# Fault Coupling Analysis and Reliability Assessment of Actuation System Based on Bond Graph Model
## 3 Fault Coupling Mechanism Analysis
In the actuation system model, the deterioration of the hydraulic rod of the motor and the control surface will be accompanied by the corresponding dissipation of energy. As one of the most common phenomena, wear has been extensively studied. The most popular definition of the wear process is [PERSON]'s wear law [22], whose equation is given as:  
\[w=\int\limits_{0}^{t}K\times\frac{Pv}{H}dt \tag{9}\]  
where \(w\) is the wear volume, \(K\) is the non-dimensional wear coefficient, \(P\) is the load on the contact surface, \(v\) is the relative motion velocity, and \(H\) is the surface hardness of the wearing material. In general, it might not be possible to determine the hardness of the topmost material in the contact, so the ratio \(K/H\) is viewed as a parameter known as the dimensional wear coefficient or the specific wear rate, which can be expressed as:  
\[w=\int\limits_{0}^{t}K_{w}\times p\times vdt \tag{10}\]  
The wear volume of a component is proportional to degradation, so it represents the degradation effect of wear on the component itself. The dissipated energy generated by wear is proportional to its volume, so it can be expressed as:  
\[E=\int\limits_{0}^{t}F\times Vdt \tag{11}\]  
where \(E\) is the dissipated energy generated by accumulation, \(F\) is the friction force, and \(V\) is the relative displacement velocity. * Fatigue  
In mechanical and electrical systems, fatigue cracking occurs in metallic materials under the influence of cyclic stresses. Fatigue is one of the most common failure phenomena in metallic materials of system components. In particular, the mutual stress exerted on surfaces that are in contact with one another might initiate a crack. Therefore, the fatigue phenomenon of the motor and hydraulic rod is mainly considered in the actuation system model. The fatigue is typically expressed by the Paris theorem [23] as:  
\[r_{f}(\Delta K)=\Delta K^{m},\,\Delta K_{th}<\Delta K<K_{lc} \tag{12}\]  
where \(r_{f}\) is the crack growth rate per unit period, \(C\) and \(m\) are referred to as Paris constant, \(\Delta K\) is the stress intensity factor, \(\Delta K_{th}\) is the fatigue threshold, and \(K_{lc}\) is the fracture toughness, with \(C\), \(m\), \(\Delta K_{th}\), and \(K_{lc}\) determined by the material properties. The Paris constant, fatigue threshold, and fracture toughness selected in this paper are based on the steel alloy material. Assuming that the energy required to generate a unit crack length is a constant value, then the necessary dissipated energy \(\psi_{f}(t)\) to produce a crack is a function of direct proportion to the cumulative damage amount \(L(t)\), as shown in the expression:Figure 4:0
* Aging The O-ring gasket is widely used in aircraft hydraulic systems due to its wide range of applications, good stability, low light pollution, and other advantages. It also plays a key sealing role in hydraulic pumps and hydraulic cylinders. However, when the actuation system is in normal operation, its temperature will also increase and the plastic materials will appear to age at high temperatures. Under the combined action of various environmental factors, rubber molecules will produce various free bases and change according to the free base courses. The aging process will cause changes in the molecular structure of rubber, which will lead to stickiness, deformation, cracking, and other phenomena, eventually resulting in liquid leakage [24; 25]. Hydraulic leakage is one of the most common failures of the hydraulic system. Increasing leakage can result in aberrant hydraulic system pressure and flow, which can cause significant energy loss and actuation system failure. Arrhenius model is usually adopted to represent the aging process of plastic materials [26], which can be expressed as: Figure 4:1 (14) where \(r_{a}\) is the aging rate, \(A_{0}\) is the Arrhenius constant, \(R_{  
Figure 4:2  
The results of the above mechanism analysis are combined with the bond graph model of the MEA actuation system using encoding.