Finding an optimal shape of a curved mechanical beam for enhanced internal resonance

One of the most interesting nonlinear phenomena of mode coupling is internal resonances, which can promote directed energy transfer from one eigenmode to another, even at small amplitudes in free or forced oscillations. Internal resonances can be highly beneficial for many engineering applications. However, in most cases, internal resonances are encountered either accidentally or by proper tuning of different control parameters during experiments without prior planning. Therefore, the ability to a priori design a mechanical resonator with intentional internal resonance at given amplitudes holds great promise. Here, we show a simple methodological way to manipulate the eigenfrequencies and the coupling between the eigenmodes of a doubly clamped mechanical beam using a genetic algorithm for shape optimization of the initial curvature of the beam. We demonstrate that our methodology can be applied to both 1-to-2 and 1-to-3 internal resonances of micro-beams. Our results pave the way to a new class of design techniques for internal resonance enhancement based on shape optimization.