## 3 Modal Simulation Analysis
### Influence of Suspension Faults on Modal Parameters  
During the long-term operation of the vehicle, load conditions and environmental factors can cause fault phenomena such as cracks and breaks in leaf springs, resulting in changes in the performance parameters of the suspension springs and a decrease in spring stiffness. To realize the suspension fault identification based on modal parameter analysis, it is essential to first understand the specific influence of the fault on the modal parameters. Therefore, it is necessary to calculate the modal parameters of the suspension system, i.e., natural frequency, damping ratio and mode shape, under different spring stiffnesses through simulation in MATLAB (R2017a, MathWorks, Natick, MA, USA). Firstly, the stiffness of the left rear spring is changed from 0% to 20% in the model to simulate spring failure and to study its effect on modal parameters. Figure 8 shows the changes in natural frequency, damping ratio and mode shapes caused by the drop in spring stiffness when the left rear leaf spring fails; the blue line represents the bounce mode, the green line represents the pitch mode and the red line represents the roll mode. It can be seen from the first three modes that when the stiffness of the left rear spring is weakened, the natural frequencies of all three decreases to varying degrees, among which the pitch mode changes the most. Because of the small amplitude of the natural frequency variation, it is difficult to obtain accurate measurements in practical applications. The damping ratios of all three modes increased significantly, but the above suggested that the identification error of the damping ratio is larger than that of the other two. That is, the identified result can hardly reflect the actual change in damping ratio, so it is not suitable as an indicator of fault diagnosis. From the change of the vibration mode shapes in the three orders of modes in Figure 8, due to the mode shapes being considered as eigenvectors, the values are dimensionless, and the changes are obvious. This obvious feature can be used to judge the fault, and from the previous analysis in Section 2.2, the identification error of the modal mode is relatively small and thus suitable as a monitoring indicator. It can be seen from the changes in the mode shape in Figure 8 that as a feature vector, the value of the mode shape is dimensionless and its variation is more obvious. This distinctive feature can be used to diagnose the fault and is suitable as a monitoring indicator. Then, the variation called modal energy difference (MED) is proposed to reveal the faults of suspension system more straightforwardly. In order to convey the imbalance of the mode shapes between the left and right sides, the rolling trends of mode shape are denoted as the differential of square of two mode shape, which is called modal energy difference (MED) in the following section. The MED between the left and right suspensions in bounce mode can be defined as Equation (9). Figure 8: Changes in natural frequency, damping ratio and mode shape caused by a left front spring fault. \[ \begin{array}{l}dE_{bf}=2[(v_{a}^{2}-v_{b}^{2})/(v_{a}^{2}+v_{b}^{2})]\\ dE_{br}=2[(v_{c}^{2}-v_{d}^{2})/(v_{c}^{2}+v_{d}^{2})]\end{array} , \tag{9}\]  
where in, \(dE_{bf}\) represents the difference between the vibration modes of left and right suspensions in front of the vehicle in the bounce mode, and \(dE_{br}\) represents the MED in rear suspensions. In Figure 9, the solid line is the MED between the two front suspensions and the dashed line represents the corresponding MED at the rear of the vehicle. The left and right sides of each figure respectively represent the modal energy changes in the first-order mode (bounce mode) and the third-order mode (pitch mode). As shown in Figure 9a, when the left front wheel leaf spring fails, the difference in modal energy in the bounce mode increases by about 15%, and the difference in modal energy in the pitch mode drops by about 10%.