Accurate computation of frequency response functions of dual Craig-Bampton reduced systems

The dual Craig-Bampton method (DCBM) employs modes with free interfaces to build the reduction bases of the substructures, but assembles the substructures using interface forces, which is called dual assembly. The DCBM preserves the sparsity of the final reduced mass and stiffness matrices similar to the classical Craig-Bampton reduced matrices, which makes the reduced matrices simpler and less populated than the reduced matrices of other free interface methods. However, the DCBM is not a Rayleigh-Ritz transformation only on the displacement degrees of freedom, but reduces the dually assembled system. Thereby, interface kinematic conditions are transformed, allowing incompatibilities between the substructures unlike other substructuring techniques. This interface kinematic transformation enforces only a weak displacement compatibility between the substructures. As a consequence, the reduced-order model always has as many negative eigenvalues as Lagrange multipliers used for the dual assembly. When applying a DCBM reduction, the reduced system is no longer positive (semi)definite. This means that the reduced system is unstable. Although rendering direct time integration impossible, this does not affect the computation of frequency response functions (FRFs), which is demonstrated in this paper. The feasibility of a reliable and accurate FRF computation of DCBM reduced systems is investigated in detail. Hereby, DCBM reduced system means that a reduced system after a DCBM reduction is used without any modifications. It is demonstrated that the unstable behavior of the DCBM reduced system does not negatively influence the computation of the FRFs. Following this, methods for the further reduction are evaluated: modal reduction and interface reduction are applied to the DCBM reduced system. Moreover, the reduced system is enhanced by incorporating modal truncation vectors. This guarantees static correctness and enables a very accurate approximation of antiresonances. This is illustrated by three different examples. Thereby, the resulting FRFs of the DCBM reduced systems are compared to the FRFs of the unreduced systems.