# Translational Mechanical Converter (G)

Interface between gas and mechanical translational networks

• Library:
• Simscape / Foundation Library / Gas / Elements

• ## Description

The Translational Mechanical Converter (G) block models an interface between a gas network and a mechanical translational network. The block converts gas pressure into mechanical force and vice versa. It can be used as a building block for linear 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 displacement 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 translational 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 displacement of the moving interface:

`$V={V}_{dead}+{S}_{\mathrm{int}}{x}_{\mathrm{int}}{\epsilon }_{\mathrm{int}}$`

where:

• Sint is the interface cross-sectional area.

• xint is the interface displacement.

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

If you connect the converter to a Multibody joint, use the physical signal input port p to specify the displacement of port R relative to port C. Otherwise, the block calculates the interface displacement from relative port velocities. The interface displacement 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 displacement of R relative to C```, the interface displacement increases when the gas volume increases from dead volume.

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

### Force Balance

Force balance across the moving interface on the gas volume is

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

where:

• Fint is the force 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 Variables tab in the block dialog box (or the Variables section in the block Property Inspector). For more information, see Set Priority and Initial Target for Block Variables and Initial Conditions for Blocks with Finite Gas Volume.

### 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 p of the joint. For more information, see Connecting Simscape Networks to Simscape Multibody Joints.

#### Dependencies

To enable this port, set the Interface displacement 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 translational conserving port associated with the moving interface.

Mechanical translational conserving port associated with the converter casing.

## Parameters

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Select the relative orientation of the converter with respect to the converter gas volume:

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

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

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

• ```Calculate from velocity of port R relative to port C``` — Calculate displacement from relative port velocities, based on the block equations. This is the default method.

• ```Provide input signal from Multibody joint``` — Enable the input physical signal port p to pass the displacement information from a Multibody joint. Use this method only when you connect the converter to a Multibody joint by using a Translational Multibody Interface block. For more information, see How to Pass Position Information.

Translational 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 displacement parameter is set to ```Calculate from velocity of port R relative to port C```.

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

The area on which the gas exerts pressure to generate the translational force.

Volume of gas when the interface displacement 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`.