# Study on the Dynamic Characteristics of a Hydraulic Continuous Variable Compression Ratio System
## 3 Mathematical Modeling
The flow continuity equation in the left cylinder is as follows:  
\[\mathrm{Q_{L}}=\left\{ \begin{array}{ll}\frac{\pi\mathrm{d}_{\mathrm{L}}^{2}}{4} \frac{\mathrm{ds}_{\mathrm{dL}}}{\mathrm{dt}}-\beta_{\mathrm{e}}\bigg{(} \mathrm{V_{L0}}-\frac{\pi\mathrm{d}_{\mathrm{L}}^{2}}{4}\mathrm{S_{dL}}\bigg{)} \frac{\mathrm{d}_{\mathrm{P}_{\mathrm{L}}}}{\mathrm{dt}}&\mathrm{the~{}left~{} \mathrm{globe~{}value~{}opened}}\\ 0&\mathrm{the~{}left~{}\mathrm{globe~{}value~{}closed}}\end{array} \right.\right. \tag{6}\]  
where \(\mathrm{\beta_{e}}\) is the elastic coefficient of the hydraulic medium and \(\mathrm{V_{L0}}\) is the initial volume of the left cylinder. The flow continuity equation in the right cylinder is as follows:  
\[\mathrm{Q_{R}}=\left\{ \begin{array}{ll}\frac{\pi\mathrm{d}_{\mathrm{R}}^{2}} {4}\frac{\mathrm{ds}_{\mathrm{dR}}}{\mathrm{dt}}+\beta_{\mathrm{e}}\bigg{(} \mathrm{V_{R0}}+\frac{\pi\mathrm{d}_{\mathrm{R}}^{2}}{4}\mathrm{S_{dR}} \bigg{)}\frac{\mathrm{d}_{\mathrm{P}_{\mathrm{R}}}}{\mathrm{dt}}&\mathrm{the~{} right~{}globe~{}value~{}opened}\\ 0&\mathrm{the~{}right~{}globe~{}value~{}closed}\end{array} \right. \tag{7}\]  
where \(\mathrm{V_{R0}}\) is the initial volume of the right cylinder. The flow continuity equation in the volume regulator is as follows:  
\[\mathrm{Q_{R}}=\left\{ \begin{array}{ll}\frac{\pi\mathrm{d}_{\mathrm{V}}^{2} }{4}\frac{\mathrm{ds}_{\mathrm{dV}}}{\mathrm{dt}}+\beta_{\mathrm{e}}\bigg{(} \mathrm{V_{V0}}+\frac{\pi\mathrm{d}_{\mathrm{V}}^{2}}{4}\mathrm{S_{dV}} \bigg{)}\frac{\mathrm{d}_{\mathrm{P}_{\mathrm{V}}}}{\mathrm{dt}}&\mathrm{the~{} solenoid~{}value~{}opened}\\ 0&\mathrm{the~{}solenoid~{}value~{}closed}\end{array} \right. \tag{8}\]  
where \(\mathrm{s_{dV}}\) is the piston displacement of the volume regulator, \(\mathrm{V_{V0}}\) is the initial volume of the volume regulator, \(\mathrm{d_{V}}\) is the diameter of the volume regulator piston, and \(\mathrm{P_{V}}\) is the operating pressure in the volume regulator. The equilibrium equation of the volume regulator piston is as follows:  
\[\frac{\pi\mathrm{d}_{\mathrm{V}}^{2}}{4}\mathrm{P_{V}}-\mathrm{m_{V}}\frac{ \mathrm{d}^{2}\mathrm{s_{dV}}}{\mathrm{dt}^{2}}-\mathrm{c}\frac{\mathrm{ds}_{ \mathrm{dV}}}{\mathrm{dt}}-\mathrm{F_{V0}}-\mathrm{k_{V0}}\mathrm{s_{dV}}=0 \tag{9}\]  
where \(\mathrm{F_{V0}}\) is pre-tightening force of volume regulator spring, and \(\mathrm{k_{V0}}\) is the stiffness of volume regulator spring. The flow continuity equation of the check valve is as follows:  
\[\mathrm{Q_{C}}=\left\{ \begin{array}{ll}\mathrm{A_{cd}}\sqrt{\frac{2\times \left(0.25-\mathrm{P_{L}}\right)}{\rho}},&\mathrm{P_{L}}<0.25~{}\mathrm{and~{} the~{}left~{}\mathrm{globe~{}value~{}opened}}\\ 0,&\mathrm{P_{L}}>0.25~{}\mathrm{or~{}the~{}left~{}\mathrm{globe~{}value~{} closed}}\end{array} \right.\right. \tag{10}\]  
where \(\mathrm{A_{cd}}\) is the effective flow area of the check valve and \(\rho\) is the density of the hydraulic medium. System flow balance equation:  
According to the working principle of VCR, the system flow balance equation can be divided into two cases. The first case is when the resultant force is upwards, and the flow of the left cylinder is equal to the total of the flow of the right cylinder and the flow of check valve:  
\[\mathrm{Q_{L}}=\mathrm{Q_{R}}+\mathrm{Q_{C}} \tag{11}\]  
Another case is when the resultant force is downwards, and the flow of the left cylinder is equal to the total of the flow of the right cylinder and the flow of the volume regulator:  
\[\mathrm{Q_{L}}=\mathrm{Q_{R}}+\mathrm{Q_{V}} \tag{12}\]