# Fixed-Displacement Motor (TL)

Hydraulic-mechanical power conversion device

**Libraries:**

Simscape /
Fluids /
Thermal Liquid /
Pumps & Motors

## Description

The Fixed-Displacement Motor (TL) block represents a device that
extracts power from a thermal 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 =
p*_{A} – *p*_{B}
and the angular velocity, *ω = ω*_{R} –
*ω*_{C}:

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**.

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.

### Energy Balance

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

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

where:

*Φ*_{A}and*Φ*_{B}are energy flow rates at ports**A**and**B**, respectively.*P*_{hydro}is the motor hydraulic power. It is a function of the pressure difference between the motor ports: $${P}_{hydro}=\Delta p\frac{\dot{m}}{\rho}$$

The motor mechanical power is generated from the motor torque,
*τ* and angular velocity, *ω*:

$${P}_{mech}=T\omega .$$

### Flow Rate and Driving Torque

The mass flow rate generated at the motor is

$$\dot{m}={\dot{m}}_{\text{Ideal}}+{\dot{m}}_{\text{Leak}},$$

where:

$$\dot{m}$$ is the actual mass flow rate.

$${\dot{m}}_{\text{Ideal}}$$ is the ideal mass flow rate.

$${\dot{m}}_{\text{Leak}}$$ 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.*τ*_{Ideal}is the ideal torque.*τ*_{Friction}is the friction torque.

**Ideal Flow Rate and Ideal Torque**

The ideal mass flow rate is

$${\dot{m}}_{\text{Ideal}}=\rho D\omega ,$$

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.

### Analytical Leakage and Friction Parameterization

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

, the block calculates leakage and friction
from constant values of shaft velocity, pressure drop, and friction 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*is*p*–_{A}*p*._{B}*ρ*_{avg}is the average fluid density.*K*is the Hagen-Poiseuille coefficient for analytical loss,$$K=\frac{D{\omega}_{nom}\left(\frac{1}{{\eta}_{v,}{}_{nom}}-1\right)}{\Delta {p}_{nom}},$$

where:

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

*Δp*is the value of the_{nom}**Nominal pressure drop**parameter.

The friction 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 value of the**No-load torque**parameter.*k*is the friction torque vs. pressure gain coefficient at nominal displacement, which is determined from the value of the**Mechanical efficiency at nominal conditions**parameter,*η*:_{m}$$k=\frac{{\tau}_{fr,nom}-{\tau}_{0}}{\Delta {p}_{nom}}.$$

*τ*is the friction torque at nominal conditions:_{fric}$${\tau}_{fr,nom}=\left(1-{\eta}_{m,nom}\right)D\Delta {p}_{nom}.$$

*Δp*is the pressure drop between ports**A**and**B**.*ω*is the relative shaft angular velocity, or $${\omega}_{R}-{\omega}_{C}$$.

### Tabulated Leakage and Friction Parameterizations

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 or modes by extending the given data into the remaining quadrants.

**Tabulated Data - Volumetric and Mechanical Efficiencies**

When you set **Leakage and friction parameterization** to
```
Tabulated data - volumetric and mechanical
efficiencies
```

, the leakage flow rate is

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

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

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

*τ*_{Friction,Motor}is the friction torque in motor mode.*τ*_{Friction,Pump}is the friction torque in pump mode.

The smoothing parameter *α* is given by the hyperbolic function

$$\alpha =\text{tanh}\left(\frac{4\Delta p}{\Delta {p}_{\text{Threshold}}}\right)\xb7\text{tanh}\left(\frac{4\omega}{{\omega}_{\text{Threshold}}}\right)\xb7\mathrm{tanh}\left(\frac{4D}{{D}_{\text{Threshold}}}\right),$$

where:

*Δp*_{Threshold}is the specified value of the**Pressure drop threshold for motor-pump transition**block parameter.*ω*_{Threshold}is the specified value of the**Angular velocity threshold for motor-pump transition**block parameter.*D*_{Threshold}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
*Δp*–*ɷ*–*D* domain via
the **Volumetric efficiency table** block parameter. When
operating in motor mode (quadrants **1** and
**3** of the
*Δp*–*ɷ*–*D* chart shown
in the Operation Modes figure), the
leakage flow rate is:

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

where *η*_{v} 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
*Δp*–*ɷ*–*D* chart), the
leakage flow rate is:

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

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

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

where *η*_{m} 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
*Δp*–*ɷ*–*D* chart):

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

**Tabulated Data - Volumetric and Mechanical 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 *Δp*–*ɷ* 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:

$${\dot{m}}_{\text{Leak}}=\rho {q}_{\text{Leak}}.$$

The friction torque is

$${\tau}_{\text{Friction}}={\tau}_{\text{Loss}}\left(\Delta p,\omega \right)\mathrm{tanh}\left(\frac{4\omega}{{\omega}_{threshold}}\right),$$

where *q*_{Leak}(*Δp*,*ω*) and *τ*_{Loss}(*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.

**Tabulated Data - Torque and Speed Parameterization**

When you set **Leakage and friction parameterization** to
`Tabulated data - torque and speed`

, the block
calculates the volumetric loss table,
*q _{loss,TLU}* and the mechanical loss
table,

*τ*, as

_{loss,TLU}$$\begin{array}{l}{\text{q}}_{loss,TLU}={q}_{TLU}-D{\omega}_{TLU}\\ {\tau}_{loss,TLU}=D\Delta {p}_{TLU}-{T}_{TLU}\end{array}$$

where:

*q*is the value of the_{TLU}**Flow rate vector, q**parameter.*ω*is the value of the_{TLU}**Shaft speed table, w(q,dp)**parameter.*Δp*is the value of the_{TLU}**Pressure drop vector, dp**parameter.*T*is the value of the_{TLU}**Torque table, T(q,dp)**parameter.

