Exact solution to free vibration of beams partially supported by an elastic foundation

This study pertains to the free vibration problem of beams on an elastic foundation of the Winkler type, which is distributed over a particular length of the beam. Closed form solutions are developed by solving the governing differential equations of beams. Moreover, an innovative mathematical approach is proposed to find the precise analytical solution of the free vibration of beams with mixed boundary conditions. Results are discussed in detail through verification studies. Ultimately, it was concluded that the proposed mathematical method could successfully obtain the exact solution to the free vibration problem of beams on partial elastic foundations under mixed boundary conditions.