Modified Differential Transform Method for Solving Vibration Equations of MDOF Systems
Abstract
Vibration equations of discrete multi-degrees-of-freedom (MDOF) structural systems is system of differential equations. In linear systems, the differential equations are also linear. Various analytical and numerical methods are available for solving the vibration equations in structural dynamics. In this paper modified differential transform method (MDTM) as a semi-analytical approach is generalized for the system of differential equations and is utilized for solving the vibration equations of MDOF systems. The MDTM is a recursive method which is a hybrid of Differential Transform Method (DTM), Pade' approximant and Laplace Transformation. A series of examples including forced and free vibration of MDOF systems with classical and non-classical damping are also solved by this method. Comparison of the results obtained by MDTM with exact solutions shows good accuracy of the proposed method; so that in some cases the solutions of the vibration equation that found by MDTM are the exact solutions. Also, MDTM is less expensive in computational cost and simpler with compare to the other available approaches.
Keywords
References
Humar, J.L., Dynamics of Structures. second ed., A. A. Balkema Publishers, 2002.
Chopra, A.K., Dynamics of Structures. fourth ed., Prentice Hall (New Jersey), 2012.
Zhou, J. K. "Differential Transformation and Its Applications for Electronic Circuits, Huazhong Science & Technology University Press, China." (1986).
Odibat, Zaid, and Shaher Momani. "A generalized differential transform method for linear partial differential equations of fractional order." Applied Mathematics Letters 21, no. 2 (2008): 194-199.
Ertürk, Vedat Suat, and Shaher Momani. "Solving systems of fractional differential equations using differential transform method." Journal of Computational and Applied Mathematics 215, no. 1 (2008): 142-151.
Nazari, D., and S. Shahmorad. "Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions." Journal of Computational and Applied Mathematics 234, no. 3 (2010): 883-891.
Alquran, Marwan. "Applying differential transform method to nonlinear partial differential equations: a modified approach." Applications and Applied Mathematics: An International Journal 7, no. 1 (2012): 155-163.
Fatoorehchi, Hooman, and Hossein Abolghasemi. "Improving the differential transform method: a novel technique to obtain the differential transforms of nonlinearities by the Adomian polynomials." Applied Mathematical Modelling 37, no. 8 (2013): 6008-6017.
Patil, N., and Khambayat, A. "Differential Transform Method for Ordinary Differential Equations." Research Journal of Mathematical and Statistical Sciences 3(2014): 330–337.
El-Shahed, M. "Application of differential transform method to non-linear oscillatory systems." Communications in Nonlinear Science and Numerical Simulation 13(2008): 1714–1720.
Momani, S., and Ertürk, V.S. "Solutions of non-linear oscillators by the modified differential transform method." Computers & Mathematics with Applications 55(2008): 833–842.
Nourazar, S., and Mirzabeigy, A. "Approximate solution for nonlinear Duffing oscillator with damping effect using the modified differential transform method." Scientia Iranica B 20(2013): 364–368.
Chen, C.K., and Ho, S.H. "Transverse vibration of a rotating twisted Timoshenko beams under axial loading using differential transform." International Journal of Mechanical Sciences 41(1999): 1339–1356.
Ho, S.H., and Chen, C.K. "Analysis of general elastically end restrained non-uniform beams using differential transform." Applied Mathematical Modelling 22(1998): 219–234.
Özdemir, Ö., and Kaya, M.O. "Flapwise bending vibration analysis of a rotating tapered cantilever Bernoulli-Euler beam by differential transform method" Journal of Sound and Vibration 289(2006): 413–420.
Çatal, S. "Solution of free vibration equations of beam on elastic soil by using differential transform method." Applied Mathematical Modelling 32(2008): 1744–1757.
Arikoglu, A., and Ozkol, I. "Vibration analysis of composite sandwich beams with viscoelastic core by using differential transform method." Composite Structures 92(2010): 3031–3039.
Demirdag, O., and Yesilce, Y. "Solution of free vibration equation of elastically supported Timoshenko columns with a tip mass by differential transform method." Advances in Engineering Software 42(2011): 860–867.
Lal, R., and Ahlawat, N. "Axisymmetric vibrations and buckling analysis of functionally graded circular plates via differential transform method." European Journal of Mechanics - A/Solids 52(2015): 85–94.
Baker, G.A., Essentials of Pade' Approximants. Academic Press (London), 1975.
DOI: 10.28991/cej-2016-00000019
Refbacks
- There are currently no refbacks.
Copyright (c) 2016 Mohammadamir Najafgholipour, Navid Soodbakhsh
This work is licensed under a Creative Commons Attribution 4.0 International License.
