*7.2. Effect of Running Speed*

The efficiency of identification using the proposed method is directly determined by the vehicle running speed in real-world projects. Therefore, this section considers two speeds, namely, *v* = 5 and 15 m/s. The pavement grade adheres to Class C, and the other parameters align with those used in Section 3. Figure 10 compares the contact force identification results at different vehicle operating speeds. It can be observed that the contact force rises with the increasing vehicle speed. The moving force remains well identified at different vehicle running speeds using the proposed method, implying that the running speed of the vehicle exerts little effect on it. From the perspective of identification accuracy, the RMSE values for the running speeds of 5 and 15 m/s are 0.0687 and 0.2087, respectively. The results indicate that the identified precision may undergo limited reductions, while the proposed method can maintain a high precision even at a rapid running speed. To ensure the test precision and efficiency, however, an intermediate speed is recommended for field applications employing the proposed method. The primary reason for this lies in the fact that a smaller volume of data, obtained at the same sampling frequency for a limited length

of the bridge, will result in an inferior identification accuracy. Thus, for the case under consideration, a test speed of 10 m/s will be optimal.

**Figure 10.** Comparison results of the contact force for the vehicle–bridge system at various running speeds: (**a**) 5 m/s. (**b**) 15 m/s.

#### *7.3. Effect of the Environmental Noise Level*

In practice, measured data inevitably incur contamination from environmental noise. To investigate the feasibility of the proposed method, four environmental noise levels of 2%, 5%, 10%, and 20% are applied to the measured data. The expression is defined as:

$$
\ddot{y}\_{vp} = \ddot{y}\_v + E\_p N\_s \sigma\_{\bar{y}\_v} \tag{27}
$$

where .. *yv* denotes the original vehicle acceleration response. *Ep* denotes the noise level. *Ns* and *σ*.. *yv* represent the mean and standard deviation of the vehicle's acceleration response.

Figure 11 demonstrates a comparison of the moving force identification results at varying environmental noise levels. The RMSE values for *Ep* = 2%, 5%, 10%, and 20% stand at 3.1940, 4.9653, 14.5237, and 36.3568, respectively. It can be seen that the recognition results are worse than those without noise, but the recognition accuracy is generally maintained at a higher level. Even at a 20% environmental noise level, the recognition results remain highly accurate. Therefore, the proposed method for identifying moving contact force presents a certain robustness to environmental noise. However, it is suggested that the environmental noise level should be controlled within 10% to ensure test precision when applying the proposed method in field tests.

**Figure 11.** *Cont*.

**Figure 11.** Comparison results of contact force for the vehicle–bridge system under various environmental noise levels: (**a**) *Ep* = 2%. (**b**) *Ep* = 5%. (**c**) *Ep* = 10%. (**d**) *Ep* = 20%.

#### **8. Conclusions**

In this study, we introduced a novel method for moving load identification on bridges, using a vehicle–bridge coupled vibration theory and the Newmark-*β* algorithm. This approach requires sensor arrangement solely on the vehicle, recording its responses as it traverses the bridge. Based on the obtained responses, the moving force can be swiftly identified via the proposed method. Tikhonov regularisation is employed to overcome the ill-conditioned nature of the inverse problem so as to enhance the recognition accuracy. A single DoF vehicle traversing a simply supported beam is used as an example to numerically verify the proposed method's reliability by comparing the predicted values with the real ones. Additionally, we examined several typical external factors to discuss the robustness of the proposed method. Based on the theoretical derivations and the data adopted in our analysis, the following conclusions were drawn:


Though the proposed method is based on a single-DOF vehicle model moving over a simply supported bridge, it is adaptable to multi-DOF vehicle models and other bridge types, i.e., elastic, clamped, fixed, and general support boundary conditions. Consequently, our future studies will focus on identifying the moving force using multi-DOF vehicle models and conduct further experimental validations of the proposed method' accuracy.

**Author Contributions:** Conceptualization, D.L.; methodology, B.L.; software, K.S.; validation, X.L.; formal analysis, K.S.; writing—original draft preparation, K.S.; writing—review and editing, X.L.; visualization, K.S.; supervision, B.L.; project administration, D.L.; funding acquisition, D.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (grant number 52008060), Chongqing Municipal Natural Science Foundation (grant No. CSTB2022NSCQ-MSX1448) and Guizhou Provincial Science and Technology Projects (Grant No. Qiankehe Support [2022] General 026).

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
