## 3 Modeling and Experimental Validation of Truck Suspension Systems
### Experimental Testing of Truck Suspensions  
To experimentally determine the force-displacement characteristic of the front leaf springs, which are multi-leaf types, two displacement sensors were placed on the master leaf, one on the right and one on the left, at a distance of \(l_{sf}=150\) mm from its center to measure the vertical displacement. Scales were also used to measure the mass and, consequently, the force applied to the springs, as shown in Figure 6. The test began by raising the front part of the truck until the scales showed a value of 0, indicating no contact between the wheels and the ground. We measured the initial curvature of the leaf spring as a reference value. Then, the truck gradually descended, and the mass values were displayed on the scales, while the corresponding displacements indicated by the sensor were recorded. The curvature of the leaf spring was determined as the difference between the values recorded at each moment and the reference value. Figure 5: Multi-body models: (**a**) front suspension; (**b**) rear suspension. The sensor used was the WDS-500-P60-CR-P produced by the company \"MICRO-EPSILON\" (\(\mu\)e) (Ortenburg, Germany). It is an industrial draw-wire sensor with a measuring range of 0 to 500 mm. The same procedure was applied to the main rear leaf spring, which is a taper leaf spring type. However, instead of placing the displacement sensor directly onto the master leaf spring, it was positioned on the auxiliary leaf spring, at a distance of \(l_{sr}\) = 150 mm. During testing, the auxiliary leaf did not function, so it can be assumed that the recorded displacement corresponds to that of the main leaf spring. Additionally, only one displacement sensor was used on the left side of the suspension, as shown in Figure 7. We also took advantage of the front spring test to measure the vertical displacement of the rear spring and the mass values displayed when the front part of the truck was raised, using these data as additional measurements. The force-displacement curves for both the front and rear suspensions, determined through experimental tests and simulations, are represented in Figure 8.  
Figure 8 represents the force-displacement curves for the front suspension, both simulated and tested, which are closely aligned. The force-displacement curves for the rear suspension align well from 0 to 50 mm, but discrepancies increase beyond this range. This discrepancy can likely be attributed to the positioning of the displacement sensor on the auxiliary leaf spring, implying a small displacement imposed by the auxiliary leaf spring on the sensor. Despite this, the error between the values does not exceed 10%. Given  
Figure 6: Layout of displacement sensors and scales on the front suspension: (**a**) the experimental procedure; (**b**) a schematic representation; (**c**) the scale positions. these observations, we can assert that the models for both the front and rear leaf springs are validated. The leaf spring model must have the same curvature as the installed leaf spring in the truck and, for this, it is necessary to know the initial curvature of each leaf spring before their assembling in the truck. To avoid disassembling this leaf spring, which is time-consuming, the leaf spring was modeled as installed in the truck by adding the static force (preload). This force needed to be included in the ADAMS model of each leaf spring and a static simulation needed to be conducted prior to dynamic simulation of the full truck. If the mentioned static force is neglected, the leaf spring model changes its curvature due to the truck's weight, thus making the dynamic simulation invalid. By adding these forces, the curvatures of the leaf springs remained unchanged, as their effect compensated for the truck's weight. Considering the truck's symmetry relative to its roll axis (\(x\)-axis), the preload for each leaf spring, whether front or rear, is the same on both the right and left sides, as detailed in Equations (5) and (6), respectively.