# Fixed-Displacement Motor (TL)

Hydraulic-mechanical power conversion device

• Library:
• Simscape / Fluids / Thermal Liquid / Pumps & Motors

## Description

The Fixed-Displacement Motor (TL) block represents a device that extracts power from a hydraulic (isothermal liquid) network and delivers it to a mechanical rotational network. The motor displacement is fixed at a constant value that you specify through the Displacement parameter.

Ports A and B represent the motor inlets. Ports R and C represent the motor drive shaft and case. During normal operation, a pressure drop from port A to port B causes a positive flow rate from port A to port B and a positive rotation of the motor shaft relative to the motor case. This operation mode is referred to here as forward motor.

Operation Modes

The block has four modes of operation. The working mode depends on the pressure drop from port A to port B, Δp = pBpA and the angular velocity, ω = ωRωC:

• Mode 1, Forward Motor: Positive shaft angular velocity causes a pressure decrease from port A to port B and flow from port A to port B.

• Mode 2, Reverse Pump: Flow from port B to port A causes a pressure increase from B to A and negative shaft angular velocity.

• Mode 3, Reverse Motor: Negative shaft angular velocity causes a pressure decrease from port B to port A and flow from B to A.

• Mode 4, Forward Pump: Flow from port A to B causes a pressure increase from A to B and negative shaft angular velocity.

The response time of the motor is considered negligible in comparison with the system response time. The motor is assumed to reach steady state nearly instantaneously and is treated as a quasi-steady component.

### Block Variants and Loss Parameterizations

The motor model accounts for power losses due to leakage and friction. Leakage is internal and occurs between the motor inlet and outlet only. The block computes the leakage flow rate and friction torque using your choice of five loss parameterizations. You select a parameterization using block variants and, in the `Analytical or tabulated data` case, the Friction and leakage parameterization parameter.

Loss Parameterizations

The block provides three Simulink® variants to select from. To change the active block variant, right-click the block and select Simscape > Block choices. The available variants are:

• `Analytical or tabulated data` — Obtain the mechanical and volumetric efficiencies or losses from analytical models based on nominal parameters or from tabulated data. Use the Friction and leakage parameterization parameter to select the exact input type.

• `Input efficiencies` — Provide the mechanical and volumetric efficiencies directly through physical signal input ports.

• `Input losses` — Provide the mechanical and volumetric losses directly through physical signal input ports. The mechanical loss is defined as the internal friction torque. The volumetric loss is defined as the internal leakage flow rate.

### Energy Balance

Mechanical work done by the pump is associated with an energy exchange. The governing energy balance equation is:

`${\varphi }_{A}+{\varphi }_{B}-{P}_{mech}=0,$`

where:

• ΦA and ΦB are energy flow rates at ports A and B, respectively.

• Pmech is the mechanical power generated due to torque, τ, and the pump angular velocity, ω: ${P}_{mech}=\tau \omega .$

The pump hydraulic power is a function of the pressure difference between pump ports:

`${P}_{hydro}=\Delta p\frac{\stackrel{˙}{m}}{\rho }$`

### Flow Rate and Driving Torque

The mass flow rate generated at the motor is

`$\stackrel{˙}{m}={\stackrel{˙}{m}}_{\text{Ideal}}+{\stackrel{˙}{m}}_{\text{Leak}},$`

where:

• $\stackrel{˙}{m}$ is the actual mass flow rate.

• ${\stackrel{˙}{m}}_{\text{Ideal}}$ is the ideal mass flow rate.

• ${\stackrel{˙}{m}}_{\text{Leak}}$ is the internal leakage mas flow rate.

The torque generated at the motor is

`$\tau ={\tau }_{\text{Ideal}}-{\tau }_{\text{Friction}},$`

where:

• τ is the actual torque.

• τIdeal is the ideal torque.

• τFriction is the friction torque.

Ideal Flow Rate and Ideal Torque

The ideal mass flow rate is

`${\stackrel{˙}{m}}_{\text{Ideal}}=\rho D\omega ,$`

and the ideal generated torque is

`${\tau }_{\text{Ideal}}=D\Delta p,$`

where:

• ρ is the average of the fluid densities at thermal liquid ports A and B.

