# Impact of Aerodynamic Interactions on Aeroelastic Stability of Wing-Propeller Systems +
## 2 Numerical Tools  
In the present work, aeroelastic stability analyses are performed using both linear frequency-domain and nonlinear time-domain simulation environments. The frequency-domain simulations are conducted using an in-house toolbox (SDBox). This tool offers robust capabilities for efficient analysis, providing key insights into flutter points, natural frequencies, damping ratios, and mode shapes from the initial results. SDBox was developed in MATLAB (R2022b) and designed to predict flutter speeds of propellers, wings, and wings coupled with multiple propellers [11]. The wing's flexibility is modeled with beam elements based on the Timoshenko theory. Propellers are mounted to the wing's elastic axis using rotational springs, simulating a simplified flexible pylon system that allows pitch and yaw rotations. The aerodynamics of the wing are represented using the doublet lattice method (DLM). Propeller aerodynamics are modeled through stability derivatives as described in [6; 18]; however, instead of using an analytical formulation for the derivatives, an enhanced, full unsteady approach is applied to identify the propeller aerodynamic derivatives as described in [7; 8]. The aeroelastic system is described by the classical flutter equations and finally solved with the p-k method. Aerodynamic interaction is neglected in this framework. SDBox has proven to be powerful and efficient in the fast prediction of flutter speeds and was successfully validated in [11]. For the time-domain simulations, a coupled toolchain consisting of MBDyn [19] and DUST [20] is employed. MBDyn is a free general-purpose multibody dynamics tool that offers the capability to model complex mechanical systems. Nonlinear dynamics of rigid and flexible bodies can be simulated, connected by kinematic constraints. The equations of motion are formulated in differential algebraic form and solved numerically using different A/L stable integration methods. MBDyn also features an aerodynamic module that incorporates the blade element theory in conjunction with a dynamic inflow model with three inflow states [21; 22]. DUST, on the other hand, is an aerodynamic tool that relies on potential flow theory. It employs a free-wake description with vortex particles. This approach allows for more accurate modeling of complex geometries by offering multiple elements, such as lifting lines for slender lift-generating surfaces, vortex lattice elements, and surface panels. The vortex particle method provides a stable description of the free-vorticity flow field, which is suitable for numerical simulations of configurations characterized by strong aerodynamic interactions. The Lagrangian grid-free approach does not require a volume mesh of the surrounding flow. Indeed, the vortex particle method effectively captures the interaction of wakes produced by lifting surfaces and bodies, as is commonly seen in rotary-wing vehicle applications. A Cartesian fast multipole method (FMM) is employed to reduce computational costs associated with vortex particle interactions [23; 24]. To facilitate the coupling of MBDyn and DUST for aeroelastic simulations, the open-source tool PreCICE [25] is used [26]. PreCICE provides a tight coupling scheme, enabling the exchange of information between the two solvers. This coupling framework enhances the accuracy and efficiency of the simulations by allowing the seamless interaction between the multibody and the aerodynamic solvers. The coupled MBDyn-DUST tool has been extensively used for aeroelastic analysis of both fixed- and rotary-wing aircraft, including tiltrotors [26; 27; 28]. This framework has been chosen by the authors since it leads to efficient simulations compared to high-fidelity CSD-CFD tools, with the possibility to include aerodynamic interference on aeroelastic analyses; in addition, MBDyn, DUST, and the coupling library PreCICE are all open-source tools.