Civil Engineering Association

A general solution to vibrations of beams on variable winkler elastic foundation

A general solution to vibrations of beams on variable Winkler elastic foundation is presented. The exact solution of the dynamic response of the beam is obtained by considering the reaction force of the foundation on the beam as the external force acting on the beam, which is an integral equation including the displacement of the beam. The four unknown constants in the solution are decided by the boundary conditions of the beam. The integrals in the solution are approximately and numerically calculated by means of the trapezoidal rule. By letting the right-hand side of the solution equal its left-hand side only at those discrete nodes of the quadrature, the frequency equation is obtained which is described by a determinant whose order is equal to the number of the discrete nodes. The mode shape functions are represented by a series of unified analytical functions. The analysis and programing are very simple. It is possible to find the natural frequencies and mode shapes of vibrations by using a small number of the discrete nodes in the trapezoidal quadrature and it is concluded that the use of the method yields better convergence at lower computation costs. Finally, several examples are given for simply supported beams on variable Winkler elastic foundation.