• D is the Displacement parameter.

• ω is the shaft angular velocity.

• Δp is the pressure drop from inlet to outlet.

Leakage Flow Rate and Friction Torque

The internal leakage flow rate and friction torque calculations depend on the block variant selected. If the block variant is ```Analytical or tabulated data```, the calculations depend also on the Leakage and friction parameterization parameter setting. There are five possible permutations of block variant and parameterization settings.

Case 1: Analytical Efficiency Calculation

If the active block variant is ```Analytical or tabulated data``` and the Leakage and friction parameterization parameter is set to `Analytical`, the leakage flow rate is

`${\stackrel{˙}{m}}_{\text{Leak}}=\frac{{K}_{\text{HP}}{\rho }_{\text{Avg}}\Delta p}{{\mu }_{\text{Avg}}},$`

and the friction torque is

`${\tau }_{\text{Friction}}=\left({\tau }_{0}+{K}_{\text{TP}}|\Delta p|\text{tanh}\frac{4\omega }{\left(5e-5\right){\omega }_{\text{Nom}}}\right),$`

where:

• KHP is the Hagen-Poiseuille coefficient for laminar pipe flows. This coefficient is computed from the specified nominal parameters.

• μ is the dynamic viscosity of the thermal liquid, taken here as the average of its values at the thermal liquid ports.

• KTP is the specified value of the Friction torque vs pressure drop coefficient block parameter.

• τ0 is the specified value of the No-load torque block parameter.

• ωNom is the specified value of the Nominal shaft angular velocity block parameter.

The Hagen-Poiseuille coefficient is determined from nominal fluid and component parameters through the equation

`${K}_{\text{HP}}=\frac{D{\omega }_{\text{Nom}}{\mu }_{\text{Nom}}\left(\frac{1}{{\eta }_{\text{v,Nom}}}-1\right)}{\Delta {p}_{\text{Nom}}},$`

where:

• ωNom is the specified value of the Nominal shaft angular velocity parameter. This is the angular velocity at which the nominal volumetric efficiency is specified.

• μNom is the specified value of the Nominal Dynamic viscosity block parameter. This is the dynamic viscosity at which the nominal volumetric efficiency is specified.

• ΔpNom is the specified value of the Nominal pressure drop block parameter. This is the pressure drop at which the nominal volumetric efficiency is specified.

• ηv,Nom is the specified value of the Volumetric efficiency at nominal conditions block parameter. This is the volumetric efficiency corresponding to the specified nominal conditions.

Case 2: Efficiency Tabulated Data

If the active block variant is ```Analytical or tabulated data``` and the Leakage and friction parameterization parameter is set to ```Tabulated data — volumetric and mechanical efficiencies```, the leakage flow rate is

`${\stackrel{˙}{m}}_{\text{Leak}}={\stackrel{˙}{m}}_{\text{Leak,Motor}}\frac{\left(1+\alpha \right)}{2}+{\stackrel{˙}{m}}_{\text{Leak,Pump}}\frac{\left(1-\alpha \right)}{2},$`

and the friction torque is

`${\tau }_{\text{Friction}}={\tau }_{\text{Friction,Motor}}\frac{1+\alpha }{2}+{\tau }_{\text{Friction,Pump}}\frac{1-\alpha }{2},$`

where:

• α is a numerical smoothing parameter for the motor-pump transition.

• ${\stackrel{˙}{m}}_{\text{Leak,Motor}}$ is the leakage flow rate in motor mode.

• ${\stackrel{˙}{m}}_{\text{Leak,Pump}}$ is the leakage flow rate in pump mode.

• τFriction,Motor is the friction torque in motor mode.

• τFriction,Pump is the friction torque in pump mode.

The smoothing parameter α is given by the hyperbolic function

`$\alpha =\text{tanh}\left(\frac{4\Delta p}{\Delta {p}_{\text{Threshold}}}\right)·\text{tanh}\left(\frac{4\omega }{{\omega }_{\text{Threshold}}}\right),$`

where:

• ΔpThreshold is the specified value of the Pressure drop threshold for motor-pump transition block parameter.

• ωThreshold is the specified value of the Angular velocity threshold for motor-pump transition block parameter.

