Optimal mass distribution in vibrating systems

The problem of determining the distribution of mass in vibrating systems that optimize the extreme eigenvalue is considered. We give explicit solutions for the fundamental discrete models of an axially vibrating rod and a Bernoulli–Euler beam.The problems are first recast as affine sum of matrices with linear total mass constraint. This formulation allows determination of the eigenvectors corresponding to the optimal eigenvalues by solving a set of quadratic equations recursively. The optimal mass distribution is then determined by the recovered eigenvectors.