A receptance based method for the calculation of nonlinear normal modes of large ordered structures with distributed localized nonlinearities

Abstract

In recent years, the concept of nonlinear normal modes (NNMs) has gained interest for interpreting a broad range of nonlinear dynamic phenomena. Present study introduces a simple and efficient computational framework to compute NNMs of large order nonlinear structures overcoming difficulties associated with commonly used time-integration and discretization methods. In this paper, a Receptance Based Nonlinear Normal Mode Calculation Method (RBNM) is developed to determine the NNMs of large ordered realistic finite element models in frequency domain. Describing Function Method (DFM), which makes it possible to write the nonlinear internal forcing vector as a nonlinear displacement dependent stiffness matrix multiplied by displacement vector, is used to model the internal nonlinear forcing vector in frequency domain. A unique matrix manipulation based on dynamic stiffness and receptance concepts is used, as a result of which the number of governing nonlinear equations becomes independent from the total number of degrees of freedom of the model and it only depends on the number of degrees of freedom associated with the nonlinear elements. This brings a significant computational advantage over the classical reduced order modeling techniques. A simple 2-degree of freedom (DOF) system, the results of which are available in the literature, is used to verify the proposed method in determining NNMs. A 20-DOF lumped parameter system model and later a complex finite element model of a large structure are used to demonstrate the applicability of RBNM in determining NNMs of larger systems with several different types of nonlinear elements distributed among the degrees of freedom.