## 4 AirSpring Equation  
Bellows actuators are employed as actuating elements in pneumatic drive systems for machinery and equipment, as well as in the form of pneumatic springs (cushions) for the vibration isolation of machines, the cushioning of vehicles, and other applications. In pneumatic drives, bellows actuators operate as single-acting actuators, commonly referred to as push cylinders. The operation of these actuators is based on the principles of filling (supplying pressure) and discharging (venting). From the perspective of vibration isolation,  
Figure 12: The mean effective acceleration \(\ddot{y}_{f1 RMS}\) and mean maximum relative displacement \(\left(y_{f1}-y_{f0}\right)_{\text{max}}\) as a function of the mass of the inertial system under test (or the human—seat system). the action of the bellows actuator when filled with a constant pressure is of significance. In this state, it functions as an air spring. Figure 13 illustrates a schematic of the air spring vibration isolation system. The system comprises a mass (1) which loads the system (3), causing the mass to vibrate vertically. The internal volume of the air spring (2) changes in response to the oscillation of the concentrated mass \(m\), resulting in a change in pressure within the air spring (\(p\)). The equation of motion according to Figure 13 for the mass-stiffness system of the bellows can be written as follows:  
\[m\ddot{h}+k_{s}h=F_{s} \tag{7}\]  
where \(h\) is the bellows deflection, \(m\) is the bellows loading mass, \(k_{s}\) is the bellow stiffness and \(F_{s}\) is the external force on the bellows. The stiffness of a pneumatic bellows \(k_{s}\) is defined as the ratio of the load \(m\) to the deflection \(h\) of the bellows. This is expressed as \(k_{s}=dF_{s}/dh\). The relationship between the bellows' deflection \(h\) and pressure \(p\) is defined as follows:  
\[k_{s}=\frac{dF_{s}}{dh}=\frac{d(pA_{e})}{dh}=p\frac{dA_{e}}{dh}+A_{e}\frac{dp }{dh} \tag{8}\]  
where \(p\) is the pressure inside the bellows and \(A_{e}\) is the effective surface area of the pneumatic bellows. To ascertain the derivative of pressure concerning height, and assuming an adiabatic transformation \(PV^{\kappa}=const\), the change in pressure as a function of bellows deflection is given by:  
\[\frac{d(pV^{\kappa})}{dh}=\kappa\ pV^{\kappa-1}\frac{dV}{dh}+V^{\kappa}\frac{ dp}{dh}=0 \tag{9}\]  
From the relationships shown in Equation (9), the change in pressure as a function of the deflection of the pneumatic bellows is apparent. \[\frac{dp}{dh}=\frac{\kappa\ pV^{\kappa-1}}{V^{\kappa}}\frac{dV}{dh}=-\frac{ \kappa}{V}\frac{p}{dh} \tag{10}\]  
Figure 13: Scheme of air spring in suspension seats. As the pressure \(p\) increases, the height of the spring also increases, accompanied by an increase in volume \(V\). Upon consideration of the relationship \(dV/dh\approx A_{e}\) and Equation (9), the following equation is derived:  
\[\frac{dp}{dh}=-A_{e}\frac{\kappa\ p}{V} \tag{11}\]  
As a result, the stiffness of the pneumatic bellows \(k_{s}\) as a function of its deflection \(h\) and gas volume \(C_{g}\) and pressure is plotted as follows:  
\[k_{s}(h,p)=p\frac{dA_{e}}{dh}+A_{e}^{2}\frac{\kappa\ p}{V}=p\frac{dA_{e}}{dh}+ \frac{A_{e}^{2}}{C_{g}} \tag{12}\]  
where \(C_{g}\)--the gas capacity of the bellows, \(C_{g}=V/K_{g}\), \(K_{g}\)--the bulk modulus of elasticity of air, \(K_{g}=xp\), and \(\kappa=1.4\)--the adiabatic exponent. The initial component of the equation represents the alteration in the effective area of the pneumatic spring. Given the configuration of the bellows actuator (Figure 2), a constant effective area, \(A_{e}\), was postulated.