Fixed-Displacement Motor (TL)

Hydraulic-mechanical power conversion device

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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_B – p_A 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.

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, the leakage flow rate is

$${\dot{m}}_{\text{Leak}}=\frac{K_{\text{HP}}{\rho }_{\text{Avg}}\Delta p}{{\mu }_{\text{Avg}}},$$

and the friction torque is

$${\tau }_{\text{Friction}}=\left({\tau }_0+K_{\text{TP}}\left|\Delta p\right|\text{tanh}\frac{4\omega }{\left(5\cdot {10}^{-5}\right){\omega }_{\text{Nom}}}\right),$$

where:

• K_HP 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_TP is the friction torque vs. pressure gain coefficient at nominal displacement, which is determined from the Mechanical efficiency at nominal conditions, η_m:

  $$k=\frac{{\tau }_{fr,nom}-{\tau }_0}{\Delta p_{nom}}.$$

  τ_fric is the friction torque at nominal conditions:

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

• τ₀ is the specified value of the No-load torque parameter.

• ω_Nom is the specified value of the Nominal shaft angular velocity parameter.

• Δp_Nom is the specified value of the Nominal pressure drop 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

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

where:

• ω_Nom is the specified value of the Nominal shaft angular velocity parameter. This is the angular velocity at which the nominal volumetric efficiency is specified.

• μ_Nom is the specified value of the Nominal Dynamic viscosity block parameter. This is the dynamic viscosity at which the nominal volumetric efficiency is specified.

• η_v,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

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

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

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

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 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 similarly specified in tabulated form:

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

where q_Leak(Δp,ω) and τ_Friction(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.

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

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

EV — Volumetric efficiency, unitless

physical signal

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

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

EM — Mechanical efficiency, unitless

physical signal

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

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

LV — Volumetric loss, m^3/s

physical signal

Physical signal input port for the volumetric loss, defined as the internal leakage flow rate between the motor inlets.

Dependencies

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

LM — Mechanical loss, N*m

physical signal

Physical signal input port for the mechanical loss, defined as the friction torque on the rotating motor shaft.

Dependencies

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

Conserving

A — Motor inlet

thermal liquid

Thermal liquid conserving port associated with the motor inlet.

B — Motor outlet

thermal liquid

Thermal liquid conserving port associated with the motor outlet.

C — Motor Case

mechanical rotational

Mechanical rotational conserving port associated with the motor case.

R — Motor Shaft

mechanical rotational

Mechanical rotational conserving port associated with the rotational motor shaft.

Parameters

Leakage and friction parameterization — Parameterization of leakage flow rate and friction torque

Analytical (default) | Tabulated data - volumetric and mechanical efficiencies | Tabulated data - volumetric and mechanical losses | Input signal - volumetric and mechanical efficiencies | Input signal - volumetric and mechanical losses

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.

Displacement — Fluid volume displaced per shaft rotation

30 cm^3/rev (default) | positive scalar

Fluid volume displaced per shaft rotation. The block maintains this value throughout the simulation.

Nominal shaft angular velocity — Shaft angular velocity for given volumetric efficiency

1800 rpm (default) | scalar

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.

Nominal pressure drop — Pressure drop for corresponding volumetric efficiency

10 MPa (default) | scalar

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.

Nominal dynamic viscosity — Dynamic viscosity at which to specify the volumetric efficiency

0.9 cP (default) | scalar

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 at nominal conditions — Volumetric efficiency at the specified nominal conditions

0.92 (default) | scalar in the range (0,1)

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.

Mechanical efficiency at nominal conditions — Ratio of actual to ideal mechanical power

0.88 (default) | positive scalar

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

Dependencies

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

No-load torque — Minimum torque required to induce shaft rotation

0 N*m (default) | scalar

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 drop vector, dp — Pressure drops for given volumetric and mechanical efficiencies

[.1, 1, 2] MPa (default) | vector

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 quadrant, (ɷ, Δp). Refer to the block description for the operation modes corresponding to the various quadrants.