The leakage flow rate is calculated from the volumetric efficiency, a quantity that is specified in tabulated form over the Δpɷ domain via the Volumetric efficiency table block parameter. When operating in motor mode (quadrants 1 and 3 of the Δpɷ chart shown in the Operation Modes figure), the leakage flow rate is:

`${\stackrel{˙}{m}}_{\text{Leak,Motor}}=\left(1-{\eta }_{\text{v}}\right)\stackrel{˙}{m},$`

where ηv is the volumetric efficiency, obtained either by interpolation or extrapolation of the tabulated data. Similarly, when operating in pump mode (quadrants 2 and 4 of the Δpɷ chart), the leakage flow rate is:

`${\stackrel{˙}{m}}_{\text{Leak,Pump}}=-\left(1-{\eta }_{\text{v}}\right){\stackrel{˙}{m}}_{\text{Ideal}}.$`

The friction torque is similarly calculated from the mechanical efficiency, a quantity that is specified in tabulated form over the Δpɷ domain via the Mechanical efficiency table block parameter. When operating in motor mode (quadrants 1 and 3 of the Δpɷ chart):

`${\tau }_{\text{Friction,Motor}}=\left(1-{\eta }_{\text{m}}\right){\tau }_{\text{Ideal}},$`

where ηm is the mechanical efficiency, obtained either by interpolation or extrapolation of the tabulated data. Similarly, when operating in pump mode (quadrants 2 and 4 of the Δpɷ chart):

`${\tau }_{\text{Friction,Pump}}=-\left(1-{\eta }_{\text{m}}\right)\tau .$`

Case 3: Loss Tabulated Data

If the active block variant is ```Analytical or tabulated data``` and the Leakage and friction parameterization parameter is set to ```Tabulated data — volumetric and mechanical losses```, the leakage (volumetric) flow rate is specified directly in tabulated form over the Δpɷ domain:

`${q}_{\text{Leak}}={q}_{\text{Leak}}\left(\Delta p,\omega \right).$`

The mass flow rate due to leakage is calculated from the volumetric flow rate:

`${\stackrel{˙}{m}}_{\text{Leak}}=\rho {q}_{\text{Leak}}.$`

The friction torque is similarly specified in tabulated form:

`${\tau }_{\text{Friction}}={\tau }_{\text{Friction}}\left(\Delta p,\omega \right),$`

where qLeak(Δp,ω) and τFriction(Δp,ω) are the volumetric and mechanical losses, obtained through interpolation or extrapolation of the tabulated data specified via the Volumetric loss table and Mechanical loss table block parameters.

Case 4: Efficiency Physical Signal Inputs

If the active block variant is `Input efficiencies`, the leakage flow rate and friction torque calculations are as described for efficiency tabulated data (case 2). The volumetric and mechanical efficiency lookup tables are replaced with physical signal inputs that you specify through ports EV and EM.

The efficiencies are defined as positive quantities with value between zero and one. Input values outside of these bounds are set equal to the nearest bound (zero for inputs smaller than zero, one for inputs greater than one). In other words, the efficiency signals are saturated at zero and one.

Case 5: Loss Physical Signal Inputs

If the block variant is `Input losses`, the leakage flow rate and friction torque calculations are as described for loss tabulated data (case 3). The volumetric and mechanical loss lookup tables are replaced with physical signal inputs that you specify through ports LV and LM.

The signs of the inputs are ignored. The block sets the signs automatically from the operating conditions established during simulation—more precisely, from the Δpɷ quadrant in which the component happens to be operating. In other words, whether an input is positive or negative is irrelevant to the block.

### Assumptions and Limitations

• The motor is treated as a quasi-steady component.

• The effects of fluid inertia and elevation are ignored.

• The motor wall is rigid.

• External leakage is ignored.

## Ports

### Input

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Physical signal input port for the volumetric efficiency coefficient. The input signal has an upper bound at the Maximum volumetric efficiency parameter value and a lower bound at the Minimum volumetric efficiency parameter value.

#### Dependencies

This port is exposed only when the block variant is set to `Input efficiencies`.

Physical signal input port for the mechanical efficiency coefficient. The input signal has an upper bound at the Maximum mechanical efficiency parameter value and a lower bound at the Minimum mechanical efficiency parameter value.

