Fixed-Displacement Motor (IL)

Fixed-displacement motor in isothermal liquid system

Since R2020a

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

Description

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.

The block has eight 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 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.

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:

${\dot{m}}_{l e a k} = K ρ_{a v g} Δ p,$

where:

Δp is p_A – p_B.
ρ_avg is the average fluid density.
K is the Hagen-Poiseuille coefficient for analytical loss,

$K = \frac{D ω_{n o m} (\frac{1}{η_{v,}_{n o m}} - 1)}{Δ p_{n o m}},$
where:
- D is the value of the Displacement parameter.
- ω_nom is the value of the Nominal shaft angular velocity parameter.
- η_{v, nom} is the value of the Volumetric efficiency at nominal conditions parameter.
Δp_nom is the value of the Nominal pressure drop parameter.

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

$τ_{f r} = (τ_{0} + k | Δ p |) \tanh (\frac{4 ω}{5 \times 10^{- 5} ω_{n o m}}),$

where:

τ₀ 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:

$k = \frac{τ_{f r, n o m} - τ_{0}}{Δ p_{n o m}} .$
τ_fric is the friction torque at nominal conditions:

$τ_{f r, n o m} = (1 - η_{m, n o m}) D Δ p_{n o m} .$
Δp is the pressure drop between ports A and B.
ω is the relative shaft angular velocity, or $ω_{R} - ω_{C}$ .

Tabulated Data 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(s) by extending the given data into the remaining quadrants.

Tabulated Data - Volumetric and Mechanical Efficiencies Parameterization

The leakage flow rate is

${\dot{m}}_{l e a k} = {\dot{m}}_{l e a k, m o t o r} (\frac{1 + α}{2}) + {\dot{m}}_{l e a k, p u m p} (\frac{1 - α}{2}),$

where:

${\dot{m}}_{l e a k, p u m p} = (η_{υ} - 1) {\dot{m}}_{i d e a l}$
${\dot{m}}_{l e a k, m o t o r} = (1 - η_{v}) \dot{m}$

and η_v is the volumetric efficiency, which is interpolated from the user-provided tabulated data. The transition term, α, is

$α = \tanh (\frac{4 Δ p}{Δ p_{t h r e s h o l d}}) \tanh (\frac{4 ω}{ω_{t h r e s h o l d}}),$

where:

Δp is p_A – p_B.
p_threshold is the value of the Pressure drop threshold for motor-pump transition parameter.
ω is ω_R – ω_C.
ω_threshold is the value of the Angular velocity threshold for motor-pump transition parameter.

The friction torque is

$τ_{f r} = τ_{f r, m o t o r} (\frac{1 + α}{2}) + τ_{f r, p u m p} (\frac{1 - α}{2}),$

where:

$τ_{f r, p u m p} = (η_{m} - 1) τ$
$τ_{f r, m o t o r} = (1 - η_{m}) τ_{i d e a l}$

and η_m is the mechanical efficiency, which is interpolated from the user-provided tabulated data.

Tabulated Data - Volumetric and Mechanical Losses Parameterization

The leakage flow rate is

${\dot{m}}_{l e a k} = ρ_{a v g} q_{l o s s} (Δ p, ω),$

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

$τ_{f r} = τ_{l o s s} (Δ p, ω) \tanh (\frac{4 ω}{ω_{t h r e s h o l d}}),$

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.

Tabulated Data - Torque and Speed Parameterization

The block calculates the volumetric loss table, q_loss,TLU and the mechanical loss table, τ_loss,TLU, as

$\begin{array}{l} q_{l o s s, T L U} = q_{T L U} - D ω_{T L U} \\ τ_{l o s s, T L U} = D Δ p_{T L U} - T_{T L U} \end{array}$

where:

q_TLU is the value of the Flow rate vector, q parameter.
ω_TLU is the value of the Shaft speed table, w(q,dp) parameter.
Δp_TLU is the value of the Pressure drop vector, dp parameter.
T_TLU is the value of the 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_loss,TLU, to additional quadrants. The signs of these extended values match the sign Δp_TLU in each quadrant.
Calculating the extended values of the shaft speed vector, ω_TLU, from the extended values of the flow rate vector and volumetric loss table, $ω_{T L U} = \frac{q_{T L U} - q_{l o s s, T L U}}{D} .$
Symmetrically extending the values of the mechanical loss table, τ_loss,TLU, to additional quadrants. The signs of these extended values match the sign ω_TLU in each quadrant.

