Stiffness model for pneumatic spring with air-diaphragm coupling effect

An accurate pneumatic spring stiffness model is essential for achieving subhertz, quasi-zero stiffness vibration isolation in optical experiments, precision measurements, and semiconductor manufacturing involving heavy equipment. Conventional stiffness models for pneumatic springs with diaphragms often neglect the effect of the elastic diaphragm, making it difficult to accurately characterise pneumatic spring stiffness. This paper presents an innovative pneumatic spring stiffness model that incorporates the air-diaphragm coupling effect—a phenomenon in which the air and diaphragm interact synergistically to affect stiffness. The inclusion of this coupling effect alters the effective area and volume of the pneumatic spring, two critical parameters influencing stiffness. Detailed mechanical and geometrical derivations are provided to establish an accurate stiffness model. A modification coefficient is introduced to quantify the air-diaphragm coupling effect on stiffness, and various factors influencing this coefficient, as well as their impact on stiffness, are examined. An experiment was conducted to validate the proposed model, showing a relative error of less than 1.65 % between the experimental and theoretical results. Considering the air-diaphragm coupling effect, the absolute error approximation was reduced by an order of magnitude, and the relative stiffness decreased by 11.05 % and 8.26 % in Experiments 1 and 2, respectively. Owing to its high precision, the proposed model provides theoretical guidance for the engineering design of pneumatic springs and facilitates the accurate matching of quasi-zero stiffness vibration isolation systems for heavy equipment in ultra-precision applications.