In this paper, free vibration of a new type of tapered beam, with exponentially varying thickness, resting on a linear foundation is analyzed. The solution is based on a semi-analytical technique, the differential transform method (DTM). Applying DTM, nonlinear partial differential equations of the varying thickness beam are transformed into algebraic equations, which are then solved to obtain the solution. An Euler-Bernoulli beam with a number of boundary conditions and different exponential factor is taken into account. Results have been compared to the 4th order Runge-Kutta, and where possible with DQEM and analytical solution. These comparisons prove the preciseness of this method, based on which DTM can be considered as a powerful framework for eigenvalue analysis of new type of tapered beams.

In this research, free vibration of rectangular functionally graded (FG) plates with in-plane exponentially non-homogeneous material is investigated. Young's modulus and mass density are assumed to vary between a metal-rich and a ceramic-rich zone along one in-plane direction of the plate. The governing differential equation is derived for the case, and a truncated Taylor series expansion technique is utilized to calculate natural frequencies. A Levy-type solution is obtained for plates having two simply supported edges parallel with the material gradient direction. Results for normalized natural frequency are compared with the 4th order Runge-Kutta method, and when possible with exact solution, showing an accurate agreement. Furthermore, a comprehensive parametric study is carried out to determine the effects of different boundary conditions, aspect ratios, and material variations on the free vibration of FGM plates. Nomenclature