# The Simulation of an Automotive Air Spring
## 3 Pseudo-Dynamic Simulation of the Automotive Suspension
### Description of the Method and Application to the Dynamic Analysis
#### 3.3.3 Testing a Bellows Air Spring under Dynamic Load Using a Pseudo-Dynamic Procedure  
The present work consisted of simulating a suspension with a virtual mass and external force to test a pneumatic bellows using a pseudo-dynamic simulation. The term \"virtual\" is used as the external force, the damper (which is non-existent in this case) and the mass are not real dynamic entities participating in the test. The aim of this procedure is mainly to demonstrate that a pseudo-dynamic process based on a sequential procedure provides results that are in good agreement with the theoretical solutions of structural dynamics problems. The experiment consisted of subjecting the bellows to a sequence of displacements in compression in a SHIMADZU-AUTOGRAPH(r) (from Kyoto, Japan) test machine (Figure 10) with 50 KN of capacity force. For safety reasons and to avoid exceeding the machine capacity, the spring displacement was restricted to an amplitude of 20 mm. Figure 10: Mounting of the bellows pneumatic accessory in a SHIMADZU® testing machine between the working table and upper crosshead with an adaptor ram and a pressure air control unit. Figure 9: Acceleration at the wheel of a vehicle tire and suspension set hitting the soil suddenly from a small height. Points were collected from [PERSON] [13] having shifted them of about 0.1 s so that the starting point could match approximately the corresponding results here evaluated. Given the likely sharp variation of the external loads acting in the structure of vehicle suspensions, it is plausible that the enclosed air spring undergoes adiabatic compressions. As the test is carried out at a very low speed, the internal air pressure has to be adjusted according to the thermodynamic law for a real (non-perfect) gas, at each time step. A pressure control unit was fitted in the pressure feed line to the bellows chamber (visible in Figure 10) to manually control the internal pressure at each time step of the test after having computed the internal volume at each step; naturally, the specimen is perfectly held in position during the pressure adjust. In a first step, the example was entirely solved with [PERSON]'s method to illustrate the main differences observed in relation to the linear systems in dynamic analysis. Only for testing the solution, an isothermal behavior was assumed, despite the dynamic characteristics of the problem. The pneumatic spring has a toroidal surround bellows and has the same overall dimensions and pneumatic parameters as the item analyzed in Section 2:  
* Torus radius r = 50 mm
* Radius of circle passing through the transverse sections radius: \(R\) = 200 mm
* Initial pressure (preload pressure) \(P_{0}\) = 2 bar (\(\cong\)0.2 MPa)
* External force \(F\)(t) = 10 KN (suddenly applied as a step load)
* Virtual mass of the SDOF system \(M\) = 50 Kg  
The dynamic loading of the spring was approached with time steps \(\Delta t\) = 0.001 s (recommended time step less than \(\Delta t_{\text{max}}\) = 0.0066 s). The spring stiffness ranges from about \(K_{min}\) = 191280 N/m (first time step) up to \(K_{max}\) = 473911 N/m (time step for maximum displacement). It is noted that the spring stiffness here calculated refers to the effect of the axial displacement (equal to the double of the ovalization amplitude \(a\)) on the internal reaction force due to the air pressure as mentioned above:  
\[K_{mean}=332595\text{ N/m}.\]  
For a suddenly applied load F = 10 KN, the dynamic spring stroke is:  
\(U_{max}\) = 2 \(\times\)\(U_{static}\) = 2 \(\times\) (10,000/332,595) = 0.0601 m, which is a result close to the amplitude vibration calculated with the Newmark algorithm as observed in the graphical results of Figure 11. The results show the accuracy of the method, where the amplitude of the vibrating system is double that of the static deflection.