On Combined Use of POD Modes and Ritz Vectors for Model Reduction in Nonlinear Structural Dynamics

Abstract

An approach to develop Proper Orthogonal Decomposition (POD) based reduced order models for systems with local nonlinearities is presented in this paper. This technique is applied to multi-degree of freedom systems of coupled oscillators with isolated nonlinear elements. Typically, reduced order models are obtained using POD modes exclusively. In this work, we explore the suitability of using a combination of POD modes and other physically based “Ritz vectors” to produce the reduced model. The objectives are 1). to improve the accuracy of the reduced order differential equation model and 2). to expand the range of system parameters for which the reduced basis provides reasonably accurate approximations. The “Ritz vectors” used in this work are static displacement vectors that are calculated in one of the following three ways: 1). “Load – based Ritz vectors” [1, 2, 7, 8, 12, 16–18] – This is the static displacement vector due to a static loading that is proportional to the static version of the actual (assumed dynamic) loading to which the structure is subjected. 2). “Milman – Chu vectors” [3] – This is the static Ritz vector due to the imposition of equal and opposite static loads on the two masses to which the non-linear element is connected. The loading used to generate the first M – C vector is dictated by the location of the non-linearity. 3). “K – B (Kumar – Burton) vector” – This is a new Ritz vector defined in the spirit of the Milman – Chu vector. The K – B vector is the static displacement vector due to the imposition of a). equal and opposite static loads on the two masses to which the nonlinear element is connected (i.e. same as M – C loading) and b). equal and opposite static loads on the nearest neighbors. Thus, four masses are statically loaded. As for the M – C vector, the K – B loading is dictated by the location of the nonlinear element. The nonlinear model is numerically integrated to generate a full ODE model solution, which we call the “baseline solution”. We select a set of POD modes of the baseline nonlinear system response as basis functions. The POD modes are then augmented by various combinations of the three aforementioned Ritz vectors to generate reduced order models for system having parameters in vicinity of baseline system parameters. Our results indicate that the K – B augmentation vector combined with the Milman – Chu vector is an effective way to account for nonlinear effects for the system considered. The use of combined M – C/K – B augmentation also expands the range of system parameters for which the baseline POD modes provide accurate reduction. This is considered to be a significant result.