Double Acting Cylinder

Ideal transformation between hydraulic and 1-D translational work

 Description The Double Acting Cylinder component models a double-acting translational actuator. It transforms work between the hydraulic and translational domains. A boolean parameter, Reversed, allows reversing the positive flow direction. Variable port volumes calculated based on the piston relative position can be added to side A and side B. Optional end cushions restrict the stroke length of the piston.

Equations

The following equations are used when $\mathrm{Reversed}=\mathrm{false}$.

${A}_{a}^{f}=\left\{\begin{array}{cc}\frac{\mathrm{\pi }}{4}{\mathrm{D}}_{\mathrm{bore}}^{2}& \mathrm{Use diameters}\\ {A}_{a}& \mathrm{otherwise}\end{array}{A}_{b}^{f}=\left\{\begin{array}{cc}\frac{\mathrm{\pi }}{4}\left({\mathrm{D}}_{\mathrm{bore}}^{2}-{\mathrm{D}}_{\mathrm{rod}}^{2}\right)& \mathrm{Use diameters}\\ {A}_{b}& \mathrm{otherwise}\end{array}$

${q}_{a}^{f}=\left\{\begin{array}{cc}{q}_{a}-{q}_{\mathrm{va}}& \mathrm{Use diameters}\\ {q}_{a}& \mathrm{otherwise}\end{array}{q}_{b}^{f}=\left\{\begin{array}{cc}{q}_{b}-{q}_{\mathrm{vb}}& \mathrm{Use diameters}\\ {q}_{b}& \mathrm{otherwise}\end{array}$

${q}_{a}^{f}={A}_{a}^{f}{v}_{\mathrm{rel}}\phantom{\rule[-0.0ex]{3.5ex}{0.0ex}}{q}_{b}^{b}=-{A}_{b}^{b}{v}_{\mathrm{rel}}$

${s}_{\mathrm{rel}}={s}_{b}-{s}_{a}\phantom{\rule[-0.0ex]{2.5ex}{0.0ex}}{v}_{\mathrm{rel}}=\frac{\mathrm{d}{s}_{\mathrm{rel}}}{\mathrm{d}t}$

${f}_{a}=-{f}_{b}=\left\{\begin{array}{cc}f+{f}_{{e}_{1}}+{f}_{{e}_{2}}& \mathrm{Use end cushions}\\ f& \mathrm{otherwise}\end{array}$

$f={A}_{a}^{f}{p}_{a}-{A}_{b}^{f}{p}_{b}-d{v}_{\mathrm{rel}}$

${f}_{{e}_{1}}=\left\{\begin{array}{cc}0& 0\le {f}_{{c}_{1}}+{f}_{{d}_{1}}\\ {f}_{{c}_{1}}+\mathrm{max}\left({f}_{{c}_{1}},{f}_{{d}_{1}}\right)& \mathrm{otherwise}\end{array}\phantom{\rule[-0.0ex]{3.0ex}{0.0ex}}{f}_{{e}_{2}}=\left\{\begin{array}{cc}0& {f}_{{c}_{2}}+{f}_{{d}_{2}}\le 0\\ {f}_{{c}_{2}}+\mathrm{min}\left({f}_{{c}_{2}},{f}_{{d}_{2}}\right)& \mathrm{otherwise}\end{array}$

${f}_{{c}_{1}}=\left\{\begin{array}{cc}-{k}_{c}\left({s}_{\mathrm{rel}}-{L}_{\mathrm{max}}\right)& {L}_{\mathrm{max}}<{s}_{\mathrm{rel}}\\ 0& \mathrm{otherwise}\end{array}\phantom{\rule[-0.0ex]{5.0ex}{0.0ex}}{f}_{{c}_{2}}=\left\{\begin{array}{cc}-{k}_{c}\left({s}_{\mathrm{rel}}-{L}_{\mathrm{min}}\right)& {s}_{\mathrm{rel}}<{L}_{\mathrm{min}}\\ 0& \mathrm{otherwise}\end{array}$

${f}_{{d}_{1}}=\left\{\begin{array}{cc}-{c}_{c}{v}_{\mathrm{rel}}& {L}_{\mathrm{max}}<{s}_{\mathrm{rel}}\\ 0& \mathrm{otherwise}\end{array}\phantom{\rule[-0.0ex]{7.5ex}{0.0ex}}{f}_{{d}_{2}}=\left\{\begin{array}{cc}-{c}_{c}{v}_{\mathrm{rel}}& {s}_{\mathrm{rel}}<{L}_{\mathrm{min}}\\ 0& \mathrm{otherwise}\end{array}$

If $\mathrm{Reversed}=\mathrm{true}$, the direction of the arrow on the component's icon changes to indicate that the direction of the positive flow is reversed.  This means, for example, if the cylinder is fixed and a positive flow is fed to side A then the direction of positive displacement for the piston is opposite that of the $\mathrm{Reversed}=\mathrm{false}$ case.

