# Parametric Selection of Optimized Epicyclic Gearbox Layouts for Wind Power Plant Applications
## 3 Methods
### Torque Method  
The torque method is a clear, precise, and easy-to-use method for the analysis of epicyclic gear trains. Unlike the previously mentioned methods, it has the advantage that internal branching or circulation of power may be detected, and the gear train efficiency and load are calculated reliably. In contrast to the other methods using angular velocity \(\omega\) and the peripheral linear velocity \(v\), the torque method uses the torques \(T\) acting on the gear train components through a lever analogy [49; 50; 52], which is best explained on the simple planetary gearset from Figure 1.  
Equation (3) is imposed by the conditions of the balance of ideal torques \(\Sigma T_{\rm i}\) acting on the shafts, as mentioned in Section 2:  
\[\sum T_{\rm i}= T_{\rm 1}+T_{\rm 3}+T_{\rm H}=0, \tag{3}\]  
where \(T_{\rm i}\) is the sum of the torques acting on the gear train, \(T_{\rm 1}\) is the torque acting on the sun gear, \(T_{\rm 3}\) is the torque acting on the ring gear, and \(T_{\rm H}\) is the torque acting on the planet carrier (Figure 1). The power flow through the gear train in the case of operation with one fixed shaft (1-DOF) \(\Sigma P_{\rm i}\) can be expressed using Equation (4):  
\[\sum P_{\rm i}= P_{\rm A}+P_{\rm B}=T_{\rm A}\omega_{\rm A}+T_{\rm B}\omega_{\rm B }=0, \tag{4}\]  
where \(P\), \(T\), and \(\omega\), respectively, denote the power, torque, and angular velocity, while index A marks the input and index B the output shaft, and \(i\) is the summation index. Equation (4) is then used to determine the kinematic transmission ratio \(i\); Equation (5):  
\[i=\frac{\omega_{\rm A}}{\omega_{\rm B}}=-\frac{T_{\rm B}}{T_{\rm A}}. \tag{5}\]  
The kinematic transmission ratio corresponds to the ideal energy transmission ratio without losses. In the case that losses are taken into consideration using the real output torque \(T_{\rm Break}\), Equation (5) is rewritten to provide the energy transmission ratio (with losses) \(i_{\rm T}\) of the gear train; Equation (6):  
\[i_{\rm T}=\frac{T_{\rm Break}}{T_{\rm A}}=\left(\frac{T_{\rm B}}{T_{\rm A}} \right)_{\rm losses}, \tag{6}\]  
Figure 5: Gear train S26 WN(E) with power flow direction (A to B) and labeled external shafts. The external shaft N is braked and therefore marked red. The efficiency \(\eta\), Equation (7), is calculated in the following manner:  
\[\eta=-\frac{P_{\rm B}}{P_{\rm A}}=-\frac{i\rm r}{i}=\frac{\left(T_{\rm B}/T_{\rm A }\right)_{\rm losses}}{\left(T_{\rm B}/T_{\rm A}\right)_{\rm without\ losses}}, \tag{7}\]  
The S55 NE(W) gear train operating as a reducer (Figure 6) will be used to demonstrate structural analysis using the torque method. It is a two-carrier train with planet carriers as the external shafts W and E, and both sun gears connected to the external compound shaft N. The internal coupled shaft connects both ring gears. Power input is through compound shaft N to both sun gears, and power output is through external shaft E (planet carrier II). The external shaft W (planet carrier I) is locked and serves as the reaction element. The equations for the calculation of the ideal (white background) and real (gray background) specific torques acting on all PGT shafts have been provided by application of Equations (1), (2), (6), and (7) to the PGT. Red solid lines are used to indicate the absolute power flow, while the green dashed lines denote the relative power flow within the PGT [49]. Power flow is from input A to output B, with \(P_{\rm A}\) denoting input shaft power and \(P_{\rm B}\) denoting output shaft power. Analysis has also shown that this PGT exhibits power circulation. The overall transmission ratio \(i_{\rm R}\) for reducer mode is provided by Equation (8):  
\[i_{\rm R}=-\frac{T_{\rm B}}{T_{\rm A}}=\frac{t_{\rm I}+\left(t_{\rm I}/t_{\rm II }\right)}{\left(t_{\rm I}/t_{\rm II}\right)-1}=\frac{t_{\rm I}(t_{\rm II}+1)}{ t_{\rm I}-t_{\rm II}} \tag{8}\]  
where \(t_{\rm I}\) and \(t_{\rm II}\) are the ideal torque ratios of the respective component planetary gear trains.