Analytical approach for the design of convoluted air suspension and experimental validation

Abstract

An improved analytical design to investigate the static stiffness of a convoluted air spring is developed and presented in this article. An air spring provides improved ride comfort by achieving variable volume at various strokes of the suspension. An analytical relation is derived to calculate the volume and the rate of change in the volume of the convoluted bellow with respect to various suspension heights. This expression is used in the equation to calculate the variable stiffness of the bellow. The obtained analytical characteristics are validated with a detailed experiment to test the static vertical stiffness of the air spring. The convoluted air bellow is tested in an Avery spring-testing apparatus for various loads. The bellow is modeled in the ABAQUS environment to perform finite element analysis (FEA) to understand and visualize the deflection of the bellow at various elevated internal pressures and external loads. The proposed air spring model is a fiber-reinforced rubber bellow enclosed between two metal plates. The Mooney–Rivlin material model was used to model the hyperelastic rubber material for FEA. From the results, it is observed that the experimental and analytical results match with a minor error of 7.54%. The derived relations and validations would provide design guidance at the developmental stage of air bellows. These expressions would also play a major role in designing an effective active air suspension system by accurately calculating the required stiffness at various loads.

Appendices

Appendix 1: abbreviations

$\dot{m}$ :

Mass flow rate

$\rho$ :

Air density

$V$ :

Air volume

$P$ :

Air pressure

$n$ :

Number of moles of gas

$T$ :

Temperature

$h$ :

Height of bellow at a particular instance

$\vartheta$ :

Change in volume with height

$F$ :

Force acting on bellow

$A_{\text{eff}}$ :

Effective area

$K_{s}$ :

Bellow stiffness

$r_{r}$ :

Radius of bellow

$r_{s}$ :

Sagitta of circular segment

$r_{c}$ :

Bellow mouth radius

$L_{a}$ :

Arc length of circular segment

$R$ :

Arc radius of the segment

$W$ :

Strain energy density function

Appendix 2: Taylor’s series expansion of Eq. (3)

Applying Taylor’s series to Eq. (3) to obtain a linearized equation, we obtain:

$$\begin{aligned} \dot{P}_{s} &= \dot{P}_{se} + \frac{{{\text{d}}\dot{P}_{s} }}{{{\text{d}}P_{s} }}|_{e} \left( {P_{s} - P_{se} } \right) + \frac{{{\text{d}}\dot{P}_{s} }}{{{\text{d}}V_{s} }}|_{e} \left( {V_{s} - V_{se} } \right)\\ &\quad + \frac{{{\text{d}}\dot{P}_{s} }}{{{\text{d}}\dot{V}_{s} }}|_{e} \left( {\dot{V}_{s} - \dot{V}_{se} } \right) + \frac{{{\text{d}}\dot{P}_{s} }}{{{\text{d}}\dot{m}}}|_{e} \left( {\dot{m} - \dot{m}_{e} } \right).\end{aligned} $$

(A1)

The subscript e indicates evaluation at equilibrium conditions where

$$ P_{se} = \dot{m}_{e} = \dot{V}_{se} = 0. $$

(A2)