The use of the fixed points method for optimal damping of harmonically forced cantilever beams

Abstract

The use of viscous dampers for vibration attenuation in harmonically forced cantilever beams is studied. The system considered is a cantilever beam with a point harmonic force applied at a given location and a viscous damper attached to it from one end, and grounded from the other. An assumed mode model of the system is derived using the first two transverse modes of the beam. For any given positions of the point force and damper, the optimal damping constant which minimizes the maximum of the frequency response function at the tip of the beam is determined analytically. It is shown that the objective function passes through a number of points independent of the damping constant. These inevitable points are used in the determination of the maximum allowable value of the objective function. As the locations of the point force and damper are varied separately from the fixed end of the beam to its tip, a two dimensional region plot is generated illustrating the different regions where each of these points is the highest. The optimal damping constant is determined analytically by forcing the frequency response function to pass horizontally through the highest fixed point which is referred to as the active peak. Four different damping ratios are determined and depending on the positions of the force and damper, the two dimensional map is consulted in the selection of the correct optimal damping ratio. The solution obtained is unique except when the active peak is the static fixed point. In this case, the solution is made unique by modifying the objective function to further enhance the solution at high frequencies.