A New Fourier series solution for free vibration of non-uniform beams, resting on variable elastic foundation

In this research, the combination of Fourier sine series and Fourier cosine series is employed to develop an analytical method for free vibration analysis of an Euler-Bernoulli beam of varying cross- section, fully or partially supported by a variable elastic foundation. The foundation stiffness and cross section of the beam are considered as arbitrary functions in the beam length direction. The idea of the proposed method is to superpose Fourier sine and Fourier cosine series to satisfy general elastically end constraints and therefore no auxiliary functions are required to supplement the Fourier series. This method provides a simple, accurate and flexible solution for various beam problems and is also able to be extended to other cases whose governing differential equations are nonlinear. Moreover, this method is applicable for plate problems with different boundary conditions if two-dimensional Fourier sine and cosine series are taken as displacement function.
Numerical examples are carried out illustrating the accuracy and efficiency of the presented approach.