## 2 Theoretical Background
### State Space Method Theory  
To describe a linear time-invariant system, the corresponding ordinary differential equation or transfer function model of the system under study should be established. However, ordinary differential equations or transfer function models can only describe the relationship between the input and output of the system but cannot describe its internal variables, that is, they cannot contain all the information about the system under study. To make up for the deficiency of ordinary differential equations or transfer function models and show the whole motion state of the system, a finite element modeling method is proposed. However, this method depends on empirical knowledge, and there is always some error with actual test results. To describe the studied system accurately, a state space model is proposed. Using the state space description and the matrix vector method to perform efficient mathematical calculation can be convenient in MATLAB/Simulink and other simulation environments [15]. The linear system state space expression is as follows:  
\[ \begin{array}{l}\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}\\ \mathbf{y}=\mathbf{C}\mathbf{x}+\mathbf{D}\mathbf{u}\end{array}  \tag{12}\]  
Here, \(\mathbf{x}\) is the state vector, \(\dot{\mathbf{x}}\) is the derivative of the state vector, \(\mathbf{A}\) is the system matrix, \(\mathbf{B}\) is the input matrix, \(\mathbf{C}\) is the output matrix, \(\mathbf{D}\) is the feedback matrix, \(u\) is the input of the system, and \(y\) is the output of the system. According to the principle of drawing the structure diagram of a univariate system, a general linear system can be represented by the form in Figure 3.  
The state vector x can be determined as follows:  
\[\mathbf{x}=\left[ \begin{array}{c}\dot{\mathbf{z}}\\ \ddot{\mathbf{z}}\end{array} \right] \tag{13}\]  
After the transformation to the principal coordinate, it can be expanded:  
\[ \begin{array}{l}\ddot{\mathbf{z}}_{p1}=F_{P1}\\ \ddot{\mathbf{z}}_{p2}+\frac{k}{m}z_{p2}=F_{P2}\\ \ddot{\mathbf{z}}_{p3}+\frac{3k}{m}z_{p3}=F_{P3}\end{array}  \tag{14}\]  
where \(F_{pi}=z_{n}^{T}F_{i}\). Figure 3: State space system block diagram. Transformation to the principal coordinate system solves the Laplace transform:  
\[z_{p}=\left[ \begin{array}{c}z_{p1}\\ z_{p2}\\ z_{p3}\end{array} \right]=\left[ \begin{array}{c}\frac{F_{p1}}{s^{2}+u_{1}^{2}} \\ \frac{F_{p2}}{s^{2}+u_{2}^{2}}\\ \frac{F_{p3}}{s^{2}+u_{3}^{2}}\end{array} \right]=\left[ \begin{array}{c}\frac{ z_{n14}F_{1}+z_{n22}F_{1}+z_{n24}F_{2}}{s^{2}+u_{2}^{2}}\\ \frac{z_{n14}F_{1}+z_{n22}F_{1}+z_{n24}F_{2}}{s^{2}+u_{2}^{2}}\\ \frac{z_{n14}F_{1}+z_{n22}F_{1}^{2}+z_{n4}F_{2}}{s^{2}+u_{3}^{2}}\end{array} \right] \tag{15}\]  
where \(z_{nij}\) is the \(i\)th row and \(j\)th column element of the modal matrix. By replacing the inverse principal coordinate system with the physical coordinate system, the contribution values of each mode can be obtained as follows:  
\[z_{p}=z_{n}^{-1}z \tag{16}\]  
For a single-input, single-output damped system:  
\[\frac{z_{j}}{F_{k}}=\sum_{i=1}^{m}\frac{z_{nij}z_{nki}}{s^{2}+2\xi_{i}w_{i}s+w_ {i}^{2}} \tag{17}\]  
where \(k\) is the free applied degree of freedom and \(j\) is the displacement output degree of freedom and the system damping value. For a system with critical damping, the motion equation in the principal coordinate system can be expanded as follows:  
\[ \begin{array}{c}\ddot{z}_{p1}=F_{p1}\\ \ddot{z}_{p2}=F_{p2}-w_{2}^{2}z_{p2}-2\xi_{2}w_{2}\dot{z}_{p2}\\ \ddot{z}_{p3}=F_{p3}-w_{3}^{2}z_{p3}-2\xi_{2}w_{2}\dot{z}_{p3}\end{array}  \tag{18}\]  
where \(w_{1},w_{2}\), and \(w_{3}\) are the three eigenvalues and \(\xi_{1},\xi_{2}\), and \(\xi_{3}\) represent the percentages of critical damping for each of the three modes, all of which can be different and are typically obtained from experimental results. The equation of state is as follows:  
\[ \begin{array}{c}x_{1}=z_{p1}\\ x_{2}=\dot{z}_{p1}\\ x_{3}=z_{p2}\\ x_{4}=\dot{z}_{p2}\\ x_{5}=z_{p3}\\ x_{6}=\dot{z}_{p3}\end{array}