Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes

This work focuses on the placement of natural frequencies of beams to desired frequency regions. More specifically, we investigate the effects of combining mode shapes to shape a beam to change its natural frequencies, both numerically and experimentally. First, we present a parametric analysis of a shaped beam and we analyze the resultant effects for different boundary conditions and mode shapes. Second, we present an optimization procedure to find the optimum shape of the beam for desired natural frequencies. In this case, we adopt the Nelder-Mead simplex search method, which allows a broad search of the optimum shape in the solution domain. Finally, the obtained results are verified experimentally for a clamped-clamped beam in three different optimization runs. Results show that the method is effective in placing natural frequencies at desired values (experimental results lie within a 10% error to the expected theoretical ones). However, the beam must be axially constrained to have the natural frequencies changed.