## 2 Research Methods and Approach  
Considering a finite structure of degrees of freedom, the structure's partial oscillation differential equation has the form [28]  
\[M\ddot{u}(t)+Ku(t)=0 \tag{1}\]  
Here, \(M\)**--**mass matrix of the structure and \(K\)--**stiffness matrix of the structure. The two equations of oscillation of the masses have the form  
\[u_{i}(t)=A_{i}\sin(\omega t+\varphi_{i}) \tag{2}\]  
\[\ddot{u}_{i}(t)=-\omega^{2}A_{i}\sin(\omega t+\varphi_{i}) \tag{3}\]  
Substitute the equations and components into Equation (1):  
\[ \begin{bmatrix}m_{1}&&\\ &m_{2}&&\\ &&\ldots&&\\ &&&m_{n}\end{bmatrix} \begin{pmatrix}A_{1}\\ A_{2}\\ \ldots\\ A_{n}\end{bmatrix}+ \begin{bmatrix}k_{11}&k_{12}&\ldots&k_{1n}\\ k_{21}&k_{22}&\ldots&k_{2n}\\ \ldots&\ldots&\ldots&\ldots\\ k_{n1}&k_{n2}&\ldots&k_{nn}\end{bmatrix}\begin{bmatrix}A_{1}\\ A_{2}\\ A_{n}\end{bmatrix} =0 \tag{4}\]  
or  
\[ \begin{bmatrix}k_{11}&k_{12}&\ldots&k_{1n}\\ k_{21}&k_{22}&\ldots&k_{2n}\\ \ldots&\ldots&\ldots&\ldots\\ k_{n1}&k_{n2}&\ldots&k_{nn}\end{bmatrix} -\omega^{2} \begin{bmatrix}m_{1}&&\\ &m_{2}&&\\ &\ldots&m_{n}\end{bmatrix}\begin{bmatrix}1\\ \varphi_{2}\\ \ldots\\ \varphi_{n}\end{bmatrix} =0 \tag{5}\]Simplify to obtain the equation to determine the specific type of vibration:  
\[\left[\mathbf{K}-\omega_{j}^{2}\mathbf{M}\right]\mathbf{\Phi}=0 \tag{6}\]  
The natural frequency (\(f=\omega/2\pi\)) and natural vibration mode shape (\(\varphi_{i}\)) determined through Equation (5) or (6) are significant features of the structure and are often used to determine the accuracy of the FEM model. It is easy to see that, for a structure, the dynamic characteristic is unique and is influenced by \(\mathbf{K}\) (stiffness matrix) and \(\mathbf{M}\) (mass matrix). However, these two parameters are uncertain parameters due to material properties and boundary conditions. For structures using composite materials, the behavior of materials is simultaneously influenced by many factors. Therefore, finding an exact equivalent to the combination of the constituent materials often has to be determined through experimentation and difficulty. The number of variables in the stiffness and mass matrices is also numerous and complex. To provide an accurate model for other analysis purposes, it is necessary to have a method to calculate and select the best parameters of the model. The essence of the problem of parameter accuracy uncertainty is choosing a set of numbers as the best solution to the mathematical or simulation problem to be solved. For a variable parameter, the search is fast and can be done manually. However, when the number of solutions is large, this seems impossible because of the significant computation. Although computers can support the calculation, it is time consuming and resource consuming. The particle swarm optimization (PSO) algorithm was established and developed based on the ideas of swarm intelligence to find solutions for optimization problems in a particular search space [29; 30]. To understand the PSO algorithm better, observe a simple example of a flock of birds foraging. The foraging space is now the entire three-dimensional space. At the beginning of the search, the whole flock flies in a specific direction, which can be very random. However, after a while of searching, some individuals in the flock began to find a place that contains food. Depending on the amount of food found, the individual sends a signal to others searching in the vicinity. This signal propagates throughout the population. Based on the information received, each individual will adjust their flight direction and speed in the order of where there is the most food. Such communication is often viewed as a phenotype of herd intelligence. This mechanism helps the whole flock of birds to find out where there is the most food in the vast search space. Figure 1 clearly illustrates the PSO algorithm when used to search for extreme value. In swarm optimization, each particle searches a space by itself, remembering the best value and informing others.