## 3 Dynamic Modelling
### Dynamic Equations of Component Bodies  
Based on the scheme from Figure 1a, the considered wind turbine can be decomposed into the following rigid bodies:  
* body B1: the assembly consisting of the sun gear 1, the generator rotor \(GR\), and the connecting shaft, Figure 3a;
* body B2: the assembly wind rotor \(R\)--ring gear 2--generator stator \(GS\), Figure 3b;
* a number \(\mathrm{s_{3}}\) of gears 3 and their shafts (body B3), evenly distributed around the central axis, Figure 3c.  
The positive direction of the angular speed and torque vectors for all the bodies B1... B3 is defined to the right (Figure 1), while \(J_{x}\) denotes the axial moment of inertia of the body \(\mathrm{x}=1\ldots 3\), \(\boldsymbol{\omega}_{x}\)--the angular speed of the body \(x\), and \(T_{x}\)--the resultant torque acting on the body \(x\).  
Thus, considering the body B1 characterised by the fixed-rigid connection between gear 1 and the generator rotor \(RG\), the following equations can be written (Figure 3a):  
\[\boldsymbol{\omega}_{1}=\boldsymbol{\omega}_{GR} \tag{9}\]  
\[J_{1}\varepsilon_{1}=T_{GR}-T_{1} \tag{10}\]  
Similarly, the kinematic and dynamic equations of the B2 body, characterised by the connections \(R\equiv 2\equiv GS\), can be described by (Figure 3b):  
\[\boldsymbol{\omega}_{R}=\boldsymbol{\omega}_{2}=\boldsymbol{\omega}_{GS} \tag{11}\]  
\[J_{2}\varepsilon_{2}=T_{R}-T_{2}+T_{GS} \tag{12}\]  
The mechanical power is transmitted from ring gear 2 to gear 1 through \(\mathrm{s_{3}}\) theoretically equal flows, corresponding to \(\mathrm{s_{3}}\) number of gears 3 that are mounted in parallel between gears 2 and 1. As a result, the body B3 is driven theoretically by a torque \(T_{32}\) defined by relation (13):  
\[T_{32}=\frac{T_{2}}{\mathrm{s_{3}}}i_{2\_3}\eta_{2\_3} \tag{13}\]Figure 3: Dynamic scheme of: (**a**) body B1; (**b**) body B2; and (**c**) body B3.  
which comes from the energetic equilibrium equation with friction for the gear pair 2-3:  
\[T_{2}\omega_{2}\eta_{2\_3}=s_{3}T_{32}\omega_{3} \tag{14}\]  
where \(i_{2\_3}=\omega_{2}/\omega_{3}=z_{3}/z_{2}\) is the kinematic ratio of the gear pair 2-3, and \(\eta_{2\_3}\)--the efficiency of the gear pair 2-3.  
Gear 3 is also subjected to a reaction torque \(T_{31}\) from gear 1:  
\[T_{31}=\frac{T_{1}}{s_{3}}i_{1\_3}\eta_{3\_1}^{-1} \tag{15}\]  
defined as well based on the energetic equilibrium equation with friction for the gear pair 3-1:  
\[s_{3}T_{31}\omega_{3}\eta_{3\_1}=T_{1}\omega_{1} \tag{16}\]  
where \(i_{1\_3}=\omega_{1}/\omega_{3}=-z_{3}/z_{1}\) is the kinematic ratio of the gear pair 1-3, and \(\eta_{3\_1}\)--the efficiency of the gear pair 3-1.  
As a result, the dynamic equation for the body B3 is:  
\[J_{3}\varepsilon_{3}=T_{32}+T_{31} \tag{17}\]  
which by considering Equations (13) and (15) becomes:  
\[s_{3}J_{3}\varepsilon_{3}=T_{2}i_{2\_3}\eta_{2\_3}+T_{1}i_{1\_3}\eta_{3\_1}^{ -1} \tag{18}\]  
Relation (18) models the transmission of mechanical power from gear 2 to gear 1 in the general case of considering the inertial effects of the body B3 and the impact of gear friction; if the mass of body 3 is neglected (i.e., \(J_{3}=0\)) or if the turbine is operating in a steady-state regime (i.e., \(\varepsilon_{3}=0\)), Equation (18) takes the particular form:  
\[T_{2}i_{2\_1}\eta_{2\_1}+T_{1}=0 \tag{19}\]  
which expresses the energetic equilibrium condition of the speed increase \(Sl\) in the premise of considering friction in gear pairs, in which:  
\[i_{2\_1}=\frac{\omega_{2}}{\omega_{1}}=i_{2\_3}\cdot i_{3\_1}=-\frac{z_{1}}{z _{2}} \tag{20}\]  
and  
\[\eta_{2\_1}=\eta_{2\_3}\cdot\eta_{3\_1} \tag{21}\]