## 2 Problem Formulation  
Designing wind systems with counter-rotating generators is challenging owing to the branched power flow in planetary transmissions from a single input to dual outputs. The generalized modeling and numerical simulation of the dynamic response of this wind turbine (WT) type is addressed in this paper. Without reducing the generality, a case study of a 1-DOF wind system is considered (Figure 1), consisting of a wind rotor \(R\), a planetary speed increase (_SI_) with one input (satellite-carrier \(H\)) and two outputs (gears 1 and 2) connected to a counter-rotating electric generator \(G\). The rotor \(GR\) and the stator \(GS\) of the generator are both mobile and rotate in opposite directions. The block scheme in Figure 1b highlights the interactions between the three key components of the wind system: mechanical power is transmitted from the wind rotor \(R\) via the shaft \(H\) to the speed increase \(SI\), which distributes the output power to both the rotor \(GR\) and stator \(GS\). Obviously, the two power outputs are not independent, the rotor and stator of the generator being permanently characterized in operation by equal and opposite torques (i.e., \(T_{GS}=-T_{GR}\)). The speed increase is a planetary mechanical transmission with cylindrical gears (1-5, Figure 1), three of which are sun gears (1, 2, and 5 \(\equiv 0\)), and the solidarized gears 3 and 4 form a double satellite. In practical applications, the planetary transmission includes \(n_{s}\geq 2\) equiangularly arranged double satellites (3-4). The power input of the speed increase (i.e., the satellite carrier \(H\)) is solidarized with the wind rotor \(R\). Satellites 3-4 engage on the one hand with a fixed ring gear 5 and on the other hand with the sun gear 1, connected with the generator rotor \(GR\), and with the ring gear 2, coupled to the generator stator \(GS\). The angular speed of a counter-rotating generator \(G\) (\(\omega_{G}\)) is given by the relative speed of the rotor \(GR\) with respect to the stator \(GS\):  
\[\omega_{G}=\omega_{GR}-\omega_{GS}=\omega_{1}-\omega_{2}. \tag{1}\]  
As a result, the kinematic amplification ratios, which describe the transmission of the rotational speed from the wind rotor to the generator rotor (\(i_{a1}\)) and to the generator stator (\(i_{a2}\)), respectively, and the total amplification ratio (\(i_{aG}\)) achieved by the wind turbine, can be established through the following relations:  
\[i_{a1}=\frac{\omega_{1}}{\omega_{H}};\ i_{a2}=\frac{\omega_{2}}{\omega_{H}};\ i_{aG}=\frac{\omega_{1}-\omega_{2}}{ \omega_{H}}=i_{a1}-i_{a2}, \tag{2}\]  
where \(\omega_{x}\) is the angular velocity of the body \(x=1\), \(2\), \(H\); \(i_{ay}\)--the amplification ratio from the input \(R\) to the element \(y=1\), \(2\), \(G\). The dynamic modeling of the analyzed 1-DOF wind system aims to identify its equation of motion \(e_{R}=f(\omega_{R},\ J_{x},\ cst)\), where \(J_{x}\) is the mechanical axial moment of inertia of the body x = 1, 2, \(H\), and \(cst\) represents the set of other constant parameters. The motion of the input shaft (of the wind rotor \(R\)) is considered as an independent kinematic variable of the wind turbine. By solving this differential equation, the time variation of the torques and kinematic variables (velocities and angular accelerations) related to all system shafts is obtained; the numerical simulations are performed under the assumption of starting the system from rest at a specified constant wind speed. In the proposed dynamic modeling, the following working premises are considered:  
1. The rotational elements have geometric symmetry with respect to their own axis of motion, and they are rigid bodies with uniformly distributed mass; as a result, the mass center of a body is located on its own axis of rotation;
2. The inertial masses of the mobile elements in the planetary transmission are reduced to their outer shafts; thus, the correlations of the torques in the planetary units coincide with those of static conditions;
3.