Application of Differential Transform Method to Buckling Problems at Clamped-Free Boundary Conditions

Abstract:

Differential Transform Method (DTM) is a new semi-analytical, semi-numerical algorithm, which transforms differential equations to the form of Taylor series. The method derives an approximate numerical solution based on Taylor series expansion, which is a analytical solution built on polynomial form. Traditional Taylor series method is used for symbolic computation, while the Differential Transform Method obtained the solution of the polynomials through itineration calculations. Applying DTM to buckling problems, the critical length of a bar at clamped-free boundary is studied. The computational results are compared with analytical solutions and shown excellent agreement between those two algorithms. The method adds a new tool for computational engineering mechanics.

• Periodical: Advanced Materials Research (Volumes 639-640)
• Pages: 818-822
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.639-640.818
• Online since: January 2013
• Authors: Ling Feng Han, Shi Yuan Wu
• Keywords: Buckling, Differential Transform Method, Taylor Series

References

[1] JK. Zhou. Differential transformation and its applications for electrical circuits, Huazhong Tech University Pres, Wuhan, China, 1986. (in Chinese))

[2] C. K. Chen, S. H. Ho. Solving partial differential equations by two-dimensional differential transform method [J]. Applied Mathematics and Computation, 1999 (106):171-179

DOI: 10.1016/s0096-3003(98)10115-7

[3] Muhammet Kurulay, Bayram Ali Ibrahimolu and Mustafa Bayram. Solving a system of nonlinear fractional partial differential equations using three dimensional differential transform method [J]. International Journal of the Physical Sciences, 2010 5(6): 906-912

[4] A Kurnaz, G Oturanc, M Kiris. n-Dimensional differential transformation method for solving linear and nonlinear PDE's [J]. International Journal of Computer Mathematics, 2005 (82): 369–80

DOI: 10.1080/0020716042000301725

[5] MJ Jang, CL Chen, YC Liy. On solving the initial-value problems using the differential transformation method [J]. Applied Mathematics and Computation, 2000 (115): 145-160

DOI: 10.1016/s0096-3003(99)00137-x

[6] H. Zeng, C.W. Bert. Vibration analysis of a tapered bar by differential transformation [J]. Journal of Sound and Vibration, 2001 (242): 737–739

DOI: 10.1006/jsvi.2000.3372

[7] C.W. Bert, and H. Zeng. Analysis of axial vibration of compound bars by differential transformation method [J]. Journal of Sound and Vibration, 2004 (275): 641-647

DOI: 10.1016/j.jsv.2003.06.019

[8] S Momani, VS Ertürk. Solutions of non-linear oscillators by the modified differential transform method [J]. Computers & Mathematics with Applications. 2008 (55): 833-842

DOI: 10.1016/j.camwa.2007.05.009

[9] Z Odibat, et al. A Multi-step differential transform method and application to non-chaotic and chaotic systems [J]. Computer and mathematics with applications. Accepted. (2010)

DOI: 10.1016/j.camwa.2009.11.005