FREE VIBRATION ANALYSIS OF TIMOSHENKO BEAM WITH DISCONTINUITIES USING DISTRIBUTIONS

The general equations for the transverse vibration of Timoshenko beam have been used since they were derived by means of classical derivatives of the shear force, the bending moment, the rotation of a cross section and the deflection of the beam. However these derivatives are not defined at such points of a centerline between ends of the beam in which there is a concentrated support or a concentrated mass or a concentrated mass moment of inertia or an internal hinge connecting beam segments, which are discontinuities that can be met with in practice. We have applied distributional derivative for discontinuous shear force, discontinuous bending moment, and discontinuous rotation of a cross section of the beam in order to derive a generalized mathematical model for free transverse vibration as a system of partial differential equations. We have computed general solution to the generalized mathematical model for prismatic beam by means of symbolic programming approach via MAPLE. As a result of this approach, computing natural frequencies and modal shapes of the beam, we do not have to put together any continuity conditions at discontinuity points mentioned.