4.2.1. Speed Identification

Firstly, the vehicle speed was identified. Assuming that the vehicle crosses the bridge with a constant speed, the integral Equation (17) of strain influence line can be obtained by modifying Equation (6):

$$A = \int\_{-\infty}^{+\infty} \varepsilon(\mathbf{x}) d\mathbf{x} = \int\_{-\infty}^{+\infty} \varepsilon(t) v dt = v \int\_{-\infty}^{+\infty} \varepsilon(t) dt \tag{17}$$

As shown in Figure 14, it corresponds to the starting point *t*1 of the wave peak when the front axle of the vehicle contacts the bridge, and it corresponds to the ending point *t*2 when the rear axle of the vehicle leaves the bridge. Then the vehicle speed *V* can be calculated according to the time difference and the driving distance, as shown in Equation (18):

$$
\upsilon = \frac{d}{\Delta t} = \frac{L+x}{t\_2 - t\_1} \tag{18}
$$

where, *x* is the vehicle wheelbase, *L* is the bridge length. In this experiment, the vehicle weight was 16.95 kg, and the vehicle speed was divided into nine levels by changing the speed of traction motor, as shown in Table 3. Meanwhile, each experimental condition was repeated three times. According to the above method, the strain data of 3# beam bottom were used to identify the vehicle speed, and the results are shown in Figure 15. It can be seen that the average relative errors of the speed identification were smaller than ±4%, and they were within an acceptable range, which shows the grea<sup>t</sup> performance of the method.

**Table 3.** Speed levels.

**Figure 15.** Relative error of the speed identification.

#### 4.2.2. Influence of Speed on Load Identification

Based on the vehicle speed obtained from inversion, the strain time history curve was converted into strain influence line. Then, the load identification was carried out by the method considering the load transverse distribution. The samples with the weight of 28 kg and speed of 0.86 m/s were selected to calibrate the strain integral coefficient. It should be noted that the driving path of the vehicle was limited to three lanes, so the five equations that were shown in Equation (13) cannot be obtained. However, the optimal solution of the strain integral coefficient can be obtained by using three equations, as shown in Table 4.


**Table 4.** The mid-span strain integral coefficient of each beam bottom.

According to Equation (17), the integral value of the influence line is independent of the vehicle speed, but the identification accuracy of the speed has an effect on the integral value. The actual speed and inversion speed were used to identify the vehicle weight of the above 18 samples, which was mentioned in Section 4.2.1, and the results are shown in Figure 16. It can be seen that the speed had no obvious influence on vehicle weight identification, and the average identification error was smaller than ±5%. In addition, it should be noted that the results obtained by using the actual speed to calculate the vehicle weight were more accurate than those obtained by using the inversion speed. Moreover, the average identification error was smaller than 2% when using the actual speed to identify the vehicle weight.

**Figure 16.** Influence of vehicle speed on load identification: (**a**) Two-axle vehicle, (**b**) three-axle vehicle.

#### 4.2.3. The Identification Results of Vehicle Weight

As the vehicle speed has no obvious influence on the identification results of the vehicle weight, the vehicle speed was set as 1.33 m/s in the following analysis. The two-axle vehicle and three-axle vehicle were divided into four grades of weight in the experiment. Each grade of vehicle weight was tested in three lanes, and the load identification results were shown in Figure 17. It can be seen that no matter which lane the vehicle drives, the vehicle weight identification error of each sample can be controlled within ±10%. The error of more than 90% of samples was smaller than ±5%, and the error was relatively larger compared with the simulation results, which was within an acceptable range. In addition, the error fluctuation of load identification was small, and the variance was smaller than 2%. Compared with reference [26], the method proposed in this paper greatly improves the identification accuracy. Therefore, it can be considered that the load identification method considering the load transverse distribution was effective. The error sources should be analyzed in the following aspects: (1) The vehicle weight identification is based on the speed identification, so the error of vehicle speed will affect the result of vehicle weight identification, (2) although there are lane restrictions in the experiment, the vehicle's trajectory is not always in a straight line along the bridge span direction, and its trajectory is relatively random.

**Figure 17.** The relative error of vehicle weight identification: (**a**) Two-axle vehicle, (**b**) three-axle vehicle.
