# Research on Prediction Method of Bolt Tightening for Aviation Components Based on Neural Network
## 3 Aviation Component Bolt Tightening  
### Tightening Scheme Design Based on the Elastic Interaction Coefficient Method  
When a bolt is tightened, the alternating load between the bolt that is tightened later and the bolt that was tightened first is called the elastic interaction. Due to this elastic interaction, different tightening sequences will have different effects on the tightened parts in the process of tightening. In selecting a tightening sequence, the goal is to obtain a uniform final load under the same initial load conditions. The elastic interaction coefficient method (EICM) is used as a benchmark to optimize the bolt tightening sequence of aviation components. The EICM is based on an assumed linear relationship between the load variation of the tightened bolt and the final preload. This relationship can be expressed using an elastic interaction coefficient matrix. Different tightening sequences will result in different elastic interaction matrices. A detailed explanation and solution method for elastic interaction matrix A can be obtained by referring to the literature [15] and this will not be repeated in this paper. Based on the EICM method, the relationship between the initial load and the final load of bolts in different sequences is as follows:  
\[\left\{S_{f}\right\}=\left[A\right]\bullet\left\{S_{i}\right\} \tag{1}\]  
where \(\left\{S_{f}\right\}\) is the final load, \(\left\{S_{i}\right\}\) is the initial load, and \(\left[A\right]\) is the elastic interaction matrix. In a system composed of n bolts, Formula (1) can be expressed as:  
\[\left\{ \begin{array}{c}S_{f1}\\ S_{f2}\\ S_{f3}\\ \vdots\\ S_{fn-1}\\ S_{fn}\end{array} \right\}=\left[ \begin{array}{cccccccc}1&A_{1,2}&A_{1,3}& \cdots&A_{1,n-1}&A_{1,n}\\ 0&1&A_{2,3}&\cdots&A_{2,n-1}&A_{2,n}\\ 0&0&1&\cdots&A_{3,n-1}&A_{3,n}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\vdots&1&A_{n-1,n}\\ 0&0&0&\vdots&\vdots&1\end{array} \right]\left\{ \begin{array}{c}S_{i1}\\ S_{i2}\\ S_{i3}\\ S_{in-1}\\ S_{in}\end{array} \right\} \tag{2}\]  
where \(S_{fk}\) is the final load of bolt k and \(S_{ik}\) is the initial load of bolt k. Taking the final load of bolt 1 as an example, according to Formula (2), the final load of bolt 1 can be expressed as:  
\[S_{f1}=S_{i1}+A_{1,2}\bullet S_{i2}+\cdots+A_{1,n}\bullet S_{in} \tag{3}\]  
According to Formula (3), the final load of any bolt can be calculated under given conditions. When two bolts are tightened on the aviation component diagonally and at the same time, Formula (2) can be expressed as follows:  
\[\left\{ \begin{array}{c}s_{f1}\\ s_{f2}\\ \vdots\\ s_{fn-1}\\ s_{fn}\\ s_{fn+1}\\ s_{fn+2}\\ \vdots\\ s_{fn-1}\\ s_{fn-1}\\ s_{fn}\\ s_{fn+2}\\ \vdots\\ s_{fn-1}\\ s_{fn-1}\\ \vdots\\ s_{fn-1}\\ \vdots\\ s_{fn-1}\\ \vdots\\ s_{fn}\end{array} \right\}=\left[\begin{array}{cccccccc}1&A_{1,2}&\cdots&A_{1,n- 1}&A_{1,n}&0&A_{1,n+2}&\cdots&A_{1,2n-1}&A_{1,2n}\\ 0&1&\cdots&A_{2,n-1}&0&0&\cdots&A_{2,2n-1}&0\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\cdots&\vdots&\vdots\\ 0&0&\cdots&1&0&0&0&\cdots&0&0\\ 0&A_{n,2}&\cdots&A_{n,n-1}&1&0&A_{n,n+2}&\cdots&A_{n,2n-1}&0\\ 0&A_{n+1,2}&\cdots&A_{n+1,n-1}&A_{n+\[\left\{ \begin{array}{c}s_{f1}\\ s_{f2}\\ \vdots\\ s_{fn-1}\\ s_{fn}\\ s_{fn+1}\\ s_{fn+2}\\ \vdots\\ s_{f2n-1}\\ s_{f2n}\end{array} \right\}=\left[ \begin{array}{cccccccccc}1&A_{1,2}&\cdots&A_{ 1,n-1}&A_{1,n}&0&A_{1,n+2}&\cdots&A_{1,2n-1}&A_{1,2n}\\ 0&1&\cdots&0&A_{2,n}&0&0&\cdots&0&A_{2,2n}\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&A_{n-1,2}&\cdots&1&A_{n-1,n}&0&A_{n-1,n+2}&\cdots&0&A_{n-1,2n}\\ 0&0&\cdots&0&1&0&0&\cdots&0&0\\ 0&A_{n+1,2}&\cdots&A_{1,2n-1}&A_{n+1,n}&1&A_{n+1,n+2}&\cdots&A_{n+1,2n-1}&A_{n+ 1,2n}\\ 0&0&\cdots&0&A_{n+2,n}&0&1&\cdots&0&A_{n+2,2n}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&A_{2n-1,2}&\cdots&0&A_{2n-1,n}&0&A_{2n-1,n+2}&\cdots&1&A_{2n-1,2n}\\ 0&0&\cdots&0&0&0&\cdots&0&1\end{array} \right]\left\{ \begin{array}{c}s_{i1}\\ s_{i2}\\ \vdots\\ s_{in-1}\\ s_{in}\\ s_{in+1}\\ \vdots\\ s_{in-2}\\ s_{i2n-1}\\ s_{i2n}\end{array}