# The Simulation of an Automotive Air Spring
## 3 Pseudo-Dynamic Simulation of the Automotive Suspension
### Pseudo-Dynamic Test of a Real Pneumatic Spring: Procedures and Results  
In this step, a pseudo-dynamic test of a commercial pneumatic spring, from _Numatics\({}^{\circledR}\)_(Emerson Industrial Automation, www.numatics.com), is performed with the Shimadzu test machine, as shown in Figure 10. The double bellows tangent stiffness is calculated from the ratio of an incremental internal reaction \(R_{t}\) (measured with the load cell of the test machine) by the respective incremental relative displacement of both bellows flanges at each time step. The main problem data for the dynamic procedure were:  
* A pneumatic double-bellows type spring (_Numatics-Emerson\({}^{\circledR}\)_) having a rigid end flanges diameter of 160 mm. Internal pressure is \(p_{0}\) = 4 bar;
* A virtual mass \(M\) = 100 Kg (no damping considered);
* An impulsive step force \(F\)(t) = 1500 N sustained after instant \(t\) = 0\({}^{+}\) s;
* No damping was considered in the dynamic model.  
An estimation of this bellows stiffness gave an average value \(K_{mean}\) = 142,000 N/m. For the virtual mass \(M\) = 100 Kg, the natural period is \(T_{0}\) = 0.166 s.  
The test was carried out by changing the internal air pressure according to the adiabatic equation: \(P_{t}V_{t}^{\gamma}=P_{0}V_{0}^{\gamma}\), where \(\gamma\) is the adiabatic exponent, here assumed to be about 1.3 (an adequate value for real gases) and an initial internal volume of the accessory \(V_{0}\cong 6.2\,l\) (0.0062 m\({}^{3}\)). The dynamic response for this _SDOF_ system was obtained with a total Newmark's non-linear solution, where the spring tangent stiffness was calculated at each time step upon changing the internal pressure according to the adiabatic equation.  
The same problem was simulated with a pseudo-dynamic procedure, where the system stiffness at each time step was calculated by the rate of the spring internal force (measured with the machine load cell) versus each displacement increment obtained with precision linear voltage displacement transducers (LVDTs with a resolution of up to 5 \(\upmu\)m) fitted in the machine. As the equipment was not automated for this pseudo-dynamic procedure, the air pressure was manually adjusted in the compressed air distribution unit after obtaining the internal volume for each axial displacement of the spring (available in technical information by the manufacturer), and then substituting its value in the adiabatic equation that was solved for the internal pressure at the spring displacement at time step \(t\). Results for the pseudo-dynamic procedure are depicted in Figure 13, where a good agreement of both methods is observed, despite the slight mismatch among the natural periods estimated with the methods.  
Figure 13: Results for the vibration of a single degree of freedom (SDOF) system with the pneumatic spring in analysis: (**a**) with [PERSON]’s method (natural period is \(\cong\)0.180 s); and (**b**) with a pseudo-dynamic simulation (natural period is \(\cong\)0.175 s).  
The peak axial displacement obtained with the pseudo-dynamic procedure at maximum reaction force (3000 N) was about 18 mm, while a simple calculation for a linear system (with spring stiffness \(K_{mann}=\) 142,000 N/m) showed that 21 mm was the maximum value. Although acceptably close, repeated cycles (with the pseudo-dynamic procedure) revealed a progressive degradation of the agreement between the experimental and theoretical results.