## 5 Rapid Calculation Model of Dual-Chamber Air Spring Stiffness and Damping
### Determination of Rapid Calculation Model of Stiffness and Damping  
From the above analysis, each factor has a nonlinear impact on the stiffness and damping of the dual-chamber air spring. The quadratic function can not accurately simulate the variation law of the stiffness and damping of the dual-chamber air spring. When the power of an independent variable is more than three, the regression function becomes very unstable [35], thus destabilizing the application of the regression model. In order to more accurately reflect the effect law of each factor on the stiffness and damping of the dual-chamber air spring, this paper uses cubic polynomials, which can usually be converted into ordinary multiple linear regression for processing. Experimental data are used to fit the expression of the stiffness and damping of the dual-chamber air spring. Suppose the stiffness experimental value meets the following cubic polynomials:  
\[ \begin{array}{l}f_{1}=\beta_{0}+\sum\limits_{j=1}^{34}\beta_{j \beta j}\left(\varphi_{1},\varphi_{2},\varphi_{3},\varphi_{4}\right)\\ =\beta_{0}+\beta_{1}\varphi_{1}+\beta_{2}\varphi_{2}+\beta_{3}\varphi_{3}+ \beta_{4}\varphi_{4}+\beta_{5}\varphi_{1}^{2}+\beta_{6}\varphi_{2}^{2}+\beta_ {7}\varphi_{3}^{2}+\beta_{8}\varphi_{4}^{2}+\beta_{9}\varphi_{1}\varphi_{2}+ \beta_{10}\varphi_{1}\varphi_{3}+\beta_{11}\varphi_{1}\varphi_{4}\\ +\beta_{12}\varphi_{2}\varphi_{3}+\beta_{13}\varphi_{2}\varphi_{4}+\beta_{14} \varphi_{3}\varphi_{4}+\beta_{15}\varphi_{1}^{4}+\beta_{16}\varphi_{2}^{2}+ \beta_{17}\varphi_{3}^{3}+\beta_{18}\varphi_{4}^{4}+\beta_{19}\varphi_{1}^{2} \varphi_{2}+\beta_{20}\varphi_{1}^{2}\varphi_{3}+\beta_{21}\varphi_{1}^{2} \varphi_{4}\\ +\beta_{22}\varphi_{1}\varphi_{2}^{2}+\beta_{23}\varphi_{1}\varphi_{3}^{2}+ \beta_{24}\varphi_{1}\varphi_{4}^{2}+\beta_{25}\varphi_{2}\varphi_{3}+\beta_{ 26}\varphi_{2}\varphi_{4}^{2}+\beta_{27}\varphi_{2}\varphi_{3}^{2}+\beta_{28} \varphi_{2}\varphi_{4}^{2}+\beta_{29}\varphi_{3}^{2}\varphi_{4}+\beta_{30} \varphi_{3}\varphi_{4}^{2}\\ +\beta_{31}\varphi_{1}\varphi_{2}\varphi_{3}+\beta_{32}\varphi_{1}\varphi_{2} \varphi_{4}+\beta_{33}\varphi_{1}\varphi_{3}\varphi_{4}+\beta_{34}\varphi_{2} \varphi_{3}\varphi_{4}+c_{i}(\sigma^{\delta})\end{array}  \tag{12}\]  
where \(\beta_{0}\), \(\beta_{1}\), \(\beta_{2}\),..., \(\beta_{34}\) are undetermined coefficients; \(\varphi_{1}\), \(\varphi_{2}\), \(\varphi_{3}\), \(\varphi_{4}\) are controllable variables in the experiment; \(g_{j}\left(\varphi_{1},\varphi_{2},\varphi_{3},\varphi_{4}\right)\) is function expression corresponding to \(\beta_{j}\); and \(f_{1}\) is the stiffness of dual-chamber air spring. This paper uses the least squares method to estimate the undetermined coefficients in the above formula to obtain the regression equation for stiffness:\[ \begin{array}{l}\frac{\hat{f}_{1}}{E13}=-0.0952+1.4139\varphi_{1}-0.6852\varphi_{2 }-1.3314\varphi_{3}+1.2366\varphi_{4}-0.9296\varphi_{1}^{2}+0.5123\varphi_{2}^{2} \\ +0.3438\varphi_{2}^{3}-0.7531\varphi_{4}^{2}-0.3542\varphi_{1}\varphi_{2}-0.797 5\varphi_{1}\varphi_{3}-1.5502\varphi_{1}\varphi_{4}+2.7841\varphi_{2}\varphi_ {3}\\ -0.5386\varphi_{3}^{2}+0.2573\varphi_{4}-0.1684\varphi_{3}^{2}-0.5071\varphi_{4} ^{3}+1.4815\varphi_{3}^{3}-0.2310\varphi_{4}^{3}\\ +0.8761\varphi_{1}^{2}\varphi_{2}+1.3092\varphi_{3}+0.1184\varphi_{1}^{2} \varphi_{4}-0.0996\varphi_{1}\varphi_{2}^{2}-0.3868\varphi_{1}\varphi_{3}^{2}+ 0.0888\varphi_{1}\varphi_{4}^{2}\\ +1.1500\varphi_{2}^{2}\varphi_{3}-0.6931\varphi_{2}^{2}\varphi_{4}-2.5784\varphi _{2}\varphi_{3}^{2}+0.1512\varphi_{2}\varphi_{4}^{2}-2.715\varphi_{1}^{3} \varphi_{4}+1.6950\varphi_{3}\varphi_{4}^{2}\\ -2.1074\varphi_{1}\varphi_{2}\varphi_{3}+1.1258\varphi_{1}\varphi_{2}\varphi_{4 }+1.3134\varphi_{1}\varphi_{3}\varphi_{4}+0.6595\varphi_{2}\varphi_{3}\varphi_ {4}\end{array}  \tag{13}\]  
Similarly, the damping regression equation can be acquired based on the stiffness calculation method:  
\[