Differential Cylinder $—$ Model of a differential cylinder

This model is a double acting cylinder with a rod on one side and damping at the ends modeled with springs and dampers. This model has two active chambers on either side of the piston and internal leakage but only external leakage from chamberB. External leakage can be turned off by clearing the hasExternalLeakage setting (found under the Properties tab ( ) in the Parameters $\to$  Leakage section.

Cylinder Geometry This is a simplified overview of the geometric parameters defining the cylinder geometry:

Mechanical Friction This model includes Stribeck Friction (see the following figure).

Variables

 Name Value Units Description Modelica ID $\mathrm{internalLeakage}$ Internal Leakage internalLeakage $\mathrm{chamberA}$ A-side chamber chamberA $\mathrm{chamberB}$ B-side chamber chamberB $\mathrm{PistonInCylinder}$ Single rod piston in a cylinder PistonInCylinder $\mathrm{rodA}$ From flange_a to piston head rodA $\mathrm{piston}$ Piston length piston $\mathrm{cylinderFriction}$ Pressure dependent friction between piston and cylinder cylinderFriction $\mathrm{friction}$ Friction between piston and cylinder friction $\mathrm{environment}$ environment external leakage External leakage externalLeakage ${p}_{A\left(\mathrm{summary}\right)}$ ${p}_{A}$ $\mathrm{Pa}$ Pressure at port A summary_pA ${p}_{B\left(\mathrm{summary}\right)}$ ${p}_{B}$ $\mathrm{Pa}$ Pressure at port B summary_pB ${P}_{\mathrm{mech}\left(\mathrm{summary}\right)}$ [1] $W$ Mechanical translational power summary_MP ${f}_{b\left(\mathrm{summary}\right)}$ ${f}_{b}$ $N$ Force acting on flange b summary_fb ${f}_{\mathrm{bearing}\left(\mathrm{summary}\right)}$ $\mathrm{mount.f}$ $N$ Force acting on bearing summary_fbearing ${q}_{\mathrm{chamA}\left(\mathrm{summary}\right)}$ $\mathrm{chamberA.qcham}$ $\frac{{m}^{3}}{s}$ Flow rate flowing into Chamber A summary_qChamA ${q}_{\mathrm{chamB}\left(\mathrm{summary}\right)}$ $\mathrm{chamberB.qcham}$ $\frac{{m}^{3}}{s}$ Flow rate flowing into Chamber B summary_qChamB ${\mathrm{stroke}}_{\mathrm{summary}}$ [2] $m$ Cylinder stroke summary_stroke ${v}_{\mathrm{summary}}$ [3] $\frac{m}{s}$ Cylinder velocity summary_v ${P}_{\mathrm{hyd}\left(A\right)\left(\mathrm{summary}\right)}$ $\mathrm{chamberA.summary_HP}$ $W$ Hydraulic Power in chamber A summary_HP_A ${P}_{\mathrm{hyd}\left(B\right)\left(\mathrm{summary}\right)}$ $\mathrm{chamberB.summary_HP}$ $W$ Hydraulic Power in chamber A summary_HP_B

[1] ${\stackrel{.}{s}}_{b}{f}_{b}+{\stackrel{.}{s}}_{\mathrm{mount}}{f}_{\mathrm{mount}}$

[2] $\mathrm{PistonInCylinder.motion.s_rel}$

[3] $\mathrm{PistonInCylinder.motion.v_rel}$

Connections

 Name Description Modelica ID ${\mathrm{port}}_{A}$ Port A, where oil flows into the component (positive q, port_A.p > chamberA.p means positive dp) port_A ${\mathrm{port}}_{B}$ Port B, where oil flows into the component (positive q, port_B.p > chamberB.p means positive dp) port_B ${\mathrm{flange}}_{b}$ Piston rod mounting point b flange_b $\mathrm{mount}$ Cylinder housing mounting point mount

Parameters

General Parameters

 Name Default Units Description Modelica ID ${\ell }_{\mathrm{rod}\left(b\right)}$ $\frac{1}{2}$ $m$ Length of rod, side b RodLength_b ${\ell }_{\mathrm{housing}}$ $2$ $m$ Length of housing HousingLength ${\ell }_{\mathrm{piston}}$ $0.05$ $m$ Length of piston PistonLength ${m}_{\mathrm{piston}}$ $100$ $\mathrm{kg}$ Effective mass of piston and rigidly coupled masses PistonMass ${c}_{A}$ ${10}^{8}$ $\frac{N}{m}$ Stiffness of stop at stroke end A c_A ${d}_{A}$ ${10}^{6}$ $\frac{Ns}{m}$ Damping of stop at stroke end A d_A ${c}_{B}$ ${10}^{8}$ $\frac{N}{m}$ Stiffness of stop at stroke end B c_B ${d}_{B}$ ${10}^{6}$ $\frac{Ns}{m}$ Damping of stop at stroke end B d_B ${A}_{\mathrm{piston}\left(A\right)}$ $0.001$ ${m}^{2}$ Effective area of left side of piston piston_areaA ${A}_{\mathrm{piston}\left(B\right)}$ $0.001$ ${m}^{2}$ Effective area of right side of piston piston_areaB ${V}_{\mathrm{dead}\left(A\right)}$ $0.001$ ${m}^{3}$ Dead volume of chamber if piston is at left end of housing (side A: stroke = 0) dead_volumeA ${V}_{\mathrm{dead}\left(B\right)}$ $0.001$ ${m}^{3}$ Dead volume of chamber if piston is at right end of housing (side B: stroke = max) dead_volumeB ${k}_{p}$ $0$ Pressure damping coefficient kp ${\mathrm{ΔT}}_{\mathrm{system}}$ $0$ $K$ Temperature offset from system temperature dT_system $\mathrm{reverse}$ $\mathrm{false}$ Reverse the positive flow in the component reverse

Friction Parameters

 Name Default Units Description Modelica ID ${f}_{\mathrm{breakaway}}$ $100$ $N$ breakaway_friction ${f}_{\mathrm{mixed}}$ $10$ $N$ mixed_friction ${v}_{f\left(\mathrm{mixed}\right)}$ $\frac{1}{10}$ $\frac{m}{s}$ mixed_friction_velocity ${d}_{\mathrm{viscous}}$ $2.5·{10}^{3}$ $\frac{Ns}{m}$ viscous_damping $\mathrm{tabular}$ $\mathrm{false}$ If true, use tabular models, else an analytic one Tabular ${f}_{\mathrm{pos}}$ $\left[0.,1.\right]$ [v, f] Positive sliding friction characteristic (v>=0) f_pos

Initialization Parameters

 Name Default Units Description Modelica ID ${\mathrm{InitialStroke}}_{A}$ [1] $m$ Initial stroke of cylinder chamber A, default is middle position InitialStroke_A

[1] $\frac{1}{2}{\ell }_{\mathrm{housing}}-\frac{1}{2}{\ell }_{\mathrm{piston}}$

Leakage Parameters

 Name Default Units Description Modelica ID has external leakage $\mathrm{true}$ If true, external leakage at rod seal is included hasExternalLeakage ${G}_{\mathrm{ext}}$ $4.2·{10}^{-13}$ $\frac{{m}^{3}}{s\mathrm{Pa}}$ Hydraulic conductance for external leakage G_ext ${G}_{\mathrm{leak}\left(\mathrm{int}\right)}$ $0$ $\frac{{m}^{3}}{s\mathrm{Pa}}$ Hydraulic conductance $G=\frac{\mathrm{∂q}}{\mathrm{∂p}}$ G_intLeak