Comparison of craig-bampton approaches for systems with arbitrary viscous damping in dynamic substructuring

The classical Craig-Bampton method does not take any damping effects into account for the model order reduction of damped systems. There is generally no justification to neglect damping effects. If damping significantly influences the dynamic behavior of the system, the approximation accuracy can be very poor. One procedure to handle arbitrary damped systems is to transform the second-order differential equations into twice the number of first-order differential equations resulting in state-space representation of the system. Solving the corresponding eigenvalue problem allows the damped equations for the internal degrees of freedom of the substructures to be decoupled, but complex eigenmodes and eigenvalues occur. Hasselman and Kaplan presented a coupling procedure for damped systems that employs complex component modes. Beliveau and Soucy proposed another version that modifies the classical Craig-Bampton method to include damping by replacing the real fixed interface normal modes of the second-order system with the corresponding complex modes of the first-order system. Additionally, they suggest an adaption of the method of Hasselman and Kaplan. A report of de Kraker gives another description of the Craig-Bampton method using complex normal modes and modified static modes. The derivation of all the different Craig-Bampton substructuring methods for viscously damped systems is presented in a comprehensible consistent manner. A comparison between the different formulations will be given. The presented theory and the comparison between the methods are illustrated by an example.