## 2 Stochastic Dynamic Response Analysis of Mechanisms
### Random Stress Analysis  
Stresses at specified observation points for reliability assessment can be obtained through system dynamics analysis. In the working process of the mechanism, the stochastic characteristics that exist in the environmental loads and working loads make the system dynamic stresses have significant stochastic characteristics. Since the mechanism stresses have time-varying characteristics, their randomness gives the stresses a time-varying stochastic characteristic. Hence, the random stresses need to be reasonably characterized according to the system reliability modeling requirements. For reliability calculations, it is necessary to establish a stochastic stress characterization method. The working time of a mechanism is divided into \(N\) time periods (TP). In the \(i\)th TP \(N_{i}(i=1,2,\ldots,N)\), the external load of the mechanism \(L_{i}\) follows a certain probability distribution with the probability density function (PDF) \(f(L_{i})\). Theoretically, the load distribution has infinite possible values and cannot be used directly in reliability calculations. However, in practice, the load distribution can be discretized according to the numerical calculation method and the reliability calculation can be performed based on the corresponding discrete values. In order to obtain the stresses based on the above deterministic dynamics calculation method, it is necessary to discretize the continuous distribution, expressed as \(L_{i1},L_{i2},...,L_{ij}(j=)\), where \(n_{L}\) is the total number of discretely distributed values of random loads within each TP. Then, the load matrix during the whole working time can be expressed as:  
\[\mathbf{L}=\left[\mathbf{L}_{1},\mathbf{L}_{2},...,\mathbf{L}_{\mathbf{N}}\right]=\left[ \begin{array} []{cccc}L_{11}&L_{21}&\cdots&L_{\text{N1}}\\ L_{12}&L_{22}&\cdots&L_{\text{N2}}\\ \vdots&\vdots&\ddots&\vdots\\ L_{1n_{L}}&L_{2n_{L}}&\cdots&L_{\text{N}n_{L}}\end{array} \right] \tag{9}\]In addition, the mechanism clearance may have different values at different TPs due to the manufacturing and installation errors of the mechanism and the continuous wear of the clearance during operation. Meanwhile, because of the continuous action of the load, the local stiffness of the structure may be degraded due to the presence of material defects, thus presenting different states in different TPs. It is impractical to use experimental methods to obtain the stress response under random loading. Therefore, it is necessary to obtain the stresses of the mechanism under different states of the clearance and local stiffness and use fitting techniques such as neural networks to realize the random stress characterization and time-varying reliability calculation of the mechanisms. Considering the sample values of the clearance vectors \(\varphi(N_{i})\) consisting of each clearance value of the mechanism in the \(i\)th TP denoted by \([\varphi_{1}(N_{i}),\varphi_{2}(N_{i}),...,\varphi_{n1}(N_{i})]\), where \(n1\) is the total number of samples for \(\varphi(N_{i})\) and considering the sample values of the elastic modulus \(E(N_{i})\) of the locally flexible structure in the \(i\)th TP denoted by \([E_{1}(N_{i}),E_{2}(N_{i}),...,E_{n2}(N_{i})]\), where \(n2\) is the total number of samples for \(E(N_{i})\), then the stress matrix of the mechanism under stochastic load \(L\) can be obtained from Equation (8) as follows:  
\[\mathbf{S}(\mathbf{t})=[\mathbf{S}_{1}(N_{1}),\mathbf{S}_{2}(N_{2}),...,\mathbf{S}_{N}(N_{N})] \tag{10}\]  
where  
\[\mathbf{S}_{i}(L_{ij})=