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 and cosine series is employed to develop an analytical method to conduct the free vibration analysis of an euler- bernoulli beam with varying cross-sections, 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's length direction. the idea behind the proposed method is to superpose fourier sine and cosine series to satisfy the general elastically end constraints; therefore, no auxiliary functions are required to supplement the fourier series. this method provides a simple, accurate and exible solution to various beam problems and is also able to be extended to other cases whose governing differential equations are nonlinear. moreover, this method is applicable to plate problems with different boundary conditions if twodimensional fourier sine and cosine series are taken as a displacement function. numerical examples are carried out, illustrating the accuracy and eciency of the presented approach.