# Study on the Dynamic Characteristics of a Hydraulic Continuous Variable Compression Ratio System
## 3 Mathematical Modeling  
According to the working principle of VCR, the eccentric piston pin suspension rotates around the center of the eccentric sleeve. As shown in Figure 4, the eccentric piston pin suspension, the left cylinder, and the right cylinder form two crank slider mechanisms by using the engine-connecting rod as the reference for motion. L0 is the eccentric size, L1 is the left arm length of eccentric piston pin suspension, L2 is the left link length, L3 is the right arm length of eccentric piston pin suspension, L4 is the right link length, \(\theta\) is the angular displacement of eccentric piston pin suspension, \(\alpha\) is the angular displacement of the left link, \(\beta\) is the angular displacement of the right link, \(\delta\) is the installation angle of the left cylinder, \(\varphi\) is the installation angle of the right cylinder, F is the force of the engine piston acting on the eccentric piston pin suspension, F\({}_{\rm L1}\) is the force of the left link acting on the eccentric piston pin suspension, F\({}_{\rm L2}\) is the force of the left link acting on the piston of left cylinder, F\({}_{\rm LP}\) is the force of oil pressure on the piston of the left cylinder, F\({}_{\rm FR1}\) is the force of the right link acting on the eccentric piston pin suspension, F\({}_{\rm R2}\) is the force of the right link acting on the piston of the right cylinder, and F\({}_{\rm RP}\) is the force of oil pressure on the piston of the right cylinder. Figure 3: Schematic diagram of the adjustment process. (**a**) First stage of adjustment; (**b**) second stage of adjustment; (**c**) adjustment completion. According to the working principle of the VCR system and Figure 4, the system equations are derived one by one, as follows. The piston displacement of the left cylinder \(\mathrm{s_{dL}}\) can be expressed as  
\[\mathrm{s_{dL}}=\sqrt{\left(\mathrm{L1 cos}\,\theta-\mathrm{L2 sin}\,\alpha \right)^{2}+\left(\mathrm{L1 sin}\,\theta+\mathrm{L2 cos}\,\alpha\right)^{2}} \tag{1}\]  
The piston displacement of the right cylinder \(\mathrm{s_{dR}}\) is expressed as  
\[\mathrm{s_{dR}}=\sqrt{\left(\mathrm{L3 cos}\,\theta-\mathrm{L4 sin}\,\beta \right)^{2}+\left(\mathrm{L3 sin}\,\theta+\mathrm{L4 cos}\,\beta\right)^{2}} \tag{2}\]  
The equilibrium equation of eccentric piston pin suspension can be expressed as  
\[\mathrm{F}\times\mathrm{L0 cos}\,\theta+\mathrm{F_{R1}}\times\mathrm{L3 cos}( \beta+\theta)-\mathrm{F_{L1}}\times\mathrm{L2 cos}(\alpha-\theta)=\mathrm{J} \frac{\mathrm{d}^{2}\theta}{\mathrm{d}t^{2}} \tag{3}\]  
where \(\mathrm{J}\) is the moment of inertia of eccentric piston pin suspension. The equilibrium equation of the left cylinder piston is expressed as  
\[\mathrm{F_{L2}}\sin(\alpha-\delta)-\mathrm{F_{LP}}-\mathrm{m_{LP}}\frac{ \mathrm{d}^{2}\mathrm{s_{dL}}}{\mathrm{d}t^{2}}-\mathrm{c}\frac{\mathrm{d} \mathrm{s_{dL}}}{\mathrm{d}t}=0 \tag{4}\]  
where \(\mathrm{c}\) is the damping coefficient, \(\mathrm{m_{LP}}\) is the mass of the left cylinder piston, \(\delta\) is the installation angle of the left cylinder, \(\mathrm{F_{L2}}=-\mathrm{F_{L1}}\), \(\mathrm{F_{LP}}=\frac{\pi\mathrm{d}_{\mathrm{L}}^{2}}{4}\mathrm{P_{L}}\), \(\mathrm{d_{L}}\) is the diameter of the left cylinder piston, and \(\mathrm{p_{L}}\) is the operating pressure of the left cylinder. The equilibrium equation of the right cylinder piston is expressed as  
\[\mathrm{F_{R2}}\sin(\beta-\varphi)-\mathrm{F_{RP}}-\mathrm{m_{RP}}\frac{ \mathrm{d}^{2}\mathrm{s_{dR}}}{\mathrm{d}t^{2}}-\mathrm{c}\frac{\mathrm{d} \mathrm{s_{dR}}}{\mathrm{d}t}=0 \tag{5}\]  
where \(\mathrm{c}\) is the damping coefficient, \(\mathrm{m_{RP}}\) is the mass of the right cylinder piston, \(\varphi\) is the installation angle of the left cylinder, \(\mathrm{F_{R2}}=-\mathrm{F_{R1}}\), \(\mathrm{F_{R2}}=\frac{\pi\mathrm{d}_{\mathrm{R}}^{2}}{4}\mathrm{P_{R}}\), \(\mathrm{d_{R}}\) is the diameter of the right cylinder piston, and \(\mathrm{p_{R}}\) is the operating pressure of the left cylinder. Figure 4: Schematic diagram of force.