Servo Valve $—$ Servovalve with second-order spool dynamics and nonlinearities

The Servo Valve component describes a servovalve with second-order spool dynamics and nonlinearities.

Implementation

Valve

The ${q}_{\mathrm{nom}}$ parameter gives the nominal flow rate of the fully opened valve at the pressure drop ${\mathrm{Δp}}_{\mathrm{nom}}$. The ${d}_{\mathrm{leak}}$ parameter gives the equivalent diameter of an orifice to describe the leakage flow if the valve is closed.

${A}_{\mathrm{max}}={q}_{\mathrm{nom}}\sqrt{\frac{1}{2}\frac{\mathrm{\rho }{k}_{2}}{{\mathrm{Δp}}_{\mathrm{nom}}}}$

${A}_{\mathrm{leak}}=\frac{1}{4}\mathrm{\pi }{d}_{\mathrm{leak}}^{2}$

Spool

Input signal: $\mathrm{command}$

Output signal: $\mathrm{position}$

The position of the spool is modeled as a second-order system with natural frequency ${\mathrm{\omega }}_{0}$, damping damp, limits for the velocity, and hysteresis for the position. The flow area depends linearly on the spool position.

The position of the spool is normalized in the interval:

$-1<\mathrm{position}$ $<$ $1$

and the parameter ${v}_{\mathrm{max}}$ is the maximum value that the normalized speed, $\frac{\partial }{\partial t}\mathrm{position}$, can obtain. If the physical position of the spool is used (denoted here by $x$), we have:

$0 $<$ $L$

where $L$ is the maximum distance the spool can move (in $\left[m\right]$).

The relationship between the normalized position and speed is then:

$\mathrm{position}=\frac{2x}{L}-1$

$\frac{\partial }{\partial t}\mathrm{position}=\frac{2\left(\frac{\partial }{\partial t}x\right)}{L}$

This can be used to express the maximum speed as:

${v}_{\mathrm{max}}=\frac{2{v}_{{x}_{\mathrm{max}}}}{L}$

where ${v}_{{x}_{\mathrm{max}}}$ is the maximum velocity of the spool in $\left[\frac{m}{s}\right]$.

Assumptions

The laminar/turbulent flow through the valve is modeled as flow through orifices without cavitation. Flow forces are not modeled.

Flow scheme

See component Servo Valve No States.

Variables

 Name Value Units Description Modelica ID ${V}_{A}$ VolumeA ${V}_{B}$ VolumeB ${V}_{T}$ VolumeT ${V}_{P}$ VolumeP $\mathrm{spoolDynamics}$ spoolDynamics $\mathrm{valveNoStates}$ valveNoStates ${p}_{A\left(\mathrm{summary}\right)}$ $\mathrm{valveNoStates.port_A.p}$ $\mathrm{Pa}$ Pressure at port A summary_pA ${p}_{B\left(\mathrm{summary}\right)}$ $\mathrm{valveNoStates.port_B.p}$ $\mathrm{Pa}$ Pressure at port B summary_pB ${p}_{P\left(\mathrm{summary}\right)}$ $\mathrm{valveNoStates.p\left[P\right]}$ $\mathrm{Pa}$ Pressure at port P summary_pP ${p}_{T\left(\mathrm{summary}\right)}$ $\mathrm{valveNoStates.port_T.p}$ $\mathrm{Pa}$ Pressure at port T summary_pT ${q}_{\mathrm{PA}\left(\mathrm{summary}\right)}$ $\mathrm{valveNoStates.morpa.q}$ $\frac{{m}^{3}}{s}$ Flow rate flowing port_P to port_A summary_qPA ${q}_{\mathrm{PB}\left(\mathrm{summary}\right)}$ $\mathrm{valveNoStates.morpb.q}$ $\frac{{m}^{3}}{s}$ Flow rate flowing port_P to port_B summary_qPB ${q}_{\mathrm{AT}\left(\mathrm{summary}\right)}$ $\mathrm{valveNoStates.morat.q}$ $\frac{{m}^{3}}{s}$ Flow rate flowing port_A to port_T summary_qAT ${q}_{\mathrm{BT}\left(\mathrm{summary}\right)}$ $\mathrm{valveNoStates.morbt.q}$ $\frac{{m}^{3}}{s}$ Flow rate flowing port_B to port_T summary_qBT

Connections

 Name Description Modelica ID ${\mathrm{port}}_{A}$ Port A, one of valve connections to actuator or motor port_A ${\mathrm{port}}_{B}$ Port B, one of valve connections to actuator or motor port_B ${\mathrm{port}}_{P}$ Port P, where oil enters the component from the pump port_P ${\mathrm{port}}_{T}$ Port T, where oil flows to the tank port_T $\mathrm{command}$ Command signal for valve position command $\mathrm{oil}$ oil

Parameters

General Parameters

 Name Default Units Description Modelica ID use volume A $\mathrm{true}$ If true, a volume is present at port_A useVolumeA use volume B $\mathrm{true}$ If true, a volume is present at port_B useVolumeB use volume P $\mathrm{true}$ If true, a volume is present at port_P useVolumeP use volume T $\mathrm{true}$ If true, a volume is present at port_T useVolumeT ${V}_{A}$ ${10}^{-6}$ ${m}^{3}$ Geometric volume at port A volumeA ${V}_{B}$ ${10}^{-6}$ ${m}^{3}$ Geometric volume at port B volumeB ${V}_{P}$ ${10}^{-6}$ ${m}^{3}$ Geometric volume at port P volumeP ${V}_{T}$ ${10}^{-6}$ ${m}^{3}$ Geometric volume at port T volumeT ${\mathrm{ΔT}}_{\mathrm{system}}$ $0$ $K$ Temperature offset from system temperature dT_system

