Complex modal derivatives for model reduction of nonclassically damped, geometrically nonlinear structures

Finite element simulations of structures that undergo large deformations can imply high computation costs due to the nonlinearity of the resulting equations of motion and high numbers of degrees of freedom. Model reduction can encounter this burden by approximating the nodal displacements as a linear combination of some basis vectors and applying a Galerkin projection. The main challenge for this step is a proper selection of basis vectors that are able to capture the deformation states of the system. A prevalent selection of basis vectors for geometrically nonlinear systems is the combination of modes of the linearized system and so called modal derivatives. The modal derivatives describe the change of the modes when the system is perturbed and thus contain nonlinear information. However, modal derivatives are only defined for modes that are real. This makes this approach unusable for systems that have complex eigenmodes, such as nonclassically damped systems, i.e., systems with the damping matrix not fulfilling the Caughey condition. This contribution shows how modal derivatives can be defined for nonclassically damped systems by extending the concept of modal derivatives to complex eigenmodes and how this can be used to build a reduction basis for geometrically nonlinear, nonclassically damped systems. A simple beam test case demonstrates that this approach can improve the accuracy of the reduced system slightly compared to the use of eigenmodes and modal derivatives of the underlying undamped system.