Shape optimization of a curved mechanical beam for transverse vibrations amplification via nonlinear interaction with longitudinal vibrations

Nano- and micro-mechanical beams are widely used as resonators in research and engineering applications. One of the key challenges of utilizing their immense potential is effectively amplifying their signal above the noise floor level to achieve a large signal-to-noise ratio. In this study, building upon prior work, we develop an optimization procedure for the initial curvature of a mechanical beam that maximizes the inherent amplification of the forced transverse vibrations of the beam due to a nonlinear interaction with driven longitudinal vibrations. We focus on two possible cases: (i) direct amplification, which is obtained when the driving frequency of the longitudinal vibrations is close to the eigenfrequency of the transverse mode, and (ii) parametric amplification, which is obtained when the driving frequency of the longitudinal vibrations is close to twice the eigenfrequency of the transverse mode. By using a simple genetic algorithm scheme, we find that the optimal curved beams enhance the amplification by a factor of 205 for direct amplification and by a factor of 60 for parametric amplification.