*3.3. Transmission*

The transmission model was derived under the assumption that all its shafts were rigid (i.e., shaft torsional stiffness was neglected). Both the conventional stepped transmission and the transmission of the parallel hybrid topology have three operating regimes defined by the automated clutch state: whether it is disengaged, slipping, or engaged. In the first and the second states, the ICE shaft has a separate degree of freedom, which, for the hybrid transmission, yields the following system of equations:

$$\begin{cases} \mathcal{I}\_{\varepsilon} \cdot \dot{\boldsymbol{\omega}}\_{\varepsilon} = \boldsymbol{T}\_{\varepsilon} - \boldsymbol{T}\_{\operatorname{clr}\operatorname{clr}\boldsymbol{\varepsilon}} \cdot \operatorname{sgn}(\boldsymbol{\omega}\_{\varepsilon} - \boldsymbol{\omega}\_{\operatorname{em}})\_{\ast} \\ \quad \boldsymbol{T}\_{\boldsymbol{w}} = \left(\boldsymbol{T}\_{\operatorname{clr}\operatorname{clr}\boldsymbol{\varepsilon}} \cdot \operatorname{sgn}(\boldsymbol{\omega}\_{\varepsilon} - \boldsymbol{\omega}\_{\operatorname{em}}) + \boldsymbol{T}\_{\operatorname{em}} - \boldsymbol{T}\_{\operatorname{em}^{\*}\boldsymbol{\omega}} \dot{\boldsymbol{\omega}}\_{\operatorname{em}\boldsymbol{\varepsilon}}\right) \cdot \boldsymbol{\mu}\_{\operatorname{GB}} \cdot \boldsymbol{\eta}^{\operatorname{sgn}(\boldsymbol{T})}\_{\operatorname{GB}} \cdot \boldsymbol{\mu}\_{0} \cdot \boldsymbol{\eta}^{\operatorname{sgn}(\boldsymbol{T})}\_{0} \\ \quad \boldsymbol{\omega}\_{\operatorname{em}^{\*}} \boldsymbol{r}\_{\boldsymbol{w}} = \boldsymbol{\nu} \cdot \boldsymbol{\mu}\_{\operatorname{GB}} \cdot \boldsymbol{\mu}\_{0} \end{cases} \tag{3}$$

where ω*<sup>e</sup>* and ω*em* are the shaft speeds of the ICE and electric machine, respectively; *Te* and *Tem* are the shaft torques of the ICE and electric machine, respectively; *Tclutch* is the clutch torque; I*<sup>e</sup>* and I*em* are the inertias of the ICE and electric machine, respectively; *uGB* and *u*<sup>0</sup> are the ratios of the gearbox and final drive, respectively; and η *sgn*(*T*) *GB* and η *sgn*(*T*) <sup>0</sup> are the efficiencies of the gearbox and final drive, respectively. The efficiencies are raised to the power of the torque sign function in order to take into account both traction and braking torques entering the transmission.

The term *sgn*(ω*<sup>e</sup>* − ω*em*) introduces a relation between the direction of the clutch torque and the speed differential of the input and output parts of the clutch. When the clutch becomes fully engaged, the separate equation of the ICE shaft dynamics, as well as the torque *Tclutch*, can be excluded from the transmission model, thus eliminating one degree of freedom.

To derive the model of the conventional transmission from the system (Equation (3)), one should exclude the terms associated with the electric machine and replace the angular speed ω*em* with that of the gearbox input shaft.
