# Control Strategy Development of Driveline Vibration Reduction for Power-Split Hybrid Vehicles
## 2 Two-Mode Power-Split Hybrid System Model  
### Two-Mode Power-split Hybrid System Transmission  
The General Motors (GM) Two-Mode hybrid transmission system was applied in the research, as shown in Figure 1. This transmission system includes two sets of planetary gear mechanisms (P1, P2), two sets of brake clutch (CB4, CB12R), and two sets of power transfer clutch (C13, C234), as well as an internal combustion engine and two motors/generators (MG1, MG2) connected to the ring gear, the sun gear, and the planet carrier on P1, respectively. This system transferred the power from the power source to the output driving shafts. This transmission system consisted of two continuous variable speed modes (EVT) to cover the needs for low speed, high-speed cruise, and rapid acceleration which also cooperated with four sets of fixed gear ratio to smoothen the mode shift. When the CB12R clutch was engaged, the torque was output by MG2 with the driving force need. When a certain speed was reached or the state of charge (SOC) was low, the engine would turn on to provide power, which was called EVT-1 mode, mainly used at low speed. When entering the high-speed segment, the CB12R would be loosened and the C234 clutch was engaged, so that the sun gear (S1) of P1 was connected to the ring gear (R2) of P2. This allowed the internal combustion engine to provide more power to the output, which was called EVT-2 mode. This research was aimed at the vibration of the driveline, and the situation of the remaining clutch would not be included. In addition, in order to accurately grasp the situation of overall torque transmission, and present the speed adjusting function of M/G in the power system and the accelerating function of the internal combustion engine, the considerations of the moment of inertia of each component in the transmission system and the internal force between gears were added to further estimate the speed of each power source and output axis [21]. The torque angular acceleration relationship of the EVT-1 mode was exemplified in the following. In mode one, the clutch CB12R was locked, the internal combustion engine power was input to P1 and  
Figure 1: Schematic diagram of a two-mode power-split hybrid system. split, and P2 was operated with the reduction gear. At this time, the rotation speed relationship of each gear on P1 and P2 were as shown in Equations (1) and (2). \[(R_{R1}-R_{S1})\omega_{C1}^{\bullet}+R_{S1}\omega_{S1}^{\bullet}-R_{R1}\omega_{E} ^{\bullet}=0 \tag{1}\]  
\[R_{S2}\omega_{S2}^{\bullet}-(R_{R2}+R_{S2})\omega_{trans}^{\bullet}=0 \tag{2}\]  
After the internal combustion engine was connected to the damper, it was connected with the ring gear (R1) of P1. MG1 was connected with the sun gear (S1), and MG2 was connected with the planet carrier (C1) to transmit the torque into the transmission system. According to the connection relationship between each power source and the transmission in Figure 1, the torque calculation formula Equations (3)-(5) were listed. Among them, \(\mathrm{T_{MG1}}\), \(\mathrm{I_{MG1}}\), and \(\omega_{\mathrm{MG1}}\) were the torque, moment of inertia, and rotation speed, respectively. The rest, ICE Damper end (coded as ED in below) and MG2 end, are presented in the following. Sun gear end--MG1:  
\[T_{MG1}=(I_{MG1}+I_{S1})\omega_{MG1}^{\bullet}+F_{1}R_{S1} \tag{3}\]  
Ring gear end--ICE Damper:  
\[T_{ED}=I_{R1}\omega_{ED}^{\bullet}-F_{1}R_{R1} \tag{4}\]  
Planet carrier end--MG2:  
\[T_{MG2}=(I_{MG2}+I_{C1}+I_{S2})\omega_{MG2}^{\bullet}-F_{1}(R_{R1}-R_{S1})+F_{ 2}R_{S2} \tag{5}\]  
At the output of the transmission, due to the connection with the flexible shaft, a reaction force (\(\mathrm{T_{react}}\)) was fed back to here when the power output to the shaft. Therefore, the transmission output axis angular acceleration calculation is shown in Equation (6). Wherein, the \(\mathrm{I_{trans}}\) is the sum of the moment of inertia from the transmission output to the half-shaft.