###### Abstract
[17] comprehensively summarized the previous research results on the nonlinear aeroelastic system of a typical airfoil section. In recent decades, [PERSON] et al. [1] and [PERSON] et al. [18] have also reviewed the research status of nonlinear problems of control surface structures. For research using three-dimensional (3D) models, there are two main kinds of objects: wings with stores [19; 20; 21; 22; 23] and all-movable control surfaces [24; 25; 26; 27]. A primary problem in such research is to find ways to establish reduced-order models of these systems. The Lagrange equation can be directly used for simple objects. For complex objects such as folding fins or folding wings, there are two main modeling methods: the modal synthesis method [26; 27] and the fictitious mass method [23; 28]. The main analysis approaches are also the DF method and the Henon-RK45 numerical integration method. In addition to considering improvements and innovations in modeling and analysis methods, it is necessary to consider how research results can be applied to aircraft design. [PERSON] et al. [20; 21] found that friction nonlinearity will benefit a system in which there is no LCO occurrence below the linear flutter boundary; this may provide a solution for the suppression of LCOs. For an all-movable fin with freeplay nonlinearities, finding ways in which to delay or suppress LCOs through the reasonable design of structural parameters is a problem worth examining. So far, however, there has been little discussion of this problem. Regarding multiple nonlinearities, [PERSON] et al. [29] considered the flutter of a 3D rigid fin with freeplay in the flapping, plunging, and pitching degrees of freedom (DOFs). The results of the time-domain analysis showed that the freeplay nonlinearities lead to lower flutter-speed predictions compared to the linear system results. In addition to the time-domain integration approach, a frequency-domain iterative procedure based on the DF method has also been established and developed for systems with multiple structural nonlinearities. [PERSON] and [PERSON] [30] considered a missile control surface with roll and pitch freeplay nonlinearities with the DF method. However, in this study, the coupled responses between nonlinearities were not deeply considered; rather, the amplitude of one nonlinear element was fixed while the other was changed. In 1980, [PERSON] [31] proposed an iterative procedure to achieve alignment of the oscillatory amplitude of each nonlinear element. Later, [PERSON] [32] proposed a comprehensive frequency-domain iteration procedure for flutter analysis of large dynamical systems with multiple structural nonlinearities; the results of this frequency-domain approach were found to be consistent with those of the time-domain approach. [PERSON] et al. [33] simplified the above solution process and emphasized the importance of the frequency ratio when dealing with a two-dimensional control-fin model. For all-movable fins, the current iterative procedure may lead to the loss of solutions. Furthermore, the iterative procedure may not converge when the selected iteration parameters are not appropriate. In this study, we provided novel points of view on the nonlinear aeroelastic characteristics of a small aspect ratio all-movable fin with freeplay nonlinearities in its root. In the remainder of the paper, the basic modeling and analysis methods are introduced. Then, for a single freeplay nonlinearity, two different LCO characteristics are introduced, and a harmonic initial condition is proposed for time-domain integration to verify the frequency-domain LCO results. For multiple freeplay nonlinearities, a discrete scanning method based on the DF method is established to avoid the loss of solutions. Finally, the frequency- and time-domain approaches are respectively used to analyze the effects of the support stiffness and freeplay size ratio on the LCO characteristics.