#### Dependencies

This port is exposed only when the block variant is set to `Input efficiencies`.

Physical signal input port for the volumetric loss, defined as the internal leakage flow rate between the motor inlets.

#### Dependencies

This port is exposed only when the block variant is set to `Input losses`.

Physical signal input port for the mechanical loss, defined as the friction torque on the rotating motor shaft.

#### Dependencies

This port is exposed only when the block variant is set to `Input losses`.

### Conserving

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Thermal liquid conserving port representing the motor inlet.

Thermal liquid conserving port representing the motor outlet.

Mechanical rotational conserving port representing the motor case.

Mechanical rotational conserving port representing the rotational motor shaft.

## Parameters

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The exposed block parameters depend on the active block variant. See Block Variants and Loss Parameterizations.

Variant 1: `Analytical or tabulated data`

Fluid volume displaced per unit shaft rotation angle. The displacement is fixed at this value during simulation. The specified value must be greater than zero.

Parameterization used to compute flow-rate and torque losses due to internal leaks and friction. The `Analytical` parameterization relies on nominal parameters generally available from component data sheets. The remaining, tabular, options rely on lookup tables to map pressure drop, angular velocity, and displacement to component efficiencies or losses.

Angular velocity of the rotary shaft at which the component’s volumetric efficiency is known. Nominal parameters are typically published for standard operating conditions in manufacturer’s data sheets. The block uses this parameter to calculate, using simple linear functions, the leakage flow rate and friction torque.

#### Dependencies

This parameter is enabled when the Leakage and friction parameterization parameter is set to `Analytical`.

Pressure drop from inlet to outlet at which the component’s volumetric efficiency is known. Nominal parameters are typically published for standard operating conditions in manufacturer’s data sheets. The block uses this parameter to calculate, using a simple linear function, the internal leakage flow rate.

#### Dependencies

This parameter is enabled when the Leakage and friction parameterization parameter is set to `Analytical`.

Dynamic viscosity of the hydraulic fluid at which the component’s volumetric efficiency is known. Nominal parameters are typically published for standard operating conditions in manufacturer’s data sheets. The block uses this parameter to calculate, using a simple linear function, the internal leakage flow rate.

#### Dependencies

This parameter is enabled when the Leakage and friction parameterization parameter is set to `Analytical`.

Volumetric efficiency, defined as the ratio of actual to ideal volumetric flow rates, at the specified nominal conditions. Nominal parameters are typically published for standard operating conditions in manufacturer’s data sheets. The block uses this parameter to calculate, using a simple linear function, the internal leakage flow rate.

#### Dependencies

This parameter is enabled when the Leakage and friction parameterization parameter is set to `Analytical`.

Torque required to overcome seal friction and induce rotation of the mechanical shaft. This torque is the load-independent component of the total friction torque.

#### Dependencies

This parameter is enabled when the Leakage and friction parameterization parameter is set to `Analytical`.

Proportionality constant at nominal displacement between the friction torque on the mechanical shaft and the pressure drop from inlet to outlet.

#### Dependencies

This parameter is enabled when the Leakage and friction parameterization parameter is set to `Analytical`.

Flow area at the component inlet and outlet. The areas are assumed equal. This parameter must be greater than zero.

M-element vector of pressure drops at which to specify the efficiency tabular data. The vector size, M, must be two or greater. The vector elements need not be uniformly spaced. However, they must monotonically increase in value from left to right.

The tabulated data need not encompass all quadrants of operation—those of a (ɷ, Δp) chart. It suffices to specify the data for a single quadrant. Refer to the block description for the operation modes corresponding to the various quadrants.

#### Dependencies

This parameter is enabled when the Leakage and friction parameterization parameter is set to ```Tabulated data — volumetric and mechanical efficiencies```.

N-element vector of shaft angular velocities at which to specify the efficiency tabular data. The vector size, N, must be two or greater. The vector elements need not be uniformly spaced. However, they must monotonically increase in value from left to right.

The tabulated data need not encompass all quadrants of operation—those of a (ɷ, Δp) chart. It suffices to specify the data for a single quadrant. Refer to the block description for the operation modes corresponding to the various quadrants.

