# Development of a Mobile Buoy with Controllable Wings: Design, Dynamics Analysis and Experiments
## 3 Fluid-Multibody Coupling Dynamics Model  
### Fluid-Multibody Coupling Analysis Method  
For the mobile buoy, hydrodynamics and multibody dynamics are two aspects that must be considered in their dynamics analysis. In this research, a fluid-multibody coupling method that incorporates both multibody dynamics and hydrodynamics model is developed to analyze the dynamic responses of the designed mobile buoy. The detailed procedure of the fluid-multibody coupling analysis is illustrated in Figure 3. Firstly, based on the initial motion condition (the dynamics equations of the mobile buoy at the initial moment) and the boundary condition (the boundary condition of the hydrodynamics model), the original mobile buoy is set within a presumed boundary region and meshed. Subsequently, hydrodynamic analysis is conducted within the CFD software. During this process, the macro commands DEFINE_CG_MOTION and Compute_Force_And_Moment function are applied via the UDF (user-defined function) of ANSYS-Fluent (15.0 version, ANSYS Corporation, Canonsburg, PA, USA) to compute the continuity equations and Navier-Stokes continuity equations, allowing for the determination of the hydrodynamic forces and moments exerted upon the mobile buoy. Then, the hydrodynamic force and moment are incorporated into its dynamics equation via the UDF codes to derive the accel  
Figure 2: Mobile buoy with controllable wings, showing (**a**) shape and structural design, (**b**) switching motion mode by controlling wing rotation angle.  
erations and angular accelerations. By incorporating positions, attitudes, and accelerations of the mobile buoy can be obtained, enabling the extraction of the motion information for the next time step. This information is then used to update the mesh and motion through dynamic mesh technology. Subsequently, the mobile buoy is reintroduced into the hydrodynamics section and iterative calculations are performed. Through continuous motion state updates, fluid-multibody coupling analysis is conducted in the flow field. After all the time steps are calculated, the dynamic responses of the mobile buoy can be acquired. Compared to other methods that directly incorporate hydrodynamics into the multibody dynamic equations using empirical formula, the fluid-multibody coupling method takes time-varying hydrodynamics into account, enabling to reflect the motion state more objectively and realistically.  
The continuity equation and Navier-Stokes equation for calculating hydrodynamic forces and moments exerted upon the mobile buoy can be expressed as  
\[\iint\limits_{CS}\rho\mu\cdot dS+\frac{\partial}{\partial t}\iint\limits_{CV} \rho dV=0 \tag{1}\]  
\[\rho\overline{u_{i}}\frac{\partial\overline{u_{i}}}{\partial x_{j}}=\rho \overline{f_{i}}+\frac{\partial}{\partial x_{i}}\left[-\overline{p}\delta_{ij} +\mu(\frac{\partial\overline{u_{i}}}{\partial x_{j}}+\frac{\partial\overline{u _{j}}}{\partial x_{i}})\right]-\rho\overline{u_{i}}^{\prime}u_{j}^{\prime} \tag{2}\]  
where \(\rho\) is the fluid density, \(\mu\) is the velocity vector and \(S\) represents the characteristic area of the buoy. \(V\) is the volume, \(t\) is the time. \(u_{i}\) is the velocity component of the fluid, \(u_{i}^{\prime}\) is the pulsation velocity, \(f_{i}\) is the external force component acting on the fluid, \(\mu\) is the dynamic viscosity and \(p\) is the pressure.  
Figure 3: Analysis process of fluid–multibody coupling method.