Variable-displacement bidirectional hydraulic pump

Pumps and Motors

The Variable-Displacement Pump block represents a device that extracts power from a mechanical rotational network and delivers it to a hydraulic (isothermal liquid) network. The pump displacement varies in proportion to the physical signal specified at port C or D. The port used depends on the block variant selected. See Ports.

Ports T and P represent the pump inlet and outlet, respectively. Port S represents the pump drive shaft. During normal operation, the pressure gain from port T to port P is positive if the angular velocity at port S is positive also. This operation mode is referred to here as forward pump.

**Operation Modes**

Four operation modes are possible. The working mode depends
on the pressure gain from port T to port P (Δ*p*),
the angular velocity at port S (*ω*), and the
instantaneous displacement input at port D (*D*).
The Operation Modes figure
maps the modes to the octants of a Δ*p*-*ω*-*D* chart.
The modes are labeled 1–4:

Mode

**1**: forward pump — A positive shaft angular velocity generates a positive pressure gain.Mode

**2**: reverse motor — A negative pressure drop (shown in the figure as a positive pressure gain) generates a negative shaft angular velocity.Mode

**3**: reverse pump — A negative shaft angular velocity generates a negative pressure gain.Mode

**4**: forward motor — A positive pressure drop (shown in the figure as a negative pressure gain) generates a positive shaft angular velocity.

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

The pump model accounts for power losses due to leakage and
friction. Leakage is internal and occurs between the pump 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.

The displacement volume input depends on the block variant selected.
If the active block variant is `Input efficiencies`

or ```
Input
losses
```

, the block obtains the instantaneous displacement
volume directly from the physical signal input at port D.

If the active block variant is ```
Analytical or tabulated
data
```

, the block computes the instantaneous displacement
volume from the control member position specified at port C. This
computation depends on the **Displacement parameterization** parameter
setting:

`Maximum displacement and control member stroke`

— Compute the displacement volume per unit rotation as a linear function of the control member position specified at port C.`Displacement vs. control member position table`

— Compute the displacement volume per unit volume using interpolation or extrapolation of displacement tabular data specified at discrete control member positions.

The volumetric flow rate generated at the pump is

$$q={q}_{Ideal}-{q}_{Leak},$$

where:

*q*is the net volumetric flow rate.*q*_{Ideal}is the ideal volumetric flow rate.*q*_{Leak}is the internal leakage volumetric flow rate.

The driving torque required to power the pump is

$$\tau ={\tau}_{Ideal}+{\tau}_{Friction},$$

where:

*τ*is the net driving torque.*τ*_{Ideal}is the ideal driving torque.*τ*_{Friction}is the friction torque.

The ideal volumetric flow rate is

$${q}_{Ideal}={D}_{Sat}\xb7\omega ,$$

and the ideal driving torque is

$${\tau}_{Ideal}={D}_{Sat}\xb7\Delta p,$$

where:

*D*_{Sat}is a smoothed displacement computed so as to remove numerical discontinuities between negative and positive displacements.*ω*is the shaft angular velocity.*Δp*is the pressure gain from inlet to outlet.

The saturation displacement depends on the block variant selected.
If the active variant is `Analytical or tabulated data`

,

$${D}_{Sat}=\{\begin{array}{ll}{D}_{Max},\hfill & \left|D\right|\ge {D}_{Max}\hfill \\ \sqrt{{D}^{2}+{D}_{Thresh}^{2}},\hfill & D\ge 0\hfill \\ -\sqrt{{D}^{2}+{D}_{Thresh}^{2}},\hfill & D<0\hfill \end{array},$$

where:

*D*is the displacement computed from the control member position.*D*_{Max}is the**Maximum displacement**parameter.*D*_{Thresh}is the**Displacement threshold for pump-motor transition**parameter.

