# Integration of Freeplay-Induced Limit Cycles Based On a State Space Iterating Scheme
###### Abstract  
Time integration is commonly used to obtain accurate system responses, such as the limit cycle oscillations (LCOs) for an aeroelastic system with freeplay. However, the integrations that start with various initial conditions (IC.s) are usually studied case by case, so only a few system states can possibly be focused on. This paper proposes a state space iterating (SSI) scheme to find LCO solutions using time integration by using another method. First, a large number of arbitrary LC. cases are used for time integrations, but only a very short integration time is required for each I.C. case. Second, system behaviors are depicted visually through a method that combines a modified Poincare map and Lorenz map, in which the LCO solutions are found as fixed points via visual inspections. To verify the SSI scheme's ability to find LCOs, a typical plunge-pitch wing section is established numerically. Time integrations with both the classic scheme and the proposed SSI scheme are carried out. The LCO results of the SSI scheme are well-aligned with those from the classic scheme. The SSI scheme visualizes the patterns of system responses using arbitrary LC. cases and analyzes the LCO stability, which provides more mathematical insights into an aeroelastic system with freeplay. time integration; initial condition; limit cycle oscillation; freeplay; Poincare map; Lorenz map +
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integration is recognized as the benchmark for other methods due to its high accuracy and compatibility with all kinds of nonlinearity [1,4]. Time integration solves equations of motion under a set of discrete time instances using various finite difference schemes such as Newmark, Euler, and Runge-Kutta, thus providing a straightforward approach to obtain accurate solutions. The accumulating errors due to finite difference have been addressed by many researchers via schemes using adaptive time steps, such as the fourth-fifth-order Runge-Kutta (RK45) scheme and the precise integration method (PIM) [4, 5, 6]. In addition, the difficulty when dealing with non-smooth nonlinearities such as freeplay can be handled by a combined scheme of the Henon method and RK45 (Henon-RK45) [4,7]. The kernel of Henon-RK45 is to find all \"switching points\" where the non-smooth effects arise and move the system states to those switching points accurately. Therefore, time integration provides excellent access for the nonlinear responses of aeroelastic systems with freeplay. However, several weaknesses of time integration are still seen in the following aspects: (1) Time integration falls short when analyzing the effects of initial conditions (I.C.s), which are quite significant for understanding nonlinear behaviors. The most widely adopted process is to assign only one principal system state a series of non-zero values and study the LCO behaviors case by case. For example, [PERSON] and [PERSON] [8] studied an all-moving surface model and a wing-store model with various structural nonlinearities, such as cubic stiffness, cubic damping, and freeplay. In their study, less than one hundred I.C.s were tested in time integrations for the second model, while that number for the first model was only one. In each I.C. case, only the pitch angle was specified as a non-zero value, and the integrations with different I.C.s were tested case by case, which was quite time-consuming and only partially confirmatory. However, they did determine an important conclusion: some aeroelastic models can be very sensitive to the I.C., which might lead to completely different system behaviors such as decaying to zero, entering into an LCO, or achieving oscillatory divergence. [PERSON] and [PERSON] [9] studied three cases of I.C.s in Section VI.A.3 in their paper. In all three cases, only one state--the displacement at the wing-tip for the first case and at the store pitch angle for the second and the third cases--was specified as a non-zero value, and the authors found that the LCO behaviours could be very sensitive to both the I.C.s and flow velocity.