Research paper Struct Multidisc Optim 27, 435 ­ 445 (2004) DOI 10.1007/s00158-004-0388-x Optimal shapes of parametrically excited beams A.A. Mailybaev, H. Yabuno, and H. Kaneko Abstract 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. ...