## 3 The Mathematical Model of VAPP
For a VAPP with an odd number of pistons, if the conditions \(0\leq\phi_{i}\leq\frac{\pi}{z}\), \(k=\frac{z+1}{2}\) and \(\frac{\pi}{z}\leq\phi_{i}\leq\frac{2\pi}{z},k=\frac{z-1}{2}\) are met, then the instantaneous output flow \(Q_{p}\) of VAPP is given by  
\[Q_{p}=\frac{\pi d_{k}^{2}R\omega x_{p}(L_{d}+x_{p}\tan\gamma)}{4\cos\gamma L_ {d}}\sum_{i=1}^{k}\frac{\sin\phi_{i}}{(L_{d}-x_{p}\cos\phi_{i}\tan\gamma)} \tag{4}\]  
For the instantaneous cylinder pressure, \(p_{ki}\) with respect to the angular position of the cylinder block, \(\phi\), with \(\phi=\omega\cdot t\) and \(d\phi=\omega\cdot dt\), is given by Reference [12; 13]:  
\[\frac{dp_{ki}}{d\phi}=-\frac{E\Big{(}Q_{pi}-sign(p_{ki}-p_{di})C_{d}A_{op} \sqrt{\frac{2}{\rho}|p_{ki}-p_{di}|}\Big{)}}{\omega\Big{(}V_{o}-\frac{R\pi d_{ k}^{2}\tan\beta}{4}\Big{(}\frac{1}{\cos\gamma+\tan\beta\sin\gamma}-\frac{\cos \phi_{i}}{\cos\gamma-\cos\phi_{i}\tan\beta\sin\gamma}\Big{)}\Big{)}} \tag{5}\]  
where \(p_{di}\) is the pump delivery pressure, \(p_{ki}\) is the instantaneous pressure of \(i\)th piston chamber, \(\rho\) is the density of oil, \(C_{d}\) is the flow discharge coefficient, \(A_{op}\) is the instantaneous discharge area of \(i\)th piston, \(V_{o}\) is the nominal volume of a piston chamber when the swash plate angle a is zero, and \(E\) is the oil bulk modulus which is affected by pressure and temperature [14]. When swash plate angle \(\beta\) was changed, the angular acceleration of the slipper relative to the cylinder is expressed as  
Figure 3: The control relation diagram of the variable mechanism. \[ \begin{split} a_{s}=\omega^{2}Rg(\frac{2\pi}{z})\cos(\phi_{1}+\frac{2 \pi i}{z})+\frac{2\omega R\sin(\phi_{1}+\frac{2\pi i}{z})}{L_{d}}\frac{dx_{p}}{ dt}+\\ R[1-\cos(\phi_{1}+\frac{2\pi i}{z})]\{\frac{(L_{d}^{2}+x_{p}^{2})^{2 }+2x_{p}(\frac{dx_{p}}{dt})^{2}}{L_{d}(L_{d}^{2}+x_{p}^{2})}+2x_{p}L_{d}(\frac {dx_{p}}{dt})^{2}\}\end{split}  \tag{6}\]  
The variable resistance torque, \(M_{d}\), needs to be overwhelmed for the symmetric hydraulic control cylinder to control the swash plate angle. Variable resistance torque \(M_{d}\)-mainly includes torque \(M_{h}\)-acting on the swash plate by the hydraulic pressure through the piston, \(M_{bf}\)-friction resistance torque of spherical hinge, \(M_{bf}\)-resistance torque (inertia force) of slipper component acting on the swash plate, \(M_{sf}\)-sliding friction resistance torque of the swash plate, and \(M_{sf}\)-inertia resistance torque of the swash plate itself. When \(M_{sf}\) is ignored, the variable resistance torque is expressed as  
\[M_{\rm d}=\left(\pm\frac{z}{i}\frac{\pi d_{k}^{2}Rp_{d}}{4\cos^{2}\beta}\cos( \phi+ia)\right)-\left(\frac{a_{s}m_{ps}R}{\cos^{2}\beta}\frac{z-1}{i=1}\cos( \phi+ia)\right)+\frac{\pi d_{k}^{2}xp_{d}(Rf_{1}+rf_{2})}{8\cos\beta} \tag{7}\]  
where \(p_{d}\) is the pump delivery pressure, \(\alpha\) is the angle between piston hole, \(m_{ps}\) is the sum of the mass of piston and shoe, \(r\) is the radius of ball head, and \(f_{1}\) is the sliding friction coefficient between the swash plate and the shoe. \(f_{2}\) is the sliding friction coefficient between the ball head and the shoe. \(f_{2}=0.08\). The variable resistance \(F_{Lb}\) controlling the swash plate angle is given by  
\[F_{Lb}=\frac{M_{\rm d}}{L_{d}} \tag{8}\]  
The flow continuity equation for the variable cylinder is expressed as  
\[Q_{L}=A_{p}\frac{dx_{p}}{dt}+C_{tb}p_{L}+\frac{V_{t}}{4E}\frac{dp_{L}}{dt} \tag{9}\]  
\(C_{tb}\) is the total leakage coefficient of the variable cylinder which is equal to the sum of the internal and external leakage coefficients of the variable cylinder \(C_{tb}=C_{ib}+\frac{C_{tb}}{2}\). It is assumed that the oil in the variable system cannot be compressed while ignoring the leakage of the oil in the system.