# Fault Coupling Analysis and Reliability Assessment of Actuation System Based on Bond Graph Model
## 2 Bond Graph Model of Actuation System
### Energy Transfer Model of Actuator  
The actuation system model established in this paper consists mainly of HA and EHA, with HA being the primary actuator and EHA the backup actuator. This is a redundant design that starts the backup actuator when the primary actuator fails. HA is widely used in aircraft hydraulic systems and is mainly composed of a hydraulic pump source, a servo valve, and a hydraulic cylinder. The hydraulic energy output from the hydraulic pump source is controlled by the servo valve, transferred to the hydraulic cylinder to drive its operation, and finally converted into the form of mechanical energy to act on the surface. The EHA is widely used in the MEA actuation system currently. It mainly consists of a generator, a brushless DC motor, a hydraulic pump, and a hydraulic cylinder. The generator provides electrical energy that is converted into rotational mechanical energy through the electric motor, which then drives the high-speed motor that turns the hydraulic pump. The hydraulic pump outputs mechanical energy to the surface after transferring hydraulic energy to the hydraulic cylinder to drive the cylinder's motion [21]. Figure 2 shows the structure diagram of the single control surface actuation system composed of HA and EHA. HA and EHA are typical energy transfer systems where the hydraulic pump source, hydraulic cylinder, motor, and surface are built as part of the actuation system. During the operation of the actuation system, there will be certain energy losses, such as resistance consumption, friction consumption, liquid leakage, and some pressure loss, which will be reflected in the following bond graph model. For EHA, the model is mainly built for its motor, hydraulic pump, and hydraulic cylinder, while for HA, the hydraulic pump source and hydraulic cylinder are mainly considered, and the on-board voltage source (VS) and hydraulic source (HS) are represented by excitation signal components. The variable potential source is represented using the MSE original in the bond graph. The internal resistance winding of the brushless DC motor is represented by the resistive component R in the bond graph, and the motor's rotational friction is represented by the resistive component, where the inertia element I is used to represent the inductance (\(L\)) and the moment of inertia (J) of the motor. The conversion between the motor's electric energy and mechanical rotation energy is completed through the gyroscope component GY, the ratio of which is the electromagnetic flux (\(k\)). The relationship between the resistive element R, the flow, and the potential in the bond graph is given by:  
\[effort=R\times flow \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Title 1** & **Title 2** & **Title 3** \\ \hline R & Motor resistance & \(\Omega\) \\ L & Motor inductance & H \\ K & Electromagnetic flux & / \\ Kfric & Wear coefficient of motor pump & kg\(\cdot\) m\({}^{2}\) \\ J & Motor pump inertia & Nm/(rad/s) \\ D & Pump displacement & m\({}^{3}\)/r \\ Kilp & Pump leakage factor & (m\({}^{3}\)/s)/Pa \\ Kilj & Leakage coefficient of hydraulic cylinder & (m\({}^{3}\)/s)/Pa \\ C\({}_{\text{Rhj}}\) & Hydraulic compressibility & Pa \\ A & Piston area & m\({}^{2}\) \\ M & Rod mass & kg \\ Rfric & Friction coefficient of piston & Ns/m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model parameter definition. The work of the motor can be expressed as:  
\[u=Ri+L\frac{di}{dt}+\omega k \tag{3}\]  
where \(u\) is the voltage in the motor (effort in electric energy), \(i\) is the current in the motor (flow in electric energy), and \(\omega\) is the speed of the motor. In the case of the hydraulic pump, the conversion element TF is used to convert the motor output speed and torque into flow rate and pressure in the hydraulic system.