Analytic Modal Analyses of a Partially Strengthened Timoshenko Beam

In this paper, the effect of an actuator on the natural frequencies and mode shapes of a cantilever Timoshenko beam is studied. In this regard, the equations of motion of a cantilever Timoshenko beam are considered. In order to incorporate the effect of the actuator in these equations, it is assumed that the beam is comprised of three uniform Timoshenko beams connected to one another at their boundaries. The continuity and equilibrium conditions are then utilized to find the sub-boundary conditions for each sub-beam. The effect of the actuator is considered as an external torque on the sub-boundary conditions. This leads to a set of partial differential equations for each sub-beam with coupled non-homogenous boundary conditions. A solution for the resulting equations is obtained and a state-space model of the beam strengthened with the actuator is derived. To evaluate the accuracy of the model, simulation and experimental results are presented.