A frequency domain nonparametric identification method for nonlinear structures: Describing surface method

In this paper a new method called 'Describing Surface Method' (DSM) is developed for nonparametric identification of a localized nonlinearity in structural dynamics. The method makes use of the Nonlinearity Matrix concept developed in the past by using classical describing function theory, which assumes that nonlinearity depends mainly on the response amplitude and frequency dependence is negligible for almost all of the standard nonlinear elements. However, this may not always be the case for complex nonlinearities. With the method proposed in this study, nonlinearities which are functions of both frequency and displacement amplitude can be identified by using response-controlled stepped-sine testing. Furthermore, the nonlinearity does not need to be mathematically expressible in terms of response amplitude and frequency, which allows us to identify more complex nonlinearities nonparametrically. The method is applicable to real engineering structures with local nonlinearity affecting the boundary conditions, where modes are not closely spaced, and sub- and super-harmonics are assumed to be negligible compared to the fundamental harmonic. Multiple nonlinearities may coexist at the same location and a priori knowledge of nonlinearity type is not necessary. The method yields the describing surface of nonlinearity, real and imaginary parts of which correspond to the equivalent nonlinear stiffness and nonlinear damping at that location in the structure. Harmonic response of a nonlinear system to any force, including any existing unstable branch, can be calculated iteratively by using the describing surface representing the nonlinearity. Unstable branches captured by using Newton's Method with arc-length continuation algorithm can be validated experimentally by using Harmonic Force Surface (HFS) concept. The validation of DSM is demonstrated with three experimental case studies: a cantilever beam with cubic stiffness at its tip point, a dummy mass on elastomeric vibration isolators, and a control fin actuation mechanism of a real missile structure which exhibits very complex (due to backlash and friction) and strong nonlinearity causing jump phenomenon in the frequency response. In each case, nonlinear FRFs calculated by using identified describing surface agree with measured FRFs at various force levels much better than FRFs calculated by using classical describing function method. It is observed in the experimental case studies that frequency dependence of nonlinearity occurs mostly in the imaginary part of the describing surface which represents nonlinear damping