Dependencies

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

Shaft angular velocity vector, w — Angular velocities at which to specify the volumetric and mechanical efficiencies

[250, 500, 1000, 2000, 3000, 4000] rpm | vector

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 quadrant, (ɷ, Δp). Refer to the block description for the operation modes corresponding to the various quadrants.

Dependencies

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

Volumetric efficiency table, e_v(dp,w) — Volumetric efficiencies for given pressure drops and angular velocities

[.816, .908, .954, .977, .981, .984; .325, .663, .831, .916, .925, .946; .137, .568, .78, .892, .893, .91] | matrix

Volumetric efficiencies for the given fluid pressure drops and shaft angular velocities.. The efficiencies must be in the range (0,1]. 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.

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

Dependencies

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

Mechanical efficiency table, e_m(dp,w) — Mechanical efficiencies for given pressure drops and angular velocities

[.996, .996, .996, .996, .996, .996; .988, .989, .989, .989, .989, .99; .981, .981, .982, .982, .983, .984] | matrix

Mechanical efficiencies for the given fluid pressure drops and shaft angular velocities. The efficiencies must be in the range (0,1]. 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.

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

Dependencies

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

Pressure drop vector, dp — Pressure drop for corresponding losses

[.1, 1, 2] MPa (default)

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 quadrant, (ɷ, Δp). Refer to the block description for the operation modes corresponding to the various quadrants.

Dependencies

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

Volumetric loss table, q_loss(dp,w) — Internal leakage flow rates for corresponding pressure drops and angular velocities

[.023, .023, .023, .023, .0285, .032; .0844, .0842, .0845, .084, .1125, .108; .1079, .108, .11, .108, .1605, .18] * 1e-3 m^3/s

Volumetric losses at the specified fluid pressure drops and shaft angular velocities. Volumetric loss is defined here as 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.

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

Dependencies

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

Mechanical loss table, torque_loss(dp,w) — Friction torques for the corresponding pressure drops and angular velocities

[.002, .002, .002, .002, .002, .002; .0607, .0556, .0556, .0556, .0556, .0505; .1937, .1937, .1833, .1833, .1729, .1626] N*m

Mechanical losses for the given pressure drops and shaft angular velocities. 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.

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

Minimum volumetric efficiency — Lower saturation bound on the volumetric efficiency input signal

1e-3 (default) | scalar in the range (0,1)

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.

Maximum volumetric efficiency — Upper saturation bound on the volumetric efficiency input signal

1 (default) | scalar in the range (0,1)

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.

Minimum mechanical efficiency — Lower saturation bound on the volumetric efficiency input signal

1e-3 (default) | scalar in the range (0,1)

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.

Maximum mechanical efficiency — Upper saturation bound on the volumetric efficiency input signal

1 (default) | scalar in the range (0,1)

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 threshold for motor-pump transition — Pressure drop at which to initiate a smooth transition between motor and pump modes

1e-3 MPa (default) | scalar

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.

Angular velocity threshold for motor-pump transition — Angular velocity at which to initiate a smooth transition between pump and motor modes

100 rpm (default) | scalar

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.

Cross-sectional area at ports A and B — Flow area at the component inlet and outlet

0.01 m^2 (default) | positive scalar

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

Check if operating beyond the range of supplied tabulated data — Simulation warning mode for operating conditions outside the range of tabulated data

None (default) | Warning

Simulation warning mode for operating conditions outside the range of tabulated data. Select Warning to be notified when the fluid pressure drop or shaft angular velocity 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

Check if operating outside of motor mode — Simulation warning mode for operating conditions outside motoring mode

None (default) | Warning

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.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2016a

R2024a: Check if block operates outside range of calculated data

The Check if operating beyond the quadrants of supplied tabulated data parameter has been renamed to Check if operating beyond the range of supplied tabulated data. You can enable the Check if operating beyond the range of supplied tabulated data parameter to check if the block is operating beyond the data specified in the block parameters, that is, operating in a region where the block has extended the data.