A NEW DERIVATION OF THE FREQUENCY RESPONSE FUNCTION MATRIX FOR VIBRATING NON-LINEAR SYSTEMS

Abstract Frequency response function matrices relate the inputs and the outputs of structural dynamic systems. If a system is linear the frequency response function matrix is the same for any combination or types of inputs over the entire operating range. Furthermore, the frequency response matrix of a linear vibrating system is a simple combination of temporal and spatial characteristics, the modal frequencies, modal vectors and modal scale factors. When a system is non-linear, the inputs interact through an exchange of energy between the linear and non-linear elements in the system. No general combination of the temporal and spatial non-linear characteristics has to date been proposed to describe these linear–non-linear interactions. This article introduces a unifying perspective of non-linearities as internal feedback forces that act together with the external forces to generate the response of the non-linear system. This perspective of the non-linearities is spatial in nature and leads to two simple but conceptually powerful relationships between the frequency response function matrix of a non-linear multiple-degree-of-freedom system and its linear counterpart. Several single- and multiple-degree-of-freedom systems are used to demonstrate the use and interpretation of these relationships. The broad implication of the new input–output frequency response representation for both linear and non-linear systems are also addressed. In particular, the merits of the spatial perspective of non-linear systems and the new frequency response relationships are stated in the context of linear and non-linear system characterization and identification. One implication is that these relationships suggest there is an input–output-dependent temporal-spatial (modal) decomposition of the frequency response function matrix for non-linear systems.