A homotopy transformation method for interval-based model updating of uncertain vibrating systems

This paper proposes a novel indirect, two-stage approach for model updating in linear vibrating systems, exploiting measured natural frequencies, antiresonances and uncertain or incomplete data of the mode shapes. In the first stage, the technique relies on the partial eigenstructure assignment paradigm and recasts model updating into a non-linear, non-convex minimization that simultaneously updates the mass and stiffness matrices. Uncertainty on mode shapes is formulated through interval (bounds) where they should belong. The inverse eigenvalue problem is solved through homotopy optimization, improved through variables lifting and McCormick's constraints. The unknown parameters are normalized introducing Jacobian matrices, computed through the complex step derivatives, to improve the numerical conditioning of the problem and speed up the computation. In the second stage, the damping matrix is identified through the generalized formulation of proportional damping provided by the Caughey's series. Two challenging experimental test-cases are solved and prove the method effectiveness: the linearized multibody model of a flexible manipulator and a structure made by a cantilever beam plus a lumped spring-mass system.