## 3 Mathematical Modeling
### Model of Pneumatic Bellows  
A pneumatic bellows is a linear pneumatic actuator consisting of a bellows type body and mounting flanges. The free length of the bellow is dependent on the difference between the ambient pressure and the pressure inside the bellows. From a physical point of view, the pneumatic bellows is a pneumatic spring with variable equilibrium length. The equilibrium length is dependent on the geometric and material properties of the bellows and the internal pressure within the bellows Figure 6.  
The bellows can be modelled using the standard mass, spring, damper model represented by Equation (3), see Figure 7. The dominant force on the system is the spring force. The spring force is dependent on the pneumatic spring stiffness and the equilibrium height of the bellows at the current internal pressure. Equation (4) shows this relationship. Without being pressurized, the bellow behaves like a spring whose stiffness depends on the shape of the bellow. The current material properties that are also dependent on other factors like ambient temperature. Therefore, if the bellows are deformed, a spring force appears in the direction opposing the deformation. The equilibrium length is the length of the bellows at which the deformation force from the internal pressure is at equilibrium with the spring force. It can be seen that, the equilibrium height is a nonlinear parameter that depends on multiple other coupled parameters. Therefore, instead of a physical modelling approach, the model for the equilibrium height was derived from experimental data by  
Figure 5: Transformation steps between frame \(x_{a}y_{a}z_{a}\) and \(x_{b}y_{b}z_{b}\)  
Figure 6: Bellow length in dependence on internal pressure. measuring the equilibrium height at different internal pressures; see Figure 8. The data was then approximated by a third-order polynomial function resulting in Equation (5). \[F_{M}+F_{b}+F_{k}(k_{p},P_{b})+F_{o}=0 \tag{3}\]  
\[F_{k}=k_{p}\left(z_{e}\left(F_{k},P_{b}\right)-z_{r}\right) \tag{4}\]  
\[z_{e}=0.45P_{b}^{3}+5.6P_{b}^{2}+23P_{b}+1200 \tag{5}\]  
where \(F_{M}\) is the inertial force, \(F_{b}\) the is damping force, \(F_{k}\) the is pneumatic spring force, \(F_{o}\) is outside the disturbance force, \(k_{p}\) is the pneumatic spring stiffness, \(P_{b}\) is internal the pressure, \(z_{e}\) is the equilibrium length, \(z_{r}\) is the actual length and \(F_{m}\) is the material spring force. To create a simulation model of a pneumatic spring, it is necessary to determine the stiffness of the spring. This parameter can be derived from the Equations (6)-(8). \[k_{p}=\frac{dF_{k}}{dz} \tag{6}\]  
where  
\[F_{k}=(P_{0}-P_{A})A \tag{7}\]  
Assuming that the change in bellows internal volume is polytropic we get  
Figure 7.0  
Figure 7.1 Relationship between internal pressure and Equilibrium height. Figure 7.2 Mass-spring-damper model of pneumatic bellow. Combining the above equations  
Figure 7.3  
where \(V\) is total volume of air within the bellows and corresponding pneumatic tube, \(P_{0}\) is the absolute pressure inside the bellows, \(A\) is an effective surface of the bellow, \(n\) is polytropic constant. The movement of the platform is expected to be slow, therefore we can approximate the process to be isothermal, hence \(n=1\). According to Equations (3)-(9) a simulation model in MATLAB version R2021b was developed. The results of this model were compared with experimental data, where the bellow was pressurized to different pressures, a positive extension force was applied to the bellow and the total extension was measured. The results are in Figure 7.4  
The model of a pneumatic bellow gives satisfactory results. The maximum deviation for pressures between 1 bar and 6 bar does not exceed 1 mm, while for pressure 0 bar the deviation is nearly 4 mm, which points to either a measurement error or to some unknown effect that is much less pronounced in higher pressures. Figure 7.5 Experiment vs. simulation (1 st and 3 rd row) and the difference between simulation and experiment (2 nd and 4 th row).