Optimal shapes of parametrically excited beams

Straight elastically supported beams of variable width under the action of a periodic axial force are considered. Two shape optimization problems for reducing parametric resonance zones are studied. In the first problem, the minimal (critical) amplitude of the excitation force is maximized. In the second problem, the range of resonant frequencies is minimized for a given parametric resonance zone and a fixed amplitude of excitation. These two optimization problems are proved to be equivalent in the case of small external damping and small excitation force amplitude. It is shown that optimal designs have strong universal character, i.e. they depend only on the natural modes involved in the parametric resonance and boundary conditions. An efficient numerical method of optimization is developed. Optimal beam shapes are found for different boundary conditions and resonant modes. Experiments for uniform and optimal simply supported elastic beams have been conducted demonstrating a very good agreement with theoretical prediction.