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.
Highlights
Most of the existing structural systems are multi-degrees-of-freedom (MDOF) that can be categorized into discrete and continuous systems
The modified differential transform method (MDTM) as a recursive semi-analytical method which is a hybrid of Differential Transform Method (DTM), Pade' approximant and Laplace transformation is generalized for solving system of differential equations
A series of examples which include free and forced vibration of some structural systems with different number of degrees of freedom are solved by MDTM and the following conclusions are obtained: 1) The results which obtained by DTM don’t have acceptable accuracy in large time intervals, so this method is not appropriate for solving the vibration equations of oscillatory systems
Summary
Most of the existing structural systems are multi-degrees-of-freedom (MDOF) that can be categorized into discrete and continuous systems. Shahed [10] applied DTM for solving vibration equation of non-linear SDOF oscillatory systems He proposed the modified differential transform method (MDTM) with Pade Approximation for this purpose. Lal and Ahlawat [19] applied DTM for axisymmetric vibration and buckling analysis of functionally graded circular plates subjected to uniform in-plane forces As it is seen, numerous researches on application of DTM and MDTM for solving the vibration equation of SDOF systems has been carried out, where the equation of motion is a single differential equation. At first DTM is introduced for solving the differential equations with initial values Since this method gives the solutions in restricted intervals, it is not suitable for solving the vibration equations of oscillatory systems. Some examples which include free and forced vibration of damped and un-damped systems subjected to different loading functions are solved by MDTM and the results are compared with the other common methods
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