Boundary element solution method of free vibration of Timoshenko beam

The fundamental solution for free vibration of Timoshenko beam was derived using the Fourier transform. Weighted residual method was adapted to deduce the boundary integral formulation from the control differential equations, from which and using boundary conditions one could find the frequency equation which could be solved through algebraic eigenvalue method and influence coefficient method. For any boundary condition, natural frequencies via boundary element method (BEM) for uniform 1D structure are exact, which is validated via the case of rod. The frequencies by BEM of Timoshenko beam were compared with those of finite element method(FEM) and the exact ones.