Identification of structural non-linearities using describing functions and the Sherman–Morrison method

In this study, a new method for type and parametric identification of a non-linear element in an otherwise linear structure is introduced. This work is an extension of a previous study in which a method was developed to localize non-linearity in multi-degree of freedom systems and to identify type and parameters of the non-linear element when it is located at a ground connection of the system. The method uses a describing function approach for representing the non-linearity in the structure. The describing function contains only the first harmonic terms. The Sherman–Morrison matrix inversion method is used in the present study to put the response expression in a form where the non-linearity term can be isolated. Using measured responses one can calculate the value of the describing function representation of the non-linear element and thus perform the identification. This new method can be used for type and parametric identification of a non-linear element between any two coordinates of the system. Case studies are given to demonstrate the applicability of the method.