Model order reduction of second-order systems with nonlinear stiffness using Krylov subspace methods and their symmetric transfer functions

In this investigation, Model Order Reduction (MOR) of second-order systems having cubic nonlinearity in stiffness is developed for the first time using Krylov subspace methods and the associated symmetric transfer functions. In doing so, new second-order Krylov subspaces will be defined for MOR procedure which avoids the need to transform the second-order system to its state space form and thus the main characteristics of the second-order system such as symmetry and positive definiteness of mass and stiffness matrices will be preserved. To show the efficacy of the presented method, three examples will be considered as practical case studies. The first example is a nonlinear shear-beam building model subjected to a seismic disturbance. The second and third examples are nonlinear longitudinal vibration of a rod and vibration of a cantilever beam resting on a nonlinear elastic foundation, respectively. Simulation results in all cases show good accuracy of the vibrational response of the reduced order models when compared with the original ones while reducing the computational load.