## 1 Introduction
Then, a comparison between the capabilities of TNSI and FNSI was addressed by [PERSON] [29] considering a series of numerical experiments towards a nonlinear SmallSat spacecraft along with stochastic analysis. Smooth structural nonlinearities were chosen as demonstrations of system identification methods. However, the capability of identifying non-smooth nonlinearities was seldom discussed. As one typical non-smooth structural stiffness nonlinearity, freeplay has often been witnessed in joints, bearings, hinges, and contact surfaces due to the manufacturing tolerance, wearing and tearing, or impacting. In practice, it is sometimes difficult to measure the actual gap directly. Furthermore, the nominal linear system of structure with freeplay plays an essential rule in the later structural and dynamic analysis since most of the analysis is performed under linear assumptions in practical engineering. To obtain an insight into the nonlinear behaviors of a structure with freeplay, both the nominal linear system and freeplay parameters have to be accurately estimated for better understanding. [PERSON] et al. exploited the Hilbert transform [30] and CRP [31] method to freeplay identification. [PERSON] [29] also addressed the identification of clearance. TNSI had been utilized on a 5-DOF structure with a one-side freeplay by [PERSON] [32]. However, little attention has been paid to the identification of freeplay with offset, which occurs more generally in real-life structures. Moreover, the existence of freeplay divides the original nonlinear system into three linear systems into different regions, and there remains a paucity of investigations on identifying them. In the present paper, we perform a series of numerical experiments on an all movable fin considering two types of rotational freeplay. Both the modal parameters of the nominal linear system and nonlinear coefficients will be estimated by the subspace-based method. Subspace identification techniques are built upon the reformulation of the state-space relations Equation (4) in a matrix form [29]. There are several advantages of the subspace-based method. First, the approach can naturally deal with MIMO systems and has been successfully applied to a wide variety of real-life systems in control or biomedical as powerful identification tools [29]. Because the method identifies the _input-state-output_ models instead of _input-output_ models. Moreover, the robustness and high numerical performance of the method are guaranteed by making use of the geometric framework and numerical linear algebra decompositions. Therefore, the subspace system identification is a non-iterative method. The paper is organized as follows. Section 2 briefly introduces the methodology of the TNSI method. The all movable fin is described in Section 3, along with its finite element model and reduced-order model. Furthermore, the definition of the underlying linear system and overlying linear system, and the description of two typical kinds of freeplay are also detailed. Section 4 details the identification of the all movable fin with either central freeplay or offset freeplay in its pitch degree of freedom. The performance of the TNSI method for estimating freeplay is also discussed. The frequency response function (FRF) matrix of both the underlying and overlying linear system of the nonlinear all movable fin were identified during these two sets of numerical experiments. Finally, Section 5 summarizes the conclusions.