# Research on Propagation Characteristics of Tire Cavity Resonance Noise in the Automobile Suspension
## 3 Establishment and Response Calculation of a McPherson Suspension System Model
### Modal Analysis and Response Calculation of a McPherson Suspension System  
As a key link of TCRN propagation to the vehicle, the natural frequency of a suspension system is close to the tire cavity resonance frequency, which directly affects the propagation of this noise energy. We performed a modal analysis of a suspension system model using the Lanczos method in Abaqus 6.14. We focused only on a low-order modal analysis in the vast majority of cases and generally selected the first six orders of modes. Then, the first six orders of natural frequencies and vibration modes of the suspension system were taken, as shown in Figure 5. The finite element type was C3D10 HS, which is a 10-node, general purpose quadratic tetrahedron with improved surface stress visualization. The DOF of C3D10 HS is ten, and the behavior of the element type is in the form of a Lagrange description. The number of elements was 393,719.  
It can be concluded that the steering knuckle bends in the longitudinal plane from the third order to the fourth order, and the lower swing arm bends in the longitudinal plane from the first to the sixth order. The natural frequencies of the suspension system are shown in Table 2. It can be seen that the sixth natural frequency was close to the resonance frequency of the tire cavity [17]. So, the FE suspension results were verified.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Location** & **Direction** & **Stiffness (N/Mm)** & **Damping (N-s/mm)** \\ \hline \multirow{3}{*}{Shock absorber upper end} & X & 10,000 & \\ \cline{2-4}  & Y & 10,000 & 1.5 \\ \cline{2-4}  & Z & 10,000 & \\ \hline Spring and shock absorber & & 25 & 1 \\ \hline \multirow{3}{*}{Lower transverse arm front bushing} & X & 10,000 & \\ \cline{2-4}  & Y & 10,000 & 2 \\ \cline{2-4}  & Z & 1000 & \\ \hline \multirow{3}{*}{Lower transverse arm rear bushing} & X & 500 & \\ \cline{2-4}  & Y & 500 & 0.8 \\ \cline{2-4}  & Z & 100 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Stiffness and damping of the bushing and spring unit.  
Figure 4: Simplified McPherson suspension system and transverse arm bushings. (**a**) Simplified McPherson suspension. (**b**) Transverse arm front bushing. (**c**) Transverse arm rear bushing.  
The Y-spoke aluminum alloy wheel and suspension system were simulated and analyzed. The suspension was made of steel and the type of elastic was isotropic. Our analysis was linear, using the implicit direct method of integration, where the steady state response of the system was calculated by directly integrating the original equations of the model. The dynamic response of tire cavity resonance noise was calculated, and the propagation characteristics of tire cavity resonance noise in suspension were obtained, providing a basis for the suppression of this noise. In our direct steady-state dynamics analysis of the suspension system, the response at the first frequency of tire cavity resonance was relevant. The defined sweep frequency range was 210-260 Hz, and 101 frequency points were defined, with one frequency excitation point every 0.5 Hz. Using this method, the stress and strain of the suspension system at the first natural frequency of the tire cavity resonance noise (238 Hz) were evaluated. The frequency (238 Hz) came from an experiment described in the literature [17], i.e., this value was not the result of other simulations. The stress and strain contours of the suspension system, obtained in the frequency domain, are shown in Figure 6. The unit of stress is MPa.  
Figure 5: The first six orders of vibration mode shapes of the suspension system.  
It can be seen from Figure 6 that the stress and strain of the suspension system were relatively large at the lower swing arm, steering knuckle, and shock absorber. Therefore, it can be speculated that the energy of the tire cavity resonance noise propagating to this area would also be large.