Total Volume

$$\class{X01}{V_T}=\left[{\frac{\class{X06}{\Delta V}}{{\left(\class{X02}{p_0}\right)^{{\frac{1}{\class{X05}{\gamma}}}}}}}\right]\cdot \left[{\frac{1}{{{\left({\frac{1}{\class{X04}{p_{Min}}}}\right)^{{\frac{1}{\class{X05}{\gamma}}}}}-{\left({\frac{1}{\class{X03}{p_{Max}}}}\right)^{{\frac{1}{\class{X05}{\gamma}}}}}}}}\right]$$

Spring Stiffness

$$\class{X05}{k}=\frac{{\class{X07}{P_{max}}\cdot \left(\frac{\pi}{4}\right)\cdot {\class{X01}{d_p}}^2}}{{\class{X06}{L_s}+\class{X04}{L}}}$$

Maximum Stored Energy

$$\class{X12}{E_{acc;Max}}=\frac{1}{2}\cdot \class{X05}{k}\cdot \left ({2\cdot \class{X06}{\class{X04}{L}_s}\cdot \class{X04}{L}+\left(\class{X04}{L}\right)^{2}} \right )$$

Component Sizing

Mass

$$\class{X01}{W}=\frac{{\class{X05}{P}\cdot {(\frac{\pi}{4}})\cdot {\class{X02}{D_p}}^2}}{\class{X10}{g}}$$

Maximum Stored Energy

$$\class{X12}{E_{acc;Max}}=\class{X01}{W}\cdot \class{X10}{g}\cdot \class{X11}{L}$$

Component Sizing

Piston Side Area

$$\class{X04}{A_p}=\pi\cdot\left(\frac{\class{X01}{d_p}}{2}\right)^2$$

Rod Side Area

$$\class{X05}{A_r}=\pi\cdot \left[\left(\frac{\class{X01}{d_p}}{2}\right)^2-\left(\frac{\class{X02}{d_r}}{2}\right)^2\right]$$

Area Ratio

$$\class{X27}{R}=\frac{{\class{X01}{d_p}}^2}{{\class{X01}{d_p}}^2-{\class{X02}{d_r}}^2}$$

Piston side pressure - Spring return

$$\class{X26}{p_p}=\frac{\class{X12}{K_s} \cdot (\class{X03}{L}-\class{X13}{L_{ko}})}{\class{X04}{A_p}}$$

Stem pressure - Resistive force

$$\class{X21}{P_r}=\frac{\class{X17}{F_r}}{\class{X05}{A_r}}$$

Stem pressure - Driving mass

$$\class{X21}{P_r}=\frac{-\class{X10}{M} \cdot \class{X24}{g} \cdot \sin (\class{X11}{\alpha})}{\class{X05}{A_r}}$$

Stem pressure - Output spring

$$\class{X21}{P_r}=\frac{\class{X12}{K_s} \cdot (\class{X03}{L} - \class{X13}{L_{ko}})}{\class{X05}{A_r}}$$

Piston side pressure - Driving mass

$$\class{X16}{P_p}=\frac{\class{X10}{M}\cdot \class{X26}{g}\cdot \sin(\class{X11}{\alpha})}{\class{X04}{A_p}}$$

Piston side pressure - Resistive force

$$\class{X14}{P_e}=\frac{\class{X10}{F_e}}{\class{X04}{A_p}}$$

Total Extend Load

$$\class{X10}{F_e}=\class{X14}{P_e}\cdot {\frac{\pi\cdot {\class{X01}{d_p}}^2}{4}}$$

Total Retract Load

$$\class{X11}{F_r}=\class{X15}{P_r}\cdot\frac{\pi \cdot \left({\class{X01}{d_p}}^2 - {\class{X02}{d_r}}^2\right)}{4}$$

Extend Pressure

$$\class{X20}{P_e}=\frac{4\cdot(\class{X14}{f_e}+\class{X10}{M}\cdot \class{X24}{g}\cdot\sin(\class{X11}{\alpha})+\class{X12}{k_s}\cdot(\class{X03}{L}+\class{X13}{L_{ko}}))}{\pi\cdot({\class{X01}{d_p})}^2}$$

Piston Side Pressure

$$\class{X16}{p_p}=\frac{{\class{X10}{M}\cdot \class{X26}{g}\cdot \sin{(\class{X11}{\alpha})}+\class{X05}{A_r}\cdot \class{X17}{p_r}+\class{X12}{f_e}}}{\class{X04}{A_p}}$$

Rod Critical Force

$$\class{X25}{F_c}= \frac{{\pi^3\cdot \class{X21}{E}}}{{4\cdot {\class{X24}{\lambda}}^2\cdot {\class{X23}{L_e}}^2\cdot \class{X22}{S}}} \cdot \frac{{\class{X02}{d_r}}^4-{\class{X20}{d_i}}^4}{16}$$

Rod Side Pressure

$$\class{X17}{p_r}=\frac{{\class{X04}{A_p}\cdot \class{X16}{p_p}+\class{X13}{f_r}-\class{X10}{M}\cdot \class{X26}{g}\cdot \sin{(\class{X11}{\alpha})}}}{\class{X05}{A_r}}$$

Average extend speed (with Stroke)

$$\class{X18}{v_e}=\frac{\class{X03}{L}}{\class{X08}{t_e}}$$

Average retract speed (with Stroke)

$$\class{X19}{v_r}=\frac{\class{X03}{L}}{\class{X09}{t_r}}$$

Average extend speed (with Piston Side Area)

$$\class{X18}{v_e}=\frac{\class{X14}{Q_{e;p}}}{\class{X04}{A_p}}$$

Average retract speed (with Rod Side Area)

$$\class{X19}{v_r}=\frac{\class{X15}{Q_{r;r}}}{\class{X05}{A_r}}$$

Extend Time

$${\class{X08}{t_e}}=\frac{\pi \cdot {\class{X01}{d_p}}^2 \cdot \class{X03}{L}}{4\cdot \class{X14}{Q_{e;p}}}$$

