## 3 Numerical Simulation and Experimental Analysis  
It can be seen that the transfer matrix expressed by Equation (3) is related with the crack modes, the crack location and depth, as well as the rotating speed, and all of these factors might have an influence on the propagation characteristics of the elastic waves. In this section, the crack mode will be analyzed firstly, the most suitable mode will be chosen out, and then, the influence of the location and the depth of the transverse crack as well as the rotating speed to the propagation characteristics is then thoroughly studied with the chosen mode.  
The numerical method is used to obtain the eigenvalues of the transfer matrix, making clear the relationship between the crack mode and the crack depth with the changing of the propagation characteristics. An experimental rig is built up and experiments are performed accordingly. The rig is shown in Figure 4.  
As shown in Figure 4, the shaft to be tested is supported by two sliding bearings which are fixed on a steel base, and the shaft is driven by a servo motor (2.5 Kw, 3000 RPM). The acceleration sensors (PCB LW16 series) are fixed on the bearings and the steel base. An exciter is used to stimulate the elastic waves in the shaft by acting a pulse force to the shaft. The pulse force is measured by a force sensor too.  
Suppose that the elastic waves propagate from location \(x_{0}\) to \(x_{l}\), and there is only one transverse crack, located at \(x_{c}\), for simplicity. The transverse crack is artificially generated by sawing a normal shaft with a hacksaw, as shown in Figure 5.  
The linear variation form of the shaft \(S(x)=S(0)(1+x/L)\) is chosen in this paper. The radius of the little end (\(x_{0}=0\)) of the selected shaft is 15.00 mm and the area ratio of the shaft is \(S(x_{l})/S(x_{0})=2/1\). As for other variation forms and area ratios, similar analysis can be performed in the same manner. A uniform shaft having the same mass with the non-uniform shaft is also studied for better comparison, the diameter of the uniform shaft is 36.74 mm, which can be easily calculated. The material of shaft is 45# steel, the density \(\rho\) is \(7.8\times 10^{3}\) kg \(\cdot\) m\({}^{-3}\), the elastic modulus \(E\) is 2.06 \(\times\) 10\({}^{11}\) Pa, the Poisson's ratio \(\
u\) is 0.3 and the shear modulus G is 80 GPa. The study is divided into 4 parts, as shown in Table 1.  
Figure 4: The rig built up for the experiments.  
Figure 5: The transverse crack in the non-uniform shaft.  
Table 1 shows the scheme for the numerical simulation and the experimental analysis. In the table, \(R_{c}\) is the radius of the shaft at location \(x_{c}\) and \(L\) is the length of the shaft. Part 1 is for choosing the suitable crack mode, the transverse crack is set to locate at \(x_{c}=0.3L\) with a depth of \(0.5R_{c}\). Part 2 is to study the relationship between the crack depth and the change of the propagation characteristics, the crack is set to locate at \(x_{c}=0.3L\) with the crack depth \(0.1R_{c}\), \(0.2R_{c}\), \(0.3R_{c}\), \(0.4R_{c}\) and \(0.5R_{c}\). Part 3 is for finding the relationship between the crack location and the change of the propagation characteristics, the crack is set to locate at \(x_{c}=0.1L\), \(0.2L\), \(0.3L\), \(0.4L\), \(0.5L\), \(0.6L\), \(0.7L\), \(0.8L\) and \(0.9L\) with crack depth \(0.5R_{c}\). To avoid possible influence from others, the rotating speeds of the shaft in the first three parts are set to be an ordinary speed 3000 rpm (50 Hz), and similar analysis can be performed for other speeds too. Part 4 is for analyzing the relationship between the rotating speed and the change of the propagation characteristics, the crack locates at \(0.3L\) from \(x_{0}\) with depth \(0.5R_{c}\), and the rotating speeds are set to be 0, 10, 20, 30, 40 and 50 Hz, according to the restriction of the experimental conditions.