# Variable-Displacement Motor (TL)

Variable-displacement bidirectional thermal liquid motor

Libraries:
Simscape / Fluids / Thermal Liquid / Pumps & Motors

## Description

The Variable-Displacement Motor block represents a device that extracts power from a thermal liquid network and delivers it to a mechanical rotational network. The motor displacement varies during simulation according to the physical signal input specified at port D.

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 eight modes of operation. The working mode depends on the pressure gain from port A to port B, Δp = pBpA; the angular velocity, ω = ωRωC; and the fluid volumetric displacement at port D. The figure above maps these modes to the octants of a Δp-ω-D chart:

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

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

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

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

• Mode 5, Reverse Pump: Positive shaft angular velocity causes a pressure increase from port B to port A and flow from B to A.

• Mode 6, Forward Motor: Flow from port A to port B causes a pressure decrease from A to B and a positive shaft angular velocity.

• Mode 7, Forward Pump: Negative shaft angular velocity causes a pressure increase from port A to port B and flow from A to B.

• Mode 8, Reverse Motor: Flow from B to A causes a pressure decrease from B to A and positive 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.

### 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 specified at physical signal port D.

• ω is the shaft angular velocity.

• Δp is the pressure drop from inlet to outlet.

### Leakage and friction parameterization

You can parameterize leakage and friction analytically, using tabulated efficiencies or losses, or by input efficiencies or input losses.

Analytical

When you set Leakage and friction parameterization to `Analytical` 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

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

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.

• k is the friction torque vs. pressure gain coefficient at nominal displacement, which is determined from the , ηm,nom:

`$k=\frac{{\tau }_{fr,nom}-{\tau }_{0}}{\Delta {p}_{nom}}.$`

τfr,nom is the friction torque at nominal conditions:

`${\tau }_{fr,nom}=\left(1-{\eta }_{m,nom}\right){D}_{nom}\Delta {p}_{nom}.$`

• DNom is the specified value of the Nominal Displacement 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.

• 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.

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

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

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.

• 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.

Tabulated Efficiencies

When you set Leakage and friction parameterization to ```Tabulated data - volumetric and mechanical efficiencies```, the leakage flow rate is

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

and the friction torque is

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

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 tangent function

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

where:

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

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

• DThreshold 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 ΔpɷD domain via the Volumetric efficiency table block parameter. When operating in pump mode (octants 1 and 3 of the ΔpɷD chart shown in the Operation Modes figure), the leakage flow rate is:

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

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

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

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

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

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

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

Tabulated Losses

When you set Leakage and friction parameterization to ```Tabulated data - volumetric and mechanical losses```, the leakage (volumetric) flow rate is specified directly in tabulated form over the ΔpɷD domain:

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

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:

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

where qLeak(Δp,ω,D) and τFriction(Δp,ω,D) 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.

Input Efficiencies

When you set Leakage and friction parameterization to ```Input signal - volumetric and mechanical efficiencies```, the leakage flow rate and friction torque calculations are identical to the ```Tabulated data - volumetric and mechanical efficiencies``` setting. The volumetric and mechanical efficiency lookup tables are replaced with physical signal inputs that you specify through ports EV and EM.

The efficiencies are positive quantities with value between `0` and `1`. Input values outside of these bounds are set equal to the nearest bound (`0` for inputs smaller than `0`, `1` for inputs greater than `1`). The efficiency signals are saturated at the Minimum volumetric efficiency or Minimum mechanical efficiency and Maximum volumetric efficiency or Maximum mechanical efficiency .

Input Losses

When you set Leakage and friction parameterization to ```Input signal - volumetric and mechanical efficiencies```, the leakage flow rate and friction torque calculations are identical to the ```Tabulated data - volumetric and mechanical efficiencies``` setting. The volumetric and mechanical loss lookup tables are replaced with physical signal inputs that you specify through ports LV and LM.

The block expects the inputs to be positive. It 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.

