# The Simulation of an Automotive Air Spring
## 3 Pseudo-Dynamic Simulation of the Automotive Suspension  
### Applications to the Dynamic Analysis of Automotive Suspensions  
Automotive suspensions attenuate the vibration of the vehicle body due to road irregularities on run. The dynamics of conventional automotive suspensions refer to a passive transmission model, where the base (road) prescribes a time-dependent displacement and velocity to the equivalent system. This set, comprising tyre, spring, and damper, is the vehicle wheel suspension model having its displacement velocity and acceleration vectors as kinematic parameters. Figure 6 shows a sketch of a suspension with the spring/damper set, but here, the elastic element is a bellows-type air spring in an equivalent dynamic system as a single degree of freedom (SDOF) model. The damper is a device that restricts the suspension amplitude on run with a force that is velocity dependent, as explained below. Figure 5: Internal reaction _vs_ axial displacement (in compression) of the pneumatic spring with isothermal and adiabatic (\(\gamma\) = 1.3) processes. Initial internal pressure \(p_{0}\) = 2 bar (displacements indicated are half of each prescribed spring stroke for symmetry). In this simplification, the equivalent mass is a part of the total vehicle mass depending on its distribution by the axles. For each wheel, the dynamic model of the automotive suspension can be represented with good accuracy as a SDOF system, provided that the displacement field can be assigned to a dominant value at a defined point. This is the case of the wheel hub, considering the tire stiffness is much larger than the suspension spring, practically transmitting the entire displacement from the road profile to the wheel hub and to the spring end. As indicated in Figure 6b, the base (road) prescribes a displacement \(U_{0}(t)\) to the wheel axle (assuming here that the elasticity of the tire is much larger than the spring one), while the equivalent mass \(M\) of the vehicle body is displaced \(U_{1}(t)\). The dynamic equilibrium of the spring/mass model of Figure 6b is:  
\[F_{spring}(t)+F_{damper}(t)=M\ddot{U}_{1} \tag{18}\]  
where \(F_{spring}\) and \(F_{damper}\) are the spring and the damper reaction forces due to the relative displacement and velocity between the road profile and the wheel hub. Supposing that the spring has a linear elastic behavior and the damper reaction force is also linear-dependent on the relative velocity, then Equation (18) can be written as:  
\[K_{t}(U_{0}-U_{1})+C(\dot{U}_{0}-\dot{U}_{1})=M\ddot{U}_{1} \tag{19}\]  
in this equation, \(K_{t}\) is the spring stiffness (in N/m), and \(C\) is the damping constant (in Ns/m), which is a parameter assigned to viscous-type dampers. Constant \(C\) has a stable value when the damper is in good condition and is operated at a stabilized temperature. In the present study, a constant value of \(C\) is assumed. Separating in each hand of Equation (19) the force terms of the displacement and velocity of the base and wheel hub, respectively, the passive transmission \(SDOF\) equation of the dynamic system is obtained:  
\[K_{t}U_{0}+C\dot{U}_{0}=F_{equiv}(t)=K_{t}U_{1}+C\dot{U}_{1}+M\ddot{U}_{1} \tag{20}\]  
The left-hand side of Equation (20) can be interpreted as an external time-dependent equivalent force \(F_{equiv}(t)\) on assuming a virtual profile for a road where the wheel/suspension set travels at a prescribed horizontal velocity. This equation must be integrated numerically with a time integration algorithm as explained below, but for now, it is important to note that the following term on the  
Figure 6: Sketch of a vehicle suspension and the equivalent dynamic model for a case of passive transmission: (**a**) drawing; and (**b**) symbology.