# A Multibody Model Calibration Method for Vehicle Misuse Testing
## 4 Nonlinear Parameter Calibration Using the Modified Quarter-Vehicle Model Analysis
### The Modified Quarter-Vehicle Model  
Calibration of the shock absorber's stiffness and the wheel's center mass is crucial for vehicle misuse tests. These parameters directly influence the responses of the shock absorber compression \(d(t)\) and the wheel's center acceleration \(\tilde{z}_{w}(t)\) (in units of standard gravitational acceleration \(g\)), as depicted in Figure 5(b). In the driving attitude, the compression reflects the relative position from the wheel center to the vehicle body, while the acceleration indicates the absolute position of the wheel center determined through two rounds of time integration. When the vehicle passes over relatively large obstacles at high speeds, the shock absorber's compression and the wheel center's acceleration exhibit strong nonlinearity. This is caused by discontinuous changes in the tire-ground contact conditions, which are subsequently reflected in the vehicle's responses. The contact process is naturally divided into three stages on the timeline, as depicted in Figure 5(a): 1\(t_{1}\leq t<t_{2}\), the obstacle contact stage; 2\(t_{2}\leq t<t_{3}\), the tire detachment stage; and 3\(t_{3}\leq t\), the ground recontact stage. The contact status between the tire and the road varies distinctly in these three stages, presenting different dynamic characteristics that result in complex, nonlinear responses. Figure 5: Calibration of the vehicle’s mass and centroid position using the wheel load responses on a flat road segment: the target, the initial responses, the parameters’ numerical sensitivity, the results predicted using the least squares method, and the simulated results. To analyze the three distinct stages in the process of passing over the speed bump, a modified quarter-vehicle model is proposed in this paper. As depicted in Figure 6c, compared to the traditional quarter-vehicle model used in durability and smoothness tests, an equivalent spring \(k_{\rm e}\) is added between the sprung mass and an imagined ceiling. The force generated by the equivalent spring represents a \"recovery force\". This force originates from the suspension springs and tires at the opposite end of the vehicle and is transferred through the chassis via pitch motion. For instance, when the front wheels of a vehicle traverse over a convex obstacle, the front end of the vehicle is lifted, creating an angle of elevation. If the height of the chassis center is assumed to remain constant, the rear end will descend, and the rear suspension will produce a recovery force to counteract the pitch motion. The converse is also true. Consequently, the equivalent spring maintains the influence of the suspension on the opposite end in the modified quarter-vehicle model. Furthermore, in this manner, the front and rear suspensions are decoupled. With only two degrees of freedom (DOFs) in the modified quarter-vehicle model, it is much easier to theoretically analyze the three distinct stages simultaneously, a task that is significantly more challenging for a half-vehicle model [9] with four DOFs. Given the brief duration of the contact process, the gravitational force can be disregarded in the theoretical analyses. This model accounts for tire damping, consistent with the references [11,12], whereas it is neglected in others [13,14]. Indeed, the subsequent numerical simulation in Section 4.5 will demonstrate that the tire's damping ratio is essential to the comparison with the actual vehicle experiment data (test data). In the context of the current misuse scenario, the tire damping is significant, particularly because of the pronounced vehicle response at high driving speeds. The scheme between \(t_{2}\) and \(t_{3}\) indicates that the tire detaches from the road's surface during this interval, where the two-DOF system responds in the form of excited free vibration. Since the tire's mass is neglected in the present paper, the tire spring is inactive during this stage and is represented in gray.