### Energy Balance

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

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

where:

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

• Phydro is a function of the pressure difference between motor ports: ${P}_{hydro}=\Delta p\frac{\stackrel{˙}{m}}{\rho }$.

The motor mechanical power is generated due to torque, τ, and angular velocity, ω:

`${P}_{mech}=\tau \omega .$`

### Assumptions

• Fluid compressibility is negligible.

• Loading on the motor shaft due to inertia, friction, and spring forces is negligible.

## Ports

### Input

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Physical signal input port for the volume of fluid displaced per rotation. A smoothing function eases the transition between positive and negative input values.

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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Method to compute flow-rate and torque losses due to internal leaks and friction. When you select `Analytical`, the block parameters are generally available from component data sheets. When you select ```Tabulated data - volumetric and mechanical efficiencies``` or ```Tabulated data - volumetric and mechanical losses```, the block uses lookup tables to map pressure drop, angular velocity, and displacement to component efficiencies or losses.

When you select ```Input signal - volumetric and mechanical efficiencies``` or ```Input signal - volumetric and mechanical losses```, the block performs the leakage flow rate and friction torque calculations the same as the ```Tabulated data - volumetric and mechanical efficiencies``` or ```Tabulated data - volumetric and mechanical losses``` settings, respectively. When you select ```Input signal - volumetric and mechanical efficiencies``` the block enables the physical signal ports, EV and EM. You use these ports to specify the volumetric and mechanical efficiency. When you select ```Input signal - volumetric and mechanical losses``` the block enables the physical signal ports, LV and LM. You use these ports to specify the volumetric and mechanical losses.

Fluid displacement for a given volumetric efficiency. These values are typically available at standard operating conditions in the manufacturer data sheet. The block uses this parameter to calculate the leakage flow rate and friction torque.

Angular velocity of the rotary shaft that corresponds to the given volumetric efficiency. These values are typically available at standard operating conditions in the manufacturer data sheet. The block uses this parameter to calculate the leakage flow rate and friction torque.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to `Analytical`.

Pressure drop that corresponds to the given volumetric efficiency. These values are typically available at standard operating conditions in manufacturer data sheet. The block uses this parameter to calculate the internal leakage flow rate.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to `Analytical`.

Dynamic viscosity of the hydraulic fluid for the given volumetric efficiency. These values are typically available at standard operating conditions in manufacturer data sheet. The block uses this parameter to calculate the internal leakage flow rate.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to `Analytical`.

Volumetric efficiency for the given conditions. The block defines the volumetric efficiency as the ratio of actual to ideal volumetric flow rates. These values are typically available at standard operating conditions in manufacturer data sheet. The block uses this parameter to calculate the internal leakage flow rate.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to `Analytical`.

Ratio of actual mechanical power to ideal mechanical power at nominal conditions.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to `Analytical`.

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

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to `Analytical`.

Pressure drops for the corresponding tabular efficiency data. The vector must be at least two elements in strictly increasing order.

You can specify the data for a single octant, (ɷ, Δp, D). Refer to the block description for the operation modes corresponding to the various octant.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to ```Tabulated data - volumetric and mechanical efficiencies```.

Shaft angular velocities for the corresponding tabular efficiency data. The vector must be at least two elements in strictly increasing order.

You can specify the data for a single octant, (ɷ, Δp,D). Refer to the block description for the operation modes corresponding to the various octants.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to ```Tabulated data - volumetric and mechanical efficiencies``` or ```Tabulated data - volumetric and mechanical losses```.

Displacements at which to specify the efficiency tabular data. The vector must be at least two elements in strictly increasing order.

You can specify the data for a single octant, (ɷ, Δp, D). Refer to the block description for the operation modes corresponding to the various octants.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to ```Tabulated data - volumetric and mechanical efficiencies```.

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

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

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

• L is the number of vector elements in the Displacement vector, D parameter.

