Hydraulic accumulator with spring-loaded back pressure

 Description The Spring Loaded Accumulator component consists of a preloaded spring, a piston, and a hydraulic chamber connected to a hydraulic port. When the fluid pressure at the port exceeds the preloaded spring pressure, fluid flows into the chamber, moving the piston and compressing the spring. The amount of fluid stored in the accumulator depends on the size and spring load. The stroke of the piston is restricted by end cushion parameters used to set the boundary of the component respectively.
 Equations ${a}_{\mathrm{rel}}=\frac{\mathrm{d}{v}_{\mathrm{rel}}}{\mathrm{d}t}\phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}{v}_{\mathrm{rel}}=\frac{\mathrm{d}{s}_{\mathrm{rel}}}{\mathrm{d}t}$ ${f}_{\mathrm{spring}}=-A{p}_{\mathrm{preload}}-c{s}_{\mathrm{rel}}-d\cdot {v}_{\mathrm{rel}}$ ${f}_{{c}_{1}}=\left\{\begin{array}{cc}0& {s}_{\mathrm{rel}}<{L}_{\mathrm{max}}\\ {c}_{\mathrm{ES}}\left({s}_{\mathrm{rel}}-{L}_{\mathrm{max}}\right)& \mathrm{otherwise}\end{array}{f}_{{c}_{2}}=\left\{\begin{array}{cc}0& 0\le {s}_{\mathrm{rel}}\\ {c}_{\mathrm{ES}}{s}_{\mathrm{rel}}& \mathrm{otherwise}\end{array}$ ${f}_{{d}_{1}}=\left\{\begin{array}{cc}0& {s}_{\mathrm{rel}}<{L}_{\mathrm{max}}\\ {d}_{\mathrm{ES}}\frac{\mathrm{d}{s}_{\mathrm{rel}}}{\mathrm{d}t}& \mathrm{otherwise}\end{array}{f}_{{d}_{2}}=\left\{\begin{array}{cc}0& 0\le {s}_{\mathrm{rel}}\\ {d}_{\mathrm{ES}}\frac{\mathrm{d}{s}_{\mathrm{rel}}}{\mathrm{d}t}& \mathrm{otherwise}\end{array}$ ${f}_{{e}_{1}}=\left\{\begin{array}{cc}0& {f}_{{c}_{1}}+{f}_{{d}_{1}}\le 0\\ {f}_{{c}_{1}}+\mathrm{min}\left({f}_{{c}_{1}},{f}_{{d}_{1}}\right)& \mathrm{otherwise}\end{array}{f}_{{e}_{2}}=\left\{\begin{array}{cc}0& 0\le {f}_{{c}_{2}}+{f}_{{d}_{2}}\\ {f}_{{c}_{2}}+\mathrm{max}\left({f}_{{c}_{2}},{f}_{{d}_{2}}\right)& \mathrm{otherwise}\end{array}$ $p={p}_{A}$ $q={q}_{A}=A{v}_{\mathrm{rel}}$ ${v}_{\mathrm{rel}}=\frac{\mathrm{d}{s}_{\mathrm{rel}}}{\mathrm{d}t}$ ${f}_{\mathrm{chamber}}=pA$ $m{a}_{\mathrm{rel}}={f}_{\mathrm{chamber}}+{f}_{\mathrm{spring}}-{f}_{{e}_{1}}-{f}_{{e}_{2}}$

Variables

 Name Units Description Modelica ID $p$ $\mathrm{Pa}$ p $q$ $\frac{{m}^{3}}{s}$ q ${s}_{\mathrm{rel}}$ $m$ Relative distance of the piston from the initial position s_rel ${v}_{\mathrm{rel}}$ $\frac{m}{s}$ Relative velocity of the piston v_rel ${a}_{\mathrm{rel}}$ $\frac{m}{{s}^{2}}$ Relative acceleration of the piston a_rel

Connections

 Name Description Modelica ID $\mathrm{portA}$ Hydraulic port portA

Parameters

Geometry

 Name Default Units Description Modelica ID $A$ $0.01$ ${m}^{2}$ Wetted area of piston A $m$ $0.1$ $\mathrm{kg}$ Mass of the piston m ${L}_{\mathrm{max}}$ $0.4$ $m$ Maximum stroke length of the accumulator Lmax

Hard Stop

 Name Default Units Description Modelica ID ${c}_{\mathrm{ES}}$ $1·{10}^{10}$ $\frac{N}{m}$ Hard stop stiffness c_ES ${d}_{\mathrm{ES}}$ $1·{10}^{6}$ $N\frac{s}{m}$ Hard stop damping d_ES

Spring

 Name Default Units Description Modelica ID $c$ $1·{10}^{4}$ $\frac{N}{m}$ Spring constant c $d$ $0$ $N\frac{s}{m}$ Piston viscous friction coefficient d

 Name Default Units Description Modelica ID ${p}_{\mathrm{preload}}$ $1·{10}^{5}$ $\mathrm{Pa}$ Spring preload pressure p_preload