 Volumes Variable port volumes calculated based on the piston relative position, ${s}_{\mathrm{rel}}$, can be added to side A and side B using $\mathrm{Use}\mathrm{volumes}=\mathrm{true}$. The equations for the variable port volumes are as follows. ${V}_{a}={V}_{a}^{0}+{A}_{a}^{f}\mathrm{min}\left(\mathrm{max}\left({s}_{\mathrm{rel}},0\right),{L}_{h}-{L}_{p}\right)$ ${V}_{b}={V}_{b}^{0}+{A}_{b}^{f}\mathrm{min}\left(\mathrm{max}\left({L}_{h}-{L}_{p}-{s}_{\mathrm{rel}},0\right),{L}_{h}-{L}_{p}\right)$ ${q}_{\mathrm{va}}=E\frac{\mathrm{d}{p}_{a}}{\mathrm{d}t}\phantom{\rule[-0.0ex]{2.0ex}{0.0ex}}{q}_{\mathrm{vb}}=E\frac{\mathrm{d}{p}_{b}}{\mathrm{d}t}$ The equations for calculating $E$ are discussed in Constant Volume. If $\mathrm{Use volumes}=\mathrm{true}$, the component icon changes to indicate that variable port volumes are being used. $\mathrm{Use volumes}=\mathrm{false}$         $\mathrm{Use volumes}=\mathrm{true}$

Variables

 Name Units Description Modelica ID ${p}_{x}$ $\mathrm{Pa}$ Pressure port $x,x\in \left\{A,B\right\}$ portx.p ${q}_{x}$ $\frac{{m}^{3}}{s}$ Flow rate through port $x,x\in \left\{A,B\right\}$ portx.q ${s}_{\mathrm{rel}}$ $m$ Relative distance from flange a to b s_rel ${v}_{\mathrm{rel}}$ $\frac{m}{s}$ Relative velocity of flanges v_rel

Connections

 Name Description Modelica ID $\mathrm{portA}$ Hydraulic port portA $\mathrm{portB}$ Hydraulic port portB ${\mathrm{flange}}_{a}$ Left flange of compliant 1-dim. translational component flange_a ${\mathrm{flange}}_{b}$ Right flange of compliant 1-dim. translational component flange_b

Parameters

Basic

 Name Default Units Description Modelica ID Reversed $\mathrm{false}$ When checked (true), positive flow direction is reversed reversed Use diameters $\mathrm{false}$ When checked (true), diameters are used to specify the area useDiameter ${A}_{a}$ $0.01$ ${m}^{2}$ Effective area of piston - side A Aa ${A}_{b}$ $0.08$ ${m}^{2}$ Effective area of piston - side B Ab ${\mathrm{D}}_{\mathrm{bore}}$ $0.01$ $m$ Diameter of bore - side A Dbore ${\mathrm{D}}_{\mathrm{rod}}$ $0.08$ $m$ Diameter of rod - side B Drod $d$ $0$ $N\frac{s}{m}$ Piston viscous friction coefficient d

Volumes

 Name Default Units Description Modelica ID Use volumes $\mathrm{false}$ When checked (true), variable hydraulic volume chambers are added to sides A and B useVolume ${L}_{h}$ $1$ $m$ Housing length LH ${L}_{p}$ $0.05$ $m$ Piston length LP ${V}_{\mathrm{a0}}$ $0.0001$ ${m}^{2}$ Dead volume of side A VA0 ${V}_{\mathrm{b0}}$ $0.0001$ ${m}^{2}$ Dead volume of side B VB0

End Cushions

 Name Default Units Description Modelica ID Use end cushions $\mathrm{false}$ True adds cushions to the cylinder and enables ${L}_{\mathrm{max}}$, ${L}_{\mathrm{min}}$, ${k}_{c}$, and ${c}_{c}$ parameters useEndCushions ${L}_{\mathrm{max}}$ $1$ $m$ Maximum piston position Lmax ${L}_{\mathrm{min}}$ $0$ $m$ Minimum piston position Lmin ${k}_{c}$ $1·{10}^{10}$ $\frac{N}{m}$ End cushion stiffness kc ${c}_{c}$ $1·{10}^{6}$ $N\frac{s}{m}$ End cushion damping cc