Designing auxiliary systems for the inverse eigenstructure assignment in vibrating systems

The inverse eigenstructure assignment aims at computing the mass and stiffness parameters, leading to the desired dynamic behaviour expressed in terms of some desired eigenpairs. Attention in the literature has been usually paid to the modification of the mass and stiffness parameters of the original system. In contrast, it often occurs that the elastic and inertial parameters cannot be directly modified. To solve this issue, this paper proposes a method based on the addition of auxiliary systems made by masses and springs with arbitrary, but pre-defined, topology. The eigenstructure assignment problem is cast as an optimization problem solved numerically through a homotopy-based approach, in order to overcome the presence of some bilinear terms in the objective function and to boost the convergence to a global minimum. The use of some auxiliary variables and the McCormick’s relaxation of the bilinear terms are also employed as effective tools to obtain useful modifications of the primary system itself, whenever allowed. Numerical validation is provided to show the method effectiveness.