# Structural Analysis and Application of Non-Standard Components Based on Genetic Algorithm
## 3 Theoretical Analysis
### Establishment of Modal Theory Model  
As shown in Figure 2 for the five key parts, simplify the quality system as shown in Figure 4.  
According to the simplified mass system diagram shown in Figure 4, the motion equation of the system is established by using Lagrange method, and the expression is as follows:  
\[\frac{d}{dt}\bigg{(}\frac{\partial T}{\partial\dot{q}_{i}}\bigg{)}-\frac{ \partial T}{\partial q_{i}}+\frac{\partial U}{\partial q_{i}}+\frac{\partial D }{\partial q_{i}}=Q_{i}\;(i=1,2,\cdot\cdot,n) \tag{1}\]  
where \(q_{i}\) and \(\dot{q}_{i}\) are generalized coordinates and generalized velocities, respectively; \(T\) and \(U\) are the total kinetic energy and potential energy of the system respectively. \(D\) is that energy dissipation function; \(Q_{i}\) is the generalized disturbance force [15; 16]. As shown in Figure 3, that kinetic energy function of the system is establish as follows:  
\[T=\frac{1}{2}\Big{(}m_{1}\dot{x_{1}}^{2}+m_{2}\dot{x_{2}}^{2}+m_{3}\dot{x_{3} }^{2}+m_{4}\dot{x_{4}}^{2}+m_{5}\dot{x_{5}}^{2}+m_{6}\dot{x_{6}}^{2}\Big{)} \tag{2}\]  
Figure 4: Simplified quality system diagram of a turret chuck load and unloading machine. Figure 3: GA algorithm flow chart. The potential energy function is:  
\[U=\frac{1}{2}\Big{[}k_{1}x_{1}^{2}+k_{2}(x_{2}-x_{1})^{2}+k_{3}(x_{3}-x_{2})^{2}+ k_{4}(x_{4}-x_{3})^{2}+k_{5}(x_{5}-x_{4})^{2}+k_{6}(x_{6}-x_{5})^{2}\Big{]} \tag{3}\]  
The energy dissipation function of the system is:  
\[D=\frac{1}{2}\Big{[}c_{1}\dot{x}_{1}^{2}+c_{2}\Big{(}\dot{x}_{2}-\dot{x}_{1} \Big{)}^{2}+c_{3}\Big{(}\dot{x}_{3}-\dot{x}_{2}\Big{)}^{2}+c_{4}\Big{(}\dot{x}_ {4}-\dot{x}_{3}\Big{)}^{2}+c_{5}\Big{(}\dot{x}_{5}-\dot{x}_{4}\Big{)}^{2}+c_{6} \Big{(}\dot{x}_{6}-\dot{x}_{5}\Big{)}^{2}\Big{]} \tag{4}\]  
To this end, the partial derivatives of the functions (2), (3), and (4) in the Lagrange equation and the differential equations of motion of the mechanical system of the turret chuck loader can be obtained by introducing \(m_{1},m_{2},m_{3},m_{4},m_{5},m_{6}\) in Figure 3, expressed in matrices as:  
\[[m]\Big{\{}\ddot{x}\Big{\}}+[c]\Big{\{}\dot{x}_{1}\Big{\}}+[k]\{x\}=\{P\} \tag{5}\]  
where \(\{x\}\)\(\Big{\{}\dot{x}\Big{\}}\)\(\Big{\{}\ddot{x}\Big{\}}\)\(\{P\}\)\([m]\)\([c]\) and \([k]\) denote displacement array, velocity array, acceleration array, interference force array, mass matrix, damping matrix and stiffness matrix, respectively. When there is no damping, its function (5) is expressed as:  
\[[m]\Big{\{}\ddot{x}\Big{\}}+[k]\{x\}=\{0\} \tag{6}\]  
Let the solution of function (6) be:  
\[\{x\}=\{A\}e^{ix_{n}x} \tag{7}\]  
Of which:  
\[\{A\}=\left\{ \begin{array}{c}A_{1}\\ A_{2}\\ \vdots\\ A_{n}\end{array} \right\},\{x\}=\left\{ \begin{array}{c}x_{1}\\ x_{2}\\ \vdots\\ x_{n}\end{array} \right\}\]  
The first derivative and the second derivative of the Equation (7) are obtained, respectively, and the equation of the main vibration mode obtained by introducing the Equation (6) into the equation is as follows:  
\[\Big{(}[k]-\omega_{n}^{2}[m]\Big{)}\{A\}=\{0\} \tag{8}\]  
Obviously, for \(\{A\}\) to have a non-zero solution, the coefficient determinant in Equation (8) must be zero, i.e., the frequency equation:  
\[\text{det}\Big{(}[k]-\omega_{n}^{2}[m]\Big{)}=\left| \begin{array}{cccc}k_{1 1}-m_{11}\omega_{n}^{2}&k_{12}-m_{12}\omega_{n}^{2}&\ldots&k_{1n}-m_{1n} \omega_{n}^{2}\\ k_{21}-m_{21}\omega_{n}^{2}&k_{22}-m_{22}\omega_{n}^{2}&\ldots&k_{2n}-m_{2n} \omega_{n}^{2}\\ \vdots&\vdots&\vdots&\vdots\\ k_{n1}-m_{n1}\omega_{n}^{2}&k_{n2}-m_{n2}\omega_{n}^{2}&\ldots&k_{nn}-m_{nn} \omega_{n}^{2}\end{array} \right| \tag{9}\]  
The \(n\)-th order algebraic equation of \(\omega_{n}^{2}\) is obtained by expanding it:  
\[\omega_{n}^{2n}+a_{1}\omega_{n}^{2(n-1)}+a_{2}\omega_{n}^{2(n-2)}+\cdots+a_{ n-1}\omega_{n}^{2}+a_{n}=0 \tag{10}\]  
where \(a_{1},a_{2},\cdots,a_{n}\) is a combination of \(k_{ij}\) and \(m_{ij}\).