## 5 Conclusions  
The influence of support height, support pressure, and support stiffness of an FSS with variation in the ambient temperature and air content in hydraulic oil was studied. The mathematical models of the FSS and single support cylinder affected by the air content in the hydraulic oil and the ambient temperature were established, and the mathematical model was theoretically analyzed using MATLAB. From the simulation, it can be seen that: (i) The volume of the working medium increases as the temperature increases, and when the volume fraction of air in the working medium increases, the trend of its volume changes with temperature also shows an increase. (ii) Under different support pressures, the support height error of the FSS shows an increasing trend with the increase in ambient temperature and shows a decreasing trend with the increase in the air volume fraction in the working medium, and the change in support pressure has relatively little influence on support height. (iii) Under different support pressures, as the volume fraction of air content in the working medium and the ambient temperature increases, the pressure change in the upper and lower chambers of the FSS shows an increasing trend. When the support pressure of the FSS increases, the support pressure error of the upper and lower cylinders of the FSS is more sensitive to the influence of ambient temperature and air volume fraction. According to the above analysis results, the FSS and support cylinder were tested in two different cases where the air in the hydraulic oil was filtered and unfiltered. Through experimental analysis, it is concluded that the change trend of the support stiffness, support pressure, and support stiffness of the support cylinder is roughly the same as the theoretical values of the established mathematical model, and the validity of the established mathematical model is further verified within the allowable error range.  
This study adopts the method of mathematical model-simulation-experiment to study the non-linear influence relationship between the ambient temperature and the air content in the working medium. The parameters of the model are modified by simulation analysis and combined with experimental parameters to verify that the established mathematical model can be consistent with the actual working conditions. The established mathematical model provides a theoretical basis for the subsequent modal analysis of the FSS and provides an experimental basis for the design of the controller and the study of error compensation control strategies. At the same time, this method can be extended to studies on the error mechanisms of other nonlinear, time-varying, uncertain, and externally disturbed systems.  
Methodology, software and writing, Z.J.; formal analysis, resources and supervision, G.C.; data curation, S.X.; project administration, W.G. All authors have read and agreed to the published version of the manuscript.  
Financial support for this work, provided by the Priority Academic Program Development of Jiangsu Higher Education Institutions and the National Natural Science Foundation of China (Grant No. 91648105), are gratefully acknowledged.  
This study does not involve humans or animals.  
This study does not involve humans.  
**Data Availability Statement:** The simulation and experimental data used to support the findings of this study are available from the corresponding author upon request.  
**Conflicts of Interest:** The authors declare that there are no conflicts of interest regarding the publication of this paper.