Rotational Mechanical Converter (G)

Interface between gas and mechanical rotational networks

Libraries:
Simscape / Foundation Library / Gas / Elements

Description

The Rotational Mechanical Converter (G) block models an interface between a gas network and a mechanical rotational network. The block converts gas pressure into mechanical torque and vice versa. It can be used as a building block for rotary actuators.

The converter contains a variable volume of gas. The pressure and temperature evolve based on the compressibility and thermal capacity of this gas volume. The Mechanical orientation parameter lets you specify whether an increase in the gas volume results in a positive or negative rotation of port R relative to port C.

Port A is the gas conserving port associated with the converter inlet. Port H is the thermal conserving port associated with the temperature of the gas inside the converter. Ports R and C are the mechanical rotational conserving ports associated with the moving interface and converter casing, respectively.

Mass Balance

Mass conservation equation is similar to that for the Constant Volume Chamber (G) block, with an additional term related to the change in gas volume:

$\frac{\partial M}{\partial p} \cdot \frac{d p_{I}}{d t} + \frac{\partial M}{\partial T} \cdot \frac{d T_{I}}{d t} + ρ_{I} \frac{d V}{d t} = {\dot{m}}_{A}$

where:

$\frac{\partial M}{\partial p}$ is the partial derivative of the mass of the gas volume with respect to pressure at constant temperature and volume.
$\frac{\partial M}{\partial T}$ is the partial derivative of the mass of the gas volume with respect to temperature at constant pressure and volume.
p_I is the pressure of the gas volume. Pressure at port A is assumed equal to this pressure, p_A = p_I.
T_I is the temperature of the gas volume. Temperature at port H is assumed equal to this temperature, T_H = T_I.
ρ_I is the density of the gas volume.
V is the volume of gas.
t is time.
$\dot{m}$ _A is the mass flow rate at port A. Flow rate associated with a port is positive when it flows into the block.

Energy Balance

Energy conservation equation is also similar to that for the Constant Volume Chamber (G) block. The additional term accounts for the change in gas volume, as well as the pressure-volume work done by the gas on the moving interface:

$\frac{\partial U}{\partial p} \cdot \frac{d p_{I}}{d t} + \frac{\partial U}{\partial T} \cdot \frac{d T_{I}}{d t} + ρ_{I} h_{I} \frac{d V}{d t} = Φ_{A} + Q_{H}$

where:

$\frac{\partial U}{\partial p}$ is the partial derivative of the internal energy of the gas volume with respect to pressure at constant temperature and volume.
$\frac{\partial U}{\partial T}$ is the partial derivative of the internal energy of the gas volume with respect to temperature at constant pressure and volume.
Ф_A is the energy flow rate at port A.
Q_H is the heat flow rate at port H.
h_I is the specific enthalpy of the gas volume.

Partial Derivatives for Perfect and Semiperfect Gas Models

The partial derivatives of the mass M and the internal energy U of the gas volume, with respect to pressure and temperature at constant volume, depend on the gas property model. For perfect and semiperfect gas models, the equations are:

$\begin{array}{l} \frac{\partial M}{\partial p} = V \frac{ρ_{I}}{p_{I}} \\ \frac{\partial M}{\partial T} = - V \frac{ρ_{I}}{T_{I}} \\ \frac{\partial U}{\partial p} = V (\frac{h_{I}}{Z R T_{I}} - 1) \\ \frac{\partial U}{\partial T} = V ρ_{I} (c_{p I} - \frac{h_{I}}{T_{I}}) \end{array}$

where:

ρ_I is the density of the gas volume.
V is the volume of gas.
h_I is the specific enthalpy of the gas volume.
Z is the compressibility factor.
R is the specific gas constant.
c_pI is the specific heat at constant pressure of the gas volume.

Partial Derivatives for Real Gas Model

For real gas model, the partial derivatives of the mass M and the internal energy U of the gas volume, with respect to pressure and temperature at constant volume, are:

$\begin{array}{l} \frac{\partial M}{\partial p} = V \frac{ρ_{I}}{β_{I}} \\ \frac{\partial M}{\partial T} = - V ρ_{I} α_{I} \\ \frac{\partial U}{\partial p} = V (\frac{ρ_{I} h_{I}}{β_{I}} - T_{I} α_{I}) \\ \frac{\partial U}{\partial T} = V ρ_{I} (c_{p I} - h_{I} α_{I}) \end{array}$

where:

β is the isothermal bulk modulus of the gas volume.
α is the isobaric thermal expansion coefficient of the gas volume.

