# 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:

`$\frac{\partial M}{\partial p}\cdot \frac{d{p}_{I}}{dt}+\frac{\partial M}{\partial T}\cdot \frac{d{T}_{I}}{dt}+{\rho }_{I}\frac{dV}{dt}={\stackrel{˙}{m}}_{A}$`

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.

• pI is the pressure of the gas volume. Pressure at port A is assumed equal to this pressure, pA = pI.

• TI is the temperature of the gas volume. Temperature at port H is assumed equal to this temperature, TH = TI.

• ρI is the density of the gas volume.

• V is the volume of gas.

• t is time.

• $\stackrel{˙}{m}$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:

`$\frac{\partial U}{\partial p}\cdot \frac{d{p}_{I}}{dt}+\frac{\partial U}{\partial T}\cdot \frac{d{T}_{I}}{dt}+{\rho }_{I}{h}_{I}\frac{dV}{dt}={\Phi }_{A}+{Q}_{H}$`

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.

• ФA is the energy flow rate at port A.

• QH is the heat flow rate at port H.

• hI 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:

`$\begin{array}{l}\frac{\partial M}{\partial p}=V\frac{{\rho }_{I}}{{p}_{I}}\\ \frac{\partial M}{\partial T}=-V\frac{{\rho }_{I}}{{T}_{I}}\\ \frac{\partial U}{\partial p}=V\left(\frac{{h}_{I}}{ZR{T}_{I}}-1\right)\\ \frac{\partial U}{\partial T}=V{\rho }_{I}\left({c}_{pI}-\frac{{h}_{I}}{{T}_{I}}\right)\end{array}$`

where:

• ρI is the density of the gas volume.

• V is the volume of gas.

• hI is the specific enthalpy of the gas volume.

• Z is the compressibility factor.

• R is the specific gas constant.

• cpI 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:

`$\begin{array}{l}\frac{\partial M}{\partial p}=V\frac{{\rho }_{I}}{{\beta }_{I}}\\ \frac{\partial M}{\partial T}=-V{\rho }_{I}{\alpha }_{I}\\ \frac{\partial U}{\partial p}=V\left(\frac{{\rho }_{I}{h}_{I}}{{\beta }_{I}}-{T}_{I}{\alpha }_{I}\right)\\ \frac{\partial U}{\partial T}=V{\rho }_{I}\left({c}_{pI}-{h}_{I}{\alpha }_{I}\right)\end{array}$`

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:

`$V={V}_{dead}+{D}_{\mathrm{int}}{\theta }_{\mathrm{int}}{\epsilon }_{\mathrm{int}}$`

where:

• Dint is the interface volume displacement.

• θint is the interface rotation.

• εint is the mechanical orientation coefficient. If Mechanical orientation is ```Pressure at A causes positive rotation of R relative to C```, εint = 1. If Mechanical orientation is ```Pressure at A causes negative rotation of R relative to C```, εint = –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

`${\tau }_{\mathrm{int}}=\left({p}_{env}-{p}_{I}\right){D}_{\mathrm{int}}{\epsilon }_{\mathrm{int}}$`

where:

• τint is the torque from port R to port C.

• penv 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

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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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Gas conserving port associated with the converter inlet.

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

Mechanical rotational conserving port associated with the moving interface.

Mechanical rotational conserving port associated with the converter casing.

## Parameters

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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.

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.

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.

Displaced gas volume per unit rotation of the moving interface.

Volume of gas when the interface rotation is 0.

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

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.

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`.

## Version History

Introduced in R2016b