# Fault Coupling Analysis and Reliability Assessment of Actuation System Based on Bond Graph Model
## 3 Fault Coupling Mechanism Analysis  
The role of a component in the system is to complete the transfer and transformation of energy. During the actuation system's operation, the components can be degraded due to various reasons, such as wear, fatigue, and aging. In this section, the fault coupling phenomenon and component degradation will be described in detail. The input energy is defined as \(E_{in}(t)\), the output energy as \(E_{out}(t)\), and the dissipated energy as \(\psi(t)\). Its working expression can be then expressed as:  
\[E_{in}(t)=E_{out}(t)+\psi(t) \tag{6}\]  
The fault coupling phenomena here can be divided into two types. The first is the local fault coupling. Figure 5 shows a node diagram of the single control surface. The different colors of the wires represent different energy transfers, including electric, hydraulic, and  
Figure 4: Schematic diagram of actuation system nodes. Figure 3: Bond graph model of single control surface actuation system. mechanical energy. For example, component node 6 degenerates or even fails during operation, increasing energy loss during the energy transfer and reducing output at component node 7. As a result, the input at all subsequent nodes is reduced and local chain fault coupling occurs. The second is the fault coupling of the entire actuation system. Each node will be experiencing various degradation phenomena in operation, and the energy lost as a result of these degradations will slowly accumulate in space as dissipated energy over time to form an energy field. This energy field will have an impact on all component nodes and thus aggravate the degradation of each node. The fault coupling of the entire system is formed. In this paper, based on the actual fault causes of the actuators and the bond graph model, the three degradations of wear, fatigue, and aging, as well as the fault coupling caused by the dissipated energy generated by them are mainly considered, along with the change in internal resistance windings of the motor with temperature. These degradations and the resulting dissipated energy are described in detail in the sequel. * Resistance varies with temperature  
First, concerning the variation of the internal resistance of the motor winding with temperature, the resistivity of the copper alloy resistance increases as the operating temperature of the motor increases. The resistivity of different doped copper alloys will vary with the temperature. The curve of resistance material changing with temperature typically resembles a straight line, and the operational temperature of the motor is in the low-temperature segment for the copper alloy resistance material itself. Therefore, it can be roughly considered that the resistance linearly changes with temperature, and its coefficient of change is defined as \(\alpha\), which is expressed as:  
\[\alpha=\frac{dR}{dT} \tag{7}\]  
where \(\alpha\) represents the change of resistance value corresponding to the temperature increase of 1 \({}^{\circ}\)C.  
Currently, the resistance temperature coefficient of copper resistance commonly used is \(\alpha_{cu}=0.00393a_{R}/^{\circ}\)C. Thus, the variation of the resistance with respect to temperature can be determined:  
\[R_{1}=R_{0}+\alpha_{Cu}R_{0}(T_{1}-T_{0}) \tag{8}\]  
Figure 5: Node diagram of single surface actuation system. where \(R_{1}\) and \(R_{0}\) are the resistance values corresponding to temperatures \(T_{1}\) and \(T_{0}\), respectively. The initial temperature is set to 0 \({}^{\circ}\)C, and the temperature during stable operation is 80 \({}^{\circ}\)C. As a result, the resistivity of the motor resistance will change with the temperature of the motor, making the motor consume more energy. * Wear and tear  
Wear failure is one of the most common failures of modern mechanical equipment. There is some friction created during the motor's operation and the hydraulic rod's rotation, and as wear time passes, particles are produced on the worm surface, directly causing material loss from the worn surface and progressive damage.