# Variable-Displacement Pump

Variable-displacement bidirectional hydraulic pump

Pumps and Motors

## Description

The Variable-Displacement Pump block represents a variable-displacement bidirectional pump of any type as a data-sheet-based model. The pump delivery is proportional to the control signal provided through the physical signal port C. The pump efficiency is determined based on volumetric and total efficiencies, nominal pressure, and angular velocity. All these parameters are generally provided in the data sheets or catalogs.

Two block parameterization options are available:

• By the pump maximum displacement and stroke — The displacement is assumed to be linearly dependent on the control member position.

• By table-specified relationship between the control member position and pump displacement — The displacement is determined by one-dimensional table lookup based on the control member position. You have a choice of three interpolation methods and two extrapolation methods.

The variable-displacement pump is represented with the following equations:

$q=D\cdot \omega -{k}_{leak}\cdot p$

$T=D\cdot p/{\eta }_{mech}$

${k}_{leak}=\frac{{k}_{HP}}{\nu \cdot \rho }$

${k}_{HP}=\frac{{D}_{\mathrm{max}}\cdot {\omega }_{nom}\left(1-{\eta }_{V}\right)\cdot {\nu }_{nom}\cdot {\rho }_{nom}}{{p}_{nom}}$

$p={p}_{P}-{p}_{T}$

where

 q Pump delivery p Pressure differential across the pump pP,pT Gauge pressures at the block terminals D Pump instantaneous displacement Dmax Pump maximum displacement x Control member displacement xmax Control member maximum stroke T Torque at the pump driving shaft ω Pump angular velocity kleak Leakage coefficient kHP Hagen-Poiseuille coefficient ηV Pump volumetric efficiency ηmech Pump mechanical efficiency ν Fluid kinematic viscosity ρ Fluid density ρnom Nominal fluid density pnom Pump nominal pressure ωnom Pump nominal angular velocity νnom Nominal fluid kinematic viscosity

The leakage flow is determined based on the assumption that it is linearly proportional to the pressure differential across the pump and can be computed by using the Hagen-Poiseuille formula

$p=\frac{128\mu l}{\pi {d}^{4}}{q}_{leak}=\frac{\mu }{{k}_{HP}}{q}_{leak}$

where

 qleak Leakage flow d, l Geometric parameters of the leakage path μ Fluid dynamic viscosity, μ = ν.ρ

The leakage flow at p = pnom and ν = νnom can be determined from the catalog data

${q}_{leak}={D}_{\mathrm{max}}\cdot {\omega }_{nom}\left(1-{\eta }_{V}\right)$

which provides the formula to determine the Hagen-Poiseuille coefficient

${k}_{HP}=\frac{{D}_{\mathrm{max}}\cdot {\omega }_{nom}\left(1-{\eta }_{V}\right)\cdot {\nu }_{nom}\cdot {\rho }_{nom}}{{p}_{nom}}$

The pump mechanical efficiency is not usually available in data sheets, therefore it is determined from the total and volumetric efficiencies by assuming that the hydraulic efficiency is negligibly small

${\eta }_{mech}={\eta }_{total}/{\eta }_{V}$

The block positive direction is from port T to port P. This means that the pump transfers fluid from T to P as its driving shaft S rotates in the globally assigned positive direction and a positive signal is applied to port C.

## Basic Assumptions and Limitations

• Fluid compressibility is neglected.

• No loading on the pump shaft, such as inertia, friction, spring, and so on, is considered.

• Leakage inside the pump is assumed to be linearly proportional to its pressure differential.

## Dialog Box and Parameters

Model parameterization

Select one of the following block parameterization options:

• By maximum displacement and control member stroke — Provide values for maximum pump displacement and maximum control member stroke. The displacement is assumed to be linearly dependent on the control member position. This is the default method.

• By displacement vs. control member position table — Provide tabulated data of pump displacements and control member positions. The displacement is determined by one-dimensional table lookup. You have a choice of three interpolation methods and two extrapolation methods.

Maximum displacement

Pump maximum displacement. The default value is 5e-6 m^3/rad.

Maximum stroke

Maximum control member stroke. The default value is 0.005 m. This parameter is visible if Model parameterization is set to By maximum displacement and control member stroke.

Pump displacements table

Specify the vector of pump displacements as a one-dimensional array. The pump displacements vector must be of the same size as the control member positions vector. The default values, in m^3/rad, are [-5e-06 -3e-06 0 3e-06 5e-06]. This parameter is visible if Model parameterization is set to By displacement vs. control member position table.

Control member positions table

Specify the vector of input values for control member position as a one-dimensional array. The input values vector must be strictly increasing. The values can be nonuniformly spaced. The minimum number of values depends on the interpolation method: you must provide at least two values for linear interpolation, at least three values for cubic or spline interpolation. The default values, in meters, are [-0.0075 -0.0025 0 0.0025 0.0075]. This parameter is visible if Model parameterization is set to By displacement vs. control member position table.

Interpolation method

Select one of the following interpolation methods for approximating the output value when the input value is between two consecutive grid points:

• Linear — Uses a linear interpolation function.

• Cubic — Uses the Piecewise Cubic Hermite Interpolation Polynomial (PCHIP).

• Spline — Uses the cubic spline interpolation algorithm.

For more information on interpolation algorithms, see the PS Lookup Table (1D) block reference page. This parameter is visible if Model parameterization is set to By displacement vs. control member position table.

Extrapolation method

Select one of the following extrapolation methods for determining the output value when the input value is outside the range specified in the argument list:

• From last 2 points — Extrapolates using the linear method (regardless of the interpolation method specified), based on the last two output values at the appropriate end of the range. That is, the block uses the first and second specified output values if the input value is below the specified range, and the two last specified output values if the input value is above the specified range.

• From last point — Uses the last specified output value at the appropriate end of the range. That is, the block uses the last specified output value for all input values greater than the last specified input argument, and the first specified output value for all input values less than the first specified input argument.

For more information on extrapolation algorithms, see the PS Lookup Table (1D) block reference page. This parameter is visible if Model parameterization is set to By displacement vs. control member position table.

Volumetric efficiency

Pump volumetric efficiency specified at nominal pressure, angular velocity, and fluid viscosity. The default value is 0.85.

Total efficiency

Pump total efficiency, which is determined as a ratio between the hydraulic power at the pump outlet and mechanical power at the driving shaft at nominal pressure, angular velocity, and fluid viscosity. The default value is 0.75.

Nominal pressure

Pressure differential across the pump, at which both the volumetric and total efficiencies are specified. The default value is 1e7 Pa.

Nominal angular velocity

Angular velocity of the driving shaft, at which both the volumetric and total efficiencies are specified. The default value is 188 rad/s.

Nominal kinematic viscosity

Working fluid kinematic viscosity, at which both the volumetric and total efficiencies are specified. The default value is 18 cSt.

Nominal fluid density

Working fluid density, at which both the volumetric and total efficiencies are specified. The default value is 900 kg/m^3.

## Global Parameters

Parameter determined by the type of working fluid:

• Fluid kinematic viscosity

Use the Hydraulic Fluid block or the Custom Hydraulic Fluid block to specify the fluid properties.

## Ports

The block has the following ports:

T

Hydraulic conserving port associated with the pump suction, or inlet.

P

Hydraulic conserving port associated with the pump outlet.

C

Physical signal port that controls pump displacement.

S

Mechanical rotational conserving port associated with the pump driving shaft.