Nonparametric identification of multi-degree-of-freedom nonlinear systems from partially measured responses under uncertain dynamic excitations

With the rapid advent of new materials and novel structures, it becomes difficult, if not impossible, to accurately model and simulate the nonlinear response of complex systems under uncertain dynamic excitations based on close-formed nonlinear functions and parametric identification. In this study, a two-step structural nonlinearity localization and identification approach for multi-degree-of-freedom (MDOF) nonlinear systems under uncertain dynamic excitations is developed by integrating an extended Kalman filter with unknown inputs into the equivalent linearized systems. In the first step, unmeasured responses and excitations are estimated as well as unknown structural parameters and nonlinearity locations are identified by fusing acceleration with displacement time histories at the observed degrees of freedom (DOFs). In the second step, the nonlinear restoring force of the detected nonlinear structural members is identified nonparametrically using three polynomial models, including a power series polynomial model (PSPM), a double Chebyshev polynomial model (DCPM), and a Legendre polynomial model (LPM). Linear multi-story shear frames controlled by nonlinear magnetorheological (MR) dampers are modelled computationally to demonstrate the generality of the proposed methodology. The multi-source uncertainties considered in these representative examples include the location and the type of nonlinearities represented by a Bingham model and a modified Dahl model of the dampers, the location of response measurements, the location and intensity of dynamic excitations, the level of measurement noise, and the initial assignment of structural parameters. The acceleration, velocity, and displacement time histories of structures can be evaluated accurately with a maximum error of 2.62% even with the presence of 8% measurement noise, while the external excitations can be estimated within an error of 1.77%. The location of nonlinear elements can be detected correctly. The structural parameters, the NRFs provided by MR dampers and the corresponding energy dissipation can be identified with a maximum error of 2.08%, 1.19% and 0.39%, respectively, even 8% measurement noise and very rough initial assignment of structure parameters (−70%) are considered. Moreover, the numerical results change little (<0.20%) even the initial assignment of structural parameters varies from 50% to 30% of their original values, no matter which nonparametric model is employed. Results indicate that the presented algorithm can effectively identify unmeasured dynamic responses, structural parameters, unknown excitations, nonlinear locations, and NRF of nonlinear elements in a nonparametric way.