A New Method for Passive Partial Eigenstructure Assignment in Vibrating Systems

Passive eigenstructure assignment on vibrating systems is aimed at computing the modifications of the inertial and elastic parameters ensuring the desired dynamical behavior, in terms of natural frequency and mode shape. In many cases, it is desirable assigning only the response of some critical parts rather than the response of the whole system. However, the currently available methods do not provide a way to distinguish the degrees of freedom with different levels of interest. To solve this relevant open issue, this paper proposes a method intended for the assignment of partial parts of an arbitrary number of eigenvectors. In order to determine the optimal structural modifications satisfying constraints on the feasible values, the assignment problem is cast as an optimization problem solved numerically. The presence of some not-imposed eigenvector entries leads to a non-convex optimization problem. Therefore, to boost the convergence to a global minimum, a homotopic continuation is implemented by morphing from a convex relaxation of the problem to the original non-convex one. The convex approximation is performed through some auxiliary variables and the McCormick’s relaxation of the bilinear terms. The proposed approach can handle general assignment tasks, with an arbitrary number of modification parameters and prescribed eigenpairs. Interrelated modifications can be also accounted for. The method is numerically validated on a 5 degree-offreedom system, showing that partial assignment significantly improves the fulfillment of the specifications on the eigenvector entries of greater concern.