# Simulation Model and Method for Active Torsional Vibration Control of an HEV
## 2 Materials and Methods
### Dynamic Parameters Determination
#### 2.3.3 Engine Moment of Inertia Determination  
One way to identify the moment of inertia is to measure the angular acceleration value of the rotating component during acceleration or deceleration and the torque it receives, then use the formula \(\mathrm{I}_{i}\cdot\ddot{\varphi}_{i}=\mathrm{T}_{i}\) to calculate it. The drive system is subject to various time-varying resistance torques, such as engine frictional resistance torque, during actual operation. To ensure the successful implementation of the above test method, the acceleration curve of the coast down process is subtracted from the acceleration process at the same speed, resulting in the acceleration curve under constant torque.  
Assuming that the moment of inertia of the engine is \(\mathrm{I}_{1}\), the total moment of inertia of the motor and other rotating components is \(\mathrm{I}_{2}\), thus, dynamic equations of the acceleration and deceleration of the powertrain can be expressed as:  
\[\mathrm{T}_{m\_1}(t)+\mathrm{T}_{res\_1}(t)=\mathrm{I}_{1}\cdot\ddot{\varphi} _{1\_1}(t)+\mathrm{I}_{2}\cdot\ddot{\varphi}_{2\_1}(t) \tag{6}\]  
\[\mathrm{T}_{res\_2}(t)=\mathrm{I}_{1}\cdot\ddot{\varphi}_{1\_2}(t)+\mathrm{I} _{2}\cdot\ddot{\varphi}_{2\_1}(t) \tag{7}\]  
where \(t1\) and \(t2\) are the lengths of time for acceleration and natural-stopping deceleration processes; \(\mathrm{T}_{m\_1}(t)\), \(\mathrm{T}_{res\_1}(t)\), \(\ddot{\varphi}_{1\_1}(t)\), \(\ddot{\varphi}_{2\_1}(t)\) are the motor output torque in the acceleration process, the total resistance torque of the system, and the angular acceleration of the engine and the motor; \(\ddot{\varphi}_{1\_12}(t)\) and \(\ddot{\varphi}_{2\_12}(t)\) are the angular acceleration of the engine and motor in the deceleration process, respectively.  
Whether in the acceleration or deceleration phase, the real-time resistive torque experienced by the system can be refined to:  
\[\mathrm{T}_{res}=\mathrm{T}_{ic\_gas}+\mathrm{T}_{ic\_iner}+\mathrm{T}_{ic\_ fric}+\mathrm{c}_{2}\cdot\dot{\varphi}_{2} \tag{8}\]  
where \(\mathrm{T}_{res}\) is the total resistance torque of the system, including the resistance torque of the engine and the motor and \(\mathrm{c}_{2}\) is the friction coefficient of the motor, which is approximated as 0 according to the assumptions in this section.  
The independent variables of the state quantities in Equations (6) to (8) are converted from time to rotational speed, and Equation (7) is subtracted from Equation (6) to get:  
\[\mathrm{T}_{m\_1}(\dot{\varphi})=\mathrm{I}_{1}\cdot\left(\ddot{\varphi}_{1 \_1}(\dot{\varphi})-\ddot{\varphi}_{1\_12}(\dot{\varphi})\right)+\mathrm{I}_{2 }\cdot\left(\ddot{\varphi}_{2\_11}(\dot{\varphi})-\ddot{\varphi}_{2\_12}( \dot{\varphi})\right) \tag{9}\]  
where \(\mathrm{T}_{m\_1}(\dot{\varphi})\), \(\ddot{\varphi}_{1\_11}(\dot{\varphi})\), \(\ddot{\varphi}_{1\_12}(\dot{\varphi})\), \(\ddot{\varphi}_{2\_11}(\dot{\varphi})\), and \(\ddot{\varphi}_{2\_12}(\dot{\varphi})\) are the motor output torque and the engine's and the motor's angular accelerations during both the acceleration and deceleration process in the speed coordinate. These parameters can be obtained by experimental measurement. According to the assumptions, \(\mathrm{I}_{2}\) is known, so \(\mathrm{I}_{1}\) can be found by Equation (9).