Application of differential transform method to determine natural frequencies of variable width Euler- Bernoulli beam with various support conditions

In this paper، differential transform method (DTM) has been used to determine natural frequencies of free vibration of variable width Euler- Bernoulli isotropic beam. The width of beam varies exponentially with arbitrary non-uniformity parameter. After extracting and nondimensionalization، the governing partial differential equation discretized to two ordinary differential equations in terms of both spatial coordinate and time based on the separation of variables method. Then، for different non-uniformity parameters and boundary conditions، natural frequencies of system have been obtained by using analytical methods and DTM. Boundary conditions are considered as: simple- simple (SS)، clamped- clamped (CC)، clamped- Simple (CS)، free- free (FF)، clamped- free (CF) and simple- free (SF). Comparison of natural frequencies obtained from DTM with analytical solution shows that DTM has a good accuracy in determination of natural frequencies، especially in the lower modes. Since that this method is based on Taylor’s series expansion، more accurate natural frequencies and also natural frequencies of higher modes can be obtain by increasing series size. Results show that for a constant series size، difference between the natural frequencies obtained from DTM with the exact solution increase totally by increasing of mode number.