Comparison of order reduction methodologies and identification of NNMs in structural dynamic systems with isolated nonlinearities

Two techniques, the local equivalent linear stiffness (LELSM) and the method of proper orthogonal decomposition (POD), are employed for order reduction and in locating the nonlinear normal modes (NNMs) of structural dynamic systems with isolated nonlinearities. The POD method requires that the solution response matrix in space and time should be known first, while LELSM has no such requirements. By utilizing these methods, NNMs can be specified and reduced order models are constructed for both cubic and dead-zone nonlinearities. Two approaches, based on the linear modal coordinates and POD, and on LELSM, are used to locate NNMs of large-order systems with isolated nonlinearities. In addition, LELSM and POD are compared for accuracy at a wide range of initial conditions around the equipotential boundary. It was found that the LELSM modes approximate the POD modes with high accuracy especially at initial conditions corresponding to the first and second NNMs. The LELSM modes are found more accurate in order reduction and give an in-phase time history with the exact numerical solution of the full model for longer time periods compared with POD. The two methods are applied to illustrative 2-DOF systems and to a cantilever beam element with nonlinear boundary conditions. Some important advantages of LELSM compared with POD will be noticed through this paper.