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 $ω_{T L U} = \frac{q_{T L U} - q_{l o s s, T L U}}{D} .$

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

$\begin{array}{l} q_{l o s s} = t a b l e l o o k u p (q_{T L U}, Δ p_{T L U}, q_{l o s s, T L U}, Q, Δ p, i n t e r p o l a t i o n = l i n e a r, e x t r a p o l a t i o n = n e a r e s t) \\ τ_{l o s s} = t a b l e l o o k u p (q_{T L U}, Δ p_{T L U}, τ_{l o s s, T L U}, Q, Δ p, i n t e r p o l a t i o n = l i n e a r, e x t r a p o l a t i o n = n e a r e s t) \end{array}$

where $Q = \frac{{\dot{m}}_{A}}{ρ_{a v g}} .$

Input Signal Parameterization

When Leakage and friction parameterization is 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 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:

${\dot{m}}_{l e a k} = ρ_{a v g} q_{L V} \tanh (\frac{4 Δ p}{p_{t h r e s h}}),$

where:

q_LV is the leakage flow received at port LV.
p_thresh is the value of the Pressure drop threshold for motor-pump transition parameter.

The friction torque is calculated as:

$τ_{f r} = τ_{L M} \tanh (\frac{4 ω}{ω_{t h r e s h}}),$

where

τ_LM is the friction torque received at port LM.
ω_thresh is the value of 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.

Pump Operation

The motor flow rate is:

$\dot{m} = {\dot{m}}_{i d e a l} + {\dot{m}}_{l e a k},$

where ${\dot{m}}_{i d e a l} = ρ_{a v g} D \cdot ω .$

The motor torque is:

$τ = τ_{i d e a l} - τ_{f r},$

where $τ_{i d e a l} = D \cdot Δ p .$

The mechanical power extracted by the motor shaft is:

$φ_{m e c h} = τ ω,$

and the motor hydraulic power is:

$φ_{h y d} = \frac{Δ p \dot{m}}{ρ_{a v g}} .$

If you would like to know if the block is operating beyond the supplied tabulated data, you can set Check if operating beyond the range 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.

Predefined Parameterization

Pre-parameterized manufacturer data is available for this block. This data allows you to model a specific supplier component.

To load a predefined parameterization,

Click the "Select a predefined parameterization" hyperlink in the block dialog description.
Select a part from the drop-down menu and click Update block with selected part.
If you change any parameter settings after loading a parameterization, you can check your changes by clicking Compare block settings with selected part. Any difference in settings between the block and pre-defined parameterization will display in the MATLAB command window.

Note

Predefined block parameterizations use available data sources to supply parameter values. The block substitutes engineering judgement and simplifying assumptions for missing data. As a result, expect some deviation between simulated and actual physical behavior. To ensure accuracy, validate the simulated behavior against experimental data and refine your component models as necessary.

To learn more, see List of Pre-Parameterized Components.

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,

$D_{F a u l t} = f_{D} D,$

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

$η_{v, F a u l t} = \frac{η_{v}}{f_{L e a k}},$

where f_Leak is the value of the Faulted leakage factor parameter and η_v is the volumetric efficiency. When 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

$q_{L e a k, F a u l t} = f_{L e a k} q_{L e a k} .$

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

$η_{m, F a u l t} = \frac{η_{m}}{f_{F r i c t i o n}},$

where f_Friction is the value of the Shaft friction torque fault parameter and η_m is the mechanical efficiency. When Leakage and friction parameterization is Analytical, the block uses the faulted mechanical efficiency to calculate the friction torque.

$τ_{F r i c t i o n, F a u l t} = f_{F r i c t i o n} τ_{F r i c t i o n} .$

When Leakage and friction parameterization is Tabulated data - torque and speed, the block calculates τ_Leak from the shaft speed and torque parameters.

