Overview of Free Interface Substructuring Approaches for Systems with Arbitrary Viscous Damping in Dynamic Substructuring

Most classical substructuring methods yield great approximation accuracy if the underlying system is not damped. One approach is a fixed interface method, the Craig-Bampton method. In contrast, many other methods (e.g., MacNeal method, Rubin method, Craig-Chang method) employ free interface modes, (residual) attachment modes, and rigid body modes. None of the aforementioned methods takes any damping effects into account when performing the reduction. If damping significantly influences the dynamic behavior of the system, the approximation accuracy can be very poor. One procedure to handle arbitrarily viscously damped systems and to take damping effects into account 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 to be decoupled; however, complex eigenmodes and eigenvalues occur. The complex modes are used to build a reduction basis that includes damping properties. The derivation of different Craig-Bampton substructuring methods (fixed interface) for viscously damped systems was presented in Gruber et al. (Comparison of Craig-Bampton approaches for systems with arbitrary viscous damping in dynamic substructuring). In contrast, we present here the derivation of different free interface substructuring methods for viscously damped systems in a comprehensible consistent manner. Craig and Ni suggested a method that employs complex free interface vibration modes (1989). De Kraker and van Campen give an extension of Rubin’s method for general state-space models (1996). Liu and Zheng proposed an improved component modes synthesis method for nonclassically damped systems (2008), which is an extension of Craig and Ni’s method. A detailed comparison between the different formulations will be given. Liu and Zheng’s method can be considered as a second-order extension of Craig and Ni’s method. We propose a third-order extension and a generalization to any given higher order. Moreover, a new method combining the reduction basis of Liu and Zheng’s approach with the primal assembly procedure applied by de Kraker and van Campen is proposed. The presented theory and the comparison between the methods will be illustrated in different examples.