# Integration of Freeplay-Induced Limit Cycles Based On a State Space Iterating Scheme
###### Abstract
[PERSON] and [PERSON] [10] also found that the flutter instability boundary was significantly dependent on the initial pitch angle. However, the authors only assessed a few non-zero initial pitch angles. Many other studies of the I.C. effects on LCO behaviors followed a similar method to that mentioned above [11,12]. (2) Time integration inevitably loses efficiency to some extent when dealing with freeplay due to the need to frequently search and verify the switching points of the system. Moreover, aperiodic/quasi-periodic motions and chaos are commonly seen in aeroelastic systems, even for simple wing section models [13,14], and a lengthy transient phase may also appear before a steady LCO. As a result, a wide range of time intervals could be required for confirming an LCO in time integration [4]. This presents a dilemma, where accurate LCO results require all switching points to be located along with a long integration time, while a large number of I.C. cases and airspeed cases demand high efficiency. To balance the accuracy and efficiency is, therefore, not an easy task. This dilemma spurred us to seek another way of using time integration when dealing with aeroelastic systems with freeplay. First, the feature of high accuracy in time integration should remain since this feature is the basis upon which time integration serves as the benchmark for other methods. Therefore, calculations were carried out using the Henon-RK45 method in this paper. Second, inspired by the application of the Poincare map ([15] and Section 3.4 in [16]) on aeroelastic systems with freeplay [10,17,18,19,20], as well as the heuristic and visualized method, using a Lorenz map (Section 9.4 in [21]), we propose a novel scheme to implement a modified Poincare map using a Lorenz map. In this way, we can visualize the spatial patterns of all system states through massive I.C. cases and use a short integration time for each I.C. case. Moreover, fixed points of the modified Poincare map that represent the LCO solutions can be easily obtained through a visual inspection of the graph of state patterns, which is very efficient. The LCO stability could also be confirmed in the same way. Because this new scheme requires an iteration procedure in state space, we named it the state space iterating (SSI) scheme and applied it to a typical plunge-pitch wing section subject to unsteady aerodynamic loads. The proposed SSI scheme yields two LCO branches for the wing section, which are well aligned with those derived from the classic time integration using the Henon-RK45 method. The highlights of the SSI scheme are as follows: (1) The spatial patterns of system states are clearly pictured, so one can easily find LCO solutions and confirm the stabilities directly and visually; (2) the integration time for each I.C. case is reduced to a few LCO periods, so a large number of I.C. cases would not decrease the calculation efficiency; and (3) the proposed method can be easily extended to all other kinds of structural nonlinearities and applied to an aeroelastic system with higher dimensions. This paper is organized as follows: Section 2 presents the state space equations for a general aeroelastic system with freeplay and introduces the classic Henon-RK45 method. Section 3 introduces the proposed SSI method and explains how the Poincare map and Lorenz map are modified, applied, and finally form the SSI scheme together. A plunge-pitch aeroelastic model is established in Section 4. Then, an LCO analysis based on the Henon-RK45 method is implemented in Section 5. Time integrations based on the SSI scheme are carried out in Section 6, along with the discussions and a comparison to the Henon-RK45 method.