Examples

Hydrostatic Transmission

A hydraulic transmission system built of a variable-displacement pump and a fixed-displacement motor. The pipes between the pump and the motor are connected by two replenishing valves (check valves) and a charge pump on the pump side and two pressure relief valves on the motor side. The motor drives a mechanical load consisting of an inertia, viscous friction, and time-varying torque. The system is tested with both positive and negative pump flow rate by varying the pump displacement.

Open Model

Priority Valve Controlling Two Hydraulic Motors

A pressure-compensated 3-way flow control valve. This valve maintains constant flow rate through the main hydraulic motor, which is connected to the pressure-compensated outlet of the flow control valve. It acts as a priority valve, diverting the excess flow to the auxiliary hydraulic motor if the main hydraulic motor receives enough fluid to maintain a preset angular velocity. The auxiliary motor is shut off completely if there is insufficient flow to power the main hydraulic motor.

Open Model

Ports

Conserving

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A — Liquid port
isothermal liquid

Entry or exit port to the motor.

B — Liquid port
isothermal liquid

Entry or exit port to the motor.

R — Mechanical port
mechanical rotational

Rotating shaft angular velocity and torque.

C — Mechanical port
mechanical rotational

Motor casing reference angular velocity and torque.

Input

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EV — Volumetric efficiency
physical signal

Motor efficiency for fluid displacement, specified as a physical signal. The value must be between 0 and 1.

Dependencies

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

EM — Mechanical efficiency
physical signal

Motor efficiency for the mechanical extraction of energy, specified as a physical signal. The value must be between 0 and 1.

Dependencies

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

LV — Leakage flow rate, `m^3/s`
physical signal

Motor losses associated with fluid displacement in m^3/s, specified as a physical signal.

Dependencies

To enable this port, set Leakage and friction parameterization to Input signal - volumetric and mechanical losses.

LM — Friction torque, `N*m`
physical signal

Motor losses associated with the mechanical extraction of energy in N*m, specified as a physical signal.

Dependencies

To enable this port, set Leakage and friction parameterization to Input signal - volumetric and mechanical losses.

Parameters

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Parameters

Leakage and friction parameterization — Method of calculating leakage flow rate and friction torque
`Analytical` (default) | `Tabulated data - volumetric and mechanical efficiencies` | `Tabulated data - volumetric and mechanical losses` | `Tabulated data - torque and speed` | `Input signal - volumetric and mechanical efficiencies` | `Input signal - volumetric and mechanical losses`

Parameterization of the leakage and friction characteristics of the pump.

In the Analytical parameterization, the leakage flow rate and the friction torque are calculated by analytical equations.
In the Tabulated data - volumetric and mechanical efficiencies parameterization, the volumetric and mechanical efficiencies are calculated from the user-supplied Pressure drop vector, dp and Shaft angular velocity vector, w parameters and interpolated from the 2-D dependent Volumetric efficiency table, e_v(dp,w) and Mechanical efficiency table, e_m(dp,w) tables.
In the Tabulated data - volumetric and mechanical loss parameterization, the leakage flow rate and torque friction are calculated from user-supplied Pressure drop vector, dp and Shaft angular velocity vector, w parameters and interpolated from the 2-D dependent Volumetric loss table, q_loss(dp,w) and Mechanical loss table, torque_loss(dp,w) tables.
In the Tabulated data - torque and speed parameterization, you can enter the torque and shaft speed as functions of flow rate and pressure drop by using the Torque table T(q,dp) and Shaft speed table w(q,dp) parameters.
In the Input signal - volumetric and mechanical efficiencies parameterization, the volumetric and mechanical efficiencies are received as physical signals at ports EV and EM, respectively.
In the Input signal - volumetric and mechanical loss parameterization, the leakage flow rate and torque friction are received as physical signals at ports LV and LM, respectively.

Displacement — Fixed-volume fluid displacement
`30 cm^3/rev` (default) | scalar

Amount of fixed-volume fluid displacement.