**1. Introduction**

In recent years, heavy-duty vehicle transportation has become important all around the world [1,2]. The concept of dynamic interactions between the vehicle and the road surface has been widely studied, and a series of problems caused by the coupling vibrations of the vehicle structure and road have become very prominent. Thanks to the in-depth study of this issue by scholars, research on the dynamic interaction between vehicles and road systems [3,4], and the vehicle–road dynamics model of coupled systems [5–11], our understanding in this area is continuously developing. The key problems that hinder the long-term development of heavy haul transportation have been partially solved, including the impact coefficient of vehicles, the vibration of vehicles under the uneven road surface, and the instantaneous response of road vibrations.

As mentioned above, the dynamic characteristics of the roadbed are regarded as one of the important components of the system. At present, a vehicle–road system can be divided into a vehicle vibration model and a road subsystem model. The vehicle is the moving part, which is the main excitation part that causes the vehicle and road vibration. The pavement and the following parts are the road subsystem. Due to the long-term bearing of the dynamic load of the upper vehicle, including the repeated load of heavy vehicle movement, subgrade diseases form easily. In particular, the deformation of flexible asphalt pavement is often affected by many factors such as traffic congestion, joints, load capacity, and repeated vehicle loads [12,13]. All these factors affect the normal service of the road and make the road surface irregular, seriously affecting the normal operation of the road and

**Citation:** Liang, B.; Xiao, J.; Shi, S. Establishment of an Eleven-Freedom-Degree Coupling Dynamic Model of Heavy Vehicle-Pavement. *Symmetry* **2022**, *14*, 250. https://doi.org/ 10.3390/sym14020250

Academic Editors: Yang Yang, Ying Lei, Xiaolin Meng, Jun Li and Juan Luis García Guirao

Received: 31 October 2021 Accepted: 11 January 2022 Published: 27 January 2022

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even damaging it. The above analysis shows that the damage of roads not only involves damage to the road surface but also the excessive deformation of the roadbed. Therefore, when analyzing the dynamic interaction between vehicle and road, the vehicle vibration excitation is the key problem, which is, therefore, one of the main concerns of this work.

Scholars have studied the dynamic problems of vehicle and highway subgrades from different perspectives [14,15]. However, few of the existing models consider the vehicle–road system as a whole, and most models analyze the road dynamic performance under the dynamic load of vehicles. The literature [16] describes the lateral dynamics model of vehicles under road environment excitation. The authors of [17,18] studied the dynamic response of the vehicle–pavement coupling system based on the nonlinear Timoshenko beam method and the vehicle–road dynamic response of the multi-degree-of-freedom vehicle with a double-layer, rectangular, thin plate. Zhang et al. [19] studied the road deformation and crack propagation path under the action of a quarter of vehicles using a modified two-parameter foundation plate. The above analysis fails to fully reflect the interaction of a vehicle–subgrade system in the process of vehicle moving, and this is another main objective of the present study.

In the present study, we establish a seven-degree-of-freedom vehicle dynamic model and use the harmonic superposition method to simulate the process of road random vibration. The dynamic interaction between vehicle and road is analyzed theoretically. Then, the validity of the model is verified by numerical analysis, and the dynamic response is analyzed. The organization of the paper is as follows. In Section 2, the dynamic model and equilibrium equation of heavy load vehicles are established. In Section 3, the coupling dynamic equation of heavy load vehicle–road is established. Then, the vehicle–road system is analyzed from multiple angles through numerical simulation, and the validity of the established equation is verified in Section 4.

### **2. Vehicle Dynamic Model and Equilibrium Equation**
