## 5 Model Reconstruction and Secondary Optimization
### Simulation Analysis and Experimental Verification of the Subframe After Secondary Optimization
#### 5.3.2 Subframe Modal Test  
To verify the accuracy of the simulation analysis, in this study, the rear subframe was inspected through modal tests. Based on the limitations of the experimental conditions, a specific subframe was selected for modal testing and simulation comparison. In the early stage of the experiment, structural feature points were selected for signal collection, and then external forces were applied through excitation equipment or impact hammers. Due to the simple operation and rapid response of the hammering method, the PCB force hammer was selected as the excitation tool in this study. When operating, it is necessary to ensure that the tapping direction is orthogonal to the excitation point, the force should be stable and moderate, and the interval time should be reasonably arranged. The installation of sensors can refer to the specifications, with a focus on deployment at the junctions of  
Figure 14: Vibration pattern diagram of the first three modes’ analysis. (**a**) First-order modal vibration mode; (**b**) second-order modal vibration mode; (**c**) third-order modal vibration mode.  
Figure 13: Optimization results of subframe size (\(X_{1}\)—The thickness of the front crossbeam of the subframe. \(X_{2}\)—The thickness of the rear crossbeam of the subframe. \(X_{3}\)—The thickness of the side beams of the subframe. \(X_{4}\)—The thickness of the subframe tube string. \(X_{5}\)—Lower control arm connection plate).  
components, key excitation areas, and the core parts of the research object. At the same time, it is necessary to avoid the node positions of each mode [18; 19].  
To improve the test accuracy, five hammering operations were carried out at each excitation point and sensors were installed. After the experiment was completed, the collected data was processed by using the signal analysis system. Figure 15 shows the modal test process of the force hammer.  
In this experiment, LMS Mobile SCM05 was used for signal analysis. Figure 16 shows the steps of modal detection. The input force signal is converted into the excitation self-spectrum \(\mathcal{S}_{\text{FF}}(\omega)\), and the acceleration signal is converted into the response self-spectrum \(\mathcal{S}_{\text{XX}}(\omega)\). The correlation between them is represented by cross-spectrum \(\mathcal{S}_{\text{FX}}(\omega)\). The calculation formulas of the system frequency response function \(H(\omega)\) and the coherence coefficient \(\gamma_{FX}^{2}(\omega)\) are as follows [20; 21]:  
\[\begin{array}{ll}H(\omega)&=\frac{\mathcal{S}_{\text{XX}}(\omega)}{\mathcal{ S}_{\text{FX}}(\omega)}\\ \gamma_{F\text{X}}^{2}(\omega)&=\frac{[\mathcal{S}_{\text{FX}}(\omega)]^{2}}{ \mathcal{S}_{\text{XX}}(\omega)\mathcal{S}_{\text{FF}}(\omega)}\end{array} \tag{18}\]  
Figure 16: Modal test flowchart.  
Figure 15: Modal experiment of the force hammer method ((**a–d**) are pictures of the experimental process).  
To eliminate invalid data caused by experimental errors, the signal coherence needs to be calculated before analysis. Usually, 0.8 is taken as the threshold to eliminate the data that do not conform to the linear relationship. After frequency response fitting processing, the comprehensive FRF was obtained. For details, please refer to Figure 17.  
Modal analysis was carried out with the aid of Hypermesh, and then the first three orders of free modal frequencies and vibration modes of the optimized subframe were obtained. It can be seen in Table 5 that the maximum error between the two is \(-2.7\%\), which meets the engineering requirements. The accuracy of the finite element modeling was verified. In terms of modal performance, the design standards and requirements have been met.