Retract Time

$${\class{X09}{t_r}}=\frac{\pi \cdot (\class{X01}{d_p}^2-\class{X02}{d_r}^2) \cdot \class{X03}{L}}{4\cdot \class{X15}{Q_{r;r}}}$$

Spring Stiffness

$$\class{X12}{K_S}=\frac{\class{X20}{P_e}\cdot\pi\cdot({\class{X01}{d_p})}^2-4\cdot \class{X14}{f_e}-4\cdot \class{X10}{M}\cdot \class{X24}{g}\cdot \sin(\class{X11}{\alpha})}{4\cdot(\class{X03}{L}+\class{X13}{L_{ko}})}$$

Spring Stiffness

$$\class{X12}{K_s}=\frac{\class{X21}{P_r}\cdot\pi\cdot({\class{X01}{d_p}}^2-{\class{X02}{d_r}}^2)-4\cdot \class{X15}{f_r}+4\cdot \class{X10}{M}\cdot \class{X24}{g}\cdot\sin(\class{X11}{\alpha}))}{4\cdot(\class{X03}{L}+\class{X13}{L_{ko}})}$$

Mechanical Output Power (Extension)

$$\class{X29}{P_{e;o}}=\left ( {\class{X12}{f_e}+\class{X10}{M}\cdot \class{X26}{g}\cdot \sin{(\class{X11}{\alpha})}} \right )\cdot \class{X18}{v_e}$$

Mechanical Output Power (Retraction)

$$\class{X31}{P_{r;o}}=\left ( {\class{X13}{f_r}-\class{X10}{M}\cdot \class{X26}{g}\cdot \sin{(\class{X11}{\alpha})}}\right) \cdot \class{X19}{v_r}$$

Component Sizing

Loss Coefficient

$$\class{X06}{K_L}=\frac{{2\cdot \class{X05}{\Delta p}}}{{\class{X03}{\rho}\cdot \left(\frac{{4\cdot \class{X02}{Q}}}{{\pi\cdot {\class{X01}{D_i}^{2}}}}\right)^{2}}}$$

Laminar Flow Pressure Loss

$$\class{X13}{\Delta p_l}=\frac{128 \cdot \class{X02}{L}\cdot \class{X08}{\nu}\cdot\class{X06}{\rho} \cdot \class{X05}{Q}}{\pi\cdot \class{X01}{D}^4}$$

Turbulent Flow Pressure Loss

$$\class{X14}{\Delta p_t}=\frac{8 \cdot \class{X02}{L} \cdot \class{X06}{\rho} \cdot \class{X05}{Q}^2}{\pi^2 \cdot \class{X01}{D}^5\cdot \left[-1.8\cdot \log_{10}\left(\frac{1.725\cdot \pi\cdot \class{X01}{D} \cdot\class{X08}{\nu}}{\class{X05}{Q}}+ \left(\frac{\class{X03}{\epsilon}}{3.7}\right)^{1.11}\right)\right]^2}$$

Fluid Line Velocity

$$\class{X09}{v_f}=\frac{4\cdot \class{X05}{Q}}{\pi\cdot \class{X01}{D}^2}$$

Component Sizing

Overall Efficiency (with Mechanical Efficiency)

$$\class{X11}{\eta_t}=\class{X09}{\eta_v}\cdot \class{X10}{\eta_m}$$

Overall Efficiency (with Shaft Input Power)

$$\class{X11}{\eta_t}=\frac{\class{X12}{P_h}}{\class{X08}{P_i}}$$

Displacement

$$\class{X01}{C}=\frac{\class{X03}{Q}}{{\class{X02}{\omega}\cdot \class{X09}{\eta_v}}}$$

Torque

$$\class{X07}{T}=\frac{\class{X01}{C}\cdot (\class{X05}{p_o}-\class{X04}{p_i})}{\class{X10}{\eta_m}}$$

Effective Hydraulic Power

$$\class{X12}{P_h}=\class{X03}{Q}\cdot \class{X06}{\Delta p}$$

Hydraulic Output Power

$$\class{X16}{P_o}=\class{X03}{Q}\cdot \class{X05}{p_o}$$

Shaft Input Power

$$\class{X08}{\class{X08}{P_i}}=\class{X07}{\class{X07}{T}}\cdot \class{X02}{\class{X02}{\omega}}$$

Component Sizing

Displacement

$$\class{X01}{C}=\frac{\class{X07}{T}\cdot\class{X09}{\eta_v}}{\class{X06}{\Delta p}\cdot\class{X11}{\eta_t}}$$

Input Flow

$$\class{X03}{Q} = \frac{{\class{X01}{C}\cdot \class{X02}{\omega}}}{\class{X09}{\eta_v}}$$

Shaft Output Power

$$\class{X08}{P_o}=\class{X03}{Q}\cdot\class{X06}{\eta_v}\cdot\class{X09}{\Delta p}\cdot\class{X10}{\eta_m}$$

Nominal Speed

$$\class{X02}{\omega}=\left({\frac{\class{X03}{Q}}{\class{X01}{C}}}\right)\cdot \class{X09}{\eta_v}$$

Component Sizing

Discharge Coefficient

$$\class{X09}{C_d}=\left(\frac{\class{X03}{Q}}{{\left (\frac{\pi}{4}\right)}\cdot {\class{X01}{d_o}}^2} \right)\cdot \sqrt{\frac{\class{X04}{\rho}\cdot (1-\class{X12}{\beta}^4)}{2\cdot\class{X10}{\Delta p}}}$$