Structural nonlinear behavior under the excitation of strong dynamic loadings should be considered for structural damage initiation and propagation identification of engineering structures. In this study， a power series polynomial of relative displacement and velocity is employed to model the nonlinear restoring force （NRF） of a structure in a nonparametric way and structural mass，stiffness， damping coefficients and NRF are identified based on an updated cubature Kalman filter （UCKF） algorithm using acceleration response at limited degrees of freedoms （DOFs） of the structure during dynamic excitation. Then， a multi-degree-of-freedom （MDOF） numerical model equipped with a magnetorheological （MR） damper mimicking structural nonlinearity is employed to validate the proposed approach numerically. By adding 20% measurement noise to the acceleration measurements， the stiffness，damping coefficients and mass of the structure， the unmeasured response and the NRF are identified. The effectiveness of the proposed method is validated by comparing the theoretical values with the identified values. Moreover， the identification results of the proposed approach are also compared with them of the approach with the traditional extended Kalman filter （EKF）， cubature Kalman filter （CKF）， and extended Kalman filter with memory fading （EKF-MF）. Dynamic test on a four-story shear frame model with a MR damper is carried out. The structural parameters of the frame structure itself， the unused dynamic response in identification and the NRF provided by the MR damper are identified with the proposed approach using acceleration responses at certain floors. The identified results are compared with the test measurements directly and the performance of the proposed identification approach is experimentally validated.