#### Dependencies

This parameter is enabled when the Leakage and friction parameterization parameter is set to ```Tabulated data — volumetric and mechanical efficiencies```.

M-by-N matrix with the volumetric efficiencies at the specified fluid pressure drops, shaft angular velocities, and displacements. The efficiencies must be in the range of `0``1`. M and N are the sizes of the specified lookup-table vectors:

• M is the number of vector elements in the Pressure drop vector for efficiencies, dp parameter.

• N is the number of vector elements in the parameter.

The tabulated data need not encompass all quadrants of operation—those of a (ɷ, Δp) chart. It suffices to specify the data for a single quadrant. Refer to the block description for the operation modes corresponding to the various quadrants.

#### Dependencies

This parameter is enabled when the Leakage and friction parameterization parameter is set to ```Tabulated data — volumetric and mechanical efficiencies```.

M-by-N matrix with the mechanical efficiencies corresponding to the specified fluid pressure drops and shaft angular velocities. The efficiencies must be in the range of `0``1`. M and N are the sizes of the specified lookup-table vectors:

• M is the number of vector elements in the Pressure drop vector for efficiencies, dp parameter.

• N is the number of vector elements in the parameter.

The tabulated data need not encompass all quadrants of operation—those of a (ɷ, Δp) chart. It suffices to specify the data for a single quadrant. Refer to the block description for the operation modes corresponding to the various quadrants.

#### Dependencies

This parameter is enabled when the Leakage and friction parameterization parameter is set to ```Tabulated data — volumetric and mechanical efficiencies```.

Pressure drop from inlet to outlet below which the component begins to transition between motoring and pumping modes. A hyperbolic `Tanh` function transforms the leakage flow rate and friction torque such that the transition is continuous and smooth.

#### Dependencies

This parameter is enabled when the Leakage and friction parameterization parameter is set to ```Tabulated data — volumetric and mechanical efficiencies```.

Shaft angular velocity below which the component begins to transition between motoring and pumping modes. A hyperbolic `Tanh` function transforms the leakage flow rate and friction torque such that the transition is continuous and smooth.

#### Dependencies

This parameter is enabled when the Leakage and friction parameterization parameter is set to ```Tabulated data — volumetric and mechanical efficiencies```.

Simulation warning mode for operating conditions outside the range of tabulated data. Select `Warning` to be notified when the fluid pressure drop, shaft angular velocity, or instantaneous displacement cross outside the specified tabular data. The warning does not cause simulation to stop.

#### Dependencies

This parameter is enabled when the Leakage and friction parameterization parameter is set to ```Tabulated data — volumetric and mechanical efficiencies``` or ```Tabulated data — volumetric and mechanical losses```.

M-element vector of pressure drops at which to specify the loss tabular data. The vector size, M, must be two or greater. The vector elements need not be uniformly spaced. However, they must monotonically increase in value from left to right.

The tabulated data need not encompass all quadrants of operation—those of a (ɷ, Δp) chart. It suffices to specify the data for a single quadrant. Refer to the block description for the operation modes corresponding to the various quadrants.

#### Dependencies

This parameter is enabled when the Leakage and friction parameterization parameter is set to ```Tabulated data — volumetric and mechanical losses```.

N-element vector of shaft angular velocities at which to specify the loss tabular data. The vector size, N, must be two or greater. The vector elements need not be uniformly spaced. However, they must monotonically increase in value from left to right.

The tabulated data need not encompass all quadrants of operation—those of a (ɷ, Δp) chart. It suffices to specify the data for a single quadrant. Refer to the block description for the operation modes corresponding to the various quadrants.

#### Dependencies

This parameter is enabled when the Leakage and friction parameterization parameter is set to ```Tabulated data — volumetric and mechanical losses```.

M-by-N matrix with the volumetric losses at the specified fluid pressure drops and shaft angular velocities. Volumetric loss is defined here as the internal leakage volumetric flow rate between port A and port B. M and N are the sizes of the specified lookup-table vectors:

• M is the number of vector elements in the Pressure drop vector for losses, dp parameter.

• N is the number of vector elements in the parameter.

