# The Simulation of an Automotive Air Spring
## 2 The Air Spring Stiffness  
### The Displacement Field of the Toroidal Shell  
Pneumatic springs for automotive applications work mainly under compression forces. Conventionally, such forces and the respective axial displacements are considered positive. The pneumatic stiffness relates the spring response to the prescribed displacements along the axis. A numerical solution is developed to define the constitutive relation of the shell deformation versus the axially applied load. As the elastic element is compressed air, the pneumatic stiffness will depend mainly on thermodynamic factors, e.g., the deformation rate, given the quasi-adiabatic transformations expected. The deformed shape of the shell is obtained upon assuming membrane inextensibility along the shell principal curvature directions. Figure 3 sketches how the toroidal bellows behaves under the external load. Two modules of area are affected by the internal pressure in the spring deformation work: the toroidal bellows surface and the flat circular piston (marked as CD in Figure 3), which is eventually bolted to the vehicle suspension arms or frame body. Figure 2: Double cell bellows air spring. The following parameters characterize the shell geometry:  
* \(r\) is the meridian radius
* \(\theta\) is the meridian angle
* \(R\) is the longitudinal (or circumferential) radius
* \(w\) is the transverse (or normal) displacement
* \(v\) is the meridian (membrane) displacement. This displacement does not participate in the simplified model to calculate the work done by the internal pressure in the shell deformation. Considering \(w(\theta)\) as the relevant displacement to characterize the shell distortion, the incremental work \(\Delta U\) performed by the internal pressure against a prescribed displacement \(\Delta a\) at both flanged ends is evaluated by the variation of the shell volume. For this, the static moment of an elementary triangle defined by points {\(A\)\(A^{{}^{\prime}}\)\(O\)} with reference to the axis of revolution \(e\) must be calculated. Then, the volume of the shell is obtained with the theorem of [PERSON] (in the book of [PERSON] [6]) and integrating the following function for the meridian angular variable \(\theta\) in interval [0, \(\pi\)]:  
\[Vol=\underbrace{2\pi\int\limits_{0}^{\pi}\frac{1}{2}(r+w)^{2}\times\left(\frac {2}{3}r\sin\ \theta+R\right)d\theta}_{\text{\it Torus}}+\underbrace{\pi R^{2}\times 2 (r+w)}_{\text{\it Cylindrical core}} \tag{1}\]  
The static moment of the elementary triangle {\(A\)\(A^{{}^{\prime}}\)\(O\)} in Figure 3 is obtained from the product of its area by the distance of the mass centre (located at 2/3 of the triangle height) to the axis of  
Figure 3: Axisymmetric toroidal shell element under internal pressure: displacement field by axial loads. revolution \(e\). Volume increment is due only to transverse displacement \(w\) obtained after subtracting from Equation (1) a similar expression but with \(w\) = 0:  
\[\Delta Vol =\underbrace{2\pi\int\limits_{0}^{\pi}\left(wr+\frac{1}{2}w^{2} \right)\times\left(\frac{2}{3}r\sin\;\theta+R\right)d\theta}_{Torus}+ \underbrace{\frac{\pi R^{2}\times 2w}{\textit{Cylindrical core}}}_{\textit{Cylindrical core}} \tag{2}\]  
Considering that the toroidal membrane shell enclosing the compressed air has a circular meridian line and, after being subjected to external forces, the distorted shape of that meridian is approximately trigonometric, then, a simple approximate solution for the transverse and meridian displacements is as follows:  
\[w(\theta)=a\times\cos\;(2\theta) \tag{3}\] \[v(\theta)=\frac{-a}{2}\times\sin\;(2\theta)\]  
(Displacement \(v(\theta)\) in Equation (3) ensures the inextensibility along the meridian arch, but is not used in the formulation of the pneumatic stiffness). where \(a\) is a parameter associated with the amplitude of displacement \(w\) to be obtained by techniques of variation calculus.