If the active variant is `Input efficiencies`

or ```
Input
losses
```

, there is no upper bound on the displacement
input and the saturation displacement reduces to:

$${D}_{Sat}=\{\begin{array}{ll}\sqrt{{D}^{2}+{D}_{Thresh}^{2}},\hfill & D\ge 0\hfill \\ -\sqrt{{D}^{2}+{D}_{Thresh}^{2}},\hfill & D<0\hfill \end{array}.$$

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

$${q}_{Leak}={K}_{HP}\Delta p,$$

and the friction torque is

$${\tau}_{Friction}=\left({\tau}_{0}+K{\text{}}_{TP}\left|\Delta p\right|\right)\mathrm{tanh}\left(\frac{4\omega}{{\omega}_{Thresh}}\right)$$

where:

*K*_{HP}is the Hagen-Poiseuille coefficient for laminar pipe flows. This coefficient is computed from the specified nominal parameters.*K*_{TP}is the**Friction torque vs pressure gain coefficient**parameter.*τ*_{0}is the**No-load torque**parameter.*ω*_{Thresh}is the threshold angular velocity for the pump-motor transition. The threshold angular velocity is an internally set fraction of the**Nominal shaft angular velocity**parameter.

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

$${K}_{HP}=\frac{{\nu}_{Nom}}{\rho v}\frac{\text{\hspace{0.17em}}{\rho}_{Nom}{\omega}_{Nom}{D}_{Max}}{\Delta {p}_{Nom}}\left(1-{\eta}_{v,Nom}\right),$$

where:

*ν*_{Nom}is the**Nominal kinematic viscosity**parameter. This is the kinematic viscosity at which the nominal volumetric efficiency is specified.*ρ*_{Nom}is the**Nominal fluid density**parameter. This is the density at which the nominal volumetric efficiency is specified.*ρ*is the actual fluid density in the attached hydraulic (isothermal liquid) network. This density can differ from the**Nominal fluid density**parameter.*ω*_{Nom}is the**Nominal shaft angular velocity**parameter. This is the angular velocity at which the nominal volumetric efficiency is specified.*v*is the fluid kinematic viscosity in the attached hydraulic fluid network.*Δp*_{Nom}is the**Nominal pressure gain**parameter. This is the pressure gain at which the nominal volumetric efficiency is specified.*η*_{v,Nom}is the**Volumetric efficiency at nominal conditions**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

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

and the friction torque is

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

where:

*α*is a numerical smoothing parameter for the motor-pump transition.*q*_{Leak,Pump}is the leakage flow rate in pump mode.*q*_{Leak,Motor}is the leakage flow rate in motor mode.*τ*_{Friction,Pump}is the friction torque in pump mode.*τ*_{Friction,Motor}is the friction torque in motor mode.

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

$$\alpha =\mathrm{tanh}\left(\frac{3\Delta p}{\Delta {p}_{Thresh}}\right)\xb7\mathrm{tanh}\left(\frac{3\omega}{{\omega}_{Thresh}}\right)\xb7\mathrm{tanh}\left(\frac{3D}{{D}_{Thresh}}\right),$$

where:

*Δp*_{Thresh}is the**Pressure gain threshold for pump-motor transition**parameter.*ω*_{Thresh}is the**Angular velocity threshold for pump-motor transition**.

The leakage flow rate is computed from efficiency tabulated data through the equation

$${q}_{Leak,Pump}=\left(1-{\eta}_{v}\right){q}_{Ideal}$$

in pump mode and through the equation

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

in motor mode, where:

*η*_{v}is the volumetric efficiency obtained through interpolation or extrapolation of the**Volumetric efficiency table, e_v(dp,w,D)**parameter data.

Similarly, the friction torque is computed from efficiency tabulated data through the equation

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

in pump mode and through the equation

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

in motor mode, where:

*η*_{m}is the mechanical efficiency obtained through interpolation or extrapolation of the**Mechanical efficiency table, e_m(dp,w,D)**parameter data.

**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 flow rate equation is

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

and the friction torque equation is

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

where *q*_{Leak}(*Δp*,*ω*,*D*_{Sat}) and *τ*_{Friction}(*Δp*,*ω*,*D*_{Sat}) are
the volumetric and mechanical losses, obtained through interpolation
or extrapolation of the **Volumetric loss table, q_loss(dp,w,D)** and **Mechanical
loss table, torque_loss (dp,w,D)** parameter data.

**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.

**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.

Fluid compressibility is negligible.

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

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