Identification of a Nonproportional Damping Matrix from Incomplete Modal Information

In modeling structures the damping matrix is the most difficult to represent. This is even more difficult in complicated structures that are not lightly damped. The work presented here provides a method of modeling the damping matrix of a structure from incomplete experimental data combined with a reasonable representation of the mass and stiffness matrices developed by finite element methods and reduced by standard model reduction techniques. The proposed technique uses the reduced mass and stiffness matrices and the experimentally obtained eigenvalues and eigenvectors in a weighted least squares or a pseudo-inverse scheme (depending on the number of the equations that are available) to solve for the damping matrix. The results are illustrated through several examples. As an indication of the accuracy of the method, fictitious examples where the damping matrix is originally known are considered. The proposed method identifies the exact viscous or hysteretic damping matrix by only using a partial set, half of the system’s eigenvalues and eigenvectors. The damping matrix is assumed to be real, symmetric, and positive semidefinite.