Nonparametric nonlinear restoring force and excitation identification with Legendre polynomial model and data fusion

Abstract

Identification of nonlinear restoring force and dynamic loadings provides critical information for post-event damage diagnosis of structures. Due to high complexity and individuality of structural nonlinearities, it is difficult to provide an exact parametric mathematical model in advance to describe the nonlinear behavior of a structural member or a substructure under strong dynamic loadings in practice. Moreover, external dynamic loading applied to an engineering structure is usually unknown and only acceleration responses at limited degrees of freedom of the structure are available for identification. In this study, a nonparametric nonlinear restoring force and excitation identification approach combining the Legendre polynomial model and extended Kalman filter with unknown input is proposed using limited acceleration measurements fused with limited displacement measurements. Then, the performance of the proposed approach is first illustrated via numerical simulation with multi-degree-of-freedom frame structures equipped with magnetorheological dampers mimicking nonlinearity under direct dynamic excitation or base excitation using noise-polluted measurements. Finally, a dynamic experimental study on a four-story steel frame model equipped with a magnetorheological damper is carried out and dynamic response measurement is employed to validate the effectiveness of the proposed method by comparing the identified dynamic responses, nonlinear restoring force, and excitation force with the test measurements. The convergence and the effect of initial estimation errors of structural parameters on the final identification results are investigated. The effect of data fusion on improving the identification accuracy is also investigated.

Hysteresis and dynamic loading nonparametric identification for multi‐degree‐of‐freedom structures using an updated general extended Kalman filter and a Legendre polynomial model

“…dimensions and structural parameters of the model are given in the literature. 44 The MR damper is employed to mimic the nonlinear behavior during the vibration test. A random excitation is applied on the second floor by a shaker, which will be identified by the approach.…”

“…The NRF provided by an MR damper located between the ith and the (i À 1)th floor, in this case, is expressed as the Dahl model shown as follows 44 :…”

“…The NRF provided by an MR damper located between the i th and the ( i − 1)th floor, in this case, is expressed as the Dahl model shown as follows 44 : where , , , and are the parameters expressing the characteristics of the Dahl model; and are the relative displacement and relative velocity between the i th and ( i − 1)th floor, respectively; z is a dimensionless coefficient determined by the following equation: in which is the coefficient that can determine the shape of the hysteresis loop. The parameters of the Dahl model here take the following values: = 60 N/m, = 2000 Ns/m, , , and …”

“…In physical science and mathematics, the LPM is composed of a system of complete and orthogonal polynomials with significant advantages and applications. It has recently been used to describe the NRF of engineering structures in a nonparametric form 44 . In this study, the LPM is used to describe the NRF of engineering structures without using any parametric models.…”

“…Numerical study and an experimental test are conducted to verify the effectiveness and robustness of the proposed method. Subsequently, considering the difficulties in measuring complete excitation information, Xu et al 44 also put forward a nonparametric nonlinear restoring force (NRF) and excitation identification approach with EKF and data fusion, where NRF was expressed in the form of a Legendre polynomial model (LPM). Aiming at the identification of the hysteretic forces of a nonlinear structure in a nonparametric way, Lei et al 45 proposed a two stage model‐free hysteresis force identification approach for MR dampers with GEKF‐UI and GKF‐UI, where the hysteresis force is treated as a “unknown fictitious forces” applied to the corresponding linear bare structure identified in the first stage.…”