# A Study on the Feasibility of Natural Frequency-Based Crack Detection
## 3 The Free Vibration of a Loaded Single-Edge Cracked Rectangular Plate
\tag{8}\]  
Substituting Equation (4) into Equations (1)-(3) and minimizing the functional \(\Pi\) with regard to undetermined coefficients \(a_{ij}\), \(B_{nl}\) and \(C_{nl}\) yields \(IJ+N(N+3)\) linear algebraic equations. The solution to the eigenproblem defined by these equations gives the eigenvalue of the system and their corresponding eigenvectors. Natural frequencies are the square root of the eigenvalues, while the eigenvectors define the vibration mode shape. When deriving the terms in the stiffness matrix, a mesh of triangles is used to perform Gaussian integration (as shown in Figure 4a). The mesh can be refined iteratively by connecting the midpoints of the edges of each triangle. The Gaussian quadrature rule of degree 19 is applied to the triangles, which leads to 73 integration points within each triangle (as shown in Figure 4b). To ensure the accuracy of frequency results, the in-plane stress resultants \(N_{xx}^{(0)}\), \(N_{xy}^{(0)}\) and \(N_{yy}^{(0)}\) are calculated using the XFEM. The XFEM allows the finite element mesh to be completely independent of the morphology of the crack [38]. It reduces the significant mesh refinement near the crack tip, thus rendering efficient computation. The computation is facilitated by compiling an APDL routine in FEA software ANSYS 18.0. The finite  
Figure 4: The mesh for Gaussian integration and the distribution of integration points. element model discretization is shown in Figure 5. The shadowed area shows the region of enrichment, where the enriched displacement approximation [39] is written as follows:  
\[u(x,y)=\sum_{I\in\mathcal{N}}N_{I}(x,y)[\mathbf{u_{I}}+H(x,y)\mathbf{a_{I}}+\sum_{\alpha =1}^{4}\Phi_{\alpha}(x,y)\mathbf{b_{I}^{\alpha}}] \tag{9}\]  
In Equation (9), \(\mathbf{u_{I}}\) is the nodal displacement vector associated with the continuous part of the solution; \(\mathbf{a_{I}}\) is the nodal enriched degree of freedom vector associated with the Heaviside (discontinuous) function; \(\mathbf{b_{I}^{\alpha}}\) is the nodal enriched degree of freedom vector associated with the elastic asymptotic crack-tip functions; \(N_{I}(x,y)\) and \(H(x,y)\) are the finite element shape function and generalized Heaviside function, respectively; and \(\Phi_{\alpha}(x,y)\) is the crack-tip enrichment function which can be expressed as the following:  
\[[\Phi_{\alpha}(x),\ \alpha=1\ \sim\ 4]=\left[\sqrt{\tau}sin\frac{\theta}{2},\ \sqrt{\tau}cos\frac{\theta}{2},\ \sqrt{\tau}sin\theta sin\frac{\theta}{2},\ \sqrt{\tau}sin\theta cos \frac{\theta}{2}\right] \tag{10}\]  
where \(r\) and \(\theta\) are defined as in Figure 3. By specifying a radius around the crack tip, the effects of the crack-tip singularity can be accounted for within the circled region defined by the radius (as shown in Figure 5). The in-plane stress resultants \(N_{xx}^{(0)}\), \(N_{xy}^{(0)}\) and \(N_{yy}^{(0)}\) obtained via the XFEM are sets of values known only at discrete points rather than at positions optimum for integration efficiency. Therefore, the stress resultants at Gaussian integration points are interpolated using triangulation-based natural neighbour interpolation [32]. This is achieved by using _scatteredInterpolant_ in MATLAB R2017b. As the objective of this study is to determine whether the natural frequency drop caused by a crack is measurable, ensuring the reliability of the natural frequency calculation is very important. In the next section, convergence studies are carried out to verify the in-plane stress and natural frequency results.