# Variable-Displacement Motor

Variable-displacement bidirectional hydraulic motor

Pumps and Motors

## Description

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

Two block parameterization options are available:

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

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

`$q=D\cdot \omega -{k}_{leak}\cdot p$`
`$T=D\cdot p\cdot {\eta }_{mech}$`

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

where

 `q` Motor flow rate `p` Pressure differential across the motor `pA,pB` Gauge pressures at the block terminals `D` Motor instantaneous displacement `Dmax` Motor maximum displacement `x` Control member displacement `xmax` Control member maximum stroke `T` Torque at the motor output shaft ω Output shaft angular velocity `kleak` Leakage coefficient `kHP` Hagen-Poiseuille coefficient ηV Motor volumetric efficiency ηmech Motor mechanical efficiency ν Fluid kinematic viscosity ρ Fluid density ρnom Nominal fluid density `pnom` Motor nominal pressure ωnom Motor 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 motor 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 A to port B. This means that the motor rotates its shaft in the globally assigned positive direction if the fluid flows from port A to port B and a positive signal is applied to port C.

## Basic Assumptions and Limitations

• Fluid compressibility is neglected.

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

• Leakage inside the motor 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 motor displacement and maximum 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 motor displacements and control member positions. The displacement is determined by one-dimensional table lookup. You have a choice of two interpolation methods and two extrapolation methods.

Maximum displacement

Motor 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`.

Motor displacements table

Specify the vector of motor displacements as a one-dimensional array. The motor 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 smooth 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` — Select this option to get the best performance.

• `Smooth` — Select this option to produce a continuous curve with continuous first-order derivatives.

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:

• `Linear` — Select this option to produce a curve with continuous first-order derivatives in the extrapolation region and at the boundary with the interpolation region.

• `Nearest` — Select this option to produce an extrapolation that does not go above the highest point in the data or below the lowest point in the data.

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

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

Total efficiency

Motor total efficiency, which is determined as a ratio between the hydraulic power at the motor inlet and mechanical power at the output shaft at nominal pressure, angular velocity, and fluid viscosity. The default value is `0.75`.

Nominal pressure

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

Nominal angular velocity

Angular velocity of the output 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:

`A`

Hydraulic conserving port associated with the motor inlet.

`B`

Hydraulic conserving port associated with the motor outlet.

`C`

Physical signal port that controls motor displacement.

`S`

Mechanical rotational conserving port associated with the motor output shaft.