 511–517. https://doi.org/10.1080/00207160802247646.
Information & Authors
Information
Published In
Copyright
© 2020 American Society of Civil Engineers.
History
Received: Jun 7, 2019
Accepted: Jun 12, 2020
Published online: Sep 30, 2020
Published in print: Dec 1, 2020
Discussion open until: Feb 28, 2021
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.