The tabulated data need not encompass all quadrants of operation—those of a (ɷ, Δp) chart. It suffices to specify the data for a single quadrant. Refer to the block description for the operation modes corresponding to the various quadrants. The tabulated data for the volumetric losses must obey the convention shown in the figure, with positive values at positive pressure drops and negative values at negative pressure drops.

#### Dependencies

This parameter is enabled when the Leakage and friction parameterization parameter is set to ```Tabulated data — volumetric and mechanical losses```.

M-by-N matrix with the mechanical losses at the specified fluid pressure drops and shaft angular velocities. Mechanical loss is defined here as the friction torque due to seals and internal components. M and N are the sizes of the specified lookup-table vectors:

• M is the number of vector elements in the Pressure drop vector for losses, dp parameter.

• N is the number of vector elements in the parameter.

The tabulated data need not encompass all quadrants of operation—those of a (ɷ, Δp) chart. It suffices to specify the data for a single quadrant. Refer to the block description for the operation modes corresponding to the various quadrants. The tabulated data for the mechanical losses must obey the convention shown in the figure, with positive values at positive angular velocities and negative values at negative angular velocities.

#### Dependencies

This parameter is enabled when the Leakage and friction parameterization parameter is set to ```Tabulated data — volumetric and mechanical losses```.

Variant 2: `Input efficiencies`

Fluid volume displaced per unit shaft rotation angle. The displacement is fixed at this value during simulation. The specified value must be greater than zero.

Smallest allowed value of the volumetric efficiency. The input from physical signal port EV saturates at the specified value. If the input signal falls below the minimum volumetric efficiency, the volumetric efficiency is set to the minimum volumetric efficiency.

#### Dependencies

This parameter is enabled when the block variant is set to `Input efficiencies`.

Largest allowed value of the volumetric efficiency. The input from physical signal port EV saturates at the specified value. If the input signal rises above the maximum volumetric efficiency, the volumetric efficiency is set to the maximum volumetric efficiency.

#### Dependencies

This parameter is enabled when the block variant is set to `Input efficiencies`.

Smallest allowed value of the mechanical efficiency. The input from physical signal port EM saturates at the specified value. If the input signal falls below the minimum mechanical efficiency, the mechanical efficiency is set to the minimum mechanical efficiency.

#### Dependencies

This parameter is enabled when the block variant is set to `Input efficiencies`.

Largest allowed value of the mechanical efficiency. The input from physical signal port EM saturates at this value. If the input signal rises above the maximum mechanical efficiency, the mechanical efficiency is set to the maximum mechanical efficiency.

#### Dependencies

This parameter is enabled when the block variant is set to `Input efficiencies`.

Pressure drop from inlet to outlet below which the component begins to transition between motoring and pumping modes. A hyperbolic `Tanh` function transforms the leakage flow rate and friction torque such that the transition is continuous and smooth.

#### Dependencies

This parameter is enabled when the block variant is set to `Input efficiencies`.

Shaft angular velocity below which the component begins to transition between motoring and pumping modes. A hyperbolic `Tanh` function transforms the leakage flow rate and friction torque such that the transition is continuous and smooth.

#### Dependencies

This parameter is enabled when the block variant is set to `Input efficiencies`.

Flow area at the component inlet and outlet. The areas are assumed equal. This parameter must be greater than zero.

Variant 3: `Input losses`

Fluid volume displaced per unit shaft rotation angle. The displacement is fixed at this value during simulation. The specified value must be greater than zero.

Pressure drop from inlet to outlet below which the component begins to transition between motoring and pumping modes. A hyperbolic `Tanh` function transforms the leakage flow rate and friction torque such that the transition is continuous and smooth.

Shaft angular velocity below which the component begins to transition between motoring and pumping modes. A hyperbolic `Tanh` function transforms the leakage flow rate and friction torque such that the transition is continuous and smooth.

Flow area at the component inlet and outlet. The areas are assumed equal. This parameter must be greater than zero.

Simulation warning mode for operating conditions outside the motoring mode. A warning is issued if the motor transitions to pumping mode. Select `Warning` to be notified when this transition occurs. The warning does not cause simulation to stop.

Variables

Mass of fluid entering the component through the inlet per unit time at the start of simulation.