If the supplied values for the **Shaft speed table, w(q,dp)**
and **Torque table, T(q,dp)** parameters do not cover all four
quadrants, the block extends the data by

Symmetrically mirroring the values of the

**Pressure drop vector, dp**and**Flow rate vector, q**parameters to contain negative values.Symmetrically extending the values of the volumetric loss table,

*q*, to additional quadrants. The signs of these extended values match the sign_{loss,TLU}*Δp*in each quadrant._{TLU}Calculating the extended values of the shaft speed vector,

*ω*, from the extended values of the flow rate vector and volumetric loss table, $${\omega}_{TLU}=\frac{{q}_{TLU}-{q}_{loss,TLU}}{D}.$$_{TLU}Symmetrically extending the values of the mechanical loss table,

*τ*, to additional quadrants. The signs of these extended values match the sign_{loss,TLU}*ω*in each quadrant._{TLU}

If your data tables have unknown data points in any of the four corners or the
**Shaft speed table, w(q,dp)** or **Torque table,
T(q,dp)** parameters, use `NaN`

in place of these
values. The block fills in the `NaN`

elements in the resulting
volumetric loss table and mechanical loss table with nearest extrapolation with
respect to pressure drop. The block adjusts the signs in the extrapolated
mechanical loss table to match the sign of the corresponding elements in the
shaft speed vector, *ω _{TLU}*, where $${\omega}_{TLU}=\frac{{q}_{TLU}-{q}_{loss,TLU}}{D}.$$

After extending or filling in the unknown data, the block uses linear interpolation and nearest extrapolation to calculate the volumetric and mechanical loss tables during simulation

$$\begin{array}{l}{q}_{loss}=tablelookup\left({q}_{TLU},\Delta {p}_{TLU},{q}_{loss,TLU},Q,\Delta p,interpolation=linear,extrapolation=nearest\right)\\ {\tau}_{loss}=tablelookup\left({q}_{TLU},\Delta {p}_{TLU},{\tau}_{loss,TLU},Q,\Delta p,interpolation=linear,extrapolation=nearest\right)\end{array}$$

where $$Q=\frac{{\dot{m}}_{A}}{{\rho}_{avg}}.$$

### Input Signal Parameterizations

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 **.

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

, the
leakage flow rate and friction torque calculations are identical to the
`Tabulated data - volumetric and mechanical losses`

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
*Δp*–*ɷ* quadrant in which the component
happens to be operating.

### Faults

To model a fault, in the **Faults** section,
click the **Add fault** hyperlink next to the fault that you want to model. Use
the fault parameters to specify the fault properties. For more information about fault modeling,
see Introduction to Simscape Faults.

You can model a displacement fault, leakage, or a shaft friction torque fault.

When you enable the **Displacement fault** parameter, the block scales
the displacement by the value of the **Faulted displacement factor**
parameter when the fault triggers,

$${\text{D}}_{Fault}={f}_{D}D,$$

where *f _{D}* is the value of the

**Faulted displacement factor**parameter. When the

**Leakage and friction parameterization**parameter is

`Analytical`

, the
block does not use the faulted displacement value to calculate the Hagen-Poiseuille
coefficient or the friction torque.When you enable the **Leakage fault** parameter and **Leakage and
friction parameterization** is `Analytical`

,
`Tabulated data - volumetric and mechanical efficiencies`

, or
`Input signal - volumetric and mechanical efficiencies`

, the
faulted volumetric efficiency is

$${\eta}_{v,Fault}=\frac{{\eta}_{v}}{{f}_{Leak}},$$

where *f _{Leak}* is the value of the

**Faulted leakage factor**parameter and

*η*is the volumetric efficiency. When

_{v}**Leakage and friction parameterization**is

`Analytical`

, the block uses the faulted volumetric efficiency
to calculate the Hagen-Poiseuille coefficient.When **Leakage and friction parameterization** is ```
Tabulated
data - volumetric and mechanical losses
```

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

, or ```
Tabulated data - torque
and speed
```

, the faulted leakage volumetric flow rate is

$${q}_{Leak,Fault}={f}_{Leak}{q}_{Leak}.$$

When **Leakage and friction parameterization** is
`Tabulated data - torque and speed`

, the block calculates
*q _{Leak}* from the shaft speed and torque
parameters.

When you enable the **Shaft friction torque fault** parameter and
**Leakage and friction parameterization** is
`Analytical`

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

, or ```
Input signal - volumetric and
mechanical efficiencies
```

, the faulted mechanical efficiency is

$${\eta}_{m,Fault}=\frac{{\eta}_{m}}{{f}_{Friction}},$$

where *f _{Friction}* is the value
of the

**Shaft friction torque fault**parameter and

*η*is the mechanical efficiency. When

_{m}**Leakage and friction parameterization**is

`Analytical`

, the block uses the faulted mechanical efficiency
to calculate the friction torque.When **Leakage and friction parameterization** is ```
Tabulated
data - volumetric and mechanical losses
```

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

, or ```
Tabulated data - torque
and speed
```

, the faulted friction torque is

$${\tau}_{Friction,Fault}={f}_{Friction}{\tau}_{Friction}.$$

When **Leakage and friction parameterization** is
`Tabulated data - torque and speed`

, the block calculates
*τ _{Leak}* from the shaft speed and torque
parameters.

### 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

### Conserving

## Parameters

## Extended Capabilities

## Version History

**Introduced in R2016a**