Using analytical equations, the paper aims to solve the dynamic behavior of beams where a clamped end of the beam does not respect the ideal boundary conditions by introducing a weakening coefficient. In the paper, the characteristic equation for determining the eigenvalues and the relationship of the modal function and strain energy are derived. The results show the first six vibration modes for different values of the weakening coefficient which is considered in the clamped end and the evolution of the strain energy.

Keywords: weak clamped end, mode shapes, strain energy