Gas Volume

The gas volume depends on the rotation of the moving interface:

$V = V_{d e a d} + D_{int} θ_{int} ε_{int}$

where:

V_dead is the dead volume.
D_int is the interface volume displacement.
θ_int is the interface rotation.
ε_int is the mechanical orientation coefficient. If Mechanical orientation is Pressure at A causes positive rotation of R relative to C, ε_int = 1. If Mechanical orientation is Pressure at A causes negative rotation of R relative to C, ε_int = –1.

If you connect the converter to a Multibody joint, use the physical signal input port q to specify the rotation of port R relative to port C. Otherwise, the block calculates the interface rotation from relative port angular velocities. The interface rotation is zero when the gas volume is equal to the dead volume. Then, depending on the Mechanical orientation parameter value:

If Pressure at A causes positive rotation of R relative to C, the interface rotation increases when the gas volume increases from dead volume.
If Pressure at A causes negative rotation of R relative to C, the interface rotation decreases when the gas volume increases from dead volume.

Torque Balance

Torque balance across the moving interface on the gas volume is

$τ_{int} = (p_{e n v} - p_{I}) D_{int} ε_{int}$

where:

τ_int is the torque from port R to port C.
p_env is the environment pressure.

Variables

To set the priority and initial target values for the block variables prior to simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set Priority and Initial Target for Block Variables and Initial Conditions for Blocks with Finite Gas Volume.

Nominal values provide a way to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. Nominal values can come from different sources, one of which is the Nominal Values section in the block dialog box or Property Inspector. For more information, see Modify Nominal Values for a Block Variable.

Assumptions and Limitations

The converter casing is perfectly rigid.
There is no flow resistance between port A and the converter interior.
There is no thermal resistance between port H and the converter interior.
The moving interface is perfectly sealed.
The block does not model mechanical effects of the moving interface, such as hard stop, friction, and inertia.

Ports

Input

expand all

q — Rotation of port R relative to port C, rad
physical signal

Input physical signal that passes the position information from a Simscape™ Multibody™ joint. Connect this port to the position sensing port q of the joint. For more information, see Connecting Simscape Networks to Simscape Multibody Joints.

Dependencies

To enable this port, set the Interface rotation parameter to Provide input signal from Multibody joint.

Conserving

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A — Converter inlet
gas

Gas conserving port associated with the converter inlet.

H — Temperature inside converter
thermal

Thermal conserving port associated with the temperature of the gas inside the converter.

R — Rod
mechanical rotational

Mechanical rotational conserving port associated with the moving interface.

C — Case
mechanical rotational

Mechanical rotational conserving port associated with the converter casing.

Parameters

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Mechanical orientation — Select the converter orientation
`Pressure at A causes positive rotation of R relative to C` (default) | `Pressure at A causes negative rotation of R relative to C`

Select the alignment of moving interface with respect to the converter gas volume:

Pressure at A causes positive rotation of R relative to C — Increase in the gas volume results in a positive rotation of port R relative to port C.
Pressure at A causes negative rotation of R relative to C — Increase in the gas volume results in a negative rotation of port R relative to port C.

Interface rotation — Select method to determine relative port positions
`Calculate from angular velocity of port R relative to port C` (default) | `Provide input signal from Multibody joint`

Select method to determine rotation of port R relative to port C:

Calculate from angular velocity of port R relative to port C — Calculate rotation from relative port angular velocities, based on the block equations. This is the default method.
Provide input signal from Multibody joint — Enable the input physical signal port q to pass the rotation information from a Multibody joint. Use this method only when you connect the converter to a Multibody joint by using a Rotational Multibody Interface block. For more information, see How to Pass Position Information.

Initial interface rotation — Rotational offset of port R relative to port C at the start of simulation
`0 rad` (default)

Rotational offset of port R relative to port C at the start of simulation. A value of 0 corresponds to an initial gas volume equal to Dead volume.

Dependencies

Enabled when the Interface rotation parameter is set to Calculate from angular velocity of port R relative to port C.

If Mechanical orientation is Pressure at A causes positive rotation of R relative to C, the parameter value must be greater than or equal to 0.
If Mechanical orientation is Pressure at A causes negative rotation of R relative to C, the parameter value must be less than or equal to 0.

Interface volume displacement — Displaced gas volume per unit rotation
`0.01 m^3/rad` (default)

Displaced gas volume per unit rotation of the moving interface.

Dead volume — Volume of gas when the interface rotation is 0
`1e-5 m^3` (default)

Volume of gas when the interface rotation is 0.

Cross-sectional area at port A — Area normal to flow path at the converter inlet
`0.01 m^2` (default)

The cross-sectional area of the converter inlet, in the direction normal to gas flow path.

