## 3 Simulation Examples
### Mode Shape Extraction Using Displacement Approximation of Contact Points  
#### 3.3.1 Effect from Noisy Disturbance  
In the previous numerical examples, the responses of the contact points could be obtained easily from Equation (30). By contrast, in practice, an attempt is made to measure the responses of the contact points from the lower part of the vehicle's suspension. Alternatively, they can be approximated from the measured vehicle responses [13] without considering the vehicle's damping. \[\ddot{u}_{c}=\ddot{u}_{v}+\frac{\dddot{u}_{v}}{\omega_{v}^{2}(\Delta t)^{2}}, \omega_{v}^{2}=\frac{k_{v}}{m_{v}} \tag{40}\]  
where the four over-dots denote the fourth temporal derivative of the vehicle's response (\(u_{v}\)). Once the vehicle's response is utilized to calculate the response of the contact point, the noise level from the vehicle response is amplified prominently on the resulting contact point's response through the twice finite difference operation on \(\dddot{u}_{v}\) with a small \(\Delta t\). Then, its displacement response could be expressed as follows:  
\[u_{c}=\frac{\ddot{u}_{v}}{\omega_{v}^{2}}+u_{v} \tag{41}\]  
where the coefficients (\(\frac{1}{\omega_{v}^{2}}\) and \(1\)) of the acceleration \(\ddot{u}_{v}\) and the displacement \(u_{v}\) could be considered as the weights between them on the resulting contact point's displacement. Figure 12: The first three beam mode shapes using the residual STFT-based extraction under noise levels of 1%, 10%, and 20%, compared to the analytical first mode shape (**a**), the second mode shape (**b**), and the third mode shape (**c**). According to the previous works that used the acceleration [11; 12; 13; 14; 15; 16] and displacement [14] of the vehicle, both quantities could be utilized to extract the mode shapes. As shown in Equations (9)-(12), like the contact points' accelerations, their displacements could also be used for the extraction of the mode shapes. Moreover, compared to using acceleration only as in Equation (40), using both acceleration and displacement could avoid the huge amplifying effect from the finite difference operations. The performance of mode shape extraction using this displacement approximation is investigated under the noise levels of 1%, 5%, and 10% on the accelerations and displacements of the vehicles. As shown in Figure 13, the first mode shapes under all three noise levels fit very well with a MAC value close to one. However, the influence of noise becomes obvious at the second mode shapes. Under 1% noise, the MAC value of the second mode shape could be kept over 0.97. Meanwhile, the second mode shapes under 5% and 10% noise are distorted, with MAC values of 0.694 and 0.638. Only the third mode shape under 1% noise is reasonable. The third mode shape at 5% noise is also poor, with a MAC value of 0.638, and the third mode shape extracted at 10% noise is meaningless, with a MAC value of less than 0.6. Compared with the STFT of the acceleration residuals in Figure 5c, there is not much energy distributed on the higher frequencies for the displacement residual in Figure 14. Except for the peak ridge along the first natural frequency, the other peak ridges at the higher frequencies are small compared to the acceleration responses. Consequently, the mode shapes extracted using the displacement approximation of contact points exhibit significantly more noise disturbances compared to those in Figure 12, particularly in the second and third mode shapes when subjected to noise levels of 5% and 10%, as illustrated in Figure 13.  
Figure 13: The first three beam mode shapes using the approximate displacement residual STFT-based extraction under noise levels of 1%, 5%, and 10%, compared to the analytical first mode shape (**a**), the second mode shape (**b**), and the third mode shape (**c**). Considering the poor performance under noisy disturbances in the above cases, it is crucial to denoise the signal before extracting the mode shape.