# The Simulation of an Automotive Air Spring
## 2 The Air Spring Stiffness
### The Pneumatic Stiffness
A complete expression for the pneumatic stiffness can be obtained from the expression:  
\[K_{a}=\pi P\bigg{(}\frac{28r}{90}+\frac{R\pi}{4}\bigg{)}+\frac{\gamma P_{0}V_{ 0}^{\gamma}}{V^{\gamma+1}}\times\bigg{\{}\pi\bigg{[}-\frac{4}{9}r^{2}+2a\bigg{(} \frac{28r}{90}+\frac{R\pi}{4}\bigg{)}\bigg{]}+\pi R^{2}\bigg{\}}^{2} \tag{12}\]  
which depends on a current value of the axial displacement \(a\) of the bellows spring. For each displacement increment \(\Delta a\) of the spring, the current force \(F_{a}\) in Equation (7) changes accordingly with a constant value of the stiffness in Equation (12):  
\[F_{a+\Delta a}=F_{a}+K_{a}(P)\Delta a \tag{13}\]  
Although \(K_{a}\) depends on the air pressure, it is assumed to be constant during the displacement increment referred above. Equation (13) is thus a linearized form to update the structure configuration past an iteration \(\Delta a\). The next section describes the linearization process of the iterative algorithm to obtain the solution for the structural deformation. In step (b), the deformation of the torus shell is calculated to be added to the pneumatic stiffness that was previously obtained. This shell is assumed to have very low bending stiffness, while being practically inextensible under membrane forces (as expected with a steel braid-reinforced rubber shell, where only the membrane stiffness is relevant). Figure 3 is used to develop the upcoming calculation. Since the relevant displacements participating in the deformation field of the torus shell were above identified to be \(w\) and \(v\), the deformations for the axisymmetric loads are:  
\[ \begin{split}\varepsilon_{s}&=\frac{1}{(r\sin\ \theta+R)}(w\sin\ \theta+v\cos\ \theta);\\ \varepsilon_{\theta}&=\frac{1}{r}\bigg{(}w+\frac{ \partial v}{\partial\theta}\bigg{)}=0\ (\text{for meridian inextensibility})\\ k_{\theta}&=\underbrace{\frac{1}{r^{2}}\bigg{(} \frac{\partial^{2}w}{\partial\theta^{2}}-\frac{\partial v}{\partial\theta} \bigg{)}}_{Linear}+\underbrace{\frac{1}{2r^{2}}\bigg{(}\frac{\partial w}{ \partial\theta}-v\bigg{)}^{2}}_{Non-Linear}\end{split}  \tag{14}\]  
The first two terms refer to the linear small displacement deformation models, respectively the longitudinal and the meridian deformations (Ugural [7]), while the third term is the meridian curvature involving a linear term (for small bending displacements) and an additional quadratic term for moderately large bending displacements ([PERSON] [8]). The non-linearity term only in the curvature expression derives of the low bending stiffness of the shell, allowing expectable large transverse displacements, while other deformation terms are small. The corresponding [PERSON]'s law for the strain tensor in Equation (14) only involves the longitudinal membrane force and the bending moment as follows:  
\[\left\{ \begin{array}{c}N_{s}\\ M_{\theta}\end{array} \right\}=[D]\{\varepsilon\}=\frac{Eh}{(1-v^{2})}\left[ \begin{array}{cc}1&0\\ 0&\frac{h^{2}}{12}\end{array} \right]\left\{ \begin{array}{c}\varepsilon_{s} \\ k_{\theta}\end{array} \right\} \tag{15}\]  
where \([D]\) is the elasticity matrix containing the [PERSON]'s modulus \(E\) and the Poisson's ratio \(v\) of the material, and \(\{\varepsilon\}\) is the strain vector. The strain-energy resulting from the axisymmetric loading of the torus is given by:  
\[U_{shell}=\frac{1}{2}\!\!\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\{ \varepsilon\}^{T}[D]\{\varepsilon\}r(R+r\sin\ \theta)d\theta d\varphi \tag{16}\]  
where \(\theta\) is the meridian angle and \(\phi\) is the longitudinal angle, which are both parameters in the integration over the shell main curvatures. According with the [PERSON]'s theorem, the internal reaction \(R_{shell}\) and associated tangent stiffness \(K_{T-shell}\) due to a displacement of magnitude \(a\) (the only unknown in Equations (3) and (4)) of the axisymmetric shell are:  
\[