Environment pressure specification — Select a specification method for the environment pressure
`Atmospheric pressure` (default) | `Specified pressure`

Select a specification method for the environment pressure:

Atmospheric pressure — Use the atmospheric pressure, specified by the Gas Properties (G) block connected to the circuit.
Specified pressure — Specify a value by using the Environment pressure parameter.

Environment pressure — Pressure outside the converter
`0.101325 MPa` (default)

Pressure outside the converter acting against the pressure of the converter gas volume. A value of 0 indicates that the converter expands into vacuum.

Dependencies

Enabled when the Environment pressure specification parameter is set to Specified pressure.

Extended Capabilities

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

Version History

Introduced in R2016b

Rotational Mechanical Converter (G)

Description

Mass Balance

Energy Balance

Partial Derivatives for Perfect and Semiperfect Gas Models

Partial Derivatives for Real Gas Model

Gas Volume

Torque Balance

Variables

Assumptions and Limitations

Ports

Input

q — Rotation of port R relative to port C, rad
physical signal

Dependencies

Conserving

A — Converter inlet
gas

H — Temperature inside converter
thermal

R — Rod
mechanical rotational

C — Case
mechanical rotational

Parameters

Mechanical orientation — Select the converter orientation
`Pressure at A causes positive rotation of R relative to C` (default) | `Pressure at A causes negative rotation of R relative to C`

Interface rotation — Select method to determine relative port positions
`Calculate from angular velocity of port R relative to port C` (default) | `Provide input signal from Multibody joint`

Initial interface rotation — Rotational offset of port R relative to port C at the start of simulation
`0 rad` (default)

Dependencies

Interface volume displacement — Displaced gas volume per unit rotation
`0.01 m^3/rad` (default)

Dead volume — Volume of gas when the interface rotation is 0
`1e-5 m^3` (default)

Cross-sectional area at port A — Area normal to flow path at the converter inlet
`0.01 m^2` (default)

Environment pressure specification — Select a specification method for the environment pressure
`Atmospheric pressure` (default) | `Specified pressure`

Environment pressure — Pressure outside the converter
`0.101325 MPa` (default)

Dependencies

Extended Capabilities

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

Version History

See Also

Topics

Rotational Mechanical Converter (G)

Description

Mass Balance

Energy Balance

Partial Derivatives for Perfect and Semiperfect Gas Models

Partial Derivatives for Real Gas Model

Gas Volume

Torque Balance

Variables

Assumptions and Limitations

Ports

Input

q — Rotation of port R relative to port C, rad physical signal

Dependencies

Conserving

A — Converter inlet gas

H — Temperature inside converter thermal

R — Rod mechanical rotational

C — Case mechanical rotational

Parameters

Mechanical orientation — Select the converter orientation Pressure at A causes positive rotation of R relative to C (default) | Pressure at A causes negative rotation of R relative to C

Interface rotation — Select method to determine relative port positions Calculate from angular velocity of port R relative to port C (default) | Provide input signal from Multibody joint

Initial interface rotation — Rotational offset of port R relative to port C at the start of simulation 0 rad (default)

Dependencies

Interface volume displacement — Displaced gas volume per unit rotation 0.01 m^3/rad (default)

Dead volume — Volume of gas when the interface rotation is 0 1e-5 m^3 (default)

Cross-sectional area at port A — Area normal to flow path at the converter inlet 0.01 m^2 (default)

Environment pressure specification — Select a specification method for the environment pressure Atmospheric pressure (default) | Specified pressure

Environment pressure — Pressure outside the converter 0.101325 MPa (default)

Dependencies

Extended Capabilities

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

Version History

See Also

Topics

q — Rotation of port R relative to port C, rad
physical signal

A — Converter inlet
gas

H — Temperature inside converter
thermal

R — Rod
mechanical rotational

C — Case
mechanical rotational

Mechanical orientation — Select the converter orientation
`Pressure at A causes positive rotation of R relative to C` (default) | `Pressure at A causes negative rotation of R relative to C`

Interface rotation — Select method to determine relative port positions
`Calculate from angular velocity of port R relative to port C` (default) | `Provide input signal from Multibody joint`

Initial interface rotation — Rotational offset of port R relative to port C at the start of simulation
`0 rad` (default)

Interface volume displacement — Displaced gas volume per unit rotation
`0.01 m^3/rad` (default)

Dead volume — Volume of gas when the interface rotation is 0
`1e-5 m^3` (default)

Cross-sectional area at port A — Area normal to flow path at the converter inlet
`0.01 m^2` (default)

Environment pressure specification — Select a specification method for the environment pressure
`Atmospheric pressure` (default) | `Specified pressure`

Environment pressure — Pressure outside the converter
`0.101325 MPa` (default)

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