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Linear conversion of pressure differential to actuation in an isothermal liquid system

**Library:**Simscape / Fluids / Isothermal Liquid / Actuators

The Double-Acting Actuator (IL) block models the linear conversion of a pressure differential between two chambers to piston motion. The piston actuation is controlled by the pressure differential acting on the piston plate that separates the chambers. The motion of the piston when it is near full extension or full retraction is limited by one of three hard stop models. Fluid compressibility is optionally modeled in both piston chambers.

Ports **A** and **B** are isothermal liquid inlets. Port
**C** represents the actuator casing, while piston velocity is
returned at port **R**. When the piston position is calculated
internally, it is reported at port **p**, and when the position is set
by a connection to a Simscape™
Multibody™ joint, it is received as a physical signal at port
**p**.

You can define the piston displacement direction with the **Mechanical
orientation** parameter. If the mechanical orientation is set to
```
Pressure at A causes positive displacement of R relative to
C
```

, the piston extends when the pressure differential
*p*_{A} –
*p*_{B} is positive. If **Mechanical
orientation** is set to ```
Pressure at A causes negative
displacement of R relative to C
```

, the piston retracts for a positive
pressure difference between the liquid and gas chambers.

The piston displacement is measured as the position at port **R** relative
to port **C**. The **Mechanical orientation**
identifies the direction of piston displacement. The piston displacement is neutral,
or `0`

, when the chamber A volume is equal to the chamber dead
volume. When displacement is received as an input, ensure that the derivative of the
position is equal to the piston velocity. This is automatically the case when the
input is received from a Translational Multibody Interface block
connection to a Simscape Multibody joint.

To avoid mechanical damage to an actuator when it is fully extended or fully retracted, an actuator typically displays nonlinear behavior when the piston approaches these limits. The Double-Acting Actuator (IL) block models this behavior with a choice of three hard stop models, which model the material compliance through a spring-damper system. The hard stop models are:

`Stiffness and damping applied smoothly through transition region, damped rebound`

.`Full stiffness and damping applied at bounds, undamped rebound`

.`Full stiffness and damping applied at bounds, damped rebound`

.

The hard stop force is modeled when the piston is at its upper or lower bound. The
boundary region is within the **Transition region** of the
**Piston stroke** or piston initial displacement. Outside of
this region, $${F}_{HardStop}=0.$$

For more information about these settings, see the Translational Hard Stop block page.

You can optionally model cushioning toward the extremes of the piston stroke.
Modeling cylinder end cushioning slows the piston motion as it approaches its
maximum extension within its respective chamber, which is defined in
**Piston stroke**. For more information on the functionality of
a cylinder cushion, see Cylinder Cushion (IL).

You can optionally model friction against piston motion. When **Cylinder
friction effect** is set to `On`

, the
resulting friction is a combination of Stribeck, Coulomb, and viscous effects. The
pressure difference is measured between the chamber pressure and the environment
pressure. For more information on the friction model and its limitations, see the
Cylinder Friction block.

You can optionally model leakage between the liquid chambers and the piston
reservoir. When **Internal leakage** is set to
`On`

, Poiseuille flow is modeled between the piston and
cylinder. This block uses the Simscape Foundation Library Laminar Leakage (IL) block. The
flow rate is calculated as:

$$\dot{m}=\frac{\frac{\pi}{128}\left({d}_{0}^{4}-{d}_{i}^{4}-\frac{{\left({d}_{0}^{2}-{d}_{i}^{2}\right)}^{2}}{\mathrm{log}({d}_{0}/{d}_{i})}\right)}{\upsilon L}\left({p}_{A}-{p}_{env}\right),$$

where:

*ν*is the fluid kinematic viscosity.*L*is the piston length,**p**–*P*_{0}.*p*_{A}is the pressure at port**A**.*p*_{env}is the environmental pressure, which is selected in the**Environment pressure specification**parameter.

The cylinder diameter, *d*_{0},
is $${d}_{0}={d}_{i}+2c,$$ where *c* is the **Piston-cylinder
clearance**, and the piston diameter,
*d*_{i}, is $${d}_{i}=\sqrt{\frac{4{A}_{P}}{\pi}},$$ where *A*_{P} is the average
of the **Piston cross-sectional area in chamber A** and
**Piston cross-sectional area in chamber B** parameters.

At the extremes of the cushion A and B orifice areas and check valve pressure ranges, you
can maintain numerical robustness in your simulation by adjusting the block
**Smoothing factor**. A smoothing function is applied to all
calculated areas and valve pressures, but primarily influences the simulation at the
extremes of these ranges.

The normalized orifice area is calculated as:

$$\widehat{A}=\frac{\left(A-{A}_{leak}\right)}{\left({A}_{\mathrm{max}}-{A}_{leak}\right)}.$$

where:

*A*is the cushion A and cushion B_{leak}**Leakage area between plunger and cushion sleeve**.*A*is the cushion A and cushion B_{max}**Cushion plunger cross-sectional area**.

The **Smoothing factor**, *f*, is
applied to the normalized area:

$${\widehat{A}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\widehat{A}}_{}^{2}+{\left(\frac{f}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left(\widehat{A}-1\right)}^{2}+{\left(\frac{f}{4}\right)}^{2}}.$$

The smoothed orifice area is:

$${A}_{smoothed}={\widehat{A}}_{smoothed}\left({A}_{\mathrm{max}}-{A}_{leak}\right)+{A}_{leak}.$$

Similarly, the normalized valve pressure is:

$$\widehat{p}=\frac{\left(p-{p}_{cracking}\right)}{\left({p}_{\mathrm{max}}-{p}_{cracking}\right)}.$$

where:

*p*is the cushion_{cracking}**Check valve cracking pressure differential**.*p*is the cushion_{max}**Check valve maximum pressure differential**.

Smoothing applied to the normalized pressure is:

$${\widehat{p}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\widehat{p}}_{}^{2}+{\left(\frac{f}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left(\widehat{p}-1\right)}^{2}+{\left(\frac{f}{4}\right)}^{2}},$$

and the smoothed pressure is:

$${p}_{smoothed}={\widehat{p}}_{smoothed}\left({p}_{\mathrm{max}}-{p}_{cracking}\right)+{p}_{cracking}.$$

The Double-Acting Actuator (IL) block comprises four Simscape Foundation blocks:

and two Isothermal Liquid library blocks:

Single-Acting Actuator (IL) | Double-Acting Actuator (G-IL) | Double-Acting Rotary Actuator (IL) | Double-Acting Servo Valve Actuator (IL)