# Vision-Based Damage Detection Method Using Multi-Scale Local Information Entropy and Data Fusion
## 3 Numerical Investigations of the Methodology
### Analytical Model of Cracked Beam  
Figure 6a illustrates the physical system under investigation, which is a cantilever beam of length \(L\) and uniform rectangular cross-section \(b\times h\). There are \(n\) cracks in the beam, with the \(i\)-th (\(i=1\), \(2\), \(\ldots\), \(n\)) crack located at \(L_{d,i}\) from the left clamped end of the beam. It is presumed that the cracks are open and have a uniform depth \(h_{d,i}\). Utilizing the linear elastic fracture mechanics principle, the transverse crack is modeled as a rotational spring [43]. This results in the cracked beam being divided into \(n+1\) segments via \(n\) springs. The corresponding analytical model of the cracked beam is presented in Figure 6b.  
During the numerical analysis, as only bending vibrations of the thin beam are considered, it is assumed that the rotational spring constant governs the local flexibility matrix. The stiffness of the rotational spring is given by [PERSON] et al. [44] as  
\[K_{t,i}=\frac{EI}{5.346 hJ(h_{d,i}/h)} \tag{21}\]  
where \(E\) is the modulus of elasticity of the beam, \(I\) is the area moment of inertia for the beam cross-section, and \(J(h_{d,j}/h)\) is the dimensionless local compliance function.  
Figure 6: Analytical model for numerical investigations: (**a**) a cracked cantilever beam, (**b**) the corresponding rational spring model.  
To be brief, we define dimensionless parameters, relative crack size \(\alpha_{i}\) and relative crack location \(\beta_{i}\), to describe damages as  
\[\alpha_{i}=h_{d,i}/h,\ \beta_{i}=L_{d,i}/L \tag{22}\]  
According to the Euler-Bernoulli beam theory, the governing equations for the free vibrations of beam segments can be written as  
\[EI\frac{\partial^{4}w_{i}(x,t)}{\partial x^{4}}+\rho A\frac{\partial^{2}w_{i}( x,t)}{\partial t^{2}}=0 \tag{23}\]  
where \(\rho\) is the mass density, \(A\) is the cross-section area, and \(w_{i}\) is the transverse deflection shape of the \(i\)-th beam segment. By using the method of separation of variables, the general solution of Equation (23) can be obtained as  
\[w_{i}(x)=c_{i,1}cosh(\lambda x)+c_{i,2}sinh(\lambda x)+c_{i,3}cos(\lambda x)+c _{i,4}sin(\lambda x) \tag{24}\]  
with \(\lambda^{4}=\omega^{2}\rho A/EI\), where \(\omega\) is the vibration angular frequency, and \(c_{i,1}\) to \(c_{i,4}\) are constant coefficients determined from the boundary conditions. The boundary conditions at the clamped and free ends are denoted by  
\[\left\{\begin{array}{l}w_{1}(x)|x=0\ =0,\frac{\partial w_{1}(x)}{\partial x}|x=0 =0\ \ (the\ clamped\ end)\\ \frac{\partial^{2}w_{1+1}(x)}{\partial x^{2}}\Big{|}x=L=0,\frac{\partial^{2} w_{1+1}(x)}{\partial x^{3}}|x=L=0\ (the\ free\ end)\end{array}\right. \tag{25}\]  
Considering the connection between the two beam segments, constraints can be established to ensure the continuity of displacement, bending moment, and shear. In addition, the condition of equilibrium between the transmitted bending moment and the rotation of the spring should be imposed. Thus, the boundary conditions at the crack are written as follows:  
\[\left\{\begin{array}{l}w_{i}(x)\big{|}x=L_{d,i}=w_{i+1}(x)\big{|}x=L_{d,i}\\ \frac{\partial^{2}w_{i}(x)}{\partial x^{2}}\big{|}x=L_{d,i}=\frac{\partial^{2 }w_{1+1}(x)}{\partial x^{3}}\big{|}x=L_{d,i}\\ \frac{\partial^{2}w_{1}(x)}{\partial x^{2}}\big{|}x=L_{d,i}=\frac{\partial^{2 }w_{1+1}(x)}{\partial x^{3}}\big{|}x=L_{d,i}\\ -EI\frac{\partial^{2}w_{1}(x)}{\partial x^{2}}\Big{|}x=L_{d,i}=K_{t,i}\Big{(} \frac{\partial w_{1}(x)}{\partial x}\Big{|}x=L_{d,i}-\frac{\partial w_{i+1}(x )}{\partial x}\big{|}x=L_{d,i}\ \Big{)}\end{array}\right. \tag{26}\]  
Based on the boundary conditions described above, the resulting characteristic equation of the system is solved numerically, and the mode shape of each beam segment is obtained. For other constraints, such as pinned cracked beams, the solution can be performed similarly by changing the boundary conditions at both ends.