15. Efficient and Robust Solution of Inverse Structural Dynamic Problems for Vibration Health Monitoring

Vibration health monitoring of structures involves solving inverse structural dynamic problems that are often mathematically ill-conditioned. Many solution approaches have been developed, of which dynamic residual minimization yielded some of the most computationally attractive algorithms. Minimum Rank Perturbation Theory (MRPT) offers a closed form solution of model matrix perturbation using only a subset of the measured modes equal to the rank of perturbation. The initial formulation of the theory assumed symmetric model matrices in the equations of motion, but was later extended to deal with skew-symmetric matrices. The development led to the discovery of a new matrix property known as “null-symmetry”, which helped explain how MRPT preserves symmetry and skew-symmetry in its solution. To handle incomplete measurements, an iterative algorithm based on MRPT was developed using repeated substitution of the dynamic model reduction transformation to achieve convergence. However, the algorithm can become unstable, and hence does not guarantee convergence. Driven by the limitation, dynamic least squares method was developed to solve for a robust set of parameter perturbations in a least square sense regularized by the reduced dynamic residual constraint. The algorithm is capable of estimating mass and stiffness parameters simultaneously under noise and uncertainty without repeatedly solving the eigenproblem.