Multi-modal tuning of vibrating bars with simplified undercuts using an evolutionary optimization algorithm

We present a method to tune the natural frequencies of a generic bar of mallet percussion instruments (e.g. marimba, vibraphone), for a set of predefined target frequencies, using a global optimization algorithm. The bar is modelled as a one-dimensional Timoshenko beam and its natural frequencies are calculated via 1-D finite elements. The undercut is made up of a series of rectangular cuts, which aside from reducing the dimension of the optimization problem to a few pairs of variables (height and length of each cut) also generates shapes that are easy to manufacture. Moreover, two penalty terms are added to the objective function, in a weighted manner, to (1) minimize the amount of extracted material and (2) minimize abrupt changes in profile height, aimed to alleviate the complexity and accelerate the manufacturing process. An evolutionary optimization algorithm is developed and extensive computational experiments are carried out to assess the algorithm's performance and find appropriate parameter values for a faster convergence. The results illustrate the effect of the different penalties on the solutions obtained. Additionally, optimized shapes for various unorthodox and demanding tuning targets are presented and compared to previously published results, illustrating the benefits of the simplified undercut model.