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Fixed-displacement motor in isothermal liquid system

**Library:**Simscape / Fluids / Isothermal Liquid / Pumps & Motors

The Fixed-Displacement Motor (IL) block models a motor with constant-volume
displacement. The fluid may move from port **A** to port
**B**, called *forward mode*, or from port
**B** to port **A**, called *reverse
mode*. Motor mode operation occurs when there is a pressure drop in the
direction of the flow. Pump mode operation occurs when there is a pressure gain in the
direction of the flow.

Shaft rotation corresponds to the sign of the fluid volume moving through the motor. Positive fluid displacement at corresponds to positive shaft rotation in forward mode. Negative fluid displacement corresponds to negative shaft angular velocity in forward mode.

**Operation Modes**

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 motor block has analytical, lookup table, and physical signal parameterizations. When using tabulated data or an input signal for parameterization, you can choose to characterize the motor operation based on efficiency or losses.

In the tabulated data and the input signal parameterization options, the threshold
parameters **Pressure drop threshold for motor-pump transition** and
**Angular velocity threshold for motor-pump transition** identify
regions where numerically smoothed flow transition between the motor operational modes
can occur. Choose a transition region that provides some margin for the transition term,
but which is small enough relative to the pressure and angular velocity that it will not
impact calculation results.

If you set **Leakage and friction parameterization** to
`Analytical`

, the block calculates leakage and friction
from constant values of shaft velocity, pressure drop, and torque. The leakage flow
rate, which is correlated with the pressure differential over the motor, is
calculated as:

$${\dot{m}}_{leak}=K{\rho}_{avg}\Delta p,$$

where:

*Δp*_{nom}is the**Nominal pressure drop**.*ρ*_{avg}is the average fluid density.*K*is the Hagen-Poiseuille coefficient for analytical loss,$$K=\frac{{D}_{nom}{\omega}_{nom}\left(\frac{1}{{\eta}_{v,}{}_{nom}}-1\right)}{\Delta {p}_{nom}},$$

where:

*D*_{nom}is the**Displacement**.*ω*_{nom}is the**Nominal shaft angular velocity**.*η*_{v, nom}is the**Volumetric efficiency at nominal conditions**.

The torque, which is correlated with shaft angular velocity, is calculated as:

$${\tau}_{fr}=\left({\tau}_{0}+k\left|\Delta p\right|\right)\mathrm{tanh}\left(\frac{4\omega}{5\times {10}^{-5}{\omega}_{nom}}\right),$$

where:

*τ*_{0}is the**No-load torque**.*k*is the**Friction torque vs. pressure drop coefficient**.*Δp*is the pressure drop between ports**A**and**B**.*ω*is the relative shaft angular velocity, or $${\omega}_{R}-{\omega}_{C}$$.

When using tabulated data for motor efficiencies or losses, you can provide data for one or more of the motor operational modes. The signs of the tabulated data determine the operational regime of the block. When data is provided for less than four operational modes, the block calculates the complementing data for the other mode(s) by extending the given data into the remaining quadrants.

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

parameterizationThe leakage flow rate is calculated as:

$${\dot{m}}_{leak}={\dot{m}}_{leak,motor}\left(\frac{1+\alpha}{2}\right)+{\dot{m}}_{leak,pump}\left(\frac{1-\alpha}{2}\right),$$

where:

$${\dot{m}}_{leak,pump}=\left({\eta}_{\upsilon}-1\right){\dot{m}}_{ideal}$$

$${\dot{m}}_{leak,motor}=\left(1-{\eta}_{v}\right)\dot{m}$$

and *η*_{v} is the volumetric efficiency,
which is interpolated from the user-provided tabulated data. The transition
term, *α*, is

$$\alpha =\mathrm{tanh}\left(\frac{4\Delta p}{\Delta {p}_{threshold}}\right)\mathrm{tanh}\left(\frac{4\omega}{{\omega}_{threshold}}\right),$$

where:

*Δp*is*p*_{A}–*p*_{B}.*p*_{threshold}is the**Pressure drop threshold for motor-pump transition**.*ω*is*ω*_{R}–*ω*_{C}.*ω*_{threshold}is the**Angular velocity threshold for motor-pump transition**.