You can specify the data for a single octant, (ɷ, Δp, D). Refer to the block description for the operation modes corresponding to the various octants.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to ```Tabulated data - volumetric and mechanical efficiencies```.

Mechanical efficiencies corresponding to the specified fluid pressure drops, shaft angular velocities, and displacements. The efficiencies must be in the range of `0``1`. M, N, and L are the sizes of the specified lookup-table vectors:

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

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

• L is the number of vector elements in the Displacement vector, D parameter.

You can specify the data for a single octant, (ɷ, Δp, D). Refer to the block description for the operation modes corresponding to the various octants.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to ```Tabulated data - volumetric and mechanical efficiencies```.

Pressure drops for the corresponding tabular loss data. The vector must be at least two elements in strictly increasing order.

You can specify the data for a single octant, (ɷ, Δp, D). Refer to the block description for the operation modes corresponding to the various octants.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to ```Tabulated data - volumetric and mechanical losses```.

Displacements at which to specify the loss tabular data. The vector must be at least two elements in strictly increasing order.

You can specify the data for a single octant, (ɷ, Δp, D). Refer to the block description for the operation modes corresponding to the various octants.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to ```Tabulated data - volumetric and mechanical losses```.

Volumetric losses at the specified fluid pressure drops, shaft angular velocities, and displacements. The block defines volumetric loss 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, dp parameter.

• N is the number of vector elements in the Shaft angular velocity vector, w parameter.

• L is the number of vector elements in the Displacement vector, D parameter.

You can specify the data for a single octant, (ɷ, Δp, D). Refer to the block description for the operation modes corresponding to the various octants.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to ```Tabulated data - volumetric and mechanical losses```.

Mechanical losses for the given pressure drops, shaft angular velocities, and displacements. The block defines mechanical loss 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, dp parameter.

• N is the number of vector elements in the Shaft angular velocity vector, w parameter.

• L is the number of vector elements in the Displacement vector, D parameter.

You can specify the data for a single octant, (ɷ, Δp,D). Refer to the block description for the operation modes corresponding to the various octants. The tabulated data for the mechanical losses must obey the convention in the figure, with positive values at positive angular velocities and negative values at negative angular velocities.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to ```Tabulated data - volumetric and mechanical losses```.

Smallest allowed value of the volumetric efficiency. The input from the physical signal port EV saturates inputs below this value.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to ```Input signal - volumetric and mechanical efficiencies```.

Largest allowed value of the volumetric efficiency. The input from the physical signal port EV saturates inputs above this value.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to ```Input signal - volumetric and mechanical efficiencies```.

Smallest allowed value of the mechanical efficiency. The input from the physical signal port EM saturates inputs below this value.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to ```Input signal - volumetric and mechanical efficiencies```.

Largest allowed value of the mechanical efficiency. The input from the physical signal port EM saturates inputs above this value.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to ```Input signal - volumetric and mechanical efficiencies```.

Pressure drop from inlet to outlet below which the block begins to transition between motoring and pumping modes. The block uses a hyperbolic tangent function to smooth the leakage flow rate and friction torque.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to ```Input signal - volumetric and mechanical efficiencies``` or ```Input signal - volumetric and mechanical losses```.

Shaft angular velocity below which the block begins to transition between motoring and pumping modes. The block uses the hyperbolic tangent function to smooth the leakage flow rate and friction torque.

#### Dependencies

To enable this parameter, set Leakage and friction parameterization to ```Input signal - volumetric and mechanical efficiencies``` or ```Input signal - volumetric and mechanical losses```.

Flow area at the component inlet and outlet. The areas are assumed equal.

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

To enable this parameter, set Leakage and friction parameterization to:

• ```Tabulated data - volumetric and mechanical efficiencies```

• ```Tabulated data - volumetric and mechanical losses```

• ```Input signal - volumetric and mechanical efficiencies```

• ```Input signal - volumetric and mechanical losses```

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

## Version History

Introduced in R2017b

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