## 2. Mathematica Models and Load-Sharing Criteria  
###.1. Mathematic Models of the Tri-Axle Semi-Trailer and Road Roughness Excitation  
A basic half-model representing a typical tri-axle semi-trailer with longitudinal-connected air suspension in most western countries was employed, as shown in Figure 1. This model includes 5-DOF, which are vertical displacement of sprung mass and three unsprung masses, \(z\), \(x_{1}\), \(x_{2}\), \(x_{3}\), as well as the pitch angle of the sprung mass \(\varphi\). The equations of motion of the semi-trailer are given by  
\[m_{t1}\ \ddot{x}_{1}=(q_{1}-x_{1})k_{t1}+c_{1}(\dot{z}-\dot{x}_{1}-\dot{ \varphi}l)-(P_{s1}-P_{0})A_{s1}+\frac{1}{3}mg \tag{1}\]  
\[m_{t2}\ddot{x}_{2}=(q_{2}-x_{2})k_{t2}+c_{2}(\dot{z}-\dot{x}_{2})-(P_{s2}-P_{0 })A_{s2}+\frac{1}{3}mg \tag{2}\]  
\[m_{t3}\ddot{x}_{3}=(q_{3}-x_{3})k_{t3}+c_{3}(\dot{z}-\dot{x}_{3}+\dot{\varphi} l)-(P_{s3}-P_{0})A_{s3}+\frac{1}{3}mg \tag{3}\]  
\[\dot{J}\ddot{\varphi}= [(P_{s3}\ -P_{0})A_{s3}-c_{3}(\dot{z}-\dot{x}_{3}+\dot{\varphi}l)+ \frac{1}{3}mg]l \tag{4}\] \[-[(P_{s1}-P_{0})A_{s1}-c_{1}(\dot{z}-\dot{x}_{1}-\dot{\varphi}l)- \frac{1}{3}mg]l\] \[m\ddot{z}= (P_{s1}\ -P_{0})A_{s1}+(P_{s2}-P_{0})A_{s2}+(P_{s3}-P_{0})A_{s3}\] (5) \[-c_{1}(\dot{z}-\dot{x}_{1}-\dot{\varphi}l)-c_{2}(\dot{z}-\dot{x}_ {2})-c_{3}(\dot{z}-\dot{x}_{3}+\dot{\varphi}l)-mg\]  
where \(m_{t1}\), \(m_{t2}\), \(m_{t3}\) and \(q_{1}\), \(q_{2}\), \(q_{3}\) are the unsprung masses and road excitations of the three axles, respectively; \(m\) is the sprung mass of the semi-trailer; \(J\) is the moment of inertia of the gross sprung mass around the lateral axis; \(P_{s1}\), \(P_{s2}\), \(P_{s3}\) and \(A_{s1}\), \(A_{s2}\), \(A_{s3}\) are the dynamic absolute air pressures and dynamic effective areas of the three air springs, respectively; \(P_{0}\) is atmospheric pressure; \(l\) is the wheelbase; \(c_{1}\), \(c_{2}\), \(c_{3}\) are the damping coefficients of three dampers; and \(k_{t1}\), \(k_{t2}\), \(k_{t3}\) are the stiffness of the three tires. In order to solve the above equations, a detailed model of longitudinal-connected tri-axle air suspension is needed to express \(P_{s1}\), \(P_{s2}\), and \(P_{s3}\) as functions of the 5 variables (5 DOF). Specifically,  
Figure 1. Schematic of the tri-axle semi-trailer with longitudinal-connected air suspension. an adiabatic process is assumed for the air inside air springs when the vehicle is travelling, and the air flow inside the connectors is considered to be an incompressible steady flow [15; 30]. This model correlated well with the measurements. A concise spectral model is used in this study to represent the road roughness excitations. This vehicle-road integrated model takes the nonlinearity of the air transmission between the air springs into account, and thus correlates well with the experimental results of field tests. The details of the models and their experimental validation can be found in the previous work of the authors [16]. To avoid repetition, they are not described in this paper. Parameters of the longitudinal-connected tri-axle semi-trailer model are tabulated in Table 1.  
The prototype of the tri-axle semi-trailer with Kenworth Airglide 200 longitudinal-connected air suspension was tested on three typical urban road sections at speeds ranging from 60 km/h to 80 km/h [16]. Strain gauges (one per hub) were mounted on the neutral axis of each axle between the spring and the hub to record the shear force on the hubs, i.e., air spring force, and accelerometers were mounted as closely as possible to each hub and to the corresponding upper positions at the chassis to derive the height of each air spring. In addition, six air pressure transducers were employed to obtain the pressures inside the air springs. Based on the experimental data, the effective area of each air spring was obtained by dividing the respective shear force by the respective pressure inside the air spring, and the volume of each air spring was derived by multiplying the respective effective area by the respective spring height.