The friction torque is calculated as:

$${\tau}_{fr}={\tau}_{fr,pump}\left(\frac{1+\alpha}{2}\right)+{\tau}_{fr,motor}\left(\frac{1-\alpha}{2}\right),$$

where:

$${\tau}_{fr,pump}=\left({\eta}_{m}-1\right)\tau $$

$${\tau}_{fr,motor}=\left(1-{\eta}_{m}\right){\tau}_{ideal}$$

and *η*_{m} is the
mechanical efficiency, which is interpolated from the user-provided tabulated
data.

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

parameterizationThe leakage flow rate is calculated as:

$${\dot{m}}_{leak}={\rho}_{avg}{q}_{loss}\left(\Delta p,\omega \right),$$

where *q*_{loss} is interpolated from the
**Volumetric loss table, q_loss(dp,w)** parameter, which is
based on user-supplied data for pressure drop, shaft angular velocity, and fluid
volumetric displacement.

The shaft friction torque is calculated as:

$${\tau}_{fr}={\tau}_{loss}\left(\Delta p,\omega \right),$$

where *τ*_{loss} is interpolated from the
**Mechanical loss table, torque_loss(dp,w)** parameter,
which is based on user-supplied data for pressure drop and shaft angular
velocity.

When **Leakage and friction parameterization** is set
to```
Input signal - volumetric and mechanical
efficiencies
```

, ports **EV** and
**EM** are enabled. The internal leakage and shaft friction are
calculated in the same way as the ```
Tabulated data - volumetric and
mechanical efficiencies
```

parameterization, except that
*η*_{v} and
*η*_{m} are received directly at ports
**EV** and **EM**, respectively.

When **Leakage and friction parameterization** is set
to`Input signal - volumetric and mechanical losses`

,
ports **LV** and **LM** are enabled. These ports
receive leakage flow and friction torque as positive physical signals. The leakage
flow rate is calculated as:

$${\dot{m}}_{leak}={\rho}_{avg}{q}_{LV}\mathrm{tanh}\left(\frac{4\Delta p}{{p}_{thresh}}\right),$$

where:

*q*_{LV}is the leakage flow received at port**LV**.*p*_{thresh}is the**Pressure drop threshold for motor-pump transition**parameter.

The friction torque is calculated as:

$${\tau}_{fr}={\tau}_{LM}\mathrm{tanh}\left(\frac{4\omega}{{\omega}_{thresh}}\right),$$

where

*τ*_{LM}is the friction torque received at port**LM**.*ω*_{thresh}is the**Angular velocity threshold for motor-pump transition**parameter.

The volumetric and mechanical efficiencies range between the user-defined specified minimum and maximum values. Any values lower or higher than this range will take on the minimum and maximum specified values, respectively.

The motor flow rate is:

$$\dot{m}={\dot{m}}_{ideal}+{\dot{m}}_{leak},$$

where $${\dot{m}}_{ideal}={\rho}_{avg}D\cdot \omega .$$

The motor torque is:

$$\tau ={\tau}_{ideal}-{\tau}_{fr},$$

where $${\tau}_{ideal}=D\cdot \Delta p.$$

The mechanical power extracted by the motor shaft is:

$${\phi}_{mech}=\tau \omega ,$$

and the motor hydraulic power is:

$${\phi}_{hyd}=\frac{\Delta p\dot{m}}{{\rho}_{avg}}.$$

If you would like to know if the block is operating beyond the
supplied tabulated data, you can set **Check if operating beyond the
quadrants of supplied tabulated data** to
`Warning`

to receive a warning if this occurs, or
`Error`

to stop the simulation when this occurs. For
parameterization by input signal for volumetric or mechanical losses, you can be
notified if the simulation surpasses operating modes with the **Check if
operating outside of motor mode** parameter.

You can also monitor motor functionality. Set **Check if pressures are
less than motor minimum pressure** to
`Warning`

to receive a warning if this occurs, or
`Error`

to stop the simulation when this occurs.

Fixed-Displacement Motor (TL) | Fixed-Displacement Pump (IL) | Variable-Displacement Motor (IL)