## 2 Theoretical Study  
### Transfer Matrix for a Shaft with Transverse Cracks  
A rotating shaft is often with variable cross-section area and there might be several transverse cracks, see Figure 1. In the figure, \(E\) is the elastic modulus, \(\rho\) is the density of the shaft, \(v\) is the Poisson's ratio and \(L\) is the length of the shaft. As can be seen in Figure 1a, the area of the cross-section of the shaft \(S(x)\) varies with the location of the \(x\)-axis. Usually, the motion status of the elements on the shaft can be expressed by a state vector \(\mathbf{y}=[u\;\phi\;Q\;M]^{T}\), where, \(u\), \(\phi\), \(Q\), \(M\) indicate displacement, slope, force and moment for a shaft [29]. Here, two vectors \(\mathbf{y}_{0}\) and \(\mathbf{y}_{l}\) are used to represent the status of location \(x_{0}\) and \(x_{l}\), separately. Suppose that there are \(n\) transverse cracks, locating at \(x_{c1}\),... \(x_{ci}\)... \(x_{cn}\), the shaft is then divided into \(n+1\) segments, see Figure 1b. As for the transverse cracks shown in Figure 1, the deformation of the crack varies, even though for the same material and with the same external load conditions. Usually, the crack modes can be divided into three typical types: the opening mode (Mode I), the edge-sliding mode (Mode II) and the tearing mode (Mode III) according to the departure type, the external loads and the deformation, see Figure 2[22]. As shown in Figure 1, suppose that the elastic waves propagate from location \(x_{0}\) (\(x=0\)) to location \(x_{l}\) (\(x=L\)), the propagation process can then be expressed by [17]  
\[\mathbf{y}(x_{l})=\mathbf{Ty}(x_{0}) \tag{1}\]  
where, \(\mathbf{T}\) is the transfer matrix. Suppose the \(i\)-th eigenvalues of the matrix \(\mathbf{T}\) to be \(\lambda_{i}\), and the eigenvalue \(\lambda_{i}\) is usually a complex number and can be expressed as \(\lambda_{i}=e^{\mu_{i}}\), where \(\mu_{i}=\alpha_{i}+j\beta_{i}\) is known as the propagation constants, \(\mathbf{j}=\sqrt{-1}\), \(\alpha_{i}\) is known as the attenuation factor, indicating the attenuation of the amplitude of the elastic waves varying with the frequency of the wave component, \(\beta_{i}\) is the phase angle indicating the phase difference of the wave components during the propagation. When the phase angle \(\beta_{i}\) equals 0 or \(\pm\pi\), it means that the transmitted waves encounter the reflected ones cancelling out each other and the waves with corresponding frequencies could not pass through the shaft [17]. As shown in Figure 1b, the shaft is divided into \(n+1\) segments. For the \(i\)-th segment, the transfer matrix can be written as [29]  
\[\mathbf{T}_{i}=\mathbf{G}(x_{i})e^{\int_{0}^{L_{i}}\mathbf{H}(x)dx}\mathbf{G} (x_{i-1})^{-1} \tag{2}\]  
Figure 1: A schematic of a non-uniform shaft with transverse cracks, the cross-section varies with the location of \(x\), as shown in (**a**), the shaft is divided into \(n+1\) segments by \(n\) transverse cracks, as shown in (**b**). Figure 2: Three typical crack modes, (**a**) is the opening mode (Mode I), (**b**) is the edge-sliding mode (Mode II) and (**c**) is the tearing mode (Mode III). where, \(\mathbf{H}(x)\), \(\mathbf{G}(x)\) are \(\mathbf{H}(x)=\left[ \begin{array}{cccc}0&1&0&0\\ 0&0&0&1\\ \frac{\rho S(x)\omega^{2}}{EI(x)}&0&0&-\frac{\rho I(x)\Omega+EI(x)_{xx}}{EI(x)} \\ 0&0&1&0\end{array} \right]\), \(\mathbf{G}(x)=\left[ \begin{array}{cccc}1&1&0&0\\ 0&1&0&0\\ 0&0&-EI(x)&0\\ 0&0&0&EI(x)\end{array} \right]\), where, \(I(x)\) is the inertia momentum at location \(x\), \(\omega\) is the angular frequency of the components of the elastic waves. For the transverse cracks in a single non-uniform shaft, it is often treated as a local spring and the transfer matrix for the \(i\)-th crack can be written as: \(\mathbf{C}_{i}=\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&c_{i}\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]\), where, \(c_{i}\) is the local stuffiness coefficient of the transverse crack, i.e., the stiffness coefficient of the local spring, which can be calculated according to the crack modes.