Actuation Parameters

 Name Default Units Description Modelica ID ${\mathrm{\omega }}_{0}$ $500$ $\frac{\mathrm{rad}}{s}$ Natural frequency of spool omega0 $\mathrm{damp}$ $\frac{7}{10}$ Damping coefficient of spool damp $\mathrm{overlap}$ $0.02$ Overlap relative to max. displacement = 1 overlap $\mathrm{hyst}$ $0.005$ Half of hysteresis width hyst ${v}_{\mathrm{max}}$ $100$ Max. spool velocity [1/s] vmax

Flow Parameters

 Name Default Units Description Modelica ID ${q}_{\mathrm{nom}\left(\mathrm{PA}\right)}$ $8.33·{10}^{-4}$ $\frac{{m}^{3}}{s}$ Nominal flow rate qnompa ${\mathrm{Δp}}_{\mathrm{nom}\left(\mathrm{PA}\right)}$ $1.5·{10}^{6}$ $\mathrm{Pa}$ Pressure drop at qnom dpnompa ${q}_{\mathrm{nom}\left(\mathrm{BT}\right)}$ $8.33·{10}^{-4}$ $\frac{{m}^{3}}{s}$ Nominal flow rate qnombt ${\mathrm{Δp}}_{\mathrm{nom}\left(\mathrm{BT}\right)}$ $1.5·{10}^{6}$ $\mathrm{Pa}$ Pressure drop at qnom dpnombt ${q}_{\mathrm{nom}\left(\mathrm{PB}\right)}$ $8.33·{10}^{-4}$ $\frac{{m}^{3}}{s}$ Nominal flow rate qnompb ${\mathrm{Δp}}_{\mathrm{nom}\left(\mathrm{PB}\right)}$ $1.5·{10}^{6}$ $\mathrm{Pa}$ Pressure drop at qnom dpnompb ${q}_{\mathrm{nom}\left(\mathrm{AT}\right)}$ $8.33·{10}^{-4}$ $\frac{{m}^{3}}{s}$ Nominal flow rate qnomat ${\mathrm{Δp}}_{\mathrm{nom}\left(\mathrm{AT}\right)}$ $1.5·{10}^{6}$ $\mathrm{Pa}$ Pressure drop at qnom dpnomat ${k}_{1\left(\mathrm{PA}\right)}$ $10$ Laminar part of orifice model k1pa ${k}_{2\left(\mathrm{PA}\right)}$ $2$ Turbulent part of orifice model, ${k}_{2}=\frac{1}{{C}_{d}^{2}}$ k2pa ${C}_{d\left(\mathrm{PA}\right)}$ $\frac{1}{\sqrt{{k}_{2\left(\mathrm{PA}\right)}}}$ Max discharge coefficient C_dpa ${\mathrm{\lambda }}_{c\left(\mathrm{PA}\right)}$ $\frac{2{k}_{1\left(\mathrm{PA}\right)}}{\sqrt{{k}_{2\left(\mathrm{PA}\right)}}}$ Critical flow number lambdacpa ${k}_{1\left(\mathrm{PB}\right)}$ $10$ Laminar part of orifice model k1pb ${k}_{2\left(\mathrm{PB}\right)}$ $2$ Turbulent part of orifice model, ${k}_{2}=\frac{1}{{C}_{d}^{2}}$ k2pb ${C}_{d\left(\mathrm{PB}\right)}$ $\frac{1}{\sqrt{{k}_{2\left(\mathrm{PB}\right)}}}$ Max discharge coefficient C_dpb ${\mathrm{\lambda }}_{c\left(\mathrm{PB}\right)}$ $\frac{2{k}_{1\left(\mathrm{PB}\right)}}{\sqrt{{k}_{2\left(\mathrm{PB}\right)}}}$ Critical flow number lambdacpb ${k}_{1\left(\mathrm{BT}\right)}$ $10$ Laminar part of orifice model k1bt ${k}_{2\left(\mathrm{BT}\right)}$ $2$ Turbulent part of orifice model, ${k}_{2}=\frac{1}{{C}_{d}^{2}}$ k2bt ${C}_{d\left(\mathrm{BT}\right)}$ $\frac{1}{\sqrt{{k}_{2\left(\mathrm{BT}\right)}}}$ Max discharge coefficient C_dbt ${\mathrm{\lambda }}_{c\left(\mathrm{BT}\right)}$ $\frac{2{k}_{1\left(\mathrm{BT}\right)}}{\sqrt{{k}_{2\left(\mathrm{BT}\right)}}}$ Critical flow number lambdacbt ${k}_{1\left(\mathrm{AT}\right)}$ $10$ Laminar part of orifice model k1at ${k}_{2\left(\mathrm{AT}\right)}$ $2$ Turbulent part of orifice model, ${k}_{2}=\frac{1}{{C}_{d}^{2}}$ k2at ${C}_{d\left(\mathrm{AT}\right)}$ $\frac{1}{\sqrt{{k}_{2\left(\mathrm{AT}\right)}}}$ Max discharge coefficient C_dat ${\mathrm{\lambda }}_{c\left(\mathrm{AT}\right)}$ $\frac{2{k}_{1\left(\mathrm{AT}\right)}}{\sqrt{{k}_{2\left(\mathrm{AT}\right)}}}$ Critical flow number lambdacat ${\mathrm{\rho }}_{\mathrm{nom}}$ $865$ $\frac{\mathrm{kg}}{{m}^{3}}$ Nominal density rhonom ${d}_{\mathrm{leak}}$ $1.67·{10}^{-5}$ $m$ Diameter of equivalent orifice to model leakage of closed valve dleak $\mathrm{Transition}$ $1$ Transition model Transition