Free vibration analysis of circular plates by differential transformation method Hasan Serter Yalcin, Aytac Arikoglu, Ibrahim Ozkol * Istanbul Technical University, Faculty of Aeronautics and Astronautics, Aeronautical Engineering Department, Maslak, TR-34469 Istanbul, Turkey article info Keywords: Free vibration Circular plate Regularity conditions Differential transform method abstract This study analyses the free vibrations of circular thin plates for simply supported, clamped and free boundary conditions. The solution method used is differential transform method (DTM), which is a semi-numerical–analytical solution technique that can be applied to var- ious types of differential equations. By using DTM, the governing differential equations are reduced to recurrence relations and its related boundary/regularity conditions are trans- formed into a set of algebraic equations. The frequency equations are obtained for the pos- sible combinations of the outer edge boundary conditions and the regularity conditions at the center of the circular plate. Numerical results for the dimensionless natural frequencies are presented and then compared to the Bessel function solution and the numerical solu- tions that appear in literature. It is observed that DTM is a robust and powerful tool for eigenvalue analysis of circular thin plates. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction Circular plates are the most critical structural elements in high speed rotating engineering systems such as circular saws, turbine flywheels, rotors, etc. The dynamic characteristics of the plate have a considerable effect on the overall structure per- formance. When the frequency of the external load matches the natural frequency of the plate, damage or destruction may occur. With this respect, the natural frequencies of the plates have been studied extensively for more than a century. The eigenvalue analysis has been usually carried out by using a variety of weighting function methods [1], including the Galerkin method, the Ritz method and the finite element method. Also, there are voluminous studies, in which the natural frequencies of circular plates are expressed in terms of the Bessel functions [1–3]. The differential transform method (DTM) which is based on the Taylor series expansion was first proposed by Zhou [4] in 1986 for the solution of linear and nonlinear initial value problems that appear in electrical circuits. DTM is a semi-analyt- ical–numerical technique depending on Taylor series and is promising for various types of differential equations. With this technique, it is possible to obtain highly accurate results or exact solutions for differential or integro-differential equations [5]. By using this method, the governing differential equations can be reduced to a recurrence relation and the boundary con- ditions can be transformed into a set of algebraic equations as in our problem. In the solution of eigenvalue problems, DTM has been successfully applied in recent studies [6–9]. This study introduces the application of DTM to the solution of eigenvalue problem governing the free vibrations of thin circular plates. There are recent studies on the application of DTM to the dynamical analysis of non-circular plates in literature. Yeh and Jang [10] solved the dynamical problem of the rectangular plate using a hybrid method which combines the finite difference method and the differential transformation method. Malik and Allali [11] derived the characteristic equations of rectangular plates by utilizing 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.02.032 * Corresponding author. E-mail address: ozkol@itu.edu.tr (I. Ozkol). Applied Mathematics and Computation 212 (2009) 377–386 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc