## 4 Simulation Verification
### An Introduction to Simulation Systems and Data Sets
#### 4.1.2 A Density Slow-Time-Varying Cantilever Beam Structure  
The density slow-time-varying cantilever beam structure is more complicated, as shown in Figure 9. One end of the cantilever beam is a fixed support, the other end is a free end, and many actual engineering structure can be simplified as a cantilever beam. Therefore, the modal parameters identified in the cantilever beam data set are also valuable. Assuming that the shear deformation of the cantilever beam is ignored, the cantilever beam structure is evenly divided into 40 elements using the finite element method. At the same time, the axial displacement of the beam is not considered, but only the vertical displacement and rotation angle. Parameter Settings: the beam is 1 m in length, 0.02 m in width, 0.02 m in height and \(Area=\text{width}\times\text{height}=4\times 10^{-4}\,\text{m}^{2}\) in cross-sectional area, second moment of area is \(I=[\text{width}\times(\text{height})^{3}]/12=1.3\times 10^{-8}\,\text{m}^{4}\), tensile modulus is \(E=2.1\times 10^{11}\,\text{N}/\text{m}^{2}\), Poisson ratio is \(u=0.3\), the density is \(\rho_{0}=78.60\,\text{kg}/\text{m}^{3}\), \(F(t)\) is the Gaussian white noise excitation applied to the cantilever beam structure. Figure 8: The natural frequency identified by MWLPP of the mass slow-time-varying 3-DOF structure when (**a**) \(t=50.025\) s, (**b**) \(t=1200\) s and (**c**) \(t=1974.375\) s.  
Figure 9: The density slow-time-varying cantilever beam structure with external force. In the finite element model of cantilever beam, the cantilever beam is divided into 40 units and the its equation of motion such as Equation (1) is established. In addition, mass matrix \(\mathbf{M}^{e}\), damping matrix \(\mathbf{C}^{e}\) and stiffness matrix \(\mathbf{K}^{e}\) of each window can be expressed as:  
\[\mathbf{M}^{e}=\rho_{0}\times\text{{Area}}\times\mathbf{M}^{e}_{\text{c}}= \frac{\rho_{0}\times L\times\text{{Area}}}{420}\left[ \begin{array}{cccc}156&2 2L&54&-13L\\ 22L&4L^{2}&13L^{2}&-3L^{2}\\ 54&13L&156&-22L\\ -13L&-3L^{2}&-22L&4L^{2}\end{array} \right] \tag{18}\]  
\[\mathbf{K}^{e}=EIK^{e}_{\text{c}}=\frac{EI}{L^{3}}\left[ \begin{array}{cccc}1 2&6L&-12&6L\\ 6L&4L^{2}&-6L&2L^{2}\\ -12L&-6L&12&-6L\\ 6L&2L^{2}&-6L&4L^{2}\end{array} \right] \tag{19}\]  
\[\mathbf{C}^{e}=\beta_{M}\mathbf{M}^{e}+\beta_{K}\mathbf{K}^{e} \tag{20}\]  
where \(L\) represents window length, \(\beta_{M}\) and \(\beta_{K}\) are proportional coefficients. Then the mass matrix \(\mathbf{M}^{e}\), damping matrix \(\mathbf{C}^{e}\) and stiffness matrix \(\mathbf{K}^{e}\) are assembled into the total mass matrix \(\mathbf{M}\mathbf{M}\), damping matrix \(\mathbf{C}\mathbf{C}\) and stiffness matrix \(\mathbf{K}\mathbf{K}\) of the system:  
\[\left\{ \begin{array}{l}\mathbf{M}\mathbf{M}=\sum_{e}\mathbf{M}^{e}\\ \mathbf{C}\mathbf{C}=\sum_{e}\mathbf{C}^{e}\\ \mathbf{K}\mathbf{K}=\sum_{e}\mathbf{K}^{e}\end{array} \right. \tag{21}\]  
Therefore, the modal parameter can be calculated by finite element method, and the mode frequency, mode shape and modal damping ratio can be expressed as:  
\[f_{r}=\sqrt{\frac{\mathbf{K}\mathbf{K}_{r}}{\mathbf{M}\mathbf{M}_{r}}},\;r=1,2,\cdots,N \tag{22}\]  
\[(\mathbf{K}\mathbf{K}_{r}-f_{r}^{2}\mathbf{M}\mathbf{M}_{r})\overset{\to}{ \phi}_{r}=0,\;r=1,2,\cdots,N \tag{23}\]  
\[\xi_{r}=\frac{\mathbf{C}\mathbf{C}_{r}}{2\sqrt{\mathbf{M}\mathbf{M}_{r} \mathbf{K}_{r}}},\;r=1,2,\cdots,N \tag{24}\]  
where \(\mathbf{M}\mathbf{M}_{r}\), and \(\mathbf{C}\mathbf{C}_{r}\) are the \(r\)-th order of modal mass matrix, modal stiffness matrix and modal damping matrix respectively. \(f_{r}\), \(\overset{\to}{\phi}_{r}\) and \(\xi_{r}\) are the \(r\)-th order of natural frequency, modal shape, and damping ratio. \(N\) is the number of sensor. To achieve the slow-time-varying condition, the density of the cantilever beam is changed with time, as shown in Equation (25). \[\rho_{2}=\left\{ \begin{array}{l}\rho_{0},\,0\leq t\leq 0.5\,\text{s}\\ \rho_{0}\left[1-0.08(t-0.5)\right],\,0.5\,\text{s}<t\leq 